Modeling Evolution of Heterogeneous Populations: Theory and Applications 0128143681, 9780128143681

Table of contents :
Front matter
Copyright
Dedication
Using mathematical modeling to ask meaningful biological questions
Introduction
General strategy
Advantages and drawbacks of the Reduction theorem
Inhomogeneous models of Malthusian type and the HKV method
Models of Malthusian type for inhomogeneous populations and a simplified version of the HKV method
Dynamics of different initial distributions
Inhomogeneous Malthusian model
Logistic equation with distributed Malthusian parameter
Inhomogeneous Allee-type models
Distribution of growth parameter a
Distribution of carrying capacity b
Distribution of parameter m
Comparison of the three distributed Allee models
Some applications of inhomogeneous population models of Malthusian type
Example 3.1. Conceptual models of global demography
Example 3.2. Modeling the dynamics of the effect of antimicrobial agents on heterogeneous microbial populations
Example 3.3. Models of forest stand self-thinning
Selection systems and the Reduction theorem
Selection systems
Dynamics of specific distributions
Selection systems with self-regulations
Reduction theorem for inhomogeneous models of populations
Reduction theorem for inhomogeneous models of communities
Some applications of the Reduction theorem and the HKV method
How to solve selection systems
Example 5.1 Birth-and-death equation with distributed birth rate and average death rate
Example 5.2 A model of group selection and evolution of altruism
Example 5.3 The principle of limiting factors in modeling of early biological evolution
Example 5.4 Inhomogeneous Ricker equation with two distributed parameters
Example 5.5 Inhomogeneous prey-predator Volterra model with three distributed parameters
Example 5.6 Competition of two inhomogeneous populations
Example 5.7 The Fisher-Haldane-Wright equation
Example 5.8 Haldane principle for selection systems
Example 5.9 The Price equation and Fisher's fundamental theorem
Nonlinear replicator dynamics
Problem formulation: Power replicator equations
Population heterogeneity as the reason for the power law growth dynamics
Canonical form of the power model
Inhomogeneous model for exponential equation
Superexponential models: The second representation
How to choose between F- and D-models?
Summary
Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution
Problem formulation and basic models
Solution to the inhomogeneous logistic equation
Generalized logistic inhomogeneous models with distributed Malthusian parameter
Inhomogeneous Gompertz equation
Logistic equation with distributed carrying capacity
Logistic equation with two distributed parameters: Malthusian parameter and carrying capacity
Dynamics of distributions in inhomogeneous models and the speed of natural selection
Some notes on internal time and the competitive exclusion principle
Mathematical Appendix (by F. Berezovskaya): The Newton diagram method and asymptotic behavior of q(t) for inhomogeneous bir ...
Basic equation in the new form
The Newton diagram method
Asymptotics of orbits of system (A.4)
Asymptotics of probabilities
Replicator dynamics and the principle of minimal information gain
Problem formulation
MinxEnt algorithm and the Boltzmann distributions
Selection systems and dynamical principle of minimal information gain
MinxEnt principle and selection systems: Applications
Information gain in inhomogeneous Malthusian model
Information gain in the model of early biological evolution
Information gain in models of tree stand self-thinning
Quasi-species equation and linear systems
Information gain in inhomogeneous birth-and-death models
Information gain in inhomogeneous Ricker model
``Conjugative´´ approach to selection system dynamics
Information gain in models of inhomogeneous communities
Discussion
Subexponential replicator dynamics and the principle of minimal Tsallis information gain
Problem formulation: Subexponential power equations
Population of freely growing parabolic replicators
Dynamical principles of minimal information gain
Population of parabolic replicators with constant total size and the principle of minimal information gain
Discussion
Modeling extinction of inhomogeneous populations
Mathematical and nonmathematical motivations
Problem formulation
Population of subexponentially decreasing clones
Dynamical principles of minimal Tsallis information loss
Parametrically inhomogeneous models of population extinction
Power extinction models and inhomogeneous F-models
The ``internal time´´ for F-models of extinction
Dynamical principles of minimal Shannon information loss
Application of the model to time perception
Some background information on time perception
Application of the proposed model to understanding time perception in a dying brain
Discussion
From experiment to theory: What can we learn from growth curves?
Problem formulation
Approach and introductory example
Generalized logistic equation
Gompertz versus Verhulst
Example 11.1 Logistic versus Gompertz curves
Hyperbolic and hyperbolic-exponential growth
Exponential-linear growth
Virus-specific RNA replication and the three-stage model
Fitting experimental data to different models
Data set 1: Naumov et al. (2006) JNCI
Data set 2: Rogers et al. (2014)
Data set 3: Rogers et al. (2014)
Data set 4: Benzekry et al. (2014)
Simeoni model and exponential-linear growth
Discussion
Applications and implications
Appendix: Supplementary material
Traveling through phase-parameter portrait
Introduction
Example 12.1. Sustainability: Using a parametrically heterogeneous model to investigate resource depletion, transitional re ...
Question 1. How will such a system behave when the number of overconsumers in it changes?
Question 2. Can we identify transitional regimes that can serve as warning signals of approaching collapse?
Question 3. What, if any, intervention measures can be implemented to prevent the tragedy of the commons?
Example 12.2. Natural selection in resource allocation strategies
Question 1. If one allows for the possibility of resource overconsumption, which strategy is preferable for avoiding popula ...
Question 2. Which strategy (allocating shared resources toward rapid proliferation, or toward slower proliferation but incr ...
Example 12.3 Cancer and oncolytic viruses
Question 1. What are the transitional regimes that occur as the cancer cell population gains resistance to the virus? Can w ...
Question 2. Why are cytotoxic therapies effective in some patients and not others?
Distributed susceptibility
Distributed susceptibility and distributed cytotoxicity
Conclusions
Evolutionary games: Natural selection of strategies
Problem formulation
The model and the main equations
Solution to the replicator equation
Equilibria of frequencies
Dynamics of the distribution of strategies
Replicator equation
Prisoner's dilemma
Coordination game or SH game
Natural selection of strategies in a ``hawk-dove´´ game
Natural selection of strategies and the principle of minimum information gain
Discussion
Natural selection between two games with applications to game theoretical models of cancer
Model description
The model
Solution to the model
Natural selection between games: Hawk-Dove versus Prisoner's dilemma
Mutual invasibility of both strategies and games
Games tumors play
Game 1: Metabolism and resource allocation
Game 2: Motility versus stability
Some conclusions
Discrete-time selection systems
Main definitions
Malthusian inhomogeneous maps
Evolution of the main statistical characteristics of inhomogeneous maps
Self-regulated inhomogeneous maps
The Price equation and the Fisher fundamental theorem for maps
Applications and examples
Ricker model with discrete time
Inhomogeneous logistic map
Inhomogeneous Ricker map with two distributed parameters
Selection in natural rotifer community
Discussion
Conclusions
Moment-generating functions for various initial distributions
Distributions on the entire line or half line
Distributions on a finite interval
Many-dimensional distributions
Bibliogrpahy
Index

Citation preview

MODELING EVOLUTION OF HETEROGENEOUS POPULATIONS

MATHEMATICS IN SCIENCE AND ENGINEERING

MODELING EVOLUTION OF HETEROGENEOUS POPULATIONS Theory and applications IRINA KAREVA GEORGY KAREV

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-814368-1 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

The views presented in this book are the authors’ personal views and do not necessarily represent the views of any other organization. Publisher: Candice Janco Acquisition Editor: Scott J Bentley Editorial Project Manager: Susan Ikeda Production Project Manager: Debasish Ghosh Cover Designer: Greg Harris Typeset by SPi Global, India

Dedication

To Faina Berezovsky, wife, mother, friend, collaborator, and an extraordinary mathematician

C H A P T E R

1 Using mathematical modeling to ask meaningful biological questions Abstract Classical approaches to analyzing dynamical systems, including bifurcation analysis, can provide invaluable insights into the underlying structure of a mathematical model and the spectrum of all possible dynamical behaviors. However, these models frequently fail to take into account population heterogeneity. While heterogeneity is critically important to understanding and predicting the behavior of any evolving system, this characteristic is commonly omitted when analyzing many mathematical models of ecological systems. Attempts to include population heterogeneity frequently result in expanding system dimensionality, effectively preventing qualitative analysis. However, Reduction theorem, or hidden keystone variable (HKV) method, allows incorporating population heterogeneity while still permitting the use of classical bifurcation analysis. A combination of these methods allows visualization of evolutionary trajectories and permits making meaningful predictions about dynamics over time of evolving populations.

1.1 Introduction Heterogeneity is a major driving force behind the dynamics of evolving systems. When it is heritable and when it affects fitness, heterogeneity is what makes evolution possible (Bell, 2008; Darwin, 1880; Johnson, 1976; Page, 2010). This comes from the fact that the environment in which the individuals interact is composed not only of the outside world (such as the resources necessary for survival, or members of other species) but also of individuals themselves. Therefore, selective pressures that are imposed on the individuals come both from the environment and from one another. Furthermore, selective pressures that individuals experience from one another will be imposed and perceived differently depending on population composition, which in turn may be changing as a result of these selective pressures. This selection process underlies Dobzhansky’s famous thesis: nothing in biology makes sense except in light of evolution (Dobzhansky, 1973). In a majority of conceptual and often even descriptive mathematical models of population dynamics—whether it be models of predator-prey interactions, spread of infectious diseases or tumor growth—population homogeneity is the first simplification that is made. The

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1. Using mathematical modeling to ask meaningful biological questions

population is not treated as homogeneous per se; rather, one assumes an average rate of growth, death, or infectiousness as a reasonable enough approximation if the system has reached a quasi-stable state of evolutionary development. However, by ignoring population heterogeneity in such a way, one ends up either ignoring natural selection or assuming that it has already “done its work.” This assumption is often incorrect within the context of such models however, since natural selection may in fact be a key driver behind the dynamics of systems that are of most interest and importance. The process of evolution of heterogeneous populations is typically modeled using replicator equations that capture the “basic tenet of Darwinism” (Hofbauer and Sigmund, 1998; Nowak and Sigmund, 2004). These equations capture the process of selection in heterogeneous populations as they demonstrate how when an individual’s value of a heritable characteristic is above average, that individual stays in the population, but when the heritable characteristic is below average, the individual is expunged. However, such models have a major drawback: modeling high levels of heterogeneity is accompanied by an inevitable increase of system dimensionality, which makes obtaining any kind of qualitative understanding of the system nearly impossible. Assuming population homogeneity makes systems of equations computationally and sometimes even analytically manageable, but at the cost of losing many of the system dynamics caused by intraspecies interactions and natural selection. Despite their shortcomings, parametrically homogeneous systems can still provide exceptionally valuable information about the structure of the system through the use of extensively developed analytical techniques, such as bifurcation analysis (Kuznetsov, 2013). A skillfully constructed bifurcation diagram can reveal various possible dynamical regimes of a system that result from variations in parameter values and initial conditions, and also provide analytical boundaries as functions of system parameters. This information can then be used to construct a theoretical framework for understanding a biological system that could never have been obtained experimentally. In this book, we will describe in detail a method to introduce population heterogeneity back into equation-based models using the Reduction theorem. Also known as parameter distribution technique or hidden keystone variable (HKV) method, Reduction theorem can use and build on insights obtained from bifurcation analysis, while incorporating population heterogeneity. Reduction theorem allows the dynamics of an evolving system to be investigated more fully, while overcoming the problem of immense system dimensionality in a wide class of mathematical models.

1.2 General strategy The key steps for using the HKV method are as follows. Assume a population of individuals is composed at time t of clones x(t, a). Each individual clone x(t, a) is characterized by parameter value a 2 A, and each parameter corresponds to a measure of some intrinsic heritable trait, such as birth rate, death rate, resource consumption rate, etc. Ð The total population size is P given by N ðtÞ ¼ a2A xðt, aÞ if the system is discrete, and N(t) ¼ Ax(t, a)da if the system is continuous. Then, since different clones can grow and die at different rates, the distribution of

1.2 General strategy

3

clones within the population Pðt, aÞ ¼ xNðtð,taÞÞ can change over time due to system dynamics. Consequently the mean value of the parameter Et[a] now becomes a function of time and changes over time as well. In the upcoming chapters we show how to analyze a parametrically heterogeneous system using the following steps: 1. Analyze the autonomous parametrically homogeneous system to the extent possible using well-developed analytical tools, such as bifurcation analysis. 2. Replace parameter a with its mean value Et[a], which is a function of time. 3. Introduce an auxiliary system of differential equations to define keystone variables that determine the actual dynamics of the system. (Note: the term “keystone” is used here to parallel the function of keystone species in ecology. Just like keystone species have disproportionately large effect on their environment relative to their abundance, keystone variables determine the direction in which the system will evolve, without being explicitly present in the original system.) 4. Express the distribution of the distributed parameter through keystone variables. This transformation allows finding all statistical characteristics of interest, including parameter’s mean and variance, which now change over time due to system dynamics. The mean of the parameter can now “travel” through the different domains of the phaseparameter portrait of the original parametrically homogeneous system. 5. Calculate numerical solutions. Exact formulation of the Reduction theorem and the theory underlying the HKV method will be found later in the book. A summary definitions and associated notation are provided in Table 1.

TABLE 1

Definitions and notation used in the application of the HKV method.

Definition

Notation and explanation

Selection system

A mathematical model of an inhomogeneous population in which every individual is characterized by a vector-parameter a ¼ (a1, … , an) that takes on values from set 

Clone x(t, a)

Set of all individuals that are characterized by a fixed value of parameter a R NðtÞ ¼  xðt, aÞda

Total population size N(t) Growth rate of a clone x(t, a)

d xðt, aÞ dt

Fitness of an individual within the population

d xðt, aÞ dt =xðt, aÞ

Distribution of clones within the population

Pðt, aÞ ¼ xNðtð,taÞÞ

Expected value of a function on distributed parameter

Et ½ f  ¼

R 

f ðaÞxðt, aÞda N ðt Þ

4

1. Using mathematical modeling to ask meaningful biological questions

1.3 Advantages and drawbacks of the Reduction theorem One of the most important properties of this method is that it allows reducing an otherwise many- or even infinitely-dimensional system to low dimensionality. However, as with any method, this method is not universal. Most importantly, the transformation can be done (with some generalizations) only to Kolmogorov type equations of the form x(t)’¼x(t)F(t, Et[f(a)]), where: • x(t) is a vector, • a is a parameter or a vector of parameters that characterize individual heterogeneity within the population, • Et[ f (a)] is of system-specific form. Reduction theorem can also increase the dimensionality of the original parametrically homogeneous system at a possible cost of auxiliary keystone equations (although these would typically be up to only one or two extra equations, depending on the original system). Finally, the resulting system is typically non-autonomous, so one cannot perform standard bifurcation analysis. When studying numerical solutions of such parametrically heterogeneous systems, trajectories can be observed that could not previously have been seen in parametrically homogeneous systems. This phenomenon arises from the expected value of the parameter “traveling” through the phase parameter portrait, and the system undergoes corresponding qualitative phase transition as the parameter’s expected value crosses the bifurcation boundaries. Furthermore, if there exists a complete bifurcation diagram for the specific parametrically homogeneous model, the boundaries crossed during system evolution can be identified analytically. Classic techniques for analyzing dynamical systems, such as bifurcation theory (Kuznetsov, 2013), can provide critical insights into the possible dynamical regimes that a system can realize. Unfortunately, doing full bifurcation analysis is labor intensive and is not always possible due to increasing complexities of many models. However, a very rich body of literature exists of fully analyzed parametrically homogeneous models in many fields, including ecology (Bazykin, 1998; Berezovskaya et al., 2005), epidemiology (Brauer and Castillo-Chavez, 2001), among others. As the examples presented throughout this book will demonstrate, even relatively simple two-dimensional systems can reveal rich, unexpected and meaningful behaviors. Application of the HKV-method to introduce population heterogeneity in a meaningful way and utilizing previously performed analysis can reveal a new layer of understanding of many existing models that was not accessible before. In Chapter 1 of this book, we introduce the HKV method for modeling population heterogeneity. Chapter 2 shows how applications of these methods to some classical models can reveal new and unexpected dynamical behaviors. Chapter 3 demonstrates how the HKVmethod can be applied to more complex biological systems, including models of world demography, microbial resistance to antibiotics, and the dynamics of tree stand self-thinning. In Chapter 4, we go over more in-depth theory. Chapter 5 features numerous examples exploring such topics as evolution of altruism, competition between two inhomogeneous equations, a puts a new spin on the classical Lotka-Volterra predator-prey model. We also discuss

1.3 Advantages and drawbacks of the Reduction theorem

5

the Fisher-Haldane-Wright equation, the Haldane principle for selection systems, and finally Fisher’s fundamental theorem—the latter three topics we will return to again and again throughout the book. In Chapter 6, we discuss models of frequency versus density-dependent population growth, and highlight some key differences between them. Chapter 7 dives into the discussion of inhomogeneous logistic and Gompertzian growth; we look at dynamics of distributions in inhomogeneous models and discuss various types of Darwinian (“survival of the fittest”) and non-Darwinian (“survival of the common” and “survival of everybody”) selection. In Chapter 8, we introduce the Principle of minimal information gain, where we show that it can be derived from system dynamics rather than being postulated a priori. Chapter 9 discusses sub-exponential system dynamics and the Principle of minimal Tsallis information gain. The main result of this Chapter is that the Principle of minimal information gain is the underlying variational principle that governs replicator dynamics. In Chapter 10, we discuss some philosophical issues on time perception and propose several hypotheses on how a model of inhomogeneous population extinction can be applied to time perception in a dying brain. Chapter 11 explores several seemingly similar models of population growth—logistic, Gompertz and Verhulst, among others—and shows that intrinsic population composition may in fact be very different depending on which model described data best. We then apply this developed theory to cancer cell growth. In Chapter 12, we show how the HKV-method can be applied to previously analyzed parametrically homogeneous systems to reveal the phenomenon of the expected value of the distributed parameter “traveling” through the phase-parameter portrait. This analysis can reveal new, complex and sometimes unexpected dynamical behaviors that help answer many interesting and important questions. We showcase several examples of this phenomenon, including the tragedy of the commons, natural selection between resource allocation strategies, and application of oncolytic virus therapy to a population of heterogeneous cancer cells. Chapter 13 applies the HKV-method to game theory and looks at dynamics of selection of strategies within a single game. In Chapter 14, we look at selection between games, and then discuss how some of these insights can be applied to understanding the complex dynamics of cancer cells in tumors. Finally, Chapter 15 (which can be read a stand-alone chapter), demonstrates how the HKV-method can be applied to selection systems with discrete time (maps). Let us begin.

C H A P T E R

2 Inhomogeneous models of Malthusian type and the HKV method Abstract In this chapter, we present description and derivation of a new method for modeling population heterogeneity. We show why we refer to it as HKV, or “hidden keystone variables,” method and demonstrate how it allows incorporating a very high degree of heritable heterogeneity into dynamical systems. We then apply the method to several well-known growth models (namely, Malthusian, logistic, and Allee) and show how differently such populations can behave when population heterogeneity is taken into account (hint: we also see some nonDarwinian selection). The goal of this chapter is to provide initial exposure to the underlying mathematical theory, which will be deepened throughout the book, and to highlight that introducing population heterogeneity even into extremely well-studied models can reveal new, rich, and surprising dynamics.

2.1 Models of Malthusian type for inhomogeneous populations and a simplified version of the HKV method Let us start from a simplified version of a model of inhomogeneous population growth and give two examples, which may clarify the main ideas of the developed approach. These examples provide motivation for focusing specifically on a wide class of models of evolving heterogeneous populations and corresponding replicator equations (RE). Furthermore, they provide a vivid illustration for how one can reduce a system of ODEs of initially large or even infinite dimensionality to low dimensionality through introducing auxiliary “keystone” variables (as was previously mentioned, the terminology was chosen to parallel the notion of “keystone species” in ecological systems, whose impact on overall system dynamics is disproportionally large relative to their abundance). Consider an inhomogeneous population composed of individuals with different reproduction rates (Malthusian parameters) a; we refer to the set of all individuals with a given value of parameter a as an a-clone. Let l(t, a) be the size of a-clone at time moment t.

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2. Inhomogeneous models of Malthusian type and the HKV method

We assume that the growth rate of each clone depends on the total population size N(t). Dynamics of such a population can be described by the following model: dlðt, aÞ ¼ alðt, aÞgðNÞ, dt

(2.1)

where Ðg(N) is some function, chosen depending on the specifics of each model, and N ðtÞ ¼ A lðt, aÞda is the total population size. For if g(N) ¼ const, then Eq. (2.1) is  example,  an inhomogeneous Malthusian model; if gðN Þ ¼ 1  NC , then Eq. (2.1) describes an inhomogeneous logistic model, where each clone grows with an individual growth rate according to the logistic growth law up to the common carrying capacity C. (All of these and many other examples will be worked out in detail throughout the book after this mandatory introduction of the underlying theory.) Now, let us derive expressions for the expected value of the distributed parameter a as it changes over time in the evolving population described by Eq. (2.1). ðt, aÞ Denote the frequency of cell clone l(t, a) to be Pðt, aÞ ¼ lN ðtÞ ; the probability density function (pdf ) P(t, a) describes the distribution of parameter a throughout the population at each time Ð∞ moment t. Hereinafter, we use the notation Et ½ f  ¼ f ðaÞPðt, aÞda for any integrable function f(a) 0

to refer to the expected value of f(a) at time t. In a more general context described in (Hofbauer and Sigmund, 1998), it can be shown that the population size N(t) satisfies the equation dN ¼ NEt ½agðN Þ dt

(2.2)

and the pdf P(t, a) solves the replicator equation of the form   dPðt, aÞ ¼ Pðt, aÞgðN Þ a  Et ½a dt

(2.3)

where Et[a] is the mean value of parameter a. Indeed, integrating Eq. (2.1) over a, we get ð ð ð dN dlðt, aÞ ¼ da ¼ alðt, aÞgðN Þda ¼ N ðtÞgðN Þ aPðt, aÞda ¼ NgðN ÞEt ½a: dt dt A

A

A

From this, we can calculate that dPðt, aÞ d lðt, aÞ NðtÞ a lðt, aÞ gðNðtÞÞ  lðt, aÞ NðtÞgðNðtÞÞEt ½a ¼ ¼ dt dt N ðtÞ ðN ðt ÞÞ2   a lðt, aÞ gðNðtÞÞ lðt, aÞEt ½agðN ðtÞÞ  ¼ Pðt, aÞgðN Þ a  Et ½a ¼ N ðtÞ N ðt Þ Now, let us assume that the initial pdf P(0, a) of the Malthusian parameter a is given and its moment-generating function (mgf) ð M0 ½λ ¼ eλa Pð0, aÞda (2.4) A

2.1 Models of Malthusian type for inhomogeneous populations and a simplified version of the HKV method

9

is known. This information could be obtained either from data when available or in the absence of data from general theoretical considerations. In order to solve Eq. (2.1), let us define formally the “keystone” auxiliary variable q(t) as the solution to the Cauchy problem dq ¼ gðN Þ,qð0Þ ¼ 0: dt

(2.5)

This equation cannot be solved at this moment, because the population size N(t) is unknown. However, clone densities and population size can be expressed with the help of the newly introduced in Eq. (2.5) keystone variable q(t): lðt, aÞ ¼ lð0, aÞeaqðtÞ ¼ Nð0ÞPð0, aÞeaqðtÞ : Therefore, by definition, total population size becomes ð NðtÞ ¼ N ð0Þ eaqðtÞ Pð0, aÞda ¼ N ð0ÞM0 ½qðtÞ:

(2.6)

(2.7)

A

Now the equation for the auxiliary variable q(t) can be written in a closed form: dq ¼ gðN ð0ÞM0 ½qðtÞÞ, qð0Þ ¼ 0: dt

(2.8)

With this, we can completely solve the initial problem defined in Eq. (2.1) and corresponding replicator Eq. (2.3). Clone densities and population size are given by Eqs. (2.6), (2.7), respectively. More general results are summarized in the Reduction theorem, which will be proven in Chapter 4. In what follows, we assume that the Cauchy problem defined in Eq. (2.8) has a unique solution q(t) in the interval 0  t < T, where 0 < T  ∞. Proposition 2.1 Current parameter distribution P(t, a) is determined by the formula Pðt, aÞ ¼

eqðtÞa M0 ½qðtÞ

(2.9)

The mgf of the current distribution P(t, a) is given by   M0 ½λ + qðtÞ (2.10) Mt ½λ ¼ Et eλa ¼ M0 ½qðtÞ ð t, a Þ From definition of pdf Pðt, aÞ ¼ lN ðtÞ , one can see that Eq. (2.9) immediately follows from Eqs. (2.6), (2.7).   Ð Ð λa + a qðtÞ Pð0, aÞda M0 ½λ + qðtÞ Next, M ½λ ¼ Et eλa ¼ eλa Pðt, aÞda ¼ e ¼ as desired. t

A

A

M0 ½qðtÞ

M0 ½qðtÞ

It is well known that the moment-generating function can be used to compute all moments of a given probability distribution using the following formula: ð ak PðaÞda ¼ MðkÞ ½0, A

10

2. Inhomogeneous models of Malthusian type and the HKV method

where M(k)[0] denotes the kth derivative of the mgf M[λ] at the point λ ¼ 0. The first and second moments, corresponding to expected value and variance of the distribution, will be of particular interest and value for our applications. The mgf of the distribution P(t, a) is given by Eq. (2.10); hence ! dk M0 ½q ð dqk : (2.11) ak Pðt, aÞda ¼ M0 ½q A

t

The mean value E [a] at each time moment can be then computed as dM0 ½qðtÞ d ln M0 dq t ¼ ½qðtÞ: E ½a ¼ M0 ½qðtÞ dq

(2.12)

The current variance can be computed as d2 M0 ½q 2    Vart ½a ¼ Et a2  Et ½a ¼

!

2

dq M0 ½q

0

12 d M0 ½qÞ Bdq C C B @ M0 ½q A :

(2.13)

Proposition 2.2 The mean value Et[a] solves the equation dEt ½a ¼ Vart ½agðNðtÞÞ dt

(2.14)

Indeed, ð ð   dEt ½a dPðt, aÞ ¼ a da ¼ ðsee Eq:2:3Þ ¼ agðN Þ a  Et ½a Pðt, aÞda dt dt A A 0 1 ð   2 ¼ gðN Þ@ a2 Pðt, aÞ  Et ½a A ¼ gðN ÞVart ½a: A

Notice that Eq. (2.14) is a particular case of general Price’ equation (Robertson, 1968; Price, 1970, 1972); see also (Rice, 2006); this equation will be discussed in Chapter 5 within the framework of inhomogeneous population models. We refer to Eq. (2.1) as a model of Malthusian type, and the variable q(t) can be viewed as “internal time” of the model, a kind of a measure of internal rate of evolution. With respect to the new time q, the model is composed of clones that grow as if they were independent from other clones. Indeed, making the change of variables dq ! g(N)dt, we obtain from Eq. (2.6): dlðqðtÞ, aÞ dl dq ¼ ¼ algðN Þ: dt dq dt

11

2.2 Dynamics of different initial distributions

Therefore, since

dq dt ¼ gðN Þ,

then dlðq, aÞ ¼ alðq, aÞ: dq

The last equation is the standard Malthusian model. Hence, with respect to the internal time q(t), each clone grows according the Malthusian model with its own Malthusian parameter as it does not depend on any other clone or on the total population. A more extensive discussion of the notion of internal time can be found in Chapter 10.

2.2 Dynamics of different initial distributions Proposition 2.1 introduced in the previous section helps us trace the dynamics of the initial distribution of parameter a. Let us now consider some important examples. Proposition 2.3 Let us assume that the initial distribution of the parameter a is (i) normal with the mean a0 and variance σ 20; then, at any time moment, the current parameter distribution is also normal with the mean Et[a] ¼ a0 + σ 20q(t) and with the same variance σ 20; (ii) Poisson with the mean a0; then, at any time moment, the current parameter distribution is also Poisson with the mean Et[a] ¼ a0eq(t). Proof Normal distribution  1 Pð0, aÞ ¼ qffiffiffiffiffiffiffiffiffiffi e 2πσ 20

ðaa0 Þ2 2σ 20

(2.15)

has the mgf λ2 σ 20 2 + λa0 :

M0 ½λ ¼ e Then, according to Eq. (2.10),

(2.16)



   λ2 σ 20 2 + λ a + q ð t Þσ 0 0 M0 ½λ + qðtÞ Mt ½λ ¼ ¼e 2 : M0 ½qðtÞ

It is the mgf of the normal distribution with the mean Et[a] ¼ a0 + σ 20q(t) and variance σ 20. Poisson distribution Pð0, a ¼ iÞ ¼ exp ða0 Þa0 i =i!,i ¼ 0, 1,…

(2.17)

M0 ½λ ¼ ea0 ð exp ðλÞ1Þ :

(2.18)

has the mgf

12

2. Inhomogeneous models of Malthusian type and the HKV method

Then, according to Eq. (2.10), Mt ½λ ¼

qðtÞ λ M0 ½λ + qðtÞ ¼ ea0 e ðe 1Þ : M0 ½qðtÞ

It is the mgf of the Poisson distribution with the mean Et[a] ¼ a0eq(t).□ As we will see soon, gamma distribution and its special case, the exponential distribution, are among the most important initial distributions for applications. Gamma distribution with coefficients k, s, η is defined by the formula Pð a Þ ¼

sk ða  ηÞk1 eðaηÞs , Γ ðkÞ

(2.19)

a > η > 0, k > 0, and Γ(k) is the gamma function. Gamma distribution has the mean k k E½a ¼ η + , the variance Var½a ¼ 2 , s s

(2.20)

and the mgf MðλÞ ¼ 

eλη λ 1 s

k :

(2.21)

Proposition 2.4 Let us assume that the initial distribution of parameter a is Gamma distribution as given by Eq. (2.19). Let T∗ ¼ inf {t : q(t) ¼ s}. Then, parameter a is Γ-distributed at any time moment t < T* with coefficients k, s  q(t), η such that Et ½ a  ¼

η+k k ,Vart ½a ¼ s  qðtÞ ðs  qðtÞÞ2

and eλη

M t ðλÞ ¼  1

λ s  q ðt Þ

k :

(2.22)

Gamma distribution with η ¼ 0, k ¼ 1 becomes exponential distribution: PðaÞ ¼ s eas , s > 0, a  0 s M½λ ¼ sλ , E½a ¼ 1s ,

Var½a ¼ s12

(2.23)

2

with mgf and ¼ E½ a  . It is useful to notice that the mgf of the exponential distribution can be written in the form M½λ ¼

1 : 1  λE½a

(2.24)

2.3 Inhomogeneous Malthusian model

13

Corollary Let us assume that the initial distribution of parameter a is exponential as given by Eq. (2.23). Let T∗ ¼ inf {t : q(t) ¼ s}. Then, the distribution of parameter a is exponential at any time moment t < T* with coefficient s  q(t) such that Et ½ a  ¼

1 1 s  q ðt Þ 1 ,Vart ½a ¼ ¼ : and Mt ½λ ¼ 2 s  qðtÞ s  qðtÞ  λ 1  Et ½aλ ðs  qðtÞÞ

(2.25)

In applications, we will also consider versions of these distributions truncated on a bounded interval. In particular, exponential distribution truncated in the interval [0, b] has a form PðaÞ ¼ Vesa ,

(2.26)

where a 2 [0, b] and V ¼ 1es sb is a normalization constant. The mgf of the probability distribution as given by Eq. (2.26) is M½λ ¼

s 1  ebðλsÞ : s  λ 1  ebs

(2.27)

Proposition 2.5 Let us assume that parameter a takes values in the interval [0, b] and its initial distribution is truncated exponential as given by Eq. (2.26). Then, at any instant, the parameter distribution is also truncated exponential in the same interval [0, b] with coefficient s  q(t). Results of this section imply that modeling of inhomogeneous population dynamics based only on the mean value of the reproduction rate without taking into account its distribution is likely to be substantially incorrect. Indeed, the dynamics of inhomogeneous population models with the same initial mean value of the Malthusian parameter can be very different depending on the initial distribution of the parameter. And as is well known, most real populations are inhomogeneous.

2.3 Inhomogeneous Malthusian model Let us consider a case, where the reproduction rate of each clone is a constant. Then, we obtain the simplest inhomogeneous Malthusian model of the form dlðt, aÞ ¼ alðt, aÞ: dt

(2.28)

The model describes a population of clones l(t, a), each of which grows at its own rate, which does not depend on any other factors. The solution to this equation is lðt, aÞ ¼ lð0, aÞeat : dq

The keystone equation is trivial in this case, dt ¼ 1, and the keystone variable is just the time. Now, let M0[λ] be the mgf as given by Eq. (2.4) of the initial distribution of the Malthusian parameter a. Then the total size of the population (see Eq. (2.7)) is N ðtÞ ¼ N ð0ÞM0 ½t:

(2.29)

14

2. Inhomogeneous models of Malthusian type and the HKV method

The corresponding replicator equation reads   dPðt, aÞ ¼ Pðt, aÞ a  Et ½a : dt

(2.30)

The solution to the replicator equation is Pðt, aÞ ¼ Pð0, aÞ

eat : M0 ½t

(2.31)

Moreover the rate of change of the total population size is now dN ¼ NEt ½a dt

(2.32)

The mean value Et[a] can be computed by the Eq. (2.12) and solves the equation dEt ½a ¼ Vart ½a > 0, dt

(2.33)

which incidentally is the simplest version of the Fisher Fundamental theorem of selection (Fisher, 1999); see also (Roughgarden, 1979, Chapter 4; Edwards, 1994). From Eqs. (2.32), (2.33), one can see that any inhomogeneous Malthusian population increases hyperexponentially compared to the exponential growth of a parametrically homogeneous Malthusian population in the following sense. The relative growth rate of inhomogeneous population equals to Et[a], which is not a constant, as in homogeneous Malthusian models; instead, it increases over time until Vart[a] > 0 in accordance with Eq. (2.33). The examples of the dynamics of population size N(t), the change over time of the expected value of distributed parameter Et[a] and its variance Vart[a], and the change over time of the distribution of clones P(t, a) are shown in Fig. 2.1. For the purposes of this illustration, we assumed the initial distribution to be truncated exponential on the interval [0,1]. This distribuÐ1 tion is of the form P(0, a) ¼ Ce sa for 0  a  1, where C ¼ 1= esa sa ¼ 1es s is a normalization λðes eλ Þs M0 ½λ ¼ ð1es ÞðλsÞ.

0

constant. The mgf of this distribution is As one can see in Fig. 2.1A, an inhomogeneous population that grows according to the Malthusian growth law goes through a long period of seemingly slow growth, until it eventually undergoes a sudden rapid growth phase. During the slow-growth period, the expected value Et[a] actually increases until it reaches the largest possible value; in this case, it is a ¼ 1, since the distribution was taken to be truncated exponential on the interval a 2 [0, 1]. Changes in the expected value of a and its variance are shown in Fig. 2.1B and C, respectively. The rapid growth phase of total population size N(t) corresponds to the rapid increase in Et[a] (Fig. 2.1B) and rapid increase in variance (Fig. 2.1C). However, in the case of a Malthusian model, the fastest growing clone is selected to be the one with the largest value of a, eventually outcompeting all the other clones; this is reflected in the bell-shaped curve of Vart[a]. The change in the distribution of clones P(t, a) over time is demonstrated in Fig. 2.1D, showing how initially the population was skewed toward clones with the lower value of a, as determined by the initial truncated exponential distribution. However, over

15

2.3 Inhomogeneous Malthusian model

1

(A)

Et(a)

N(t)

1

0.5

0 0

50

100

150 Time

200

250

(B)

0.5

0 0

50

100

150 Time

200

250

(C)

Density P

Vart(a)

0.1

0.05

0 0

50

100

150 Time

200

250

200 100 0

200

(D)

100 Time t

0

0

0.5 Parameter a

1

FIG. 2.1 Parametrically heterogeneous Malthusian growth model with respect to distributed growth rate a. Initial distribution of a is taken to be truncated exponential with s ¼ 195. Initial population size is N(0) ¼ 0.001. (A) Dynamics of population N(t); (B) dynamics of the expected value Et[a] of Malthusian parameter a; (C) dynamics of variance Vart[a]; (D) dynamics of the distribution of clones change over time. Adapted from Kareva, I., 2016. Primary and metastatic tumor dormancy as a result of population heterogeneity. Biol. Direct 11, 37.

time, we can see selection toward the highest possible value of a, with the distribution of clones evolving to become concentrated in one point. It is of interest also to consider the inhomogeneous Malthusian model of population decrease, which has a form dlðt, aÞ ¼ alðt, aÞ: (2.34) dt The model describes a population of clones l(t, a), each of which dies at its own rate, which does not depend on any other factors. The solution to this equation is lðt, aÞ ¼ lð0, aÞeat : Then the total population size of the population described by the inhomogeneous model (2.28) is ð N ðtÞ ¼ Nð0Þ eat Pð0aÞda ¼ N ð0ÞL0 ðtÞ, (2.35) Ð

A at

where L0 ðtÞ ¼ A e Pð0, aÞda is the Laplace transform of the pdf P(0, a). Notably, the initial distribution may be unknown, so we are faced with a problem of how to estimate the initial distribution P(0, a) knowing the current distribution P(t, a). For model (2.34), this problem has a simple solution:   1 N ðtÞ Pð0, aÞ ¼ L (2.36) N ð 0Þ where L1 is the converse Laplace transform.

16

2. Inhomogeneous models of Malthusian type and the HKV method

Comparing definitions of Laplace transform L(λ) and moment-generating function M[λ] for the same pdf P(a) reveals that technically, L(λ) ¼ M[ λ]. Notice that the Laplace transform of P(a) exists for all λ 0; in contrast, the mgf M[λ] of some pdf P(a) may be indefinite for large λ, but if the support A of the pdf is bounded, then its mgf also exists for all λ 0. Inhomogeneous models of population decline and extinction will be studied in detail in Chapter 10. An example of Malthusian-type model of population decline in application to forest ecology will be presented in Chapter 3. It should be noted that even the simplest inhomogeneous Malthusian models, such as the ones given by Eqs. (2.28), (2.34), possess a variety of solutions depending on the initial distribution of clones in the population, and thus the predicted dynamics could be qualitatively different depending on the initial state of the heterogeneous population. Let us now review some important definitions and results. Definition A function φ(λ) is called absolutely monotone on the interval [a, b] if it has derivatives of all orders φ(n)(λ) and φ(n)(λ) 0 for all n and λ 2 [a, b]. A function φ(λ), λ 2 [0, ∞] is called completely monotone if it has derivatives of all orders φ(n)(λ) and (1)nφ(n)(λ) 0 for all n and λ 0. Function φ(λ) is completely monotone if φ( λ) is absolutely monotone on [0, ∞]. The following famous theorem is very useful in calculus and probability (see, e.g., (Feller, 2008)). Theorem (S. Bernstein) Function L(λ) is the Laplace transformation of some pdf if and only if it is completely monotone and L(0) ¼ 1. Thus, we can conclude that the function M[λ] is a moment-generating function of some pdf if it is absolutely monotone in [0, ∞] and M[0] ¼ 1. Now the following statement directly follows from the Bernstein theorem. Proposition 2.6 (i) Any absolutely monotone function N(t) in [0, ∞] solves a heterogeneous Malthus model of population growth given by Eq. (2.28) with corresponding initial distribution P(0, a); the solution is given by Eq. (2.29). (ii) Any completely monotone function N(t) solves a heterogeneous Malthusian model of population extinction as given by Eq. (2.34) with corresponding initial distribution P(0, a); the solution is given by Eq. (2.35). This assertion demonstrates the broad scope of applications of even simplest inhomogeneous Malthus models.

2.4 Logistic equation with distributed Malthusian parameter Inhomogeneous logistic equation, which accounts both for free exponential growth and for resource limitations, takes the form:   dlðt; aÞ N ¼ alðt; aÞ 1  , (2.37) dt C

2.4 Logistic equation with distributed Malthusian parameter

17

where a is the Malthusian reproduction rate, which is assumed to be distributed, C is the common carrying capacity, N is the total population size. Model by Eq. (2.37) is a model  described  of Malthusian type as given by Eq. (2.1), where gðNÞ ¼ 1  NC : According to the theory described above, let us define the auxiliary keystone variable using the following equation:   dq N ¼ 1 , qð0Þ ¼ 0: dt C Then, integrating Eq. (2.37) results in lðt, aÞ ¼ lð0, aÞeaqðtÞ , and the total size of the population is given by the formula N ðtÞ ¼ N ð0ÞM0 ½qðtÞ, where M0[λ] is the mgf of the initial distribution of the parameter a. Equation for the population size N(t) is given by the nonautonomous logistic equation   dN N ¼ Et ½aN 1  : dt C Through these simple transformations, we have reduced the original inhomogeneous infinitely dimensional logistic Eq. (2.37) to a single equation for q(t), dq Nð0ÞM0 ½qðtÞ ¼1 ,qð0Þ ¼ 0: dt C The last equation has a stable equilibrium q*, which solves the equation M0 ½q∗  ¼

C : N ð 0Þ

Hence, the limit equilibrium state of inhomogeneous logistic model (2.37) coincides with the current state of the inhomogeneous Malthus model at the instant q*. This and other logistic-like inhomogeneous models will be studied in detail in Chapter 7. Here, let us illustrate the behavior of an inhomogeneous logistic model (2.37) in the following Fig. 2.2. Similarly to the inhomogeneous Malthusian model, we choose truncated exponential distribution on the interval a 2 [0, 1] to enable better comparison between the models. Firstly, one can see in Fig. 2.2A that population N(t) grows up to its carrying capacity (in this case 1), as is expected for a logistic model. The expected value of parameter a also increases over time; however, unlike in the inhomogeneous Malthusian model, it does not tend to its maximal value but to a value below it (Fig. 2.2B). Furthermore, the dynamics of Vart[a] is very different as well—while it undergoes an increase, similarly to the Malthusian case, it does not decrease to zero but remains at a nonzero level. That is, an inhomogeneous population that grows according to the logistic model remains heterogeneous over time. A comparison of the two models, inhomogeneous Malthusian and logistic, is shown in Fig. 2.3.

18

2. Inhomogeneous models of Malthusian type and the HKV method

0.8 0.6

(A)

Et(a)

N(t)

1 0.5 0 0

0.4 0.2

50

100 150 Time (months)

200

(B)

0 0

50

100 150 Time (months)

200

0.1

(C)

Pt(a)

Vart(a)

100 0.05

50 0 200

0 0

50

100 150 Time (months)

200

(D)

100 Time t

0 0

0.5

1

Parameter a

FIG. 2.2 Parametrically heterogeneous logistic growth model with respect to growth rate a. Initial distribution is truncated exponential on the interval a 2 [0, 1] with s ¼ 120, initial population size is N(0) ¼ 0.001, C ¼ 1. As can be seen, (A) in this case, population undergoes a phase of rapid growth at t  120. (B) The rapid growth phase is accompanied by rapid increase in the expected value of a and (C) a rapid increase in variance, which remains at a nonzero value even when the population is at a steady state. (D) The distribution of clones over time also changes away from the initial composition, albeit less dramatically than in the parametrically heterogeneous Malthusian model. Adapted from Kareva, I., 2018. Understanding cancer from a systems biology point of view. In: From Observation to Thepry and Back. Elsevier.

FIG. 2.3 Comparison of distributed Malthusian and logistic growth models, with N(0) ¼ 0.001 and with truncated exponential initial distribution on the interval a 2 [0, 1] with parameter of the distribution being s ¼ 240. For the logistic model, C ¼ 1. As one can see, dynamics up to the rapid growth phase are identical, but at the steady state, the logistic population maintains heterogeneity and consequently lower final Et[a]. Adapted from Kareva, I., 2018. Understanding cancer from a systems biology point of view. In: From Observation to Thepry and Back. Elsevier.

2.5 Inhomogeneous Allee-type models

19

Let us once again emphasize a notable property of the inhomogeneous logistic model (2.37) with a distributed Malthusian parameter: it remains inhomogeneous at any instant and has a nontrivial limit distribution of the parameter at t ! ∞. Every clone that was present initially will be present in the limit steady state. Therefore, inhomogeneous logistic model illustrates the phenomenon of “survival of everybody” in the population, in contrast to Darwinian “survival of the fittest.” The problem of different possible outcomes of the natural selection will be discussed in more details in Chapter 7.

2.5 Inhomogeneous Allee-type models Now, let us consider a model of Allee-type growth, which can demonstrate dynamics qualitatively different from the previous two examples. The Allee equation in its simplest form is given by dx ¼ axðb  xÞðx  mÞ, dt

(2.38)

where x(t) is the population size, b is the carrying capacity of the population, and m is the unstable equilibrium point that divides areas of attraction of the two stable equilibria 0 and b. This model states that fecundity, which equals to a(b  x)(x  m), depends nonmonotonically on the population size x, in contrast to Malthusian and logistic models (see Bazykin, 1998, Chapter 2 for details). Here, we will consider Allee-type models that are heterogeneous with respect to each of the three parameters, a, b, m, and then compare their behavior. In order to study these models, we need to consider heterogeneous population models that are a bit more general than the models of Malthusian type. Namely, let a population be composed of clones that grow according to the equation dlðt,aÞ ¼ lðt, aÞðagðN Þ + f ðNÞÞ, dt

(2.39)

where g(N), f(N) are some function, chosen depending on the specifics of each model. In order to solve problem (2.39), let us define two auxiliary variables q(t), p(t) as the solution to the Cauchy problem dq ¼ gðN Þ,qð0Þ ¼ 0 dt dp ¼ pðN Þ,pð0Þ ¼ 0: dt Then lðt, aÞ ¼ Pð0, aÞNð0ÞeaqðtÞ + pðtÞ NðtÞ ¼ N ð0ÞepðtÞ M0 ½qðtÞ

(2.40)

20

2. Inhomogeneous models of Malthusian type and the HKV method

and Pðt, aÞ ¼ Pð0, aÞ

eaqðtÞ : M0 ½qðtÞ

Notice that total population size depends on the variable p(t) and hence on the function f(N), but the current pdf P(t, a) does not.

2.5.1 Distribution of growth parameter a Consider a population of clones l(t, a) that are characterized by an intrinsic heritable value of parameter a and solve the Allee-type equation dlðt, aÞ ¼ alðt, aÞðb  NðtÞÞðN ðtÞ  mÞ, dt

(2.41)

where N(t) is the total population size, and a, b, and m are constants. It is a model of Malthusian type. Introduce a keystone variable q(t) such that dqðtÞ ¼ ðb  NðtÞÞðN ðtÞ  mÞ dt

(2.42)

Then lðtaÞ ¼ lð0aÞeaqðtÞ ,N ðtÞ ¼ N ð0ÞMa ½qðtÞ, where Ma[λ] is the mgf of initial distribution of the parameter a, and Pðt, aÞ ¼ Pð0, aÞ

eaqðtÞ : Ma ½qðtÞ

(2.43)

The expected value Et[a] and variance Vart[a] of the population are defined through the moment-generating function of the initial distribution according to Eqs. (2.12), (2.13).

2.5.2 Distribution of carrying capacity b Now, consider a population of clones l(t, b) that differ in the value of the carrying capacity b, and solve the Allee-type equation dlðt, bÞ ¼ alðt, bÞðb  N ðtÞÞðN ðtÞ  mÞ: dt Rewrite this equation as dlðt, bÞ ¼ lðt, bÞðbaðNðtÞ  mÞ  aN ðtÞðN ðtÞ  mÞÞ dt

(2.44)

2.5 Inhomogeneous Allee-type models

21

It is a model of type (2.39). According to Eq. (2.40), let us define the auxiliary variables:

Then

dlðt, bÞ dt ¼ lðt, bÞ



dq ¼ aðN ðtÞ  mÞ, dt dp ¼ aNðN ðtÞ  mÞ: dt

(2.45)

lðt, bÞ ¼ lð0, bÞebqðtÞpðtÞ :

(2.46)

dp b dq and consequently dt  dt

Total population size then is given by ð ð N ðtÞ ¼ lðt,bÞdb ¼ lð0, bÞebqðtÞpðtÞ db ¼ N ð0ÞepðtÞ Mb ½qðtÞ, B

(2.47)

B

where Mb[λ] is the mgf of initial distribution of parameter b. The distribution of clones is Pðt, bÞ ¼

lðt, bÞ ebqðtÞ ¼ Pð0, bÞ : N ðtÞ Mb ½qðtÞ

(2.48)

The characteristics of the full system can thus be given by Eqs. (2.47), (2.48) together with Eq. (2.45).

2.5.3 Distribution of parameter m Finally, consider the case, where each clone is characterized by an individual value of parameter m, so that dlðt, mÞ ¼ alðt, mÞðb  N ðtÞÞðNðtÞ  mÞ ¼ lðt, mÞðaðb  N ðtÞÞm + aðb  N ðtÞÞN ðtÞÞ: dt Again, it is a model of type (2.39). Let us define the auxiliary variables dq ¼ aðb  N Þ, dt dp ¼ Naðb  N Þ: dt

(2.49)

N ðtÞ ¼ N ð0ÞepðtÞ Mm ½qðtÞб

(2.50)



dq (p(t)+mq(t)) dl Consequently, dt ¼ l dp . dt + dt , and l(t, m) ¼ l(0, m)e The total population size is given by

where Mm[λ] is the mgf of initial distribution of the parameter m. The distribution of clones is Pðt, mÞ ¼

lðt, mÞ emqðtÞ ¼ Pð0, mÞ : N ðtÞ Mm ½qðtÞ

(2.51)

22

2. Inhomogeneous models of Malthusian type and the HKV method

The dynamics of the system is now completely given by Eqs. (2.50), (2.51) together with Eq. (2.49). Note that formulas for distributions of clones have the same form for all three Allee-type models; the difference is in equations that define the keystone variables q(t) and p(t).

2.5.4 Comparison of the three distributed Allee models Consider the following Fig. 2.4, where we compare the three parametrically heterogeneous Allee-type models. All the examples in Fig. 2.4 were chosen to describe the onset of the rapid growth phase at approximately t ¼ 220. Rapid growth phase predicted by a-distributed Allee model occurs in the most gradual way out of all of the examples and is accompanied by slight increase in the expected value of a and a steady Vart[a] at equilibrium. A population that grows according to b-distributed Allee model exhibits sharper increase in population size during the escape phase; its final population composition is the most homogeneous, with selection toward the largest value of b. Finally, a population that grows according to the m-distributed Allee model exhibits more unusual dynamics, with population size N(t) dropping to near-zero and remaining

FIG. 2.4 Comparison of the three parametrically heterogeneous Allee growth models. The initial distribution is truncated exponential on the interval [0, 1] for all three models. Parameters are a ¼ 1, b ¼ 1, and m ¼ 0.1 for each respective model; the initial population size is N(0) ¼ 0.2. (A) The total population size N(t) for all three models, with a-distributed Allee model increasing most gradually, b-distributed Allee model remaining at a steady size before increasing, and m-distributed Allee model predicting a decrease in population size to near-zero until the escape phase. (B) Expected value of each of the parameters for their respective models and (C) variance of each of the parameters for their respective models. In every case, escape phase is accompanied by increase in both the expected value and variance. The a-distributed model maintains heterogeneity at a steady state; b-distributed model rapidly becomes more homogeneous at the steady state; m-distributed model slowly gradually loses heterogeneity over time after the escape phase. Adapted from Kareva, I., 2018. Understanding cancer from a systems biology point of view. In: From Observation to Thepry and Back. Elsevier.

2.5 Inhomogeneous Allee-type models

23

“dormant” for a long time before eventually rapidly increasing in size. Like in every case described, the rapid growth phase in all of the populations is accompanied by increase in both the expected value of the distributed parameter and in the variance. In this case, the population becomes less heterogeneous over time but does so much more gradually, compared to the population with distributed carrying capacity b. These three models capture the dynamics consistent with what is known as tumor dormancy, or “cancer without disease” (Folkman and Kalluri, 2004). Tumor dormancy describes a phenomenon, when a population of cancerous cells is present in the tissue but is not growing, very similarly to dynamics shown in Fig. 2.4. The applicability of these models to this topic was discussed in (Kareva, 2016); a deeper discussion of other mechanisms that could underlie tumor dormancy, and the corresponding mathematical models can further be found in (Kareva, 2018). These examples demonstrate that introducing heterogeneity even into the most wellstudied and understood models, such as Malthusian, logistic, or Allee-type growth models, can reveal new surprising and rich dynamics. In the next chapter, we will go over more complex examples as applied to specific problems in global demography, forest ecology, and to the effect of toxic agents on heterogeneous microbial populations.

C H A P T E R

3 Some applications of inhomogeneous population models of Malthusian type Abstract In this chapter, we will look at some applications of the HKV method to problems in global demography and discuss whether we will have population explosion in the coming decades as predicted by many models of human population growth. We will then look at the problem of differential susceptibility of microbial populations to antibiotics and finish with a discussion of forest self-thinning, an exciting and important problem in ecology.

3.1 Example 3.1. Conceptual models of global demography The simplest Malthusian model of population growth is rather unrealistic on a large time scale as it predicts that the population will grow exponentially and will eventually tend to infinity, a proposition that terrified Thomas Malthus in 1798 when he first realized his model’s implications. However, the estimates of world population size look even more unbelievable. Specifically the growth of the world population N over hundreds of years, up to
1980, is described with high accuracy by the hyperbolic law (Von Foerster et al., 1960): N ðt Þ ¼

C ðT  tÞk

with C  1:9∗ 1011 , T ffi 2027, k ffi 0:99:

(3.1)

Eq. (3.1) predicts that the population size will tend to infinity in finite time T. Von Hoerner (1975) suggested a simpler expression to describe the same phenomenon: N ðt Þ ¼

C Tt

(3.2)

with T ¼ 2025, C ¼ 200, where N is measured in billions. Eq. (3.2) describes hyperbolic growth and solves the quadratic growth model: dN ¼ N 2 =C, dt Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00003-5

25

(3.3)

# 2020 Elsevier Inc. All rights reserved.

26

3. Some applications to inhomogeneous models of Malthusian type

Although Eqs. (3.2), (3.3) are very simplified “toy” analytical models, they fit well historically available data on the growth of world population, and so it’s possible that they might reflect its real trend. If that is the case, human population growth looks dramatically different from that of other biological populations under the same conditions. The quadratic growth law of Eq. (3.3) was considered to be an exceptional phenomenon inherent only to humankind, and thus human population was considered to be the only one with positive feedback between average reproduction coefficient (growth rate per individual) and population size (see Odum, 1971, Chapter 21). Analysis of simple conceptual models given by Eqs. (3.2), (3.3) is a promising way to understand some basic principles of population dynamics. There exist many attempts in the literature to explain and/or modify model (3.3); see, for example, surveys by (Golosovsky, 2010; Nielsen, 2016) and references within. An interesting theory of global demography was developed by (Kapitza, 1996); see also his report to the Club of Rome (Kapitza, 2006). From “physical” point of view of this theory, Eqs. (3.2), (3.3) describe the self-similar nonlinear growth of a statistically uniform system. The theory was based on the following assumptions (Kapitza, 2010): (1) The entire world population is regarded as a unified strongly coupled evolving system. (2) Its growth rate is proportional to the size of the world population squared. This law is a product of a universal interaction that, being an internal process, is independent of external resources. (3) The interaction is based on the proliferation and dissemination of information of various types. Briefly, humankind is considered in this theory to be a strongly integrated system, whose growth is independent of external resources but is based on information processes within it. This point of view seems to be questionable at the very least. Kapitza also emphasized that the nonlinear quadratic law of growth assumes interrelations, such as cooperation between individuals within the group. “Obviously, this pattern of population growth is only applicable if population is treated as an integrated body.” Later in this chapter we show that the last statement is just incorrect. Namely, the opposite assumption—that mankind is not a strongly integrated body but is composed from many noninteracting populations—may also imply hyperbolic growth of total population that solves the quadratic growth equation given by Eq. ( 3.3). The interpretation of the “quadratic law” (3.3) in terms of the information community hypothesis and especially the underlying assumption of the humankind uniformity are, perhaps, unrealistic: humankind was and is very inhomogeneous according to the main demographic and economic parameters. Let us emphasize that paradoxically (from the standpoint of theory developed by Kapitza) hyperbolic growth was observed in the past but has substantially slowed down during the last decades when most of the world population indeed became an “information community.” Notably, from a “biological” point of view the growth rate in Eq. (3.3) is proportional to the number of “pairwise contacts” in a population. When applied to the world population, this means that the growth rate is proportional to the number of “pair contacts” in the whole of N humankind, and the reproduction rate per individual, which equals to N1 dN dt ¼ C , is proportional to the world population. This is difficult to interpret from the point of view of elementary processes, and it seems clearly incorrect even for small populations.

27

3.1 Example 3.1. Conceptual models of global demography

Thus a contradiction exists between the good numerical accuracy of Eq. (3.2) and the interpretation of Eq. (3.3). Furthermore, Eq. (3.3) and its mathematical corollary (3.2) predict a demographic explosion as we approach the year t ! T ¼ 2025. This means that N(t) ! ∞ as t ! T, and the same is true for the population growth rate and reproduction rate per individual. Let us notice that typically, Eqs. (3.2), (3.3) would be considered to be equivalent. In our opinion, Eq. (3.3) cannot be the starting point of any realistic demographic theory because it clearly makes no biological sense, at least for large populations. Then a problem arises: why does the quadratic Eq. (3.3), which has no acceptable interpretation, provide such a good fit of empirical data by the hyperbola of Eq. (3.2) for such a long period of time? The answer is C that the hyperbola N ðtÞ ¼ Tt can result not only from the quadratic growth model but also from a more plausible Malthusian inhomogeneous model (2.9) with exponential initial distribution (see (Karev, 2005) for details). It seems that the grave inadequacy of many (if not all) conceptual models of global demography is rooted in missing the great heterogeneity of human population. The growth rate of the human population has been and will increasingly be distributed unevenly across the world. According to Bloom and Canning (Bloom and Canning, 2007), today, 95% of population growth occurs in developing countries. The population of the world’s 50 least-developed countries is expected to more than double by the middle of this century, with several poor countries tripling their population over this period. By contrast, the population of the developed world is expected to remain steady with population declines in some wealthy countries; see Fig. 3.1 (see (Merrick, 1986), adapted from (Kapitza, 2006)). Given the great heterogeneity of human population, the simplest inhomogeneous Malthusian model dlðt, aÞ ¼ alðt, aÞ, dt ð

(3.4)

N ðtÞ ¼ lðt, aÞda A

100 90 Annual growth (million)

FIG. 3.1 World demographic transition 1750–2100. Annual growth averaged over a decade. 1, developed countries; 2, developing countries. Reproduced with permission from Merrick, T.W., 1986. World population in transition. Popul. Bull. 41(2). All rights reserved.

80 70 60 50 40

2

30 20 10 0 1750

1 1800

1850

1900

1950

2000

2050

2100

28

3. Some applications to inhomogeneous models of Malthusian type

is a more acceptable starting point for global demography modeling than the quadratic growth equation given by Eq. ( 3.3). Indeed the total population size for model (3.4) is given by the formula (2.7): N ðtÞ ¼ N ð0ÞM0 ½t, where M0[λ] is the mgf of the initial distribution of the Malthusian parameter a. Then the total population can be described by Eq. (3.1) if and only if M0 ½t ¼ 1 k . It is exactly the mgf of the ð1t=TÞ

Gamma distribution as given by Eq. (2.2) with η ¼ 0. In order for the total population growth to 1 be hyperbolic as prescribed by Eq. (3.2), we need to have M0 ½t ¼ ð1t=T Þ, which is the mgf of exponential distribution (see Eq. (2.23)). Then, N ðtÞ ¼ N ð0ÞM0 ½t ¼

N ð0ÞT Tt

(3.5)

As one can see, Eq. (3.5) is the hyperbola of Eq. (3.2) up to notation. Note that Eq. (3.2) allows us to estimate N ð 0Þ ¼

C ¼ 0:099 ffi 0:1 bln: T

With this, we have proven that the solution of the simplest inhomogeneous Malthusian model with exponential initial distribution of the Malthusian parameter is given by the hyperbola of Eq. (3.2). We now have an alternative mechanism underlying hyperbolic growth: it is not a result of hypothetical collective information interconnections reflected in the quadratic growth law but is just a result of real heterogeneity of the humankind. And therefore we can conclude that humanity’s protracted hyperbolic growth was not an exclusive phenomenon but a result of the same laws that govern any heterogeneous biological population. It follows from Proposition 2.4 that the population described by Eq. (3.4) grows in such a way that the distribution of the Malthusian parameter a is exponential at every instant t < T t 1 1 with the mean Et ½a ¼ ðTt Þ and variance Var ½a ¼ ðTtÞ2 . The “demographic explosion” occurs at the moment t ¼ T when not only N(T) ¼ ∞ but also ET[a] ¼ ∞ and VarT[a] ¼ ∞. Our interpretation of the hyperbolic growth helps us reveal the reasons for these unrealistic predictions: they are corollaries of another unrealistic and clearly wrong assumption (incorporated implicitly into the quadratic growth model) that the Malthusian parameter a, which is equal to the reproduction rate per individual, may take arbitrarily large values with nonzero probability. Hence a natural way to eliminate the unrealistic “demographic explosion” from the model is to take into account that possible values of the per capita reproduction rate should be limited. So, let us assume that the values of a in Eq. (3.4) are bounded, and the initial distribution is exponential but truncated in the T 1ebðλTÞ interval (0, b). This distribution is given by formula (2.26), and its mgf is M0 ½λ ¼ Tλ 1ebT (see Eq. (2.27)). Then the total population size is   N ð0ÞT 1  ebðTtÞ : N ðt Þ ¼ Tt 1  eðbTÞ

29

3.1 Example 3.1. Conceptual models of global demography

Specifically, for real demographic data b  0.114 ; in this case, e bT ¼e230 ¼ 5.5∗ 10101, which is practically equal to 0, and hence the last equation reads N ð t Þ ¼ N ð 0Þ

 T  1  ebðTtÞ Tt

(3.6)

Population size N(t) as defined by Eq. (3.6) is finite for all t. It shows hyperbolic growth for a long period of time up to the moment t1 < T such that e b(Tt1) is small; e b(Tt1) < 0.01 if T ¼ 2025 and t1  1980. A simple way to see if the growth of N(t) is close to hyperbolic is to make a graph of N1ðtÞ. Þ Indeed, if N(t) is given by hyperbola (3.2), then the graph of N(t) is just a line, N1ðtÞ ¼ ðTt C ; this approach was systematically used by Nielsen (2014). Fig. 3.2 shows that the graph of N1ðtÞ, where N(t) is defined by Eq. (3.6) with N(0) ¼ 0.1, coincides with the straight line up to the year 1980. Then, after a short transitional period, N(t) grows exponentially; to see it, let us make the graph of logN(t) (Fig. 3.3). We can see that after time moment t  2040 the plot of logN(t) is very close to being linear; it means that N(t) increases exponentially. Fig. 3.4 shows fit of the inhomogeneous Malthusian growth model with truncated initial distribution to world population data, as reported at https://ourworldindata.org/worldpopulation-growth/. Deviation of model solution from real data around 1400 could potentially be explained by the world population decline due to the Black Plague epidemic in Europe in years 1346–53. The transition from hyperbolic to exponential phase of population growth is known in demography as “demographic transition” (Kapitza, 1996, 2006). From the initial time points, the mean value Et[a], which in this case is equal to the growth rate of the total population, 1 N 0.10

0.08

0.06

0.04

0.02

500

1000

1500

FIG. 3.2 The plot of N1ðtÞ , where N(t) is defined by Eq. (3.6); N(0) ¼ 0.1.

2000

t

30

3. Some applications to inhomogeneous models of Malthusian type

logN 1010 108 106 104 100 1

1200

1400

1600

1800

2000

2200

t

FIG. 3.3 Plot of logN(t), where N(t) is defined by Eq. (3.6). × 109

0.2

7 6

Expected value

World population data Model prediction

Population size

5 4

0.1 0.05 0

(B)

3

0

Variance

1

0

500

1000 1500 Year

2000

2500

(C)

2000

3000

2000

3000

× 10–3

1

Vart[a]

0.5 0

0

1000 time

1.5

2

(A)

Et[a]

0.15

0

1000 Time

FIG. 3.4 Fit of Eq. (3.6) with b ¼ 0.012, to world population data. (A) Demographic data together with model predictions, tracking the dynamics over time of total population size N(t). (B) Change over time of the expected value of Malthusian parameter Et[a], during the transition from hyperbolic to exponential phase; (C) change over time of variance of the Malthusian parameter a, Vart[a], during the transition from hyperbolic to exponential phase. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80(1), 151–174.

increases sharply and tends to its maximal possible value; at this time the population transitions to exponential growth stage. This transition is clearly connected with the behavior of the variance of Malthusian parameter a. During this short period, one can observe a sharp bellshaped dynamics with rapid increase, followed by rapid decrease in variance (see Fig. 3.4C).

3.1 Example 3.1. Conceptual models of global demography

31

This model prediction was endorsed by analysis of real demographic data given in (Tolstikhina et al., 2013). It is instructive to study the behavior of model (3.6) around time moment t ¼ T, when the quadratic growth (Eq. 3.3) predicts “demographic explosion.” One can show that if t ! T, then N(t) ! N(0)bT. Next, according to Eq. (2.32), Et ½a ¼

d ln M0 1 b + ðt Þ ¼ ! b=2 T  t 1  ebðTtÞ dt

and Vart ½a ¼

dEt ½a ! ðEq: ð2:33ÞÞ ! b2 =12 as t ! T: dt

Hence the “demographic explosion” is completely eliminated from the inhomogeneous Malthusian model if the initial distribution of the Malthusian parameter is concentrated on a finite interval. Nevertheless, model (3.6) is also not realistic at large t as the exponential growth predicted by this model cannot persist for a long time. To take the next step towards creating a more realistic demographic model, we need to assume that there exists an upper bound for the total population size. Then we arrive at the logistic modification of the inhomogeneous model (3.6), which is written as follows:   dlðt; aÞ N k ¼ alðt; aÞ 1  dt B

(3.7)

Here, we assume that the initial distribution of parameter a is truncated exponential, B is the total carrying capacity, and k > 0 is a constant. This equation at k ¼ 1 was studied in Section 2.4, Eq. (2.35). The generalized inhomogeneous logistic equation will be considered also in Chapters 7 and 11. Now, we can reach a satisfactory agreement of the model solution with some existing forecasts (UNO and IIASA), as well as with the world population data as available for the last 2000 years. Fig. 3.5 shows that solutions of all considered models are very close to the hyperbola as described by Eq. (3.2) for a long time (up to 1990 for corresponding values of coefficients) but are dramatically different at larger t. The inhomogeneous logistic model (3.7) shows a transition from prolonged hyperbolic growth to the brief transitional phase of “nearly exponential” growth (the phase of “demographic transition”) and then to a stabilized regime. The dynamics of Vart[a] for the three models is shown in Fig. 3.6, highlighting further the key differences between the three models as the underlying population composition changes over time due to system dynamics. Let us now give a couple of examples of fitting the inhomogeneous logistic model (3.7) to some real and prognostic demographic data (Figs. 3.7 and 3.8). These examples are given mainly to demonstrate the capabilities of the developed approach and of the constructed models to fit real demographic data and existing forecasts.

N

20

15

10

5

Time 1600

1700

1800

1900

2000

2100

2200

FIG. 3.5 World population size, N (in billions), as a function of time t. Top line corresponds to Eq. (3.2), where N(t) grows hyperbolically, N(t) ! ∞ as t ! T ¼ 2025. Middle line corresponds Eq. (3.6), where N(t) shows hyperbolicexponential growth, N(t) ! ∞ as t ! ∞ . Bottom line corresponds to Eq. (3.7) with B ¼ 12 and k ¼ 2, where N(t) shows hyperbolic-saturation growth, N(t) ! const as t ! ∞.

Var t [a] 0.0025

0.0020

0.0015

0.0010

0.0005

1000

1200

1400

1600

1800

2000

2200

2400

t

FIG. 3.6 Change over time of Vart[a] for the models (3.2), (3.6), (3.7). Top line corresponds to Eq. (3.2), where N(t) grows hyperbolically, N(t) ! ∞ as t ! T ¼ 2025. Middle line corresponds Eq. (3.6), where N(t) shows hyperbolicexponential growth, N(t) ! ∞ as t ! ∞ . Bottom line corresponds to Eq. (3.7) with B ¼ 12 and k ¼ 2, where N(t) shows hyperbolic-saturation growth, N(t) ! const as t ! ∞.

3.2 Example 3.2. Modeling the dynamics of the effect of antimicrobial agents on heterogeneous microbial populations

33

FIG. 3.7 Fitting of model (3.7) to the N IIASA data and forecast of the world population size. The data, N (thin) is fit well 12,000 by the solution (thick) of the inhomogeneous logistic model with b ¼ 0.114, T ¼ 1977, 10,000 r ¼ 1.08, B ¼ 12, 900, N(0) ¼ 0.094. Adapted from Karev, G.P., 2005. Dynamics of inhomoge8000 neous populations and global demography models. 6000 J. Biol. Syst. 13(01), 83–104. 4000 2000 1200

1400

1600

1800

t

2000

FIG. 3.8 Fitting of model (3.7) to the UNO N data and forecast of the world population size. The data, N (thin) is fit well to the solution (thick) of the model with b ¼ 0.114, 10,000 T ¼ 2026, r ¼ 1.8, B ¼ 11, 600, N(0) ¼ 0.104. Adapted from Karev, G.P., 2005. Dynamics of in8000 homogeneous populations and global demography models. J. Biol. Syst. 13(01), 83–104. 6000 4000 2000

1200

1400

1600

1800

2000

2200

t

3.2 Example 3.2. Modeling the dynamics of the effect of antimicrobial agents on heterogeneous microbial populations A microbial population in an environment containing an antimicrobial agent was studied in (Nikolaou and Tam, 2006); the suggested model has (up to notation) the form of an inhomogeneous Malthusian equation: dN ¼ ðK  mðCÞÞN, dt

(3.8)

where K is the physiological growth rate and m(C) is the kill rate induced by the antimicrobial agent, which has concentration C. A more accurate version with logistic growth rate was also discussed:   dN N ¼ KN 1   mðCÞN: (3.9) dt B

34

3. Some applications to inhomogeneous models of Malthusian type

The value of m(C) as compared to K represents the resistance of microbes to a specific antimicrobial agent with concentration C. Population resistance is distributed over a multitude of values. For the Malthusian version of the model, given by Eq. (3.8), the authors derive the equation for the size of an entire population over time and then approximate it using the variance and higher-order cumulants of the distribution of the kill rate within a heterogeneous population. The theory of inhomogeneous Malthusian and logistic equations developed in Chapter 2 allows us to obtain complete solutions of Eqs. (3.8), (3.9). Letting R ¼ K  m(C) be the resistance of microbes to the antimicrobial agent at concentration C, we can consider R as the parameter distributed within the population. The distribution of R can be easily computed if the distribution of concentration C is known. The inhomogeneous model is of the form dlðt, RÞ ¼ Rlðt, RÞ, dt Ð where l(t, R) is population density with respect to resistance R. Let M0 ½λ ¼ A eλR Pð0, RÞdR be the mgf of the initial distribution of resistance. Then the population size at moment t is given eR by N(t) ¼ N(0)M [t], and the current pdf of resistance becomes Pðt, RÞ ¼ Pð0, RÞe . M 0 ½ t

0

For example, if we assume (as in (Nikolaou and Tam, 2006)) that the initial distribution of 2 2 the resistance is normal with a mean m0, variance σ 0 2 , and mgf M0 ½λ ¼ eλ σ0 =2 + λm0 , then the resistance distribution is also normal at any time point t with the mean Et ½R ¼ m0 + σ 0 2 t, var2 2 iance σ 0 2 , and total population size NðtÞ ¼ N ð0Þet σ0 =2 + tm0 . Notably, this assumption is not realistic as the value of resistance parameter should be positive. If the initial distribution of the resistance parameter is Γ-distribution, then Pð0, RÞ ¼ sk ðR  ηÞk1

eðRηÞs , ΓðkÞ

with mean E0 ½R ¼ η + ks, variance σ 0 2 ¼ sk2 , and the mgf M0 ½λ ¼

eλη . ð1λ=sÞk

k For λ < s, R is also Γ-distributed at any moment t < s with the mean Et ½R ¼ η + ðst Þ, variance

k , and total population size N ðtÞ ¼ σ t 2 ¼ ðst Þ2

N ð0Þetη . ð1t=sÞk

The model “blows up” when t ¼ s, that is,

the population size tends to infinity as t ! s, and hence the model is unrealistic. Normal and Γ-distributions are completely characterized by their mean and variance; it follows from the examples given earlier that the fate of the population can be dramatically different at the same initial mean and variance of the resistance. More realistically the actual initial distribution of the resistance should be concentrated in a finite interval; we can assume that the initial distribution is uniform, beta distribution, truncated normal or exponential in that interval. In all these cases the model can be solved explicitly. For example, in the case of truncated exponential distribution, Pð0, RÞ ¼ VesR , where 0  R  c ¼ const, s is the distribution parameter, and V ¼ 1es sc is the normalization constant. Then the current population size N(t) is defined by the formula

3.2 Example 3.2. Modeling the dynamics of the effect of antimicrobial agents on heterogeneous microbial populations

35



 1  ecðtsÞ  , N ðtÞ ¼ N ð0Þ t ð1  esc Þ 1  s and the current distribution of R is the truncated exponential distribution with parameter s-t (see Chapter 2). Next, let us consider a more realistic logistic version of the model as given by Eq. (3.9). Corresponding inhomogeneous model reads     dlðt, CÞ N ðtÞ ¼ lðt, CÞ K 1   mðCÞ , dt B where l(t, C) is the microbial subpopulation under the selective pressure of an antimicrobial agent with concentration C. The model and its solution are simplified if we consider the kill rate m rather than the concentration C as a distributed parameter. In this case the model reads     dlðt, mÞ N ðtÞ ¼ lðt, mÞ K 1  m : (3.10) dt B Notice that mathematically, both versions of the model are equivalent. To solve the model, let us introduce the following auxiliary keystone variable: dq N ðtÞ ¼1 , qð0Þ ¼ 0: dt B Then

  dlðtmÞ dq ¼ lðtmÞ K  m , dt dt

and the solution to Eq. (3.10) becomes lðt, mÞ ¼ lð0, mÞeKqðtÞmt : Let M0[λ] ¼ E0[eλm] be the mgf of the initial distribution of the kill rate m. Then the total population size is given by ð N ðtÞ ¼ lð0, mÞeKqðtÞmt dm ¼ N ð0ÞeKqðtÞ M0 ½t: M

Here, we denoted M to be the domain of the possible values of parameter m. Next, the distribution of the kill rate can be calculated as Pðt, mÞ ¼

lðt, mÞ emt ¼ : NðtÞ M0 ðtÞ

These equations completely solve Eq. (3.10).

36

3. Some applications to inhomogeneous models of Malthusian type

There exist many other examples of hyperbolic growth of population (at least during initial stage of population growth). Some examples pertaining to the growth of populations of cancer cells will be considered in Chapter 11.

3.3 Example 3.3. Models of forest stand self-thinning The problem of population extinction is one of the most important problems in modern ecology. The HKV method can be successfully applied to this problem. In particular, different peculiarities of the dynamics of a dying population may be explained by population heterogeneity. As an example, let us consider one of the oldest important problems in forest ecology, the problem of forest stand self-thinning. A forest stand is defined as a community of trees that is sufficiently uniform in composition, age, or size distribution to distinguish it from adjacent communities, making a forest a collection of stands (Nyland, 2016). Tree interactions, competition for light and other resources, variations in genetic structure and various environmental conditions affect growth and death of trees in complex ways. It seems impossible to take into account all factors that impact death rates of trees in explicit form within the framework of a unit model. A promising way to overcome these difficulties is to construct tree population models with distributed values of the mortality rate a. A forest stand can be seen as a population of nonidentical trees having different mortality rates depending on the conditions and local environment. It was shown in (Karev, 2003) that different expressions describing forest stand selfthinning can be considered as solutions of inhomogeneous Malthusian population decrease model, which we write in the form dlðt, aÞ ¼ aclðt, aÞ dt ð N ðtÞ ¼ lðt, aÞda, A

where c is a scaling constant. Then, lðt, aÞ ¼ lð0, aÞeact , ð N ðtÞ ¼ Nð0Þ eðactÞ Pð0, aÞda ¼ N ð0ÞLðctÞ, A

where L is the Laplace transform of the initial distribution P(0, a). The current pdf of the parameter is given by the formula Pðt, aÞ ¼

N ð0Þeact LðctÞ

The dynamics of population size is defined by equation

(3.11)

37

3.3 Example 3.3. Models of forest stand self-thinning

dN ¼ cEt ½aN: dt The mean value Et[a] can be computed as Et ½ a  ¼ 

1 d ln LðtÞ ; c dt

It solves the equation dEt ½a ¼ cVart ½a: dt

(3.12)

All of these formulas follow from corresponding formulas of Chapter 2, except now the keystone variable is q(t) ¼  ct. Mortality rate of an inhomogeneous population is proportional to the mean value Et[a] of the Malthusian parameter at moment t. Eq. (3.12) shows that Et[a] decreases at a rate proportional to the current variance of a. This implies that the mortality rate of an inhomogeneous population with any initial distribution decreases over time as long as Vart[a] > 0, that is, as long as the population remains inhomogeneous. Now, let us consider some known empirical formulas of tree stand self-thinning from the point of view of the developed theory. The first step is to find initial death rate distribution such that the considered empirical formula coincides with the solution of the corresponding inhomogeneous extinction model. It is an inverse problem of inhomogeneous population modeling: given the function N(t), we need to find a probability distribution P(a) such that N(t) is a solution of inhomogeneous model (3.11) with P(a) as the initial distribution of the parameter. The solution to this problem is given by Eq. (2.36). Let us start from a well-known formula suggested by (Khilmi, 1996) for tree number N(t) of even-age tree stand: N ðtÞ ¼ Nð0Þea0 ðe

ct

1Þ

:

Let us assume that the initial distribution of parameter a is the Poisson distribution Pð0, a ¼ iÞ ¼ eða0 Þ a0 i =i!, i ¼ 0,1, … λ1)

with the mean a0 and the mgf M0[a] ¼ ea0(e Then,

.

NðtÞ ¼ N ð0ÞM0 ½ct ¼ N ð0Þea0 ðe

ct

1Þ

,

which is exactly the Khilmi formula. The distribution of a at any time is also Poisson with the mean Et[a] ¼ a0e ct; see Eq. (2.20). Thus the Khilmi formula describes the Malthusian process of decline of an inhomogeneous population that is composed of a countable number of groups. Individuals from ith group, i ¼ 0, 1, 2, … have mortality rate ic, and the initial size of the group is l0 ðiÞ ¼

N ð0Þea0 a0 i : i!

(3.13)

38

3. Some applications to inhomogeneous models of Malthusian type

It is difficult to give an acceptable interpretation of this model when it allows for an infinite number of groups. Individuals from groups with large i have unrealistically large extinction rates and will be eliminated rapidly from the population. Thus it is reasonable to consider a subdivision of the tree population only to a finite number of groups with different extinction rates that correspond to the truncated Poisson distribution. Assume that parameter a can take only a finite number of values i ¼ 0, 1, … k. It means that the initial distribution is truncated Poisson of the form Pð0, a ¼ iÞ ¼

cðkÞa0 i , i ¼ 0, 1, …k, i!

 1 Pk a 0 i where cðkÞ ¼ is the normalization constant. Then for all time points t, the distrii¼0 i! bution of parameter a is again the k-truncated Poisson distribution, and the population size is given by k  X i N ð0ÞcðkÞ eða0 Þ a0 N ðtÞ ¼

i¼0

i!

:

(3.14)

This formula is more meaningful from a biological perspective than the initial Khilmi formula that corresponds to k ¼ ∞; Eq. (3.14) allows to fit real data more precisely. Moreover, it is now possible to fit the unknown number k of different a-groups to the observed time series of tree population sizes. Thus we can estimate the number of groups of trees that may have different “survival levels.” For example, calculations for pine tree stands provide an estimate of k ¼ 7  10. Fig. 3.9 shows the model solution and real data for normal pine planting (see Zagreev et al., 1992, Table 130). Data fit provides better accuracy than 5% of mean-square deviation. Notice that although the Khilmi formula gives a good approximation of real data of evenage tree stands for a long time period, it has a serious theoretical defect: according to Eq. (3.14), N(t) ! N(0)e( a0) > 0 as t ! ∞, while it should be N(t) ! 0 in reality. The reason of this defect becomes clear when we consider the underlying inhomogeneous model with FIG. 3.9 Data for pine planting (thin) and modified Khilmi formula (3.14) (thick) with k ¼ 9, α0 ¼ 3.57, c ¼ 0.02, and S ¼ 4.2%. Adapted from Karev, G., 2003. Inhomogeneous models of tree stand selfthinning. Ecol. Model. 160(1–2), 23–37.

3.3 Example 3.3. Models of forest stand self-thinning

39

Poisson (or truncated Poisson) distribution of the death rate. Indeed, this representation of the Khilmi formula reveals an implicit underlying assumption that the population contains a subpopulation of “immortal” individuals with i ¼ 0, that is, individuals, whose death rate is a ¼ 0. The initial size of this immortal clone is l0(0) ¼ N(0)e a0 (see Eq. (3.13)), and it is exactly the asymptotic value of population size given by the Khilmi formula. Hence a biologically consistent modification of the model should exclude the immortal clone form consideration. Next, let us consider what is known as the power law of population decline, and its modification. A typical data set on self-thinning of pine tree stands (Zagreev et al., 1980; Terskov and Terskova, 1980) can be well approximated by the power law relationship: N ðt Þ ¼

N ð0Þt0 k ðt + t 0 Þk

,

(3.15)

where t0 is a initial time moment (or initial age). Computing the inverse Laplace transform of function t0 k ðt + t0 Þk (with the help of computer program such as “InverseLaplaceTransform” of integrated packets “Mathematica”) we see that this function is a Laplace transform of the pdf: Pð a Þ ¼

t0 k ak1 eðat0 Þ : Γ ðkÞ

It is the Gamma distribution (2.21) with E½a ¼ tk0 , variance Var½a ¼ tk2 , and mgf 0  k λ : M½λ ¼ 1  t0 Therefore the power law Eq. (3.15) is the solution to the inhomogeneous Malthusian model of population decline, given by Eq. (3.11) if the initial distribution of the mortality rate is the Gamma distribution. At any time moment for such a population, Et ½a ¼

k , ðt 0 + t Þ

Var½a ¼

k ðt0 + tÞ2

:

In reality, there is no need to consider very large values of mortality rate, and so the simplest way to modify Eq. (3.15) is to truncate the Γ-distribution to an interval [0, v]. The truncated distribution is of the form PðaÞ ¼ CðvÞak1 eðat0 Þ , (3.16) !1 Ðv k1 ðat Þ where a 2 [0,v] and where CðvÞ ¼ a e 0 da is the normalization constant. Then, 0

N ðtÞ ¼ C ðt + t0 Þk ðΓðkÞ  Γðkðt + t0 ÞvÞÞ, Ð∞ where C ¼ const, Γ(k, x) is the incomplete Γ-function, Γðk, xÞ ¼ et tk1 dt. x

(3.17)

40

3. Some applications to inhomogeneous models of Malthusian type

These formulas become simpler if we assume that t0 ¼ 0. Then the initial distribution of the mortality rate PðaÞ ¼ kvk ak1 ,

(3.18)

where a 2 [0,v]. Total population size is given by N ðtÞ ¼ k ðvtÞk ðΓðkÞ  Γðk, vtÞÞ:

(3.19)

The dynamics of tree stand number can be described by the modified Eq. (3.19) more precisely than by the initial power law Eq. (3.15) (see Karev, 2003 for details). The current parameter distribution for the considered model with the initial distribution defined by Eq. (3.18) is given by the formula: Pðt, aÞ ¼

eat ak1 tk ΓðkÞ  Γðk, vtÞ

(3.20)

This distribution with t > 0 is exactly the Γ-distribution truncated in [0, v]. Thus the initial power distribution on a bounded interval [0, v] as given by Eq. (3.16) becomes the truncated Γ-distribution on [0, v] due to the inhomogeneous Malthus model of extinction for any t > 0. Some other formulas for tree stand self-thinning can be derived and then modified using the same approach (see (Karev, 2003) for more details). In the next chapter, we will dive deeper into the mathematical underpinnings of the HKV method. If you’ve had plenty of mathematical theory, feel free to skip to Chapter 5 for additional examples.

C H A P T E R

4 Selection systems and the Reduction theorem Abstract In Chapters 2 and 3, we saw how the HKV method can be used to expand and enrich analysis of even very wellknown mathematical models by introducing heterogeneity. Now we want to show how the method can be further developed in application to more general models of “selection systems.” In this chapter, we collect together main mathematical assertions that form the basis for HKV method for selection systems as mathematical models of inhomogeneous populations and communities. This is a theory-heavy chapter, so a reader who is interested primarily in learning how to use the HKV method in different applications can skip it and move on to the next chapter.

4.1 Selection systems A selection system is a mathematical model of a closed inhomogeneous population, in which every individual is characterized by a set of qualitative traits; the values of these traits determine the reproduction rate of the individual. A trait in this context is any characteristic that is inherent to the individual; it is fixed at the very beginning of the individual’s life and does not change over time. More formally, each individual in the model is characterized by a vector-parameter a ¼ (a1, … , an) that takes on values from some set A. Parameter a describes an individual’s inherited invariant properties; it remains unchanged for any given individual but varies from one individual to another, such that the population is nonuniform. Any changes of mean, variance, and other characteristics of the parameter distribution over time are caused only by variation of the population structure. It is assumed that the mean values of the traits are the only information that is known about the entire population. The dynamics of the joint distribution of these traits depending on the initial distribution and on correlations between the traits is the main problem of interest. The set of all individuals with a given value of the vector-parameter a in the population is called an a-clone. Let l(t, a) be the density of the population at time t with respect to parameter a; informally, l(t, a) is the number of individuals in the a-clone, which, as you may recall, is defined as a set of all individuals with a specific value of parameter a. Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00004-7

41

# 2020 Elsevier Inc. All rights reserved.

42

4. Selection systems and the Reduction theorem

Let us denote F(t, a) to be the per capita reproduction rate (Malthusian fitness) of the a-clone at time t. The reproduction rate F(t, a) of an individual from the a-clone depends on a and on the clone’s “environment,” which, in turn, may depend on other population characteristics. If we assume overlapping generations and smoothness of l(t, a) in t for each a 2 A, then the population dynamics can be described by the following general model: dlðt, aÞ ¼ lðt, aÞFðt, a Þ, dt ð N ðtÞ ¼ lðt, aÞda,

(4.1)

A

Pðt, aÞ ¼

lðt, aÞ , N ðtÞ

where N(t) is the total population size and P(t, a) is the pdf of the current distribution of parameter a at time t. The initial pdf P(0, a) and the initial population size N(0) are assumed to be given. Eq. (4.1) comprises the formal (after Price, 1995) selection system with continuous time. Let us now Ðspecify some notation. Hereafter Et means the mean value along the pdf P(t, a). Let Et ½F ¼ A Fðt, aÞPðt, aÞda. t Proposition 4.1 (i) The total population size satisfies the equation dN dt ¼ NE ½F. (ii) The current pdf P(t, a) of the selection system (4.1) solves the replicator equation

  dPðt,aÞ ¼ Pðt, aÞ Fðt, aÞ  Et ½Fðt,aÞ : dt (iii) Replicator equation (4.2) for a given initial distribution P(0, a) has a unique solution. Proof

dN d ¼ dt dt

ð

ð lðt, aÞda5 A

lðt, aÞFðt, aÞda5N ðtÞEt ½F, A

  d d lðt, aÞ lðt, aÞFðt, aÞ lðt, aÞ dN ðtÞ Pðt, aÞ ¼ ¼  2 ¼ Pðt, aÞ Fðt, aÞ  Et ½Fðt, aÞ : dt dt N ðtÞ N ðtÞ N ðtÞ dt Replicator equation (4.2) can be written in the form d lnPðtaÞ ¼ FðtaÞ  Et FðtaÞ: dt Let P1(t, a), P2(t, a) solve the replicator equation and let P1(0, a) ¼ P2(0, a). Then d P1 ðt, aÞ ln ¼ 0, dt P2 ðt, aÞ hence

□

P1 ðt, aÞ ¼ const ¼ 1 for allt: P2 ðt, aÞ

(4.2)

4.1 Selection systems

43

Replicator equations are among the basic tools in mathematical theory of selection and evolution; see, for example, (Hofbauer and Sigmund, 1998). Some examples of selection system (4.2) and solutions of corresponding replicator equations will be considered later in the text. Given the replicator equation (4.2), we can consider the associated selection system (4.1). If the reproduction rate F(t, a) is known explicitly as a function of t, then we can define the reproduction coefficient of the selection system for the time interval [0, t) as Ðt

Kt ðaÞ ¼ eBðt, aÞ ,

(4.3)

where B(t, a) ¼ 0F(u, a)du. It is easy to verify that lðt, aÞ ¼ lð0, aÞKt ðaÞ, NðtÞ ¼ N ð0ÞE0 ½Kt ,

(4.4)

and that Pðt, aÞ ¼

Pð0, aÞKt ðaÞ : E0 ½Kt 

(4.5)

We have shown that to solve the replicator equation, one can find the solution of the associated selection system; its current distribution (4.5) is equal to the desired solution of the replicator equation due to uniqueness. Conversely, if the solution of the replicator equat tion, the pdf P(t, a), is known, then one can solve the equation dN dt ¼ NE ½F and then obtain the solution of model (4.1) using the formula l(t, a) ¼ P(t, a)N(t). Hence problems (4.1) and (4.2) are equivalent. Now we can show that the set of all possible solutions of replicator equations coincides with the set of generalized Boltzmann distributions that have, by definition, the form Pðt, aÞ ¼

eBðt, aÞ Pð0, aÞ, ZðtÞ

(4.6)

Here B(t, a) is a smooth function of time, P(0, a) is a given initial distribution, and Z(t) ¼ E0[eB(t,a)] is the normalization function. To parallel physics terminology, we can call Z(t) a partition function and eB(t,a) a Boltzmann factor. Given the Boltzmann distribution or only the Boltzmann factor, we can define Fðt, aÞ5

d Bðt, aÞ: dt

Proposition 4.2 Any generalized Boltzmann distribution (4.6) solves the replicator equation (4.2). Conversely, if the distribution P(t, a) satisfies the replicator equation, then it is the generalized Boltzmann distribution. Indeed, if P(t, a) is of the form (4.6), then d lnPðt, aÞ d 1 dZ ¼ ðBðt, aÞ  lnZÞ ¼ Fðt, aÞ  , dt dt Z dt

44 and

4. Selection systems and the Reduction theorem

  1 dZ E0 eB F ¼ ¼ Et ½F, Z dt ZðtÞ

so P(t, a) solves Eq. (4.2). Conversely, if P(t,a) satisfies Eq. (4.2), then it is a distribution for associated system (4.1) and hence is of the form (4.5). That is, P(t,a) is a generalized Boltzmann distribution with the Boltzmann factor equal to the reproduction coefficient for the interval [0,t) of system (4.1), Kt(a) ¼ eB(t,a). Proposition 4.2 shows that the generalized Boltzmann distributions and their dynamics are completely described by the replicator equations. It does not mean, of course, that these distributions cannot solve other equations. Taking into account that any smooth function F(t, a) can be well approximated by a finite P sum of the form fi(t)φi(a), we will assume that the reproduction rate is of the form Fðt, aÞ ¼

n X

fi ðtÞφi ðaÞ:

(4.7)

i¼1

Biologically it means that we are considering individual fitness that depends on a given finite set of traits or “predictors” (see Frank, 1997). We refer as a trait to any characteristic that is inherent to the individual, that is, it is fixed at the very beginning of the individual’s life. The function φi(a) in Eq. (4.7) describes the quantitative contribution of a particular ith trait to the total fitness, and functions fi(t) describe relative importance of trait contributions depending on the environment, population size, etc. For example, a may be an individual genotype; then {φi(a)} is the set of phenotypical traits of interest. The function fi(t) describes relative importance and possible variation over time of the trait’s contribution. In this example, Eq. (4.7) defines a mapping from the set of all possible genotypes {a} ¼ A to the set of corresponding fitness functions. Explicit determination of this map is typically a very complex and important biological problem.

4.2 Dynamics of specific distributions Let us formulate the following assertions that capture the dynamics of the system distribution for some probability distributions of biological interest. A selection system, whose evolution is governed by selection over a single trait, φ(a), is the simplest but widely used and important case. Such system is of the form dlðt, aÞ ¼ lðt, aÞFðt, aÞ, dt Fðt, aÞ ¼ f0 ðtÞ + f1 ðtÞφðaÞ:

(4.8)

dq0 ¼ f0 ðtÞ, dt dq1 ¼ f1 ðtÞ, dt qi ð0Þ ¼ 0, i ¼ 1, 2:

(4.9)

Let

45

4.2 Dynamics of specific distributions

Then lðt, aÞ ¼ lð0, aÞeq0 ðtÞ + q1 ðtÞφðaÞ , N ðtÞ ¼ Nð0Þeq0 ðtÞ M0 ½q1 ðtÞ,

(4.10)

Pð0, aÞeq1 ðtÞφðaÞ : Pðt, aÞ ¼ M0 ½q1 ðtÞ Formally the trait φ(a) can be considered a random variable defined on probability space {A, A, P(t, a)} for any t (here A is a σ-algebra of subsets of A). It is known from textbooks that any random variable (r.v.) induces a measure on the line, defined by the formula Pφ ðt, xÞ ¼ Pðt, φðaÞ < xÞ: For simplicity, we will assume that the measure Pφ(t, x) has density pφ(t, x), which is called the probability density function (pdf ) of the r.v. φ. Then, for any function g, Ð Ð Ð A gðφðaÞÞPðt, aÞda ¼ φðAÞ gðxÞPφ ðt, dxÞ ¼ φðAÞ gðxÞpφ ðt, xÞdx: Now we can define the mgf of the distribution P(t, a) as follows: Ð Ð∞ ðλxÞ Mt ðλÞ ¼ A eðλφðaÞÞ Pðt, daÞ ¼ e pφ ðt, xÞdx: ∞

Using Eq. (4.10), we can derive the following equation, which is similar to Eq. (2.10): Mt ½λ ¼

M0 ½λ + q1 ðtÞ : M0 ½q1 ðtÞ

(4.11)

In what follows, we derive a more general equation; see Proposition 4.4. Using Eq. (4.11), we can now prove the following proposition the same way we proved Propositions 2.3–2.5. Proposition 4.3 Consider model (4.8), and assume that the initial pdf of the trait pφ(0, x) is as follows: (i) Normal with mean m0 and variance σ 0 2 . Then the trait distribution will also be normal at any t with mean mt ¼ m0 + σ 0 2 q1 ðtÞ and with the same variance σ 0 2 . (ii) Poisson with mean m0. Then the trait distribution will also be Poisson at any t with the mean mt ¼ m0eq1(t). k1 ðxηÞa k (iii) Γ-distribution with coefficients k, a, η, that is, P0 ðφ ¼ xÞ ¼ a ðxηÞΓðkÞe , where k, a > 0,  ∞ < η < ∞, and x  η; Γ(k) is the Γ-function. Define T ∗ ¼ inf(t : q1(t) ¼ a), if such t exists; otherwise T ∗ ¼ T. Then the trait φ will be Γ-distributed at any time moment t < T ∗ with coefficients k, a – q1(t), η, such that Et ½φ ¼ η + ðaqk1 ðtÞÞ and σ t 2 ¼ ðaqk ðtÞÞ2 . 1

(iv) Assume the trait can take in the interval [0, b]; assume also that its initial pdf corresponds to truncated exponential distribution (2.26). Then at any time instant the trait distribution will also be truncated exponential in the same interval [0, b] with coefficient s  q(t). A list of practically implemented distributions can of course be extended. Some formulas for more commonly used distributions can be found in Chapter 17, Math Appendix.

46

4. Selection systems and the Reduction theorem

Now let us consider a model of a selection system with many traits: dlðt, aÞ ¼ lðt, aÞFðt, aÞ, dt n X Fðt, aÞ ¼ fi ðtÞφi ðaÞ:

(4.12)

i¼1

Let φ ¼ (φ1, … , φn) be a random vector. Similar to the case of single trait, let Pφ(t, x) ¼ P(t, φ1(a) < x1, … , φn(a) < xn) Assume again that Pφ(t, x) has density pφ ðt, xÞ ¼ pφ ðt, x1 , …, xn Þ, which is the pdf of the random vector φ. Let λ ¼ (λ1, … , λn); define the mgf: " !# ! ð n n X X λ i φi ð aÞ ¼ exp λi xi pφ ðt, x1 , …, xn Þdx1 , …, dxn : Mt ½λ ¼ Et exp i¼1

i¼1

φðAÞ

The mgf of the initial pdf of the vector φ, M0(λ), plays a fundamental role in the theory. dq Let dti ¼ fi ðtÞ, qi(0) ¼ 0, i ¼ 1, … , n. Then ! n X qi ðtÞφi ðaÞ , lðt, aÞ ¼ lð0, aÞ exp i¼1

N ðtÞ ¼ Nð0ÞM0 ½φ, and Pð0, aÞ exp

Mt ½λ ¼

Mt ðλÞ ¼ exp

n X

ð exp

φðAÞ

λi φi ðaÞ exp

i¼1

A

¼

!

n X i¼1

n X

λ i φi ð aÞ

M0 ½φ

Proposition 4.4

ð

!

i¼1

Pðt, aÞ ¼

Indeed

n X

:

M 0 ½ λ + qð t Þ  : M0 ½qðtÞ

(4.13)

! qi ðtÞφi ðaÞ Pð0, aÞda=M0 ½qðtÞ

i¼1

ððλi + qi ðtÞÞxi Þpφ ðt, x1 , …, xn Þdx1 , …, dxn

!, M0 ½qðtÞ ¼

M0 ½λ + qðtÞ : M0 ½qðtÞ

If initially traits φi(a) are independent (as random variables on probabilistic space {A, A, P(0, a)}), then they remain independent indefinitely (as random variables on

4.2 Dynamics of specific distributions

47

probabilistic space {A, A, P(t, a)}) for any t; given the initial mgfs, their joint mgf can be easily computed at any time moment. Indeed, if the traits φi(a) are initially independent, then n Y ðiÞ M0 ½λ ¼ M0 ½λi , i¼1

where M(i0 )[λi] is the initial mgf of the ith trait. Then n Q ðiÞ M ½ λ i + q i ðt Þ Y n M0 ½λ + qðtÞ i¼1 0 ðiÞ ¼ n ¼ Mt ½λi : Q ðiÞ M0 ½qðtÞ i¼1 M0 ½qi ðtÞ i¼1

Hence the traits are independent at time t. In the most interesting and realistic cases, the evolution of a system is governed by simultaneous selection over many traits, whose contributions to fitness depend on each other. The evolution of the pdf of the vector φ ¼ (φ1, … , φn) in the general case of correlated traits {φi, i ¼ 1, … , n} is of great practical interest. Let us recall some definitions (see also, e.g., Kotz et al., 2000). A random vector X ¼ (X1, … , Xn) has a multivariate normal distribution with the mean EX ¼ m ¼ (m1, … , mn) and covariance matrix C ¼ {cij}, cij ¼ cov(Xi, Xj) if its mgf is M(λ) ¼ exp (λTm + 1/2λTCλ). A random vector X ¼ (X1, … , Xn) has a multivariate polynomial distribution with parameters (k; p1, … , pn), if k! p1 m1 …pn mn , PðX1 ¼ m1 , …, Xn ¼ mn Þ ¼ m !…m ! 1 n for n X mi ¼ k: i¼1

!k The mgf of the polynomial distribution is n X λi MðλÞ ¼ pi e : i¼1

A general class of multivariate natural exponential distributions is especially important for selection systems and their applications. It includes multivariate polynomial, normal and other distributions as special cases. A random n-dimension vector X ¼ (X1, … , Xn) has multivariate natural exponential distribution (NED) if its joint density function is of the form f(X) ¼ T h(X)eX θs(θ) with respect to the positive measure ν on Rn. Here Ð θX¼T θ(θ1, … , θn) is a vectorparameter, and s(θ) is the normalization function, exp ð s ð θ Þ Þ ¼ hðXÞdνðXÞ. The family Rn e  Pn  of distributions pθ with pdf exp x θ  s ð θ Þ with respect to the measure dμ(x) ¼ h(x) i i i¼1 dν(x) is called the natural exponential family generated by μ. The mgf of NED is Xn  λ x M½λ ¼ Eμ exp ¼ exp ðsðθ + λÞ  sðθÞÞ: i i i¼1 Proposition 4.5 Let us assume that at the initial time moment the vector φ ¼ (φ1, … , φn) has the following: (i) Multivariate normal distribution with the mean vector m(0) and covariance matrix C ¼ (cij). Then the vector φ also has the multivariate normal distribution at any Pmoment t < Τ with the same covariance matrix C and the mean vector mðtÞ,mi ðtÞ ¼ mi ð0Þ + 1=2 nk¼1 ðcik + cki Þqk ðtÞ.

48

4. Selection systems and the Reduction theorem

(ii) Multivariate polynomial distribution. Then the vector φ also has the multivariate polynomial p eqi ðtÞ distribution at any moment t < Τ with parameters (k; p1(t), … , pn(t)), where pi ðtÞ ¼ Pni q ðtÞ : pei j¼1 j

n

(iii) Multivariate natural exponential distribution on R with respect to the Lesbegue measure, with the density function f0(X) ¼ h(X) exp(XTθ  s(θ)) and the mgf M[λ] ¼ exp(s(θ + λ)  s(θ)). Then the vector φ also has the multivariate NED at any moment t < Τ with parameters θ + q(t), the density function ft(X) ¼ h(X) exp(XT(θ + q(t))  s(θ + q(t))), and the moment-generating function Mt(λ) ¼ exp(s(θ + λ + q(t))  s(θ + q(t))). Proof of Proposition 4.5 is based on the main formula (4.13) and is similar to the proof of Propositions 2.3–2.5. Let us prove, for example, assertion (i). According to Eq. (4.13), 0 1 n n n n   X X X X M0 ðλ + qðtÞÞ 1 1 M t ðλ Þ ¼ ðλ + q ðtÞÞcij λj + qj ðtÞ  qi m i  qc qA ¼ exp @ ðλi + qi ðtÞÞmi + M0 ðqðtÞÞ 2 i, j¼1 i i 2 i, j¼1 i ij j i¼1 i¼1 0 1 ! n n  n  X X X 1 1 ¼ exp @ λi mi + c + c q ðt Þ + λ c λ A: 2 k¼1 ik kj k 2 i, j¼1 i ij j i¼1

Let us assume now that the initial pdf of the vector φ ¼ (φ1, … , φn) belongs to the natural exponential family with parameters θ and has initial mgf M0[λ] ¼ exp(s(θ + λ)  s(θ)). Then Mt ½λ ¼

M0 ½λ + qðtÞ ¼ exp ðsðθ + λ + qðtÞÞ  sðθ + qðtÞÞÞ: M0 ½qðtÞ

25A1

4.3 Selection systems with self-regulations Functions fi(t) can theoretically be known explicitly at any time moment, but it is not the case for most realistic models that account for self-limitations of the population growth. For   example, even the simple inhomogeneous logistic model with Fðt, aÞ ¼ ϕðaÞ 1  NBðtÞ does not

satisfy this condition because the current population size N(t) is unknown a priori. Therefore we should investigate a class of models, where fi are functions of the total population size and other population characteristics, which are not known and should be computed at every time moment. In what follows, we show that this nontrivial problem can be solved effectively. We assume that the reproduction rate of every a-clone does not depend on other individual clones but can depend on the actual value of a and on some general population characteristics, such as the total population size N(t), the total biomass and the covariance between different traits within the population. These quantities evolve over time, providing some selfregulation of system dynamics. For example, the reproduction rate of the Ricker model (see, e.g., (Thieme Horst, 2003), Section 5.30) is proportional to e βN(t). Inhomogeneous versions of these models will be considered later.

4.3 Selection systems with self-regulations

49

Most of the population characteristics, which we account for in the model, are functions of some averages within a population; we refer to them here as “regulators.” Formally a regulator has the form ð GðtÞ ¼ gðaÞlðt, aÞda, A

where g(a) is some function. Total population size is the most important regulator, corresponding to g(a)  1. In many cases the reproduction rate may depend on mean values of the characteristic that is distributed within the population; it is described by equation ð t E ½g ¼ gðaÞPðt, aÞda: A

Let us emphasize that these mean values can be computed using corresponding regulators GðtÞ according to the formula Et ½g ¼ N ðtÞ. Overall, for each model we can specify a finite set of regulators G(t) ¼ {G1(t), … , Gm(t)}, which include total population size; we assume that the individual reproduction rate F(t, a) can depend on this set of regulators at each time moment. Specifically, we assume that Fðt, aÞ ¼

n X

ui ðt, GÞφi ðaÞ,

(4.14)

i¼1

where ui(t, G) are continuous functions. The mathematical form of the reproduction rate suggests (from a biological point of view) that an individual’s fitness depends on a given finite set of traits. The function φi(a) in Eq. (4.14) may describe quantitative contribution of a particular ith trait to the total fitness. Then ui(t, G) describes the relative importance (weight) of the ith trait contribution, which at every time moment can depend on the state of the environment, population size, the mean, variance, covariance, and other statistical characteristics of the traits. The dependence of growth coefficient on a few (2–3) parameters is typical for most inhomogeneous models of population and biological communities. An opposite situation appears in problems of mathematical genetics, where a whole genome could be considered as a vector-parameter a of length 106–109. Then φi (a) may be interpreted as a measure of phenotype fitness of ith gene (i ¼ 1, … , n  30, 000 for Homo Sapiens) and ui(t, G) as a “weight” of ith gene, such that the total fitness is a weighted sum of all phenotype fitnesses. Both situations could be studied within the framework of the following master model of a selection system with self-regulation: dlðt, aÞ ¼ lðt, aÞFðt, aÞ, dt n X Fðt, aÞ ¼ ui ðt, GÞφi ðaÞ:

(4.15)

i¼1 ð0, aÞ The initial distribution of the vector-parameter Pð0, aÞ ¼ lN ð0Þ and the initial population size N(0) are assumed to be known.

50

4. Selection systems and the Reduction theorem

We emphasize that the model accounts for the interactions between the traits only with the help of a given set of regulators. Nevertheless, this way we can account for the influence of different statistical characteristics of the system on the system dynamics. For example, if one needs to account for all moments up to the second order, the following set of regulators should be used: ð N ðtÞ ¼ lðt, aÞda, A

ð

Gi ðtÞ ¼ φi ðaÞlðt, aÞda,

(4.16)

A

ð Gik ðtÞ ¼ φi ðaÞφk ðaÞlðt, aÞda: A

Then the covariance between the traits φi, φk at time t is the function of these regulators: uðt, GÞ ¼ Cov½φi , φk ðtÞ ¼

Gik ðtÞ Gi ðtÞGk ðtÞ  : N ðt Þ N2 ðtÞ

Clearly, this way one can account for the dependence of fitness on mixed moments of any order; however, the approach that is described below is truly useful only when we consider just a few regulators. The current probability distribution is defined as before by the formula Pðt, aÞ ¼

lðt, aÞ : N ðtÞ

According to Proposition 4.1, dN ¼ Et ½FNðtÞ, dt where now Et ½F ¼

n X

ui ðt, GÞEt ½φi :

i¼1

Our primary interest now lies in understanding the dynamics of the distribution of the vector-parameter, the population size and the regulators; an approach to solving these problems is described in the following section.

4.4 Reduction theorem for inhomogeneous models of populations In model (4.15) the regulators and hence the reproduction rate F(t, a) are not given explicitly but should be computed using the current pdf P(t, a) at each time moment. The model in the general case is a nonlinear equation of infinite dimensionality. Nevertheless, it can be

4.4 Reduction theorem for inhomogeneous models of populations

51

reduced to a Cauchy problem for the escort system of ODE of dimensionality equal to the number of traits. To this end, introduce the generating functional: ! ð n X λi φi ðaÞ Pð0, aÞda, (4.17) Φðr; λÞ ¼ rðaÞexp i¼1

A

where λ ¼ (λ1, … , λn), and r(a) is a measurable function on A. Define auxiliary (keystone) variables as a solution to the escort system of differential equations: dqi ðtÞ ¼ ui ðt, G∗ ðtÞÞ,qi ð0Þ ¼ 0, i ¼ 1, …, n, dt where G∗(t) ¼ {G∗1(t), … , G∗m(t)} and G∗k (t) ¼ N(0)Φ(gk, q(t)), q(t) ¼ {q1(t), …qn(t)}. Denote Kt ðaÞ ¼ exp

n X

(4.18)

! qi ðtÞφi ðaÞ :

(4.19)

i¼1

Reduction Theorem 4.1 Let 0 < T  ∞ be the maximal value of t such that Cauchy problem (4.18) has a unique solution q(t) at t 2 [0, T). Then the function lðt, aÞ ¼ lð0, aÞKt ðaÞ,

(4.20)

Gk ðtÞ ¼ G∗k ðtÞ ¼ Nð0ÞΦðgk , qðtÞÞ,

(4.21)

satisfies system (4.15) at t 2 [0, T) and for all k ¼ 1, … , m Conversely, any solution of system (4.15) has forms (4.20) and (4.19), where the variables qi(t) solve the dq ðtÞ Cauchy problem dti ¼ ui ðt, GðtÞÞ, qi ð0Þ ¼ 0, i ¼ 1, …, n: Proof Let q(t) be the solution of Cauchy problem (4.18); denote G∗i ðtÞ ¼ N ð0ÞΦðgi ; qðtÞÞ, F*ðtaÞ ¼

n X

ui ðtG∗ ðtÞÞφi ðaÞ:

i¼1

Then dlðt, aÞ ¼ lð0, aÞKt ðaÞF∗t ðaÞ ¼ lðt, aÞF∗t ðaÞ: dt Next,

ð ð GðtÞ ¼ gðaÞlðtaÞda ¼ gðaÞKt ðaÞlð0aÞda ¼ N ð0ÞΦðg; qðtÞÞ ¼ G*ðtÞ: A

A

Hence F*(t, a) ¼ F(t, a); this means that functions l(t, a) defined by Eq. (4.20) satisfy system (4.15).

52

4. Selection systems and the Reduction theorem

Conversely, let l(t, a), Gk(t), and k¼ 1,…m solve system (4.15) for t 2 [0, T). For now, define the functions qi(t) and i ¼ 1, … n using the formula ðt qi ðtÞ ¼ ui ðxGðxÞÞdx: 0

Then n X dlðt, aÞ ¼ lðt, aÞFðt, aÞ ¼ lðt, aÞ φi ðaÞdqi ðtÞ=dt, dt i¼1

and lðt, aÞ ¼ lð0, aÞ exp

n X

! φi ðaÞqi ðtÞ ,

i¼1

for all t 2 [0, T). Hence ð GðtÞ ¼

ð gðaÞlðt, aÞda ¼

A

gðaÞexp A

n X

! ϕi ðaÞqi ðtÞ lð0, aÞda ¼ N ð0ÞΦðg; qðtÞÞ:

i¼1

From the definition, q(t) is the solution of Cauchy problem (4.17) for t 2 [0, T). Theorem is proven. □ Some important and useful corollaries immediately follow from the Reduction theorem, which we collect together in the following theorem. Consider again model (4.15) of a selection system. Theorem 4.2 (i) The total size of the population can be computed by the formula NðtÞ ¼ N ð0ÞΦð1, qðtÞÞ ¼ N ð0ÞE0 ½Kt :

(4.22)

(ii) The current pdf is defined by equation Pðt, aÞ ¼

Pð0, aÞKt ðaÞ : E0 ½Kt 

(4.23)

(iii) The mean values with respect to the current pdf can be computed by the formula Et ½ φ ¼

E0 ½φKt  : E0 ½Kt 

Let us emphasize that formula (4.23) that defines the pdf of the model and its dynamics is the central result of the theory. Formulas (4.19), (4.23) show that the distribution of the vector-parameter a belongs to the natural exponential family of distributions defined earlier (see also Chentsov, 1982). In other terms, the pdf (4.23) is the time-dependent Boltzmann distribution of the form (4.6) with the Boltzmann factor eB, where

53

4.5 Reduction theorem for inhomogeneous models of communities

BðqðtÞ; aÞ ¼

n X

qi ðtÞϕi ðaÞ,

(4.24)

i¼1

and the partition function is given by

 ZðqðtÞÞ ¼ E0 exp ðBðqðtÞ; aÞ ¼ E0 ½Kt :

(4.25)

Remark that within the frameworks of selection system (4.15), the partition function as a function of time is completely known given the initial pdf P(0, a) and the solution to the Cauchy problem (4.18). It is proportional to the current population size: ZðqðtÞÞ ¼

N ðt Þ , N ð 0Þ

which follows from Eq. (4.22).

4.5 Reduction theorem for inhomogeneous models of communities Consider a model of a community that consists of r interacting populations (see Karev, 2010b). We assume again that every individual is characterized by their own value of vector-parameter a. Let lj(t, a) be the density of a-clone in jth population at time t. In this section, we consider the model of an inhomogeneous community, where reproduction rates can depend on the parameter a and on current characteristics of every population in the community, composing a “regulator.” Formally, weconsider the set of m regulators, each of which is the r-dimension vector-function Gi ðtÞ ¼ Gi 1 ðtÞ, …, Gi r ðtÞ and i ¼ 1, … , n, where ð j Gi ðtÞ ¼ gi ðaÞlj ðt, aÞda: (4.26) A

Each regulator corresponds to an appropriate weight function gi. A finite set of regulators corresponds to each specific model; weÐdenote this set as G(t) ¼ (G1(t), … , Gn(t)). The current population sizes Nj ðtÞ ¼ A lj ðt, aÞda compose a regulator of particular importance, N(t) ¼ (N1(t), … , Nr(t)). We assume that N(t) is included in the set of the model’sj regulators. The distribution of jth population in the community is by definition Pj ðt, aÞ ¼ lNðtj,ðtaÞÞ. The model of inhomogeneous community considered here is of the form dlj ðt, aÞ j ¼ l ðt, aÞFj ðt, aÞ, dt Fj ðt, aÞ ¼

n X

ui j ðt, GÞϕi ðaÞ,

(4.27) (4.28)

i¼1

where the functions ui j can be specific for each trait and each population. Pj(0, a) and the initial population sizes Nj(0) are assumed to be given. The current pdf Pj(t, a) solves the replicator equation:    dPj ðt, aÞ ¼ Pj ðt, aÞ Fj ðt, aÞ  Et j Fj : dt

(4.29)

54

4. Selection systems and the Reduction theorem

The theory for inhomogeneous community model (4.26)–(4.29) is similar to the theory presented earlier for inhomogeneous populations. The following theorem about the model of inhomogeneous community reduces complex model (4.26)–(4.28) to an escort system of ordinary nonautonomous equations of dimension r  n and gives the solution to replicator Eq. (4.29). We omit the proofs because these theorems are similar to the corresponding theorems for inhomogeneous populations given in the previous section up to more complex technical details. Let us define the keystone variables qji(t) by the escort system of ordinary nonautonomous equations of dimensionality r  n: dqi j ¼ ui j ðtGi *ðtÞÞ, qi j ð0Þ ¼ 0, j ¼ 1, …, r, i ¼ 1, …, n: dt Here

(4.30)

  qj ðtÞ ¼ q1 j ðtÞ, …, qn j ðtÞ ,   Gi ∗j ðtÞ ¼ N j ð0ÞΦj gi ; qj ðtÞ ,

and Φj(r; λ) is the generating functional (4.17) for the initial distribution Pj(0, a). Specifically,   N ∗j ðtÞ ¼ Nj ð0ÞΦj 1; qj ðtÞ , Pn j  Denote Kt j ðaÞ ¼ exp i¼1 qi ðtÞϕi ðaÞ . Theorem 4.3 Assume Cauchy problem (4.30) has a unique global solution at t 2 [0, T), 0 < T, ∞ . Then the functions lj ðtaÞ ¼ lj ð0aÞKt j ðaÞ,  Gi j ðtÞ ¼ Nj ð0ÞΦj gi ; qj ðtÞ ,   Nj ðtÞ ¼ N j ð0ÞΦj 1; qj ðtÞ satisfy system (4.26)–(4.28) at t 2 [0, T). The pdf Pj ðt, aÞ ¼

Pj ð0, aÞKt j ðaÞ Pj ð0, aÞKt j ðaÞ   ¼ j , Φ ð1; qj ðtÞÞ E0 Kt j

solves the replicator equations (4.29). It follows from Theorem 4.3 that

! n X dN j j j j ¼N ui ðt, Gi ÞEt ½ϕi  , dt i¼1

where E t ½ ϕi  ¼ j

  Φj ϕi ; qj ðtÞ Φj ð1; qj ðtÞÞ

:

(4.31)

4.5 Reduction theorem for inhomogeneous models of communities

55

In the next chapter, we will look at numerous applications of the Reduction theorem. We will consider examples from ecology with distribution of two and three parameters and tie the Reduction theorem into the well-developed field of selection systems, where it adds several additional insights.

C H A P T E R

5 Some applications of the Reduction theorem and the HKV method Abstract In this chapter, we will provide more examples of applications of the HKV method to a variety of problems, including evolution of altruism, competition between inhomogeneous populations, and predator-prey type systems, among others. We will introduce examples of distribution of two and even three parameters to demonstrate applications of the theory to more complex biological problems. Finally, we will discuss in detail the work of the founding fathers of selection theory (Fisher, Price, and Haldane) and show how the Reduction theorem fits into and even contributes to the well-established field of selection theory.

5.1 How to solve selection systems The Reduction theorem proved in Chapter 4 provides the basis for the powerful hidden keystone variables (HKV) method for solving selection systems and corresponding replicator equations. Introducing auxiliary keystone variables, which solve the escort system of nonautonomous differential equations, allows us to write down the solution of a selection system of the form of Eq. (4.18) and the corresponding replicator equation and to find all the corresponding statistical characteristics of interest. According to the results proven in Chapter 4, the following steps should be performed for solving a selection system: dlðt, aÞ ¼ lðt, aÞFðt, aÞ dt , n X Fðt, aÞ ¼ ui ðt, GÞφi ðaÞ

(5.1)

i¼1

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00005-9

57

# 2020 Elsevier Inc. All rights reserved.

58

5. Some applications of the Reduction theorem and the HKV method

(1) Writing down the expression for the generating functional ! ð n X Φðr; λÞ ¼ rðaÞexp λi φi ðaÞ Pð0, aÞda; i¼1

A

(2) Solving escort system of ODE for keystone variables dqi ðtÞ ¼ ui ðt, G∗ ðtÞÞ, qi ð0Þ ¼ 0, i ¼ 1,…n , dt   G∗ ðtÞ ¼ G∗1 ðtÞ, …G∗m ðtÞ

(5.2)

where G∗k ðtÞ ¼ N ð0ÞΦðgk , qðtÞÞ, qðtÞ ¼ fq1 ðtÞ, …qn ðtÞg; (3) After that, the solution to the selection system l(t, a), the population size N(t), and the values of regulators at t moment are given by l(t, a) ¼ l(0, a)Kt(a), where Kt ðaÞ ¼ exp

n X

! qi ðtÞφi ðaÞ

i¼1

:

Gk ðtÞ ¼ Φðgk , qðtÞÞ

(5.3)

NðtÞ ¼ N ð0ÞΦð1, qðtÞÞ ¼ N ð0ÞE0 ½Kt  (4) The main result is the equation for the pdf at each time moment: Pðt, aÞ ¼

Pð0, aÞKt ðaÞ : E0 ½Kt 

(5.4)

The general HKV method is simplified in one particular but possibly the most important case when the moment-generating function of the joint initial distribution of the variables φi, " !# n X 0 λi φi , M0 ½λ ¼ E exp i¼1

can be used instead of the general generating functional. P Proposition 5.1 Assume that the reproduction rate Fðt, aÞ ¼ ni¼1 ui ðt, GÞφi ðaÞ depends only on the regulators of the form N(t), Gi(t) ¼ Et[φi]. Then the system of ODEs for keystone variables reads dqi ¼ ui ðt, GðtÞÞ, qi ð0Þ ¼ 0, i ¼ 1, …n, dt

(5.5)

Example 5.1 Birth-and-death equation with distributed birth rate and average death rate

59

where regulators are defined by N ðtÞ ¼ N ð0ÞM0 ½qðtÞ, Et ½φi  ¼ ∂i lnM0 ½qðtÞ: The solution to the corresponding replicator equation is Pðt, aÞ ¼ Pð0, aÞ

Kt ðaÞ , E0 ½Kt 

where E0[Kt] ¼ M0[q(t)]. In general, Eq. (5.1) is a complex integrodifferential equation of, perhaps, infinite dimensionality. The HKV method allows us to reduce Eq. (5.1) to the escort system of ODEs (5.2) or (5.5) and obtain the solution at any instant within the time interval, where the global solution of the escort system exists. Now, let us apply the HKV method to several examples of selection systems and give solutions of some particular equations that were previously introduced (but not solved explicitly) in the literature. A wide class of selection systems can be described by the inhomogeneous birth-and-death equations of the form dlðt, aÞ ¼ lðt, aÞFðt, aÞ , dt Fðt, aÞ ¼ f1 ðtÞbðaÞ  f2 ðtÞdðaÞ where f1(t)b(a) is the birth rate, f2(t)d(a) is the death rate, and fi are the functions of regulators of the form H(t) ¼ Et[h] or G(t) ¼ N(t)Et[g]. Different versions of the birth-and-death equation were studied and discussed in numerous works (see, for example, Gorban and Khlebopros, 1988; Ackleh et al., 1999, 2005; Perthame, 2006; Gorban, 2007).

Example 5.1 Birth-and-death equation with distributed birth rate and average death rate Let us consider an inhomogeneous population, where all individuals have an inherited value of the vector parameter a. Let b(a) be the specific birthÐ rate for these individuals, and let the death rate be determined by the common factor A mðaÞlðt, aÞda, where m(a) is the individual contribution to this death rate. Then the dynamics of the population is described by the equation (see Gorban, 2007) dlðt, aÞ ¼ lðt, aÞFðt, aÞ dt ð Fðt, aÞ ¼ bðaÞ  mðaÞlðt, aÞda ¼ bðaÞ  NðtÞEt ½m A

:

60

5. Some applications of the Reduction theorem and the HKV method

Using the method summarized earlier, we are now able to write an explicit solution of this model. Introduce the generating functional Φ(r; λ1, λ2) ¼ e λ2E0[reλ1b]. Then the escort keystone system reads dq1 ¼ 1, q1 ð0Þ ¼ 0, hence q1 ðtÞ ¼ t; dt   dq2 ¼ N ð0ÞΦðm; q1 , q2 Þ ¼ N ð0Þeq2 E0 metb , q2 ð0Þ ¼ 0: dt The last equation can be rewritten as eq2

  dq2 ¼ N ð0ÞE0 metb ; dt

in this form, the equation can be easily integrated, since E0[metb] ¼ f(t) is a known function of t at given P(0, a). Therefore, ðt q2 ðtÞ ¼ 1 + Nð0Þ f ðsÞds: e 0

Then the solution to the original model is given by the formula lðt, aÞ ¼ lð0, aÞetbðaÞq2 ðtÞ ¼

lð0, aÞetbðaÞ : ðt 1 + N ð0Þ f ðsÞds 0

Additionally, total population size can be calculated as   Nð0ÞE0 etbðaÞ N ðtÞ ¼ , ðt 1 + N ð0Þ f ðsÞds 0

and the distribution of clones in the population is given by Pðt, aÞ ¼

lðt, aÞ Pð0, aÞetbðaÞ ¼ 0 tbðaÞ : N ðt Þ E ½e 

Example 5.2 A model of group selection and evolution of altruism There exist a large number of papers devoted to the question of evolution of altruism. We do not discuss here the interesting and important problem how an altruistic trait can be selected—this complex topic is discussed extensively in the literature (see, for instance, Axelrod and Hamilton, 1981; Smith, 1998; Wilson, 2005; Fletcher and Zwick, 2006; Lehmann and Keller, 2006; Nowak, 2006). Instead, using the developed tools, we give an exact solution of a model used in Wilson and Dugatkin (1997) and show that this model possesses some unexpected dynamical properties.

Example 5.3 The principle of limiting factors in modeling of early biological evolution

61

Consider a population, in which every individual possesses a trait x that increases the fitness of everyone in the population (including itself) by mx at a personal cost  cx. Then the fitness of the individual is Fðt, xÞ ¼ cx + mN ðtÞEt ½x, x > 0,

(5.6)

which incidentally coincides with definition of fitness in the previous example up to notation but has the opposite sign. It is interesting that model (5.6) exhibits dramatically different dynamical behavior. Denoting V ¼ N(0)E0[x] as the “initial amount of the altruistic trait” in the population, we can write the model solution as lðtxÞ ¼

lð0xÞetc lð0xÞetc m when c 6¼ 0 ¼ ðt 1 V ð1  etc Þ c 1 + N ð0Þ E0 ½mxesc ds 0

and lðtxÞ ¼

lð0xÞ when c ¼ 0: ð1  mtV Þ

We can now observe an interesting phenomenon (Karev, 2010). If the initial amount of the altruistic trait V > mc , then the equation V ð1  etc Þ ¼ mc has a solution t ¼ T < ∞, so the total population size and the size of every x-clone tend to infinity at a certain instant T, and hence the model loses biological meaning. If V < mc , then the size of every x-clone tends to 0 and the population eventually goes to extinction. However, if the strong equality V ¼ mc is fulfilled, then l(t, x) ¼ l(0, x) for all t. Therefore, we can conclude that the model with fitness (5.6) is not appropriate for modeling a real population.

Example 5.3 The principle of limiting factors in modeling of early biological evolution Principle of limiting factors (according to Liebig (1876); see also Poletaev (1966) for mathematical formulation) states that at any given moment the rate of a process is determined by the factor, whose sufficiently small modification produces a change of the process rate; it is assumed that similar changes in other factors do not affect the process rate. The principle of limiting factors was actually used in a model of early biological evolution suggested in Zeldovich et al. (2007). Each organism was characterized by a vector a, where the component ai is the thermodynamic probability that protein i is in its native conformation. In order to study the connection between molecular evolution and population, the authors assume that the death rate d of the organism depends on the stability of its proteins such that b d ¼ d0(1  min ai), where d0 ¼ ð1a , b is the birth rate and a0 is the probability of a protein being 0Þ in its native state. Hence, if we for now disregard possible mutations (which were accounted for by the authors in their simulations), the model can be formalized as follows: dlðt, aÞ ¼ lðt, aÞðd0 ðmðaÞ  a0 ÞÞ, dt

(5.7)

62

5. Some applications of the Reduction theorem and the HKV method

where m(a) ¼ min[a1, … , an]. In what follows, we let d0 ¼ 1 for simplicity. It was assumed in Zeldovich et al. (2007) that the values ai are independent and fall within the Boltzmann distribution. We can Ð consider ai as the ith realization of a random variable with a common pdf f(a). Let G(a) ¼ a0f(x)dx be the cumulative distribution function. Then, it is well known (see, e.g., Casella and Berger (2002), Theorem 5.4.4) that the pdf of m(a) ¼ min[a1, … , an] is equal to g(a) ¼ n(1  G(a))n1f(a). Eq. (5.7) is a version of the inhomogeneous Malthusian equation, which can now be solved explicitly for any given pdf f(a). In particular, if a

X a e T f ðaÞ ¼ , Z¼ e T Z a is the Boltzmann distribution with a > 0, then gðaÞ ¼ nð1  GðaÞÞ

n1

f ðaÞ ¼ n

X

(5.8)

!n1 ðx=TÞ

e

eða=TÞ =

x>a

X

eðx=TÞ :

x

For distribution (5.8) with continuous range of values of a, a 2 (0, ∞),

and

Z ¼ T, 1  GðxÞ ¼ eðx=TÞ  n eðaðn1Þ=TÞ eða=TÞ neðan=TÞ ¼ : gð a Þ ¼ T T

(5.9)

If a 2 (0, E), then Ea   e T 1 Z ¼ T 1  eE=T , 1  GðaÞ ¼ E , eT  1

and gðaÞ ¼

n1 nea=T 1  eðEaÞ=T : T ð1  eE=T Þ 1  eE=T

Let M0[λ] ¼ E0[eλm]. For initial distribution (5.9), M0 ½λ ¼

1 : 1  λT=n

Hence, Pðt, aÞ ¼ Pð0, aÞemðaÞt ð1  tT=nÞ 1 N ðtÞ ¼ N ð0Þea0 t : 1  tT=n lðt, aÞ ¼ lð0, aÞeðmðaÞa0 Þt

(5.10)

Example 5.4 Inhomogeneous Ricker equation with two distributed parameters

63

At the moment tmax ¼ Tn the population “blows up”: N(t) and l(t, a) tend to infinity as t ! tmax : Let us denote p(t, a) ¼ P(t, {a : m(a) ¼ a}). Then, at t < tmax,

 n an=T + at tT n  aðtn=TÞ t e 1 : ¼ pðt, aÞ ¼ e T n T The probability p(t, {a : m(a) < a}) tends to 0 for any finite a as t < tmax. Loosely speaking, the total “probability mass” goes to infinity after a finite time interval. So, we should conclude that the model defined by Eqs. (5.7), (5.8), which allow arbitrarily large values of the parameter a with nonzero probability, has no “physical” or realistic meaning. This problem can be eliminated by taking the initial distribution (5.10), which allows for only bounded values of the parameter a. For pdf defined by Eq. (5.10), the integral ÐE M0 ½λ ¼ eλx gðxÞdx is finite for any λ; although it cannot be expressed in quadratures, we 0

can obtain a lot of information about system distribution and its dynamics. The current pdf of the system is given by  n1 eat n ðEaÞ=T ðEaÞ=T , e e  1 pðt, aÞ ¼ n M0 ½t T ðeE=T  1Þ where M0[t] is finite for all t. So, the pdf is defined and finite at any time moment, in contrast to the previous case. The total distribution concentrates over time at the point a ¼ E, which provides the maximal reproduction rate. Let us emphasize that the pdf p(t, a) does not depend on the native state probability a0.

Example 5.4 Inhomogeneous Ricker equation with two distributed parameters Let us consider an inhomogeneous version of the well-known Ricker equation, which depends on two distributed parameters:   dlðt; β, μÞ ¼ lðt; β, μÞ βecNðtÞ  μ , (5.11) dt where c is a scaling constant. Let

ð

M0 ½λ1 , λ2  ¼

eλ1 β + λ2 μ P0 ðβ, μÞdβdμ

β, μ

be the mgf of the joint initial distribution of β and μ. Denoting u1 ðN ðtÞÞ ¼ ecNðtÞ , u2 ¼ 1,

64

5. Some applications of the Reduction theorem and the HKV method

we can write the escort system (5.5) as dq2 ¼ 1, q2 ð0Þ ¼ 0, hence q2 ðtÞ ¼ t, dt dq1 ¼ ecNð0ÞM½q1 , t , q1 ð0Þ ¼ 0: dt

(5.12)

Eq. (5.12) can be solved for known mgf M[λ1, λ2], and then the solution to Eq. (5.11) is lðt; β, μÞ ¼ lð0; β, μÞeβq1 ðtÞμt : The total population size and the pdf are given by N ðtÞ ¼ N ð0ÞM½q1 ðtÞ,  t, Pðt; β, μÞ ¼

Pð0; β, μÞeβq1 ðtÞμt : M½q1 ðtÞ,  t

For example, let parameters β and μ be independent and exponentially distributed in [0, ∞) with the mean values of parameters at the initial time moment being s1 and s2, respectively. Then M½q,  t ¼

s1 s2 , ðs1  qÞðs2 + tÞ

and Eq. (5.12) for the auxiliary variable reads

dq1 s1 s2 ¼ exp cN ð0Þ : dt ðs1  q1 Þðs2 + tÞ This equation reaches a steady state at q1 ¼ s1. As t ! ∞, q1(t) ! s1 and M[q1,  t] ! ∞; therefore, over time the total population size tends to infinity; population density becomes concentrated at μ ¼ 0 for parameter μ and vanishes in any finite interval of values of parameter β. This unrealistic behavior is a corollary of two assumptions: the death rate μ can be arbitrarily small, and the growth coefficient β can be arbitrarily large. It can be proven that if the parameters β and μ are independent and there exist max β ¼ β∗ < ∞ and min μ ¼ μ∗ > 0, then the limit distribution of the parameters will be concentrated at the point (β∗, μ∗). Indeed, using the covariance equation (as you will see in Example 5.9, Eq. (5.47)) and assuming that parameters β and μ are independent we get h i dEt ½β ¼ Covt βecNðtÞ  μ, β ¼ Vart ½βecNðtÞ > 0, dt h i dEt ½μ ¼ Covt μecNðtÞ  μ, μ ¼ Vart ½μ < 0: dt Hence, the mean value Et[β] of parameter β increases until Vart[β] > 0 (that is, as long as the population is inhomogeneous) and tends to a maximal possible value Et[β] ¼ β∗. This means that the limit distribution of parameter β will be concentrated at the point β ¼ β∗. Similarly, the mean value Et[μ] decreases as long as the population is inhomogeneous and tends to a minimal possible value Et[μ] ¼ μ∗, so the limit distribution of the parameter will be concentrated at the point μ ¼ μ∗.

Example 5.5 Inhomogeneous prey-predator Volterra model with three distributed parameters

65

Example 5.5 Inhomogeneous prey-predator Volterra model with three distributed parameters The prey-predator Volterra model in its simplest form reads: dx ¼ a1 x  a2 xy dt dy ¼ a3 y + a4 xy dt

,

where x(t) and y(t) denote prey and predator densities, respectively; a1 is the reproduction rate of the prey population; a2 is the per capita rate of the consumption of prey by the predator; a3 is the death rate of the predator; and a4/a2 is the fraction of prey biomass, which is converted into predator biomass. Let us consider the inhomogeneous version of this classical model by assuming that parameters a1, a2, a3 are distributed and the ratio a4/a2 is fixed (and hence could be chosen equal to 1). Then we get a system dx=dt ¼ a1 x  a2 xy dy=dt ¼ a2 xy  a3 y

:

(5.13)

Let x(t; a1, a2), y(t; a3) be the densities of the prey and predator populations with respect to parameters a1, a2, and a3 correspondingly, and let ð ð XðtÞ ¼ xðt; a1 , :a2 Þda1 da2 , YðtÞ ¼ yðt; a3 Þda3 A

A

be the total sizes of the two populations. The initial population sizes X(0), Y(0) and initial distributions P(0; a1, a2), P3(0; a3) are assumed to be given. We also assume that the reproduction and death rates are specific for each subpopulation, while consumption is driven by the interaction of the prey (predator) subpopulation with the entire predator (prey) population. Then, assuming the “proportional distribution” of prey among the predators, we can write the inhomogeneous version of Volterra’ model in the form: dxðt; a1 , a2 Þ ¼ xðt; a1 , a2 Þða1  a2 YðtÞÞ dt dyðt; a3 Þ ¼ yðt; a3 ÞðGðtÞ  a3 Þ dt

,

(5.14)

Ð where GðtÞ ¼ A a2 xðt; a1 , a2 Þda1 da2 ¼ XðtÞEt ½a2 . Theorem 4.3 provides a method for studying this model. The key step is reduction of this model to an escort system of ODE. It is instructive to derive the escort system and the main results informally to clarify the main idea of the method in application to community models. It is reasonable to assume that parameter a3 is stochastically independent of parameters a1, a2. Let M[λ1, λ2] be the mgf of the initial joint distribution of parameters a1, a2, and let M3[λ3] be the mgf of the initial distribution of parameter a3.

66

5. Some applications of the Reduction theorem and the HKV method

Let us now formally introduce auxiliary keystone variables q1(t), q2(t) as solutions to the following Cauchy problem: dq1 ¼ YðtÞ dt dq2 ¼ GðtÞ ¼ XðtÞEt ½a2 , dt

(5.15)

q1 ð0Þ ¼ q2 ð0Þ ¼ 0 Then system (5.14) can be written as

dq1 dxðt; a1 , a2 Þ ¼ xðt; a1 , a2 Þ a1  a2 dt dt :

dq2 dyðt; a3 Þ ¼ yðt; a3 Þ  a3 dt dt

(5.16)

Its solution is given by xðt; a1 , a2 Þ ¼ xð0; a1 , a2 Þea1 ta2 q1 ðtÞ yðt; a3 Þ ¼ yð0; a3 Þeq2 ðtÞa3 t

:

(5.17)

Now, we can express all values of interest with the help of the mgf-s of the initial distributions and the auxiliary keystone variables: ð XðtÞ ¼ Xð0Þ ea1 ta2 q1 ðtÞ P1 ð0; a1 , a2 Þda1 da2 ¼ Xð0ÞM½t,  q1 ðtÞ A

ð

, q2 ðtÞa3 t

YðtÞ ¼ Yð0Þ e

q 2 ð tÞ

P3 ð0; a3 Þda3 ¼ Yð0Þe

(5.18)

M ½t 3

A

Pðt; a1 , a2 Þ ¼ P3 ðt; a3 Þ ¼

xðt; a1 , a2 Þ ea1 ta2 q1 ðtÞ ¼ Pð0; a1 , a2 Þ X ðt Þ M ½ t  q 1 ðt Þ yðt; a3 Þ ea3 t ¼ 3 P3 ð0; a3 Þ YðtÞ M ½t ð

E ½a2  ¼ t

A

:

(5.19)

a2 ea1 ta2 q1 ðtÞ ∂ Pð0; a1 , a2 Þda1 da2 ¼  ln M½t,  q1 ðtÞ ∂q1 M½t  q1 ðtÞ

Hence, XðtÞEt ½a2  ¼ Xð0Þ

∂ M½t,  q1 : ∂q1

(5.20)

Example 5.6 Competition of two inhomogeneous populations

67

Finally, we obtain a closed escort system of nonautonomous equations: dq1 ¼ Yð0Þeq2 ðtÞ M3 ½t, q1 ð0Þ ¼ 0, dt dq2 ∂ ¼ Xð0Þ M½t,  q1 , q2 ð0Þ ¼ 0: ∂q1 dt

(5.21)

Now that we have a solution to the Cauchy problem (5.21), we can obtain explicit formulas for population sizes defined in Eq. (5.18) and current distribution of the parameters defined in Eq. (5.19), which completely solve the problem. In particular, the current mean values of the parameters are given by ∂ Et ½a1  ¼ ln ðM½t,  q1 ðtÞÞ, ∂t ∂ Et ½ a 2  ¼  ln ðM½t,  q1 ðtÞÞ, (5.22) ∂q1  ∂ Et ½a3  ¼  ln M3 ½t : ∂t Integrating the equations of system (5.14) over the three parameters, we obtain the following system:  dX ¼ X Et ½a1   Et ½a2 Y , dt (5.23)  dY ¼ Y Et ½a2 X  Et ½a3  : dt These equations for total sizes of inhomogeneous populations have the same form as the initial Volterra system (5.13). The key difference is that now parameter values are not constants but vary over time according to Eq. (5.22). The phase-parameter portrait of the parametrically homogeneous Volterra’ model is well known (see, e.g., Bazykin, 1998). The dynamics of system (5.23) are determined by the parametric point (Et[a1], Et[a2], Et[a3]), which moves across the phase-parameter portrait of model (5.13) over time as the system evolves. This phenomenon, which amounts to “traveling across the phase-parameter portrait of the homogeneous model,” is a common feature of corresponding inhomogeneous models. It was well observed on the example of discrete-time models (see Chapter 15). A Volterra-type model of two inhomogeneous populations with logistic reproduction rates and ratio-dependent predator functional response will be studied in Chapter 12, together with additional examples of travel across the phase-parameter portrait.

Example 5.6 Competition of two inhomogeneous populations The dynamics of two populations competing for a common resource can be described by the following logistic-like model (see, e.g., Bazykin, 1998, ch. 4):

dx ðx + αyÞ ¼ ax 1  , dt A

(5.24) dy ðy + βxÞ ¼ by 1  , dt B

68

5. Some applications of the Reduction theorem and the HKV method

where A and B are the carrying capacities for both populations and α, β are the coefficients of interspecies competition. A more general Allee-like model has the form dx ¼ axððx  LÞðA  xÞ  αyÞ, dt dy ¼ byððy  MÞðB  yÞ  βxÞ, dt

(5.25)

where L < A, M < B. Consider inhomogeneous versions of these models, assuming that the reproduction rates a, b are distributed, and the competition is defined by the total sizes X(t), Y(t) of the population. Then, instead of the logistic model, we obtain the following model:

dxðt, aÞ XðtÞ + αYðtÞ ¼ axðt, aÞ 1  dt A

(5.26) dyðt, bÞ YðtÞ + βXðtÞ ¼ byðt, bÞ 1  dt B and correspondingly dxðt, aÞ ¼ axðt, aÞððXðtÞ  LÞðA  XðtÞÞ  αYðtÞÞ dt dyðt, bÞ ¼ byðt, bÞððYðtÞ  MÞðB  YðtÞÞ  βXðtÞÞ dt

(5.27)

instead of the Allee-type model. Both systems have the form dxðt, aÞ ¼ axðt, aÞðuðXðtÞÞ  αYðtÞÞ, dt dyðt, bÞ ¼ byðt, bÞðvðYðtÞÞ  βXðtÞÞ, dt

(5.28)

where u, v are appropriate functions. Let P1(0; a) and P2(0; b) be the initial distributions of the Malthusian rates a, b and M1[λ] and 2 M [λ] be corresponding mgf-s. In order to study model (5.28), we apply Theorem 4.3 and consider the four-dimension escort system:    dq1 1 ¼ u Xð0ÞM1 q1 1  αq2 1 , dt   dq2 1 ¼ Yð0ÞM2 q1 2  βq2 2 , dt    dq1 2 ¼ v Yð0ÞM2 q1 2  βq2 2 , dt   dq2 2 ¼ Xð0ÞM1 q1 1  αq2 1 , dt q1 1 ð0Þ ¼ q2 1 ð0Þ ¼ q1 2 ð0Þ ¼ q2 2 ð0Þ ¼ 0:

(5.29)

69

Example 5.6 Competition of two inhomogeneous populations

Assume that Cauchy problem (5.29) has a unique global solution at t 2 [0, T), 0 < T  ∞. Define 1 1 Kt 1 ðaÞ ¼ eaðq1 ðtÞαq2 ðtÞÞ , 2 2 Kt 2 ðbÞ ¼ ebðq1 ðtÞβq2 ðtÞÞ :

Then the solution to model (5.28) is xðt, aÞ ¼ xð0,aÞKt 1 ðaÞ ¼ xð0, aÞeaðq1 yðt, bÞ ¼ yð0,bÞKt 2 ðaÞ ¼ yð0, bÞebðq1   XðtÞ ¼ Xð0ÞM1 q1 1 ðtÞ  αq2 1 ðtÞ   YðtÞ ¼ Yð0ÞM2 q1 2 ðtÞ  βq2 2 ðtÞ

1

ðtÞαq2 1 ðtÞÞ

2

ðtÞβq2 2 ðtÞÞ

:

(5.30)

The mean values of the Malthusian rates at time moment t are given by    ∂ ln M1 q1 1 ðtÞ  αq2 1 ðtÞ ∂λ    : ∂ Et2 ½b ¼ ln M2 q1 2 ðtÞ  βq2 2 ðtÞ ∂λ

Et1 ½a ¼

(5.31)

The current pdf-s can be calculated as P1 ðt; aÞ ¼

1 1 P1 ð0; aÞeaðq1 ðtÞαq2 ðtÞÞ M1 ½q1 1 ðtÞ  αq2 1 ðtÞ

2 2 P2 ð0; bÞebðq1 ðtÞβq2 ðtÞÞ P2 ðt; bÞ ¼ M2 ½q1 2 ðtÞ  βq2 2 ðtÞ

:

(5.32)

A simpler logistic model (5.26) can be reduced to a two-dimension escort system: 

 1 q + αq2 dq1 1 ¼ Xð0ÞM t  A dt

 1 2  βq +q : dq2 ¼ Yð0ÞM2 t  B dt q1 ð0Þ ¼ q2 ð0Þ ¼ 0 The solution to model (5.26) is given by the following formulas:

αq2 ðtÞ 1 xðt, aÞ ¼ xð0,aÞ exp a t  q ðtÞ + A

βq1 ðtÞ + q2 ðtÞ yðt, bÞ ¼ yð0,bÞ exp b t  B  :

 1 2 q ð t Þ + αq ð tÞ 1 XðtÞ ¼ Xð0ÞM t  A 

 1 βq ðtÞ + q2 ðtÞ 2 YðtÞ ¼ Yð0ÞM t  B

(5.33)

70

5. Some applications of the Reduction theorem and the HKV method

The current pdf-s are given by

 a t

P1 ðt; aÞ ¼



P1 ð0; aÞe



 αq2 ðtÞ 1 1 M t  q ðtÞ + A   : 1 2 b

P2 ðt; bÞ ¼

q1 ðtÞ + αq2 ðtÞ A

tðβq ðtÞ + q ðtÞ B

(5.34)

P2 ð0; bÞe

 βq1 ðtÞ + q2 ðtÞ M2 t  B

The mean values of the Malthusian rates at time moment t are given by

 ∂ q1 ðtÞ + αq2 ðtÞ Et1 ½a ¼ ln M1 t  ∂λ A  :  1 βq ðtÞ + q2 ðtÞ ∂ Et2 ½b ¼ ln M2 t  ∂λ B

(5.35)

We now have explicit formulas to describe the dynamics of two competing inhomogeneous populations, where once again we will observe travel through the phase-parameter portrait of the previously analyzed parametrically homogeneous system. Calculating solutions for various initial distributions and visualizing evolutionary trajectories for this model and the parametrically heterogeneous predator-prey model are left as an exercise for the reader. Numerous additional models with fully analyzed phase-parameter portraits for parametrically homogeneous populations can be found in Bazykin (1998). Formulas for truncated mgf-s for several common and useful initial distributions are provided in Chapter 17, Math Appendix.

Example 5.7 The Fisher-Haldane-Wright equation One of the first replicator equations was introduced by R. Fisher in 1930 to describe genotype evolution as follows: dpi ¼ pi ðWi  W Þ, i ¼ 1, …n, (5.36) dt P P where Wi ¼ j Wij pj and W ¼ i, j pi Wij pj . Here, pi is the frequency of gamete i, and Wij is the absolute fitness of the zygote ij. In mathematical genetics, this equation is known as the Fisher-Haldane-Wright equation (FHWe) and sometimes is referred to as the main equation of mathematical genetics. This example was studied in detail in Karev et al. (2010). The matrix W ¼ {Wij} is symmetric and hence has the spectral representation (see, e.g., Ortega, 2013): Wij ¼

m X k¼1

ωk hk ðiÞhk ð jÞ,

(5.37)

71

Example 5.7 The Fisher-Haldane-Wright equation

where ωk are nonzero eigenvalues and hk are corresponding orthonormal eigenvectors of P W; m is the rank of W. Then, using notation Et ½hk  ¼ nj¼1 hk ð jÞpj ðtÞ, we have Wi ðtÞ ¼

n X

Wij pj ðtÞ ¼

j¼1

W¼

m X n X

ωk hk ðiÞhk ð jÞpj ðtÞ ¼

k¼1 j¼1

X

pi W i ¼

m X

ωk Et ½hk hk ðiÞ,

k¼1

m m X X X  2 pi ωk Et ½hk hk ðiÞ ¼ ωk Et ½hk  :

i

i

k¼1

k¼1

The FHW equation now reads m  X  2  dpi ¼ pi ωk hk ðiÞEt ½hk   Et ½hk  , i ¼ 1, …n: dt k¼1

Consider the associated selection system: m X dlðt, iÞ ¼ lðt, iÞ ωk hk ðiÞEt ½hk  dt k¼1

(5.38)

The range of values of the parameter i is now a finite set, unlike in the previous examples. Define the mgf of the initial distribution of parameter i to be ! " !# n m m X X X 0 exp δk hk ðiÞ Pð0; iÞ ¼ E exp δk hk , δ ¼ ðδ1 , …δm Þ: M 0 ð δÞ ¼ i¼1

k¼1

k¼1

Introduce and solve the following escort system for keystone variables qi; informally, dqi ¼ wi Et ½hi , qi ð0Þ ¼ 0: dt According to the Reduction theorem, this system can be written in a closed form as dqi ωi ∂ ln M0 ½q ¼ , ∂qi dt or in more detail

"

m X

ωi E hi exp 0

dqi ¼ dt

" E0 exp

!# q k hk

k¼1 m X

!# , i ¼ 1,…m:

q k hk

k¼1

Then, solution to the selection system (5.38) is given by the formula lðt, iÞ ¼ lð0, aÞKt ðiÞ, where Kt ðiÞ ¼ exp

m X k¼1

! qk ðtÞhk ðiÞ ;

72

5. Some applications of the Reduction theorem and the HKV method

the population size is given by N ðtÞ ¼ N ð0ÞM0 ½qðtÞ, the values of regulators at t moment are given by Hk ðtÞ ¼ Et ½hk  ¼

∂ ln M0 ½qðtÞ ∂qk

and the current distribution is given by Pðt, iÞ ¼

Pð0, iÞKt ðiÞ with E0 ½Kt  ¼ M0 ½qðtÞ: E0 ½Kt 

(5.39)

The last formula gives the solution of FHW equation (5.36). Technically the described approach is useful only if the rank of the fitness matrix W is significantly smaller than its dimension, m < n. This approach is especially useful for the infinitely dimensional system (5.36). Let us remark that, in general, the fitness matrix cannot be known exactly, but its elements can be well approximated by expression (5.37) with small m. For example, if Wij ¼ wiwj for all i, j, then the initial multidimensional (or even infinitely dimensional) system (5.36) is reduced to a single ODE. This case corresponds to a well-known example of a population in the Hardy-Weinberg equilibrium. In this case the asymptotic outcome of the dynamics is well known and trivial: all alleles but the one with the highest fitness mmax are lost. In addition to this known result, we obtain here a simple expression that can be used to capture time-dependent behavior. System (5.36) for the multiplicative fitness can be rewritten as   d pi ðtÞ ¼ pi mi Et ½m  ðEt ½mÞ2 , i ¼ 1, …, n, (5.40) dt P where Et[m] ¼ nj¼1mjpj(t). The following selection system corresponds to replicator equation (5.40): d li ðtÞ ¼ li ðtÞmi Et ½m, i ¼ 1, …, n: dt

(5.41)

As before, denote M0[λ] to be the moment-generating function of the initial distribution pi(0), X  λm  λmi e p ð 0 Þ ¼ E e : M0 ½λ ¼ i 0 i The escort system consists only of one equation: d 1 d d q ðt Þ ¼ M0 ½qðtÞ ¼ ln M0 ½qðtÞ, qð0Þ ¼ 0: dt M0 ½qðtÞ dq dq

(5.42)

The solution for the frequencies is given by emi qðtÞ pi ðtÞ ¼ pi ð0Þ , E0 ½Kt  X E0 ½Kt  ¼ eqðtÞmj p0i , j where p0i denotes the initial frequencies and p0i ¼ pi(0) > 0 for any i.

(5.43)

Example 5.8 Haldane principle for selection systems

73

From Eq. (5.42) and using the change of the variables q(t) ¼  ln u(t), we obtain the following equation for the new variable u X m p0 udi i i i u_ ¼ u X , p0 udi i i where di ¼ mmax  mi; this implies that one of di ¼ 0. From the last equation, it follows that u(t) ! 0 as t ! ∞, which yields that q(t) ! ∞. Using this last fact and the solution to Eq. (5.42), we have pi ðtÞ p0i qðtÞðmi mj Þ ¼ e !∞ pj ðtÞ p0j

P if mi > mj, which is possible only if pj(t) ! 0 due to constraint ipi(t) ¼ 1. Major advantage of the considered approach is that if one is interested in the time-dependent behavior of system (5.36), then, instead of solving n differential equations, it is sufficient to only solve one differential equation (5.42) for the auxiliary variable q(t). The case of additive fitness mij ¼ mi + mj can be analyzed in a similar vein and is left as an exercise for the reader.

Example 5.8 Haldane principle for selection systems Mathematical theory of selection has a long history, and R. Fisher, S. Wright, and J. Haldane were its founding fathers. The Haldane optimal principle (Haldane, 1932) can be considered to be one of the first general assertions about selection systems, describing the asymptotical behavior of population composition. This principle was generalized later in terms of abstract theory of systems with inheritance (see, e.g., a survey by Gorban, 2007). The “resampling down” (reduction of the support of the initial distribution) demonstrates the qualitative effects of “natural selection” at t ! ∞. Here, we consider a simple version of the Haldane principle within the framework of selection system (5.1). Let us assume for now that the reproduction rate F(t, a) of selection syst, aÞ tem dlðdt ¼ lðt, aÞFðt, aÞ is known explicitly as a function of t. Then, most questions about this system can be easily answered. We can compute the reproduction coefficient of an a-clone for the time interval [s, s + t) as 0s+t 1 ð Kss + t ðaÞ ¼ exp @ Fðu, aÞduA: s

The function Kt ðaÞ ¼ K0 ðaÞ was defined above in Eq. (5.3). It was shown in Chapter 4 that t

lðt, aÞ ¼ lð0, aÞKt ðaÞ N ðtÞ ¼ N ð0ÞE0 ½Kt  and Pðt, aÞ ¼

Pð0, aÞKt ðaÞ : E0 ½Kt 

74

5. Some applications of the Reduction theorem and the HKV method

Taking s as an initial time, we can obtain a more general and useful formula: Pðs + t, aÞ ¼

Pðs, aÞKs s + t ðaÞ   : Es Ks s + t

(5.44)

The following proposition helps understand the evolution of distribution in the long run and explains the Haldane principle within the frameworks of the selection system model. Let us define the mean reproduction rate as kðaÞ ¼ lim t!∞ ð1=tÞKt ðaÞ:

Proposition 5.2 Let us assume that the mean reproduction rate exists and is finite. Then 0t 1 ð lðt,a1 Þ lð0, a1 Þ lð0, a1 Þ tðkða1 Þkða2 ÞÞ ¼ exp @ ½Fðs, a1 Þ  Fðs,a2 ÞÞdsA ffi e : lðt,a2 Þ lð0, a2 Þ lð0, a2 Þ

(5.45)

0

This implies that the evolution of a heterogeneous population leads to an exponentially fast replacement of individuals with smaller values of k(a1) by those with greater values of k(a2), even though the fraction of the latter in the initial distribution was arbitrarily small. Let a* be a point of global maximum of k(a) and P(0, a∗) > 0; then, k(a) < k(a∗) implies that P(t, a) ! 0. Hence any stationary or limit distribution (if it exists) should be concentrated in a set of points of the global maximum of the average reproduction rate k(a) as supported by the initial distribution. Remark that the dynamical Haldane principle in its “relative” form does not require the existence of a stable or even ω-limit distribution. The Haldane principle allows us to predict behavior of selection systems “at infinity.” On the other end of the time scale, the Price equation describes the change of the mean value of individual characteristics, which may depend on time for any selection system.

Example 5.9 The Price equation and Fisher’s fundamental theorem According to Darwin’s theory, natural selection occurs when genetic variation in a population produces individuals that are best adapted to their environment. A measure of an individual’s ability to survive and reproduce is called fitness. Fisher’s fundamental theorem of natural selection (FTNS) states that “the rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time, except as affected by mutation, migration, change of environment, and the effect of random sampling” (Fisher, 1999). The FTNS was discussed in many papers and textbooks (see, e.g., Frank, 1997, 1998; Page and Nowak, 2002; Ewens, 2012). A standard interpretation of FTNS is that the rate at which natural selection acts upon character distribution within a population is controlled by the variance of that character distribution (Roughgarden, 1979). Many versions and special cases of the FTNS were proven within the framework of different mathematical models.

Example 5.9 The Price equation and Fisher’s fundamental theorem

75

Nevertheless, the actual biological content of the theorem, even if it is true mathematically within the framework of appropriate mathematical models, has been the subject of discussion in the literature for decades (Frank and Slatkin, 1992; B€ urger, 2000; Ewens, 2012). Although Fisher himself noted that FTNS only holds subject to important assumptions, he claimed that FTNS is an exact theorem that takes a superlative position among the biological sciences and demonstrates “the arrow of time”; he even compared it to the second law of thermodynamics in its importance. However, there are serious problems with interpretations of the standard FTNS. First, if the variance of the fitness is non-zero, then the mean fitness increases, and therefore population size increases hyperexponentially until the population becomes either homogeneous or infinite at a finite time moment (the so-called “population explosion”). In reality and in some more realistic models, the average fitness does not always increase. For example, the average fitness can decrease when selection acts on two linked loci that have epistatic effects on fitness (i.e., if the average effect of a substitution at one locus depends on the genotype frequencies at a second locus; B€ urger, 2000). Later, we show that the average fitness can behave in a very complex way if the fitness depends on the total population size, contrary to the standard FTNS. Let us emphasize that the FTNS in its standard form is a mathematical assertion that is valid within the framework of appropriate mathematical models of population dynamics; it may however not be applicable in some real situations. Sharp contradictions between reality and inevitable corollaries of the standard form of FTNS appear because some of its underlying conditions are not met. Fisher wrote, “An increase in number of any organism will impair its environment…The numbers must indeed be determined by the elastic quality of the resistance offered to increase in numbers, so that life is made somewhat harder to each individual when the population is larger, and easier when the population is smaller.” These words actually describe and justify a well-known transition from Malthusian models of freely growing population to the population model with the size-dependent reproduction coefficient, such as logistic, Allee, or other more complex models. G. Price (Price, 1970, 1972; Frank, 1997) gave an explanation of the contradiction between Fisher’s claim for generality and the limited scope of the usual interpretation of FTNS (see also Frank and Slatkin, 1992). G. Price was the first who tried to find a general formulation of selection that could be applied to any (not only biological) problem and to develop a formal general theory. He hoped that the concept of selection proposed in his paper (published only in 1995 many years after his death) “will contribute to the future development of ‘selection theory’ as helpfully as Hartley’s concept of information contributed to Shannon’s communication theory.…Many scientists must have felt surprise to find that at such late date there had still remained an opportunity to develop so fundamental a scientific area. Perhaps a similar opportunity exists today in respect to selection theory.” The Price equation was the first outstanding contribution to the future theory; its special cases are in fact the Fisher’s fundamental theorem (Fisher, 1999) and the Robertson covariance equation (Robertson, 1968). The Price equation is universally applicable to any selection system at any instant independently of the underlying mathematical model and its specific dynamics (see, e.g., Rice, 2006, Chapter 6 for details) because this equation is a mathematical identity. It is a reason of the theoretical universality and practical ineffectiveness (dynamical insufficiency) of the Price equation.

76

5. Some applications of the Reduction theorem and the HKV method

The problem of dynamical insufficiency is well known and discussed in numerous publications (Lewontin, 1974; Barton and Turelli, 1987; Frank, 1997; Rice, 2006). It means that the equation cannot be used alone as a propagator of the dynamics of the model forward in time. Roughly speaking, in order to compute the mean value of a trait, one needs to know its variance, and in order to compute the variance, one needs to know the moment of third order, and so on. Hence, if one needs to compute the dynamics of a trait, then additional information or other methods are required. Here, we show how one can overcome this problem and “solve” the Price equation within the frameworks of the developed theory. In terms of model (5.1), a trait or a character can be formally considered as a random variable z(t, a) (formally defined on the probabilistic space {A, A, P(t, a)} where A is a σ-algebra of Borel subsets of A); then the (complete) Price equation states that

 dEt ½z t t dz ¼ Cov ½F, z + E : (5.46) dt dt If trait z does not depend on time, then we get the “covariance equation” (Robertson, 1968): dEt ½z ¼ Covt ½F, z: dt

(5.47)

The Price equation is valid in very general situations (see, e.g., Frank, 1997, 1998; Rice, 2006); notice that within the frameworks of selection system model (5.1), this equation is a very simple mathematical assertion. Indeed, ð ð ð dEt ½z d dzðt, aÞ dPðt, aÞ ¼ Pðt, aÞda + zðt, aÞ da zðt, aÞPðt, aÞda ¼ dt dt dt dt A

A

A

 ð

  dz dz ¼ Et + zðt, aÞ Fðt, aÞ  Et ½F Pðt, aÞda ¼ Et + Covt ½F, z: dt dt A

We can write the Price equation in, perhaps, intuitively clearer integral form that shows the connection between the reproduction coefficient and the selection differential Δtz ¼ Et[z]  E0[z], which is an important characteristic of selection. Proposition 5.3 Integral form of the Price equation Let zt be any trait; then, within the framework of model (5.1), the following identity is valid for the selection differential:   Covt zt + s , Kt t + s t+s t   E ½zt + s   E ½zt  ¼ + Et ½zt + s  zt : (5.48) Et Kt t + s In particular, Es ½ z s  ¼

Cov0 ½zs , Ks  + E0 ½zs : E0 ½Ks 

(5.49)

Example 5.9 The Price equation and Fisher’s fundamental theorem

77

Indeed, according to formula (5.44),     Et z t + s K t t + s Covt zt + s , Kt t + s t+s t t      E ½zt  ¼ + Et ½zt + s  zt : E ½zt + s   E ½zt  ¼ Et K t t + s Et Kt t + s Notice that this version of the Price equation is quite similar to that for discrete-time models (which will be covered in detail in Chapter 15) compared to the differential version (5.46). Corollary (Integral form of the covariance equation) Let z be any trait that does not depend on time; then, within the framework of model (5.1),   Covt z, Kt t + s t+s t   : E ½z  E ½z ¼ Et K t t + s Taking into consideration that Kt s + t ðaÞ  1 + sFðaÞ, we can see that the integral expression (5.48) becomes the differential Price equation (5.46) at s ! 0. Formula (5.49) allows us to compute the mean value of any trait at any time moment if the initial pdf is known. We can then predict the value of a mean trait indefinitely if the initial mean values E0[Ks] and covariances Cov0[zs, Ks] can be computed; if the system distribution is known at some initial nonzero instant, then we can use formula (5.44). In this sense, Proposition 5.3 and the Corollary eliminate the problem of dynamical insufficiency of the Price equation if the reproduction coefficient Ks is known. Let us emphasize that in most realistic and interesting problems, the reproduction coefficient is unknown and hence should be computed within the framework of a specific model. Model (5.1) of a selection system with self-regulation gives us the necessary general frameworks. The theory developed in Chapter 4 helps resolve the problem of dynamical insufficiency of the Price equation, the covariance equation, and the equation of the Fisher’s fundamental theorem of natural selection (FTNS). Let z(t, a) be an arbitrary trait. Proposition 5.4 On the Price equation Within the frameworks of model (5.1), the solution to the Price equation (5.46) for a given initial distribution is given by the formula E 0 ½zt K t  : (5.50) E0 ½ K t  Pn The function Kt is defined by Eq. (5.3), Kt ðaÞ ¼ exp i¼1 qi ðtÞφi ðaÞ , where the keystone variables qi(t) solve the escort ODE system (5.2). In order to find Et[zt], we actually do not need to “solve” the Price equation because we t ðaÞ . Remark that due to this have general formulas (4.5), (5.4) for the current pdf Pðt, aÞ ¼ Pð0E, 0a½ÞK Kt  formula, all terms of the Price equation can be computed in explicit form depending on the initial distribution only: Et ½ z t  ¼

78

5. Some applications of the Reduction theorem and the HKV method

E0 ½zt Ft Kt  E0 ½zt Kt E0 ½Ft Kt   , 2 E0 ½Kt  ðE0 ½Kt Þ

 dzt

 E0 Kt dzt dt : Et ¼ E0 ½Kt  dt

Covt ½Ft , zt  ¼

Proposition 5.5 On FTNS t ½ F t Within the framework of model (5.1), the solution of thet FTNS equation dE dt ¼ Var ½F with time dE ½Ft  t t dFt independent fitness F ¼ F(a) and more general equation dt ¼ Var ½Ft  + E dt with time-dependent fitness F ¼ F(t, a) is given by the formula Et ½Ft  ¼

E0 ½Ft Kt  dð ln M0 ½qðtÞÞ , ¼ E0 ½Kt  dt

(5.51)

where M0[λ] is the moment-generating function of the initial distribution of parameter a and the keystone variables q(t) solve the escort ODE system (5.2). Indeed, Ft K t ¼ Hence

dKt , E0 ½Kt  ¼ M0 ½qðtÞ: dt

 d E0 ½ K t  E ½Ft Kt  dð ln M0 ½qðtÞÞ : Et ½Ft  ¼ 0 ¼ 0dt ¼ E ½Kt  dt E ½K t  0

We will discuss the Price equation and the FTNS in Chapter 15 within the frameworks of selection models with discrete time. In the next chapter, we will introduce the differences between density-dependent and frequency-dependent growth and discuss in detail the exciting implications of these two models.

C H A P T E R

6 Nonlinear replicator dynamics Abstract In this chapter, we will discuss in detail power law growth, as well as the dynamics of various types of power replicator equations. We will demonstrate that true exponential growth in inhomogeneous populations occurs only in the so-called frequency-dependent models (F-models), where growth of each clone is proportional to its frequency in the population. We will then compare density-dependent and frequency-dependent growth models and discuss their similarities and differences, implications of each growth law, and some ways to distinguish between the two. This chapter provides the first in-depth introduction into the discussion of both Darwinian and non-Darwinian selection, which will be continued in subsequent chapters. The chapter is based on Karev (2014).

6.1 Problem formulation: Power replicator equations The study of population growth reveals that behaviors that follow the power law appear in numerous biological, demographic, ecological and other contexts. In its simplest form the power equation for population growth generalizes the standard Malthusian equation and is written as dx ¼ kxp : dt

(6.1)

Three cases are distinguished: (1) exponential with p ¼ 1, (2) superexponential (hyperbolic) with p > 1, (3) subexponential (parabolic) with p < 1. Of course, such a simple model can describe a real system only approximately; nevertheless, both sub- and superexponential growth curves can be observed in real populations in large domains of values of model variables and parameters. There exist well established examples of superexponential growth as applied to global demography (Von Foerster et al., 1960; Kapitza, 2006; see also Chapter 3), as well as in several ecological and economic problems ( Johansen and Sornette, 2001; H€ usler and Sornette, 2011).

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00006-0

79

# 2020 Elsevier Inc. All rights reserved.

80

6. Nonlinear replicator dynamics

Relevance of superexponential growth model was discussed by Eigen and Schuster (1978), who noticed that the quasi-species model can exhibit hyperbolic growth if the growth rate is not a constant but a linear function of the quasi-species concentration. The most striking peculiarity of the solution to the superexponential equation is that it has “finite time singularity.” Indeed the solution to Eq. (6.1) at p > 1 reads xðtÞ  1=ðT  tÞ

1 p1

(see Theorem 6.3 below for details). Hence x(t) tends to infinity when t approaches critical time T < ∞, which is determined by initial conditions in Eq. (6.1). It is worth noticing that “singularities do not exist in natural and social systems, but the singularities of our approximate mathematical models are usually very good diagnostics of the change of regimes that occur in these systems” (H€ usler and Sornette, 2011). Subexponential growth models also have a long history. Schmalgausen (1930, 1935) suggested the following formula to describe the growth of a biological object (such as weight or size of an organism or separate organ) for any measurable parameter W: lnðWÞ ffi c + b lnðtÞ,

(6.2)

where b and c are constants and t is time (or age). This dependence corresponds to the power function
b W t ¼ (6.3) W0 t0 and to the equation dW b ¼ W: dt t

(6.4)

Excluding time from Eq. (6.4) via Eq. (6.3), we obtain dW 1=b ¼ bW 0 W 11=b  W 11=b : dt

(6.5)

Hence the Schmalhausen formula has the form of the subexponential version of Eq. (6.1). Growth coefficient in Eq. (6.4) is inversely proportional to time, which is the simplest and universal (but not unique and not always acceptable) explanation of the phenomenon of subexponential growth. Formula (6.3) is closely connected with the allometric principle, which postulates a stable relationship between different characteristics x and y of an organism during ontogenesis as: y ¼ αxβ :

(6.6)

Indeed, if some characteristics of an organism grow according to the Eq. (6.3) with different b, then, excluding time, we obtain Eq. (6.6). Allometric relationships allow for the possibility of calculating mass or volume of organs that are inaccessible for direct measurement, using available data for other organs.

6.1 Problem formulation: Power replicator equations

81

Several interesting attempts were made to substantiate the principle of allometry theoretically. It was shown that allometric relationships (and hence the Schmalhausen formula and subexponential equation) are connected with the self-similarity property and the dimensional theory (see, e.g., Kofman, 1986 for applications to models of plant growth). In other fields, allometric scaling is used in pharmaceutical industry, where one needs to make predictions about how a drug will behave in humans based on how it behaves in animals in preclinical studies. In 1947, Max Kleiber (1947) published an observation that many physiological processes follow allometric scaling and, specifically, that metabolic rate scales to 0.75 power of the animal’s mass for a vast majority of animals (see Fig. 6.1). Knowing this relationship, it is possible to predict how a drug might behave in humans and thus make more educated predictions about first in human doses that would go into clinical trials. It is one of the many methods that is currently used to make such predictions; for deeper introduction into mathematical modeling in pharmaceutical sciences, see, for instance, Mager et al. (2009). Eq. (6.2) describes quite well only separate stages of organism development; arbitrary stages of development correspond to different values of the constants. Consequently, the Schmalhausen generalized growth formula was rewritten as lnðWi Þ ¼ ci + bi lnðtÞ or Wi ¼ ci tbi

(6.7)

for Ti  t < Ti+1, where ci, bi are constants, i ¼ 1, … s; typically s ¼ 3. It was found that unusual wealth of experimental data on the development of a number of very different biological objects might be approximately described by the generalized Schmalhausen formula

FIG. 6.1 Allometric relationship between body weight and metabolic rate; the original figure can be found in Kleiber (1947).

82

6. Nonlinear replicator dynamics

(Schmalgausen, 1935; Poletaev, 1966; Terskova, 1970). The underlying model, which implies the generalized Schmalhausen formula (6.7), was suggested and studied in Poletaev (1966) and Karev and Berezovskaya (2003). The same relationships as described by Eq. (6.7) were found when studying the development of not only individuals but also populations and communities. A well-known example is the “3/2 power rule” of plant self-thinning and its modifications (Zeide, 1987; Weller, 1991). Part of the monograph by Zagreev et al. (1980) was devoted to systematic applications of Eq. (6.7) to different characteristics of tree populations and forest communities, such as dynamics of an average diameter of trees, wood stock, etc. Therefore, it seems that the Schmalhausen formula of subexponential growth may be an essential empirical law of developmental biology. More recent well-known example of subexponential population growth applies to some molecular replicator systems. von Kiedrowski (1986) realized that populations of most experimentally studied artificial replicators (typically oligonucleotides that replicate in vitro via binary ligation) grow approximately according to the parabolic law (Eq. (6.1) with p ¼ 1/2) rather than exponentially. The principal cause of subexponential growth of these populations appears to be product inhibition, which slows down the reproduction process compared with the exponential case (von Kiedrowski, 1993; Luther et al., 1998; Wills et al., 1998; Von Kiedrowski and Szathma´ry, 2001). Based on these results, Szathma´ry and Smith (1997) presented a general conceptual model of prebiological evolution of replicators using the power equation (6.1) to describe the concentration of molecules. Models of biological populations composed of nonexponentially growing homogeneous or monomorphic subpopulations (clones) deviate from Darwinian “survival of the fittest” (see Szathma´ry and Gladkih, 1989; Szathma´ry, 1991; Lifson and Lifson, 1999; Von Kiedrowski and Szathma´ry, 2001). Specifically the models imply “survival of the fittest” when p ¼ 1 , “survival of the common” when p > 1, and “survival of everybody” when p < 1. The reason for these predictions is that these nonexponentially growing clones possess some unrealistic properties. The birth rate per individual in model (6.1) is r ¼ kxp1. Then the birth rate for the parabolic model increases indefinitely as the population size decreases and tends to zero. It can be easily shown that the birth rate for hyperbolic models increases as the population size increases, and both become infinite at a finite time moment. Recall that the birth rate per individual must be bounded for any realistic biological population. Nevertheless, despite these unrealistic assumptions of the simplified growth equation (6.1), power law growth appears to be an essential feature of evolving populations; it could even be more directly relevant for biological and prebiological evolution than the exponential growth case. Therefore understanding the laws governing this type of growth is potentially of great interest for evolutionary studies. The question arises: what are the possible origins of nonexponential models and how can we derive them from realistic assumptions? The last problem is a nontrivial one because both sub- and superexponential growth models possess unrealistic properties described earlier. To eliminate these peculiarities, we need to know where they come from. The starting point is that heterogeneity is one of the key properties of any real evolving biological system. In the next section we will show that model (6.1) can be understood within

83

6.2 Population heterogeneity as the reason for the power law growth dynamics

the frameworks of inhomogeneous population models, and that population heterogeneity can be a reasonable explanation for the observed growth laws of the total population size.

6.2 Population heterogeneity as the reason for the power law growth dynamics Heterogeneity amounts to the existence of differences between individuals that could be subject to natural selection and drift, which can operate only if the population is nonhomogeneous. The dynamics of distributions of individuals within heterogeneous populations can be described by replicator equations that capture this “basic tenet of Darwinism” (Hofbauer and Sigmund, 1998; Nowak and Sigmund, 2004). The full theory underlying the HKV method for solving a wide class of replicator equations was described in Chapter 4. For the problems we study in this chapter, we use a simplified version of this method that was described in detail in Chapter 2 in application to Malthusian type models. Consider the following equation describing a population of individuals, where each individual is characterized by its own value of growth rate parameter a: ð dlðt, aÞ ¼ alðt,aÞgðN Þ, NðtÞ ¼ lðt,aÞda, (6.8) dt A where g(N) is an appropriate function. The solution to this model is given by the equations lðt, aÞ ¼ N ð0ÞPð0, aÞeaqðtÞ ,

(6.9)

N ðtÞ ¼ N ð0ÞM0 ½qðtÞ

(6.10)

where the keystone variable q(t) is the solution to the Cauchy problem dq ¼ gðN Þ, qð0Þ ¼ 0, dt

(6.11)

and where M0[λ] is the mgf of the initial distribution P(0, a). The population size can be calculated by solving the equation dN ¼ Et ½aNgðN Þ: dt

(6.12)

Current distribution of clones in the population P(t, a) is determined by Pðt, aÞ ¼

Pð0, aÞeqðtÞa , M0 ½qðtÞ

(6.13)

and the expected value of the growth rate parameter as it changes over time is given by Et ½ a  ¼

d ln M0 ½qðtÞ : dq

(6.14)

84

6. Nonlinear replicator dynamics

Keeping in mind the power equation (6.1) for total population size, let us now construct the inhomogeneous model (6.8) such that its total population size solves Eq. (6.1). Let gðN Þ ¼ kN p1s :

(6.15)

The additional nonnegative parameter s will be specified later. Consider the model dlðt, aÞ ¼ alðt, aÞgðN Þ ¼ aklðt, aÞN p1s : dt Given the value of the parameter s, Eq. (6.12) reads dN ¼ kEt ½aNps : dt p If we want for equation dN dt ¼ kN to hold, then we need to find an initial distribution such t s that E [a] ¼ N . According to Eqs. (6.14) and (6.10),

Et ½a ¼

d lnðM0 ½qðtÞÞ ¼ N ðtÞs ¼ ðN ð0ÞM0 ½qðtÞÞs , dq

and this equation holds only if dM0 ¼ N ð0Þs M0 ½qs + 1 : dq The solution to this equation is given by the formula M0 ½q ¼ ð1  sNð0Þs qÞ

1=s

:

It is well known that the function M0[δ] ¼ (1  βδ) ρ at β > 0 is the mgf of the Gamma distribution 

a

aρ1 e β , a > 0: Pð a Þ ¼ ρ β ΓðρÞ

(6.16)

The mean and variance of this distribution are ρβ and ρβ2, respectively. In our case, this means that the initial distribution of parameter a must be the Gamma distribution with parameters β ¼ sN(0)s and ρ ¼ 1/s: PðaÞ ¼

a1=s1 e



1

a sN ð0Þs

N ð0Þs s Γ


: 1 s

(6.17)

Gamma distribution (6.17) is completely characterized by its mean value N(0)s and the variance sN(0)2s; its mgf is M0 ½δ ¼ ð1  sNð0Þs δÞ

1=s

:

6.3 Canonical form of the power model

85

Now, we can compute the keystone variable q(t) and hence all statistical characteristics of interest. By definition, dq ¼ gðNÞ ¼ kN p1s , qð0Þ ¼ 0: dt

(6.18)

Since N(t) ¼ N(0)M0[q(t)], we obtain the equation s + 1p dq ¼ kðN ð0ÞM0 ½qÞp1s ¼ kðN ð0ÞÞp1s ð1  sNð0Þs qÞ s : dt

The solution to this equation is   s 1  1 + kN ð0Þp1 ð1  pÞt p1 qðtÞ ¼ for p 6¼ 1 sN ð0Þs and qðtÞ ¼

1  ekst for p ¼ 1: sN ð0Þs

Next, N ðtÞ ¼ Nð0ÞM0 ½qðtÞ ¼ N ð0Þð1  sN ð0Þs qÞ

1 s

  1 ¼ Nð0Þ 1 + kNð0Þp1 ð1  pÞt 1p for p 6¼ 1, (6.19)

and N ðtÞ ¼ Nð0Þekt for p ¼ 1: The current mean value of the parameter
 1=ðp1Þ s , Et ½a ¼ N ðtÞs ¼ Nð0Þ 1 + kN ð0Þp1 ð1  pÞt The total population size and its growth rate, of course, do not depend on the choice of the value of s, but the distribution of parameter a does.

6.3 Canonical form of the power model The simplest and universal representation of the power model (6.1) within the frameworks of inhomogeneous population models (6.8) can be obtained by taking s ¼ p and gðN Þ ¼ Nk in all formulas of the previous section. We arrive at the canonical form of the models, which can also be referred to as the frequency-dependent model (F-model for brevity). Let us summarize our finding in the following theorems. Denote βð0Þ ¼ pN ð0Þp , βðtÞ ¼ pN ðtÞp :

86

6. Nonlinear replicator dynamics

p Theorem 6.1 Equation dN dt ¼ kN for any p > 0 describes the total population size of inhomogeneous frequency-dependent model

dlðt,aÞ kalðt, aÞ ¼ ¼ kaPðt, aÞ, dt N ðt Þ

(6.20)

where the initial distribution of the parameter a, P(0, a), is the Gamma distribution 

a

aρ1 e βð0Þ PðaÞ ¼ , a > 0, with ρ ¼ 1=p β ð 0Þ ρ Γ ð ρÞ

(6.21)

and the mgf is given by M0[δ] ¼ (1  β(0)δ)1/p. Additional properties of the F-model (6.20) will be formulated separately for subexponential and superexponential models. Theorem 6.2 (Subexponential models) Let p < 1 in Eq. (6.1). Then, (i) the solution to F-model (6.20) is l(t, a) ¼ l(0, a)eaq(t), where " #   p 1 1p p1 1  1 + kNð0Þ ð1  pÞt q ðt Þ ¼ β ð 0Þ and l(0, a) ¼ N(0)P(a) with P(a) given by formula (6.21); in explicit form, lðt; aÞ ¼ C1

1p a p exp

!  p a  1p p1  1 + kN ð0Þ ð1  pÞt , pN ð0Þp

(6.22)

where C1 ¼

1
; 1 p

1 pp Γ

(ii) the total population size at the moment t for F-model (6.20) is   1 N ðtÞ ¼ N ð0Þ 1 + kN ð0Þp1 ð1  pÞt 1p ; (iii) the current distribution at any moment t is Gamma distribution (6.16) with ρ ¼ 1/p and βðtÞ ¼ 1qβððt0ÞβÞ ð0Þ ¼ pN ðtÞp ; the distribution has the mean Et[a] ¼ N(t)p, the variance 2

Vart ½a ¼ βðptÞ ¼ pN ðtÞ2p , and the mgf Mt[δ] ¼ (1  β(t)δ)1/p.

(6.23)

6.3 Canonical form of the power model

87

Remark that the keystone variable q(t) increases monotonically with time but tends to a finite value, qðtÞ ! βð10Þ ¼ pN1ð0Þp as t ! ∞. The density of each clone (Eq. (6.22)) tends over time to C1a(1–p)/p, i.e., is given by lðt, aÞ !

a1=p1
 as t ! ∞: 1 1 pp Γ p

(6.24)

Hence
11 lðt, a1 Þ a1 p ! : lðt, a2 Þ a2 This formula can be interpreted as “survival of everyone” within the subexponential population: the ratio of the frequencies of different clones that compose the population tends to a certain finite nonzero value. At the same time the total size of the population described by Eq. (6.23) increases indefi1 nitely according to the power law, N ðtÞ  t1p , while the density of each clone tends to a finite ð t , aÞ value according to Eq. (6.24), so the frequency of each clone Pðt, aÞ ¼ lN ðtÞ ! 0. A typical plot of the density (6.22) is shown on Fig. 6.2; the dependence of densities (6.22) on time and on parameter a at different values of p is shown in Figs. 6.3–6.5.

FIG. 6.2 A typical plot of the density given by Eq. (6.22); p ¼ 1/2, a ¼ 10. Adapted from Karev, G.P., 2014. Non-linearity and heterogeneity in modeling of population dynamics. Math. Biosci. 258, 85–92.

88

6. Nonlinear replicator dynamics

100

100

PrC

50 0 10 50

t

5 a

0

0

FIG. 6.3 Densities of clones described by Eq. (6.22) at p ¼ 0.1 as a function of time and parameter a (k ¼ 1, N(0) ¼ 1). Adapted from Karev, G.P., 2014. Non-linearity and heterogeneity in modeling of population dynamics. Math. Biosci. 258, 85–92.

10 a 5

0

20 PrC 10

0 100 50 0

t

FIG. 6.4 Densities of clones described by Eq. (6.22) at p ¼ 0.5 as a function of time and parameter a (k ¼ 1, N(0) ¼ 1). Adapted from Karev, G.P., 2014. Non-linearity and heterogeneity in modeling of population dynamics. Math. Biosci. 258, 85–92.

89

6.3 Canonical form of the power model

2 PrC

100

1 0 10 50

t

5 a

0

0

FIG. 6.5 Densities of clones described by Eq. (6.22) at p ¼ 0.8 as a function of time and parameter a (k ¼ 1, N(0) ¼ 1). Adapted from Karev, G.P., 2014. Non-linearity and heterogeneity in modeling of population dynamics. Math. Biosci. 258, 85–92.

Theorem 6.3 (Superexponential models) Let p > 1 in Eq. (6.1). Then, (i) solution to model (6.1) exists only up to the moment of “population explosion”   T ¼ 1= kðp  1ÞNð0Þp1 ; (ii) solution to F-model (6.20) for t < T is lðt, aÞ ¼ lð0, aÞeaqðtÞ , where


 p   1 t p1 qðtÞ ¼ 1 1 T pNð0Þp

and lð0, aÞ ¼ N ð0ÞPðaÞ with P(a) given by formula (6.21); in explicit form, it reads
1p p  a p p1 ð1  t=T Þ lðt, aÞ ¼ C1 a exp  , pNð0Þp

90

6. Nonlinear replicator dynamics

where C1 ¼

1
; 1 p

1 pp Γ

(iii) the total population size at the moment t < T is 1

NðtÞ ¼ N ð0Þ=ð1  t=TÞp1 : The total population size of the model together with the mean and variance of the parameter distribution increase indefinitely as t ! T, where T is the moment of “population explosion,” in contrast to subexponential and exponential cases. The keystone variable q(t) increases monotonically and tends to a finite value, qðtÞ ! pN1ð0Þp 1p

as t ! T. The density of clones lðt, aÞ ! C1 a p , so the ratio of clone frequencies remains finite and nonzero up to the moment T of “population explosion”:
1p lðt, a1 Þ a1 p ! as t ! T: lðt, a2 Þ a2 This formula can be interpreted as “survival of everyone” within the superexponential population, similar to the subexponential model. As t ! T, the total size of the population increases indefinitely according to the hyperbolic ð t , aÞ law, and the frequency of each clone Pðt, aÞ ¼ lN ðtÞ ! 0 as t ! T. One can see that the “internal” dynamics in subexponential and superexponential populations are similar, and the only (but very important) difference is that the subexponential population approaches the limit state as t ! ∞, while the superexponential population approaches its final state at the moment T < ∞ of “population explosion.”

6.4 Inhomogeneous model for exponential equation The capacity to grow exponentially under ideal conditions is a common property of populations. However, in any realistic situation, eventually limits to unrestrained growth are encountered, which generates the struggle for existence, leading to natural selection (Gause, 1934). Natural selection can operate only if the evolving population is nonhomogeneous. According to Szathma´ry (2006), an important step of prebiotic evolution must have been the emergence of replicators capable of exponential growth. It is worth emphasizing here that inhomogeneous population composed of different exponentially growing clones can never demonstrate exponential growth, but instead always grows faster, “overexponentially.” Indeed, consider a population that is composed of clones l(t, a) such that

and N(t) ¼

Ð

dlðt, aÞ ¼ alðt, aÞ dt Al(t, a)da

is its total size.

91

6.4 Inhomogeneous model for exponential equation t

dE ½a t t Then, dN dt ¼ E ½aN (see Eq. (6.13)) and dt ¼ Var ½a > 0 due to Fisher’s fundamental theorem (Fisher, 1999). It means that the population growth rate, which is equal to Et[a], is not a constant, as is required for exponential growth, but increases with time as long as Vart[a] > 0, that is, until the population stays inhomogeneous. Hence one can conclude that for exponential growth, the population should be homogeneous with respect to Malthusian growth rate parameter, when the selection has already “done its work.” It certainly should take a long period of time, during which the population will almost inevitably encounter restrictions, such as resource limitations, preventing true exponential growth. It means that on both ends of the temporal scale, the population cannot show exponential growth, either in the beginning of the population development when it grows overexponentially or at its end when it grows subexponentially, possibly tending to an equilibrium; the only time therefore when true exponential growth could be observed is during a short transitional period between these two regimes. (This problem was discussed in Chapter 2 in the context of a model of global demography.) Given that all real populations are inhomogeneous, we can pose the following question: does there exist a model of an inhomogeneous population that demonstrates true exponential growth of the total population size on the whole time scale? The answer is affirmative. Namely, the simplest classical Malthusian model in actuality describes the growth of total population size of an inhomogeneous F-model. Corresponding results collected in Theorem 6.4 can be obtained from Theorem 6.1 by limit transition as p ! 1 or by direct application of the results of Section 6.2.

Theorem 6.4 The Malthusian equation dN dt ¼ kN describes the dynamics of total size of inhomogeneous 

a N ð0Þ

F-model dlðdtt, aÞ ¼ kaPðt, aÞ with exponentially distributed parameter a, Pð0, aÞ ¼ e Nð0Þ , with the mean Nð0Þ at the initial moment. The solution to the F-model has a form lðt, aÞ ¼ lð0, aÞeaqðtÞ , where qðtÞ ¼

1  ekt , N ð 0Þ

so the final solution is given by the formula



kt  e lðt, aÞ ¼ exp a : N ð 0Þ

(6.25)

The current distribution of the parameter at time moment t is again exponential with the mean Et[a] ¼ N(t) ¼ N(0)ekt and the mgf Mt[δ] ¼ (1  N(0)ekt δ)1. To clarify the place of the exponential model within the set of parabolic models, it is convenient to use the notion of the q-exponential function: 1

exp q ðxÞ  ð1 + ð1  qÞxÞ1q

92

6. Nonlinear replicator dynamics

that reveals ordinary exponential function as q ! 1 (see, e.g., Tsallis, 2009, Chapter 3, for formulas and properties of the so-called q-calculus). Then the solution to F-model (6.20) given in Theorem 6.2 can be presented in the form
1p p lðt, aÞ ¼ Ca exp 

  a  p1 t p exp p kN ð0Þ pN ð0Þ

p

and the total population size at the moment t for F-model (6.20) is given by   N ðtÞ ¼ N ð0Þ exp p kNð0Þp1 t :

(6.26)

(6.27)

One can see that these formulas turn to corresponding formulas for exponential models given in Theorem 6.4 as p ! 1. So the simple Malthusian model has now been included into the general frameworks of inhomogeneous F-models within the class of subexponential models. Let us notice an interesting corollary that follows from Theorem 6.4. The inhomogeneous F-model, which shows exponential growth of the total population size, actually consists of clones, each of which grows according the Gompertz curve. Recall that the Gompertz function is given by the formula  GðtÞ ¼ r exp cekt , (6.28) where r is the upper asymptote, c ¼ const > 0, and k > 0 determines the growth rate, so that Eq. (6.25) coincides with the Gompertz function (6.28) up to notation. A typical plot of the Gompertz curve is shown on the Fig. 6.6. Integrating densities given by Eq. (6.25) (the Gompertz functions) over a assuming initial exponential distribution, we obtain the exponential function. Fig. 6.7 shows the plot of

FIG. 6.6 Typical Gompertz curve given by Eq. (6.28) with r ¼ 1, c ¼ 10, and k ¼ 1.

93

6.5 Superexponential models: The second representation

1.0

0.5

10 0

t

Gom

0.0

5 50 a 0

100

FIG. 6.7 Densities of clones as given by Eq. (6.25) as a function of time and parameter a other parameters are fixed at k ¼ 1, N(0) ¼ 1. Adapted from Karev, G.P., 2014. Non-linearity and heterogeneity in modeling of population dynamics. Math. Biosci. 258, 85–92.

the Gompertz functions (6.25) depending on two variables, a and t. Notice that this plot is a “limit case” of the plots in Figs. 6.3, 6.4 for parabolic F-models as p!1. Remark that the Gompertz curves are successfully used to fit data and modeling of growth of tumors (see, for instance, Kendal, 1985; Gyllenberg and Webb, 1988; Benzekry et al., 2014). A more thorough discussion about the implications of various growth curves with respect to tumor growth models will be found in Chapter 11.

6.5 Superexponential models: The second representation The frequency-dependent inhomogeneous model (6.20) is a universal representation for any power model (6.1). For superexponential (hyperbolic) models, where dN ¼ kNp , p > 1, dt

(6.29)

there exists another density-dependent form for the underlying inhomogeneous model (Dmodel for brevity), which is in fact just an inhomogeneous Malthusian model. To this end, let us denote s ¼ p  1 > 0 and apply this change to all formulas of Section 6.3. Then, it is easy to see that the auxiliary variable, defined by Eq. (6.18), is q(t) ¼ kt. This allows formulating the following theorem:

94

6. Nonlinear replicator dynamics

Theorem 6.5 (i) Any hyperbolic model (6.29) describes the dynamics of total population size for inhomogeneous Malthusian model: dlðt, aÞ ¼ kalðt,aÞ, dt

(6.30)

where the initial distribution of the Malthusian parameter a is the Gamma distribution (6.17) with s ¼ p  1, and where mgf M0[δ] ¼ (1  (p  1)N(0)p1δ)1/(p1); 1 ; (ii) The solution to model (6.29) exists up to the moment T ¼ kðp1ÞN ð0Þp1 1

a 1  ðp1ÞNð0Þp1 + kat

kat p1 e (iii) The solution to model (6.30)   is lðt, aÞ ¼ Nð0ÞPð0, aÞe ¼ C2 a 1 C2 ¼ 1= ðp  1Þp1 Γ p1 ;

, where

(iv) The total population size at time moment t < T is   1 p1 NðtÞ ¼ N ð0ÞM0 ½kt ¼ N ð0Þ 1  ðp  1ÞN ð0Þp1 kt ; (v) The current distribution of parameter a at time moment t is the Gamma distribution (6.16) with 1 ρ ¼ p1 and β(t) ¼ (p  1)N(t)p1, which has the mean Et ½a ¼ N ðtÞp1 , variance Vart ½a ¼ ðp  1ÞN ðtÞ2ðp1Þ , and mgf Mt ½δ ¼ 1 

ðp  1ÞN ð0Þp1 1  tkðp  1ÞNð0Þp1

!1=ðp1Þ δ

:

Let us emphasize that parabolic models cannot be represented in the form of an inhomogeneous Malthusian model (D-model). Indeed, in model (6.30), Et[a] increases because dEt ½a ¼ Vart ½a > 0, dt while Et[a] ¼ Np1 decreases if p < 1. Theorems 6.1 and 6.5 show where some unrealistic peculiarities of the power growth models (such as “blowing up” of hyperbolic models) may come from. They follow from the obviously unrealistic assumption (incorporated implicitly into growth models (6.1)) that the individual reproduction rate may take arbitrarily large values with nonzero probabilities, given by pdf of Eq. (6.16).

95

6.6 How to choose between F- and D-models?

6.6 How to choose between F- and D-models? Both F-models (frequency-dependent) and D-models (density-dependent) can be of use for describing hyperbolic models. Let us summarize the main formulas in Table 6.1. The following criteria can be used to distinguish circumstantial applicability of the two models: (1) If one is describing an inhomogeneous population that consists of competing clones, whose dynamics are governed by the total population size, then the frequency-dependent F-model should be used. (2) If one is describing an inhomogeneous population that consists of independently growing clones, then the inhomogeneous Malthusian or density-dependent D-model is more relevant. Notice that formally, any intermediate case corresponding to the model of Section 6.3 with 0  s  p is also possible. Both models imply identical dynamics of the total population size. Nevertheless, the question of what approach is closer to describing the dynamics of real populations is of great interest. As an example, let us consider both representations for the hyperbolic “quadratic” equa2 tion dN dt ¼ kN that appears in global demography (see Chapter 3). The aforementioned results show that the quadratic equation can be considered as an equation describing the dynamics of total population size of Malthusian inhomogeneous model with an exponentially distributed reproduction rate (D-model). This equation also describes the dynamics of total population size of the inhomogeneous F-model with Gamma-distributed reproduction rate.

p Comparison between D- and F-models for equation dN dt ¼ kN ,p > 1.

TABLE 6.1 D-model

F-model

dlðt, aÞ dt ¼ kalðt, aÞ

dlðt, aÞ lðt, aÞ dt ¼ ka N ðtÞ ¼ kaPðt, aÞ

M0[δ] ¼ (1  (p  1)N(0)p1δ)1/(p1)

M0[δ] ¼ (1  pN(0)pδ)1/p

dq dt ¼ k

dq k dt ¼ N ðtÞ


 p   t p1 qðtÞ ¼ pN1ð0Þp 1  1  T t < T ¼ 1/(k(p  1)N(0)p1)

q(t) ¼ kt t 1), the growth rate per individual and the population size increase indefinitely in finite time. So deviations from Darwinian selection in both cases start when the models become unrealistic. Interestingly, it appears to have been underappreciated that a much simpler and more realistic model, the inhomogeneous logistic equation with distributed Malthusian parameter, also shows “survival of everybody.” This equation was considered in Chapter 2, Section 2.4; it presents a simple conceptual model for Malthusian struggle for existence, which accounts for both free exponential growth and resource limitations:
 dlðt; aÞ N ¼ alðt; aÞ 1  dt C (7.3) ð N ðtÞ ¼ lðt, aÞda: A

In this model, each clone l(t, a) grows logistically with its own Malthusian parameter a 2 A, which is assumed to be distributed with initial distribution P(0, a), and common carrying capacity C; N is the total population size. Let us cite (Gause, 1934): “All populations have the capacity to grow exponentially under ideal conditions, and no population can grow exponentially forever—there are limits to growth. This generates the Malthusian struggle for existence.” One may recognize here the description of the logistic population model (7.3), but this model does not generate Malthusian struggle for existence and survival of the fittest, as it was shown in Chapter 2, Section 2.4, and as will be discussed later in this chapter. In what follows, we study different generalizations of inhomogeneous logistic models from the angle “what kind of evolution these models show.” All models belong to the class of inhomogeneous models of “Malthusian” type studied in Chapter 2, Section 2.1. These models were described by the equation ð dlðt,aÞ ¼ alðt, aÞgðN Þ, N ðtÞ ¼ lðt, aÞda, (7.4) dt A

7.2 Solution to the inhomogeneous logistic equation

101

where g(N) issome function, chosen depending on the specifics of the model. For example, if  gðN Þ ¼ 1  NC , then Eq. (7.4) describes an inhomogeneous logistic model (7.3), where each clone grows logistically up to a common carrying capacity C. We refer to models (7.4) as inhomogeneous density-dependent models (D-models for brevity), since the right hand of the equation is proportional to the density of clones l(t, a). In what follows, we will also consider frequency-dependent models (F-models for brevity) that have the form dlðt,aÞ ¼ aPðt,aÞgðN Þ, dt

(7.5)

ðt, aÞ where Pðt, aÞ ¼ lN ðtÞ is the frequency of parameter a. Additional similarities and differences between these two model types were discussed in Chapter 6. Notice that formally model (7.5) is a special case of Eq. (7.4), but in applications it shows some interesting additional properties. In Chapter 11, we show that many experimental growth curves, including nonstandard two- and three-stage growth curves, can be understood and described within the frameworks of F-models. We show also in Chapter 6 that nonlinear population growth (7.2) can be realistically explained within the frameworks of inhomogeneous frequency-dependent models. In this chapter, we show that different generalizations of the inhomogeneous logistic equation with a distributed Malthusian parameter result in non-Darwinian “survival of everybody.” In contrast, the inhomogeneous logistic equation with distributed carrying capacity shows Darwinian “survival of the fittest.”   N Notice that the standard logistic equation dN dt ¼ bN 1  B can be written as the birth-anddeath equation dN dt ¼ N ðb  dN Þ: Ackleh et al. (1999) considered an inhomogeneous logistic equation in the form of inhomogeneous birth-and-death equation:

dlðt; b,dÞ ¼ lðt; b, dÞðb  dNÞ, dt

(7.6)

where b and d are, respectively, the per capita birth and death rates for clone l(t; b, d). As one can see, Eqs. (7.5) and (7.6) are equivalent if parameters a, C in Eq. (7.5) and b, d in Eq. (7.6) are constants, but it is not so if these parameters are distributed. It was proven in Ackleh et al. (1999) and the following papers of Ackleh and coauthors that under general conditions, only “the fittest,” that is, those individuals that have the largest value of b and the smallest value of d, survive in the population (7.6). In this chapter, we also consider an inhomogeneous birth-and-death equation (7.6) and give a simple proof that this equation results in the “survival of the fittest.” In addition to known results, we show an exact limit distribution of parameters of this equation. Finally, we consider “frequency-dependent” inhomogeneous models and show that although some of these models demonstrate Darwinian survival of the fittest as time goes to infinity, realistically there is not enough time for selection of the fittest species.

7.2 Solution to the inhomogeneous logistic equation Inhomogeneous logistic equation (7.3) with distributed Malthusian parameter a was solved in Section 2.4 with the help of HKV method. Recall that solution to this equation is

102

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

given by the formula l(t, a) ¼ l(0, a)eaq(t). Here the keystone variable q is the solution to the Cauchy problem dq N ð 0Þ ¼1 M0 ½q, qð0Þ ¼ 0, dt C

(7.7)

N ðtÞ ¼ N ð0ÞM0 ½qðtÞ

(7.8)

Ð where M0[λ] ¼ AeλaP(0, a)da is the mgf of the initial distribution of the parameter a. The total size of the population is given by the formula

and it solves the logistic-like equation
 dN N ¼ Et ½aN 1  : dt C

Ð Here Et[a] ¼ AaP(t, a)da is the current mean value of the parameter a; it can be computed using the following equation: E t ½a ¼

d lnðM0 ½qðtÞÞ : dq

Hence the initial inhomogeneous multi- or infinitely dimensional logistic equation (7.3) is reduced to a single equation (7.7) for q(t). The keystone variable q(t) can be considered as the “internal time” of the population: the dynamics of inhomogeneous logistic model (7.3) with respect to the internal time is identical to the dynamics of the inhomogeneous Malthusian model with respect to regular time. The principal difference is that for the inhomogeneous logistic model, q(t) tends to a finite value q∗ < ∞ as t ! ∞, which is the single equilibrium of Eq. (7.7) and can be found as the solution to the equation M½q∗  ¼ NCð0Þ. Proposition 7.1 The single equilibrium q∗ of Eq. (7.7) is stable for any initial distribution of the Malthusian parameter a. Indeed,

dq dt ¼ 0

if and only if 1

N N ð 0Þ ¼1 M0 ½q ¼ 0: C C

The function M0[q] 1 and it increases monotonically; hence the equation 1

N ð 0Þ M0 ½q ¼ 0 C

has a single solution q∗. Function q(t) also increases monotonically because qð0Þ ¼ 0, M0 ½0 ¼ 1,

N ð0Þ < 1, C

hence dq N ð 0Þ ¼1 M0 ½qðtÞ > 0 dt C

7.2 Solution to the inhomogeneous logistic equation

103

until N ð0ÞM0 ½qðtÞ ¼ N ðtÞ < C: Next,


 d N ð 0Þ N ð0Þ dM0 1 M0 ½q ¼  < 0; dq C C dq

therefore the equilibrium q is stable, as desired. Example 7.1 Let the initial distribution of the Malthusian parameter be exponential, ma m Pð0, aÞ ¼ e m , m ¼ const > 0; its mgf M0 ½λ ¼ mλ . Then dq N ð 0Þ m ¼1 : dt C mq

(7.9)

Fig. 7.1 shows the solution to this equation for m ¼ 1, NCð0Þ ¼ 0:1. Now we can see that the limit state of inhomogeneous logistic model (7.5) coincides with the current state of the inhomogeneous Malthus model at the instant q*. The limit stable population size and the limit distribution of the parameter a are N∗ ¼ N ð0ÞM0 ½q∗ , P∗ ðaÞ ¼ Pð0; aÞeq∗ a M0 ½q∗ :

(7.10)

Let us emphasize a notable property of the inhomogeneous logistic model (7.3) with a distributed Malthusian parameter: it remains inhomogeneous at any instant and has a nontrivial limit distribution of parameter a at t ! ∞. Every clone that was present initially will be present in the limit stable state. Therefore, the inhomogeneous logistic model illustrates the phenomenon of “survival of everybody” in the population over time, in contrast to Darwinian “survival of the fittest.”

FIG. 7.1 The “internal time” q(t) plotted against regular (chronological) time t for Eq. (7.9); q(t) ! q∗ ¼ 0.9 as t ! ∞.

104

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

7.3 Generalized logistic inhomogeneous models with distributed Malthusian parameter Different generalizations of the standard logistic equation are known in literature, which were applied to many particular problems; see, for instance, Tsoularis and Wallace (2002). Most of them are special cases of the generalized logistic equation "
β #γ dN N α ¼ rN 1  , (7.11) dt C where α, β, γ are the model parameters. Examples are the Blumberg equation (Blumberg, 1968) with β ¼ 1, the Richards equation (Richards, 1959) with α ¼ γ ¼ 1, and von Bertalanffy equation (Von Bertalanffy, 1938) with α ¼ 23 , β ¼ 13 , γ ¼ 1. In what follows, we will assume that r ¼ 1 (it always can be done by changing the timescale τ ! rt). The simplest (but not unique) inhomogeneous version of model (7.11) is "
β #γ dlðt, aÞ N α1 ¼ alðt, aÞN 1 (7.12) dt C so that

"
β #γ dN N t α ¼ E ½aN 1  : dt C

(7.13)

Again, l(t, a) is the density of the clone with an intrinsic Malthusian parameter a; the expres"
β #γ N α1 sion gðN Þ ¼ N 1 describes the dependence of the growth rate of each clone on C total population size. The inhomogeneous generalized equation (7.12) can be also solved by the HKV method. Let us define the keystone variable (the internal time of the population) q(t) by the equation "
 β #γ dq N ¼ gðN Þ  Nα1 1  , qð0Þ ¼ 0: (7.14) dt C Then lðt, aÞ ¼ lð0, aÞeaqðtÞ :

(7.15)

The total size of the population N(t) and the current distribution P(t, a) of the parameter a are given by the formulas (see Eqs. 2.7 and 2.9) ð N ðtÞ ¼ lðt, aÞda ¼ N ð0ÞM0 ½qðtÞ, A

Pðt, aÞ ¼

lðt, aÞ eaqðtÞ ¼ Pð0, aÞ, NðtÞ M0 ½qðtÞ

where M0[q] is the mgf of initial distribution P(0, a).

7.3 Generalized logistic inhomogeneous models with distributed Malthusian parameter

105

FIG. 7.2 Plots of internal time q(t) for von Bertalanffy equation (blue, bottom), Richards equation with γ ¼ 3 (green, 1 middle), and Richards equation with γ ¼ 1 (red, top); in all equations, K ¼ 10, and M0 ½q ¼ 1q .

The equation for q(t) can be written in a closed form: "
 #γ dq M0 ½q β α1 ¼ M0 ½q 1 , qð0Þ ¼ 0: dt C

(7.16)

This way the inhomogeneous multi- or infinitely dimensional logistic equation (7.12) is reduced to a single equation (7.16) for q(t). Now all statistical characteristics of the model (such as the current mean value and variance of parameter a) can be effectively computed. Fig. 7.2 shows the behavior of q(t) for different versions of the generalized logistic equation; 1 the initial distribution of the Malthusian parameter a is exponential with mgf M0 ½q ¼ 1q . We can see that behaviors of q(t) are very similar for all three equations and hence the behaviors of its solutions are also very close due to Eq. (7.15). The limit value q∗ solves the M ½q equation C0 ¼ 1 and is the same for all generalized logistic equations (7.12) with a given carrying capacity C and a given mgf of the initial distribution; q∗ is the single equilibrium of Eq. (7.16). It is easy to show (similar to Proposition 7.1) that this equilibrium is stable as the derivative of the right hand side of Eq. (7.16) is negative for q ¼ q∗. The limit distribution of inhomogeneous model (7.11) coincides with the current distribution of the inhomogeneous Malthus model (7.1) at the instant q*. Hence all the clones that were present in the population at the initial time will have positive frequency in the limit state of the population. It means “survival of everybody” for all inhomogeneous generalized logistic models (7.11). Notice that in the considered models, we assumed that the population is composed of logistic-like clones (species) that grow according to Eq. (7.3) or (7.12). In all cases the equations for total population size differ from the original logistic equations—compare, for instance, Eqs. (7.11) and (7.13); therefore the dynamics of their solutions is also different.

106

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

FIG. 7.3 Dynamics of the total population size; the standard logistic equation with a ¼ 0.5 (blue, bottom); inhomogeneous logistic equation (7.3) with exponential initial distribution: E0[a] ¼ 0.001 (black, second from the bottom), E0[a] ¼ 0.5 (red, second from the top), and E0[a] ¼ 5 (green, top).

Let us consider, for example, the standard logistic equation and its inhomogeneous modification, that is, Eqs. (7.11) and (7.13) for α ¼ β ¼ γ ¼ 1. Let the initial distribution of parameter a be exponential. In Fig. 7.3, we compare the dynamics of total population size for standard and inhomogeneous logistic models with different mean values of the initial distribution. We can see that the dynamics are different, although the limit values of N(t) as t ! ∞ are the same.

7.4 Inhomogeneous Gompertz equation The Gompertz growth curve (Gompertz, 1825) is widely used in cancer modeling and ecological problems (see e.g., Bajzer and Vuk-Pavlovic, 2000; Gyllenberg and Webb, 1988; Kendal, 1985). Statistical analysis of the results of Gause experiments (Figs. 24 and 25 in Gause (1934), provided in Nedorezov (2015)) showed that in some cases the Gompertz curve describes experimental time series even better than the logistic curve. The Gompertz curve is given by the equation    N ðtÞ ¼ N ð0Þ exp A 1  ert , where A, r are positive parameters. The Gompertz curve can be also written in equivalent form
 N ð0Þ rt e N ðtÞ ¼ K exp ln , K where K, r are positive parameters.

(7.17)

7.4 Inhomogeneous Gompertz equation

107

The curve given by Eq. (7.17) is a solution to the equation
 dN K ¼ rN ln : dt N The generalized Gompertz curve is defined as a solution to the equation
 dN K γ ¼ rN ln : dt N

(7.18)

Eq. (7.18) is a limit case of the generalized logistic equation "
β #γ dN r N ¼ γ N 1 as β ! 0: dt β K An inhomogeneous version of the Gompertz model (7.18) with distributed Malthusian parameter is
 dlðt, aÞ K γ ¼ alðt, aÞ ln : (7.19) dt N Again, l(t, a) is the density of a clone, which has the Malthusian parameter equal to a. Similarly to the previous sections, we can show that inhomogeneous Gompertz model also shows “survival of everybody.” Indeed, let us define the keystone variable q(t) through the following equation:
 dq K γ ¼ ln , qð0Þ ¼ 0: dt N This equation can be written in a closed form (see Eq. 2.8):
γ dq K ¼ ln , qð0Þ ¼ 0, dt N ð0ÞM0 ½q

(7.20)

where M0[q] is the mgf of the initial distribution P(0, a) of the parameter a. Eq. (7.20) has a unique stable equilibrium q∗, which is a solution to the equation M0 ½q ¼ NKð0Þ. Then the limit stable population size and the limit distribution of the parameter a are given by the formulas N ∗ ¼ N ð0ÞM0 ½q∗  P∗ ðaÞ ¼

Pð0; aÞeq∗ a : M0 ½q∗ 

Hence the limit distribution of inhomogeneous Gompertz model (7.19) coincides with the current distribution of the inhomogeneous Malthus model (7.1) at the instant q*; every clone that was present in the initial time moment will be present in the population forever. Therefore, the inhomogeneous Gompertz model also shows non-Darwinian “survival of everybody.”

108

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

7.5 Logistic equation with distributed carrying capacity Consider the logistic model of an inhomogeneous population, where positive birth rate parameter b is fixed for the whole population but each clone has its own value of carrying capacity C. Dynamics of such a population are described by the equation dlðt, dÞ ¼ lðt, dÞðb  dN ðtÞÞ, dt

(7.21)

where N is the total population size and d ¼ C1 is distributed parameter defining the “death rate.” Assume parameter d takes values in domain D. Let us define the keystone variable q(t) by equation dq ¼ N, qð0Þ ¼ 0: dt

(7.22)

Then lðt, dÞ ¼ lð0, dÞeðbtdqðtÞÞ ; the total size of this population is given by N ðtÞ ¼ N ð0Þebt M0 ½qðtÞ,

(7.23)

where M0 is the mgf of the initial distribution of parameter d. Hence we have reduced the inhomogeneous logistic equation (7.21) to a single equation for the keystone variable dq ¼ Nð0Þebt M0 ½qðtÞ, qð0Þ ¼ 0: dt

(7.24)

With the solution to this equation, we can compute the total population size at any time by formula (7.23); current distribution of parameter d is given by Pðt, dÞ ¼

edqðtÞ Pð0, dÞ: M½qðtÞ

Now we can compute all statistical characteristics of interest at any time given initial distribution P(0, d); for example, the expected value of d can be computed as Et ½ d  ¼

dlnðM½δÞ =δ¼qðtÞ : dδ

We can also make some interesting conclusions about asymptotic composition of the population for any initial distribution. According to Eq. (7.22), q(t) monotonically increases. Lemma 7.1

lim qðtÞ ¼ ∞ as t ! ∞:

Assume that q(t) is bounded, q(t) < K ¼ const. Then

7.5 Logistic equation with distributed carrying capacity

M½q ¼

ð∞

eqx Pð0, xÞdx >

0

ð∞

109

eKx Pð0, xÞdx ¼ MðKÞ > 0,

0

so dq ¼ N ð0Þebt M½qðtÞ > N ð0Þebt M½K: dt It follows from here that q(t)! ∞ as t! ∞, in contrast to the assumption that q(t) is bounded. □ Let us assume that the minimal possible value dmin of parameter d belongs to the domain of values of the parameter, dmin 2 D. Proposition 7.2 Inhomogeneous population (7.21) asymptotically consists of a single clone that has the minimal value of parameter d. Indeed, lðt, d1 Þ lð0, d1 Þ ððd1 d2 ÞqðtÞÞ ¼ e ! 0 if d1 > d2 : lðt, d2 Þ lð0, d2 Þ We can arrive at this and some other assertions about asymptotic behavior of the model in t

another way as well. Applying the covariance equation dEdt½z ¼ Covt ½F, z (see Eq. ( 5.47) from Chapter 5) to Eq. (7.21), we obtain dEt ½d ¼ Covt ½d, b  dN  ¼ N ðtÞVart ½d < 0: dt

(7.25)

It follows from Eq. (7.25) that Et[d] decreases with time as long as Var[d] > 0, that is, as long as the population is inhomogeneous; so Et[d] tends to the minimal possible value of parameter d. This means that asymptotically the population will consist of a single clone with d ¼ dmin. Asymptotic behavior of the total population size is quite different in the cases when dmin > 0 versus when dmin ¼ 0. Namely, in the first case, lim N(t) ¼ b/dmin as t ! ∞, while in the second case, lim N(t) ¼ ∞. Example 7.2 Let the initial distribution of parameter d be exponential in [0, ∞) with mgf 1 M0 ½q ¼ 1q . Then the solution to Eq. (7.24) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðtÞ ¼ 1 + ð2 + b + 2ebt Þ=b for N(0) ¼ 1, and (see Eq. 7.23) bebt N ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bð2 + b + 2ebt Þ increases asymptotically exponentially; see Fig. 7.4. In contrast, if the initial distribution of parameter d is exponential in [0.01, ∞) with mgf 0:01q b ¼ 10 for b ¼ 0.1, and so dmin ¼ 0.01; see Fig. 7.5. M0 ½q ¼ e1q , then lim NðtÞ ¼ dmin Overall the logistic model with distributed carrying capacity and fixed Malthusian parameter shows Darwinian “survival of the fittest.”

110

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

0:32 e ffi. FIG. 7.4 Plot of log N(t), for b ¼ 0.1, dmin ¼ 0 ; NðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:9 + 2e0:1t 0:1t

0:1t0:01qðtÞ

FIG. 7.5 Plot of NðtÞ ¼ e 1 + qðtÞ ; q(t) solves Eq. (7.22).

7.6 Logistic equation with two distributed parameters: Malthusian parameter and carrying capacity We showed in the previous sections that the outcomes of evolution of populations described by the inhomogeneous logistic equations can be different. Namely, logistic-type equations with distributed Malthusian parameter show survival of everybody, while equations with distributed carrying capacity demonstrate the survival of the fittest (and the fittest have the largest carrying capacity).

7.6 Logistic equation with two distributed parameters: Malthusian parameter and carrying capacity

111

Let us now consider logistic equations, where both the Malthusian parameter and the carrying capacity are distributed; we can write it in the form of birth-and-death equation: dlðt; b, dÞ ¼ lðt; b, dÞðb  NdÞ: dt

(7.26)

This and more general birth-and-death equations were studied in Ackleh et al. (2005) and Ackleh et al. (1999). The authors proved that only “the fittest,” that is, individuals that have the largest value of b and the smallest value of d, survive in population over time; corresponding proofs are mathematically rather difficult. In the rest of this section, we apply the HKV method to model (7.26) and give a simple proof of the “survival of the fittest” in this model. In addition to known results, we show an exact limit distribution of the parameters of this model. First, let us consider the case when parameters b and d are independent at the initial time moment and take on values in domains B and D, respectively. It means that Pð0; b, dÞ ¼ P1 ð0, bÞP2 ð0, dÞ, where P1(0, b) and P2(0, d) are, respectively, the initial distributions of the parameters b and d. Let M1[λ], M2[λ] be the mgf-s of distributions P1(0, b) and P2(0, d), and let the parameters b amd d take their values from corresponding domains B and D. Introduce the auxiliary keystone variable q(t) as the solution to the Cauchy problem dq ¼ N, qð0Þ ¼ 0: dt

(7.27)

Then the solution to Eq. (7.26) reads ðtbqðtÞdÞ lðt; b, dÞ ¼ lð0; ð dÞe ð b,

eðtbqðtÞdÞ P1 ð0bÞP2 ð0dÞdbdd ¼ N ð0ÞM1 ½tM2 ½qðtÞ:

N ðtÞ ¼ N ð0Þ

(7.28)

B D

Taking N(0) ¼ 1 for simplicity, we obtain the closed equation for q(t): dq ¼ M1 ½tM2 ½q: dt

(7.29)

The current pdf of model (7.26) is given by the formula Pðt; b, dÞ ¼ Pð0; b, dÞ

eðtbqðtÞdÞ P1 ð0bÞetb P2 ð0dÞeðqðtÞdÞ ¼ : M1 ½0tM2 ½0  qðtÞ M1 ½0t M2 ½0  qðtÞ

Lemma 7.2 Let T  ∞ be such that M1[t] ! ∞ as t ! T. Then q(t) ! ∞ as t ! T. Indeed, if q(t) < C ¼ const for all t, then M2 ½q > M2 ½C and

dq ¼ M1[t]M2[ q]dt > M1[t]M2[ C]dt. Integrating this inequality and taking into account that q(0) ¼ 0, we have

(7.30)

112

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

qðtÞ > M2 ½C

ðt

M1 ½tdt:

0

Now the statement of the Lemma is evident. Proposition 7.3 lim t!T l ltð;t;bb∗ ,, ddÞ ¼ 0 for any b∗ b, d∗ < d or b∗ > b, and d∗  d. ð ∗Þ Indeed, lðt; b, dÞ lð0; b, dÞ ððbb∗ Þtðdd∗ ÞqðtÞÞ  ¼  e ! 0 as t ! T: l t; b∗ , d∗ l 0; b∗ , d∗ Hence, if dmin 2 D and bmax 2 B, then only the clone with the values of parameters b ¼ bmax and d ¼ dmin survives over time, implying “survival of the fittest.” Notice that it is natural to assume that dmin > 0; otherwise there exist “immortal” individuals in the population. In the general case, distributions of parameters d and b will concentrate in the vicinities of the points dmin and bmax (including the case bmax ¼ ∞ ), but if dmin 62 D or bmax 62 B, then the total “probability mass” leaves the domain D or B, respectively. It means that every clone will be overtaken by another clone over time. Example 7.3 See Ackleh et al. (1999). Let parameters b and d be independent and initially uniformly distributed, that is, b 2 [a1, b1] and d 2 [a2, b2] and Pð0; b, dÞ ¼ Pð0; bÞPð0; dÞ ¼

1 1 : b1  a1 b2  a2 ðλbÞ

ðλaÞ

e Recall that the mgf of the uniform distribution in the interval [a,b] is M½λ ¼ e λðba Þ . Then, according to Eq. (7.30),

Pðt; b, dÞ ¼ P1 ðt; bÞP2 ðt; dÞ, where P1 ðt; bÞ ¼ P2 ðt; dÞ ¼

etb tetb ¼ tb , ðb1  a1 ÞM0 ½t e 1  eta1

eqðtÞd qðtÞeqðtÞd ¼ qðtÞb : 2  eqðtÞa2 ðb2  a2 ÞM0 ½qðtÞ e

Here the keystone variable q(t) is defined by Eq. (7.29): dq etb1  eta1 eqb2  eqa2 ¼ N ð0ÞM1 ½qM2 ½q ¼ Nð0Þ : dt tðb1  a1 Þ qðb2  a2 Þ The current mean values of the birth and death rates are given by b1  a1 1  1  eðb1 a1 Þt t b2  a2 1 : + Et ½d ¼ a2 + ð b a Þq ð t Þ 2 2 qðtÞ 1e Et ½b ¼ b1 

7.6 Logistic equation with two distributed parameters: Malthusian parameter and carrying capacity

113

It follows from the last formulas that Et[b] ! b1 and Et[d] ! a2 as t ! ∞ (q(t) ! ∞ as t ! ∞ according to Lemma 7.2). It is another proof that in model (7.26) with independent uniformly distributed parameters, only the “fittest” clone, that is, the one with maximal birth rate b2 and minimal death rate a2, survives with time. Dynamics of model (7.26) vary dramatically depending on the initial distribution. Example 7.4 Let the positive parameters b, d be independent and the initial distributions of both parameters be exponential with respective means 1/T and 1; P1(0; b) ¼ Te bT, T ¼ const > 0, and P2(0; d) ¼ ed; 0  b, d < ∞. Then the mgfs of these distributions are M1 ½λ ¼

T 1 , M2 ½λ ¼ : Tλ 1λ

Let N(0) ¼ 1 for simplicity. Then dq T ¼ M1 ½tM2 ½q ¼ : dt ðT  tÞð1 + qÞ Solution to this equation is given by the formula qðtÞ ¼ 1 +

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 2T ln ðT=ðT  tÞÞ,

so q(t) ! ∞ as t ! T. The current system distribution P(t; b, d) ¼ P1(t; b)P2(t; d), where both marginal distributions are again exponential (see Eq. (7.30)) and are given by P1 ðt; bÞ ¼

P2 ðt; dÞ ¼

P1 ð0; bÞetb ¼ ðT  tÞebðTtÞ , M1 ½t

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 ð0; dÞeqðtÞd ¼ ð1 + qðtÞÞedð1 + qðtÞÞ ¼ 1 + 2T lnðT=ðT  tÞed 1 + 2T ln ðT=ðTtÞ, M2 ½qðtÞ Et ½b ¼ E t ½ d ¼

1 ! ∞, Tt

1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 0 as t ! T: 1 + qðtÞ 1 + 2T ln ðT=ðT  tÞÞ

Notice now that ðB 0

P1 ðt, bÞdb ¼ 1  eBðTtÞ ! 0

114

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

as t ! T for arbitrarily large B; it means that all clones with bounded birth rate at any given boundary B disappear from the population before time T. On the other hand, ðε P2 ðt, dÞdd ¼ 1  eεð1 + qðtÞÞ ! 1 0

as t ! T (and hence q(t) ! ∞) for arbitrarily small ε; it means that all clones with a lower bound for the death rate disappear from the population before time T. The total population size N ðtÞ ¼ M1 ½tM2 ½q ¼

T T pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðT  tÞð1 + q2 ðtÞÞ ðT  tÞ 1 + 2T ln ðT=ðT  tÞÞ

tends to infinity as t ! T. Hence the solution to Eq. (7.26) with the initial exponential distribution exists only up to the time t ¼ T; the model loses meaning for t T. We have shown that the stochastic independence of the birth and death rates in the inhomogeneous model (7.26) implies “survival of the fittest.” Now let us consider a general case, relaxing the assumption of independence of the birth and death rates. We restrict our analysis to a more realistic case, when each parameter takes only a finite number of values. More exactly, we consider a population consisting of n clones as defined by Eq. (7.26), such that ith clone is characterized by a pair of parameters (bi, di) and is governed by the equation dlðt; bi , di Þ ¼ lðt; bi , di Þðbi  di N Þ: dt

(7.31)

Then Eqs. (7.27) and (7.28) take the form: lðt; bi , di Þ ¼ lð0; bi , di Þeðbi tdi qðtÞÞ , N ð t Þ ¼ N ð 0Þ

n X

Pð0; bi , di Þeðbi tdi qðtÞÞ ,

i¼1

dqðtÞ ¼ NðtÞ, qð0Þ ¼ 0: dt So, to solve Eq. (7.31), we need to study the equation n dqðtÞ X ¼ pi eðbi tdi qðtÞÞ , dt i¼1

(7.32)

115

7.6 Logistic equation with two distributed parameters: Malthusian parameter and carrying capacity

P where pi ¼ P(0; bi, di) 0 are initial frequencies of the clones, ni¼1pi ¼ 1; we assume N(0) ¼ 1 for simplicity. Asymptotic behavior of q(t) is studied in the Mathematical Appendix at the end of this chapter using the Newton diagram method. The obtained results allow us to study the asymptotic behavior of current frequencies of the clones:   pj ebj tdj qðtÞ P t; bj , dj ¼ Xn , j ¼ 1, …,n: p ebi tdi qðtÞ i¼1 i

(7.33)

Now we would like to find the limit values of these probabilities, Pj ¼ limt!∞ P(t;bj,dj), and to find j, at which Pj > 0. Before formulating the main result, let us introduce some notation and definitions. Let ρ ¼ maxi dbii , and let I be the set of all indices such that dbii ¼ ρ. Define a function: f ðzÞ ¼

X

pi z d i :

i2I

Let C 6¼ 0 be a solution to the equation f(z) ¼ ρ. Function f(z) increases monotonically, so the solution to equation f(z) ¼ ρ exists and is unique. di

Theorem 7.1 Pi ¼ pi Cρ if iEI and Pj ¼ 0 for all other j. See Mathematical Appendix at the end of this chapter for proof. The statement of Theorem 7.1 (see also Theorem A.1 in Mathematical Appendix) can be interpreted as follows. Only the “fittest” clones survive in the population, but the population may have not one but several fittest clones. The fittest clone is defined by the b condition dbii ¼ ρ, for all other clones djj < ρ, and the frequencies of these clones tend to 0 over time. In other words the “fittest” clones are those, where the ratio of birth-to-death coefficients reaches the maximal value possible in the population rather than their absolute values. Example 7.5 Assume a population consists of n ¼ 100 clones, and the initial distribution is uni1 form, with pi ¼ 100 for all i. Let dbii ¼ ρ for the first 50 clones, i ¼ 1, … , 50. Then, according to Theorem A.1, the limit frequencies Pj ¼ 0 for all j > 50 and Pj > 0 for all 1  j  50. Let us assume that di ¼ si, s ¼ const., say s ¼ 0.1. Then f ðzÞ ¼

X i2I

pi zdi ¼

50 1 X z0:1i , 100 i¼1

P50 0:1i 1 and C in Theorem A.1 is the root of the equation 100 ¼ ρ. i¼1 C Consider two cases: (a) ρ ¼ 0.3, and (b) ρ ¼ 3. In case (a), C ¼ 0.8026; the plot of the final distribution is shown on panel A of Fig. 7.6. In case (b), C ¼ 1.778; the plot of the final distribution is shown on panel B of Fig. 7.6.

116

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

FIG. 7.6 Plots of the limit frequencies for model (7.30). (A) Final distribution of the population in Example 7.5, ρ ¼ 0.3, C ¼ 0.8026. (B) Final distribution of the population in Example 7.5, ρ ¼ 3, C ¼ 1.778.

7.7 Dynamics of distributions in inhomogeneous models and the speed of natural selection In the previous sections, we investigated the dynamics of the total size of various inhomogeneous populations and their final composition in terms of the final distribution of the Malthusian parameter. Here we will focus on the dynamics of the distribution of the Malthusian parameter, which will allow us to trace the effects and outcomes of natural selection in the aforementioned models. The current pdf of the Malthusian parameter given its initial distribution for models of the Malthusian type (2.1) is given by the general formula (2.9). The dynamics of the current pdf is

7.7 Dynamics of distributions in inhomogeneous models and the speed of natural selection

117

determined by the auxiliary keystone variable q(t) (an “internal time” of the population) as defined by Eq. (2.5). In most cases the function g(N) is positive, so q(t) increases monotonically. We discuss the concept of “internal time” within the context of inhomogeneous models in more detail in Chapter 10. Dynamics of q(t) depends on the initial distribution of the Malthusian parameter according to Eq. (2.8). The results reported in Chapter 3 highlight an important role of exponential initial distributions for different applications. Notice that it is unrealistic to assume that the Malthusian parameter can have an arbitrarily large value even with extremely small probability. Therefore it is more realistic to assume that the initial distribution of the Malthusian parameter is truncated exponential of the form ÐB

Pð0, aÞ ¼ Cesa for 0  a  B,

where C ¼ 1= 0 esa sa ¼ 1esBs is the normalization constant. The mgf of the exponential distribution truncated in the interval [0, B] with distribution parameter s is given by the formula  Bs  ðB e  eBλ s aλ M½λ ¼ e Pð0, aÞda ¼ : ð1  eBs Þðλ  sÞ 0 Then the current pdf is defined by the formula (see Eq. 2.9) Pðt, aÞ ¼

Pð0, aÞeqðtÞa eqðtÞs ðqðtÞ  sÞ ¼ BðqðtÞsÞ : M½qðtÞ 1 e

(7.34)

Now let us compare the dynamics of truncated exponential distribution with respect to different inhomogeneous models. The simplest models are inhomogeneous versions of the standard Malthusian model, namely, the density-dependent Malthusian model (D-model) dlðt, aÞ ¼ alðt, aÞ dt and the frequency-dependent Malthusian model (F-model). dlðt, aÞ alðt, aÞ ¼ ¼ aPðt, aÞ: dt N ðt Þ Over time, distributions in both models tend to become concentrated near the maximal possible value of the distributed parameter a, such that Et[a] ! B. Indeed, according to Eq. (2.14), dEt ½a ¼ Vart ½a > 0 for D  model dt and dEt ½a Vart ½a ¼ > 0 for F  model: dt N ðt Þ Therefore Et[a] increases as long as Vart[a] > 0, that is, until Et[a] approaches the maximal possible value (here equal to B). Note that according to Eq. (2.2),

118

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

dN ¼ NEt ½a for the D-model dt and dN ¼ Et ½a for the F-model: dt It follows from here that asymptotically, population size increases exponentially for the D-model and linearly for the F-model. Qualitatively the evolution of both models is identical up to time change t ! q(t), where dq 1 ¼ dt N . However, the evolution of the F-model in real time t is dramatically slower compared to the density-dependent model. Let us illustrate the difference in the rate of evolution in D- and F-models. A numerical example is given in Fig. 7.7, where the initial distribution is truncated exponential with s ¼ 2 on the interval [0, 1]. F-model at time t and D-model at time t∗ have identical distributions 20 15 10 t 5 0 15 ε

10 P 5 0 1.0

0.5 0.0

a

(A) 8 ´ 106 6 ´ 106 t 4 ´ 106 2 ´ 106 0

15 P

10 5 0

(B)

0.0

0.5 a

1.0

FIG. 7.7 Dynamics of distributions of the Malthusian parameter for (A) densitydependent model and (B) frequencydependent model. Initial distribution is truncated exponential with s ¼ 2 in the interval [0, 1]. Distribution of the D-model at time t ¼ 20 coincides with the distribution of F-model at the time t∗ ¼ 8970000.

7.7 Dynamics of distributions in inhomogeneous models and the speed of natural selection

119

of the parameter a, if q(t∗) ¼ t. However, qualitatively, q(t∗) ¼ 10 if t∗ ¼ 1015, q(t∗) ¼ 15 if t∗ ¼ 86030, and q(t∗) ¼ 20 if t∗ ¼ 8970000. As one can see, the rate of evolution is orders of magnitude slower in the frequency-dependent model compared to the density-dependent model. The reason for such relatively slow rate of evolution in the F-model compared to the Dmodel is that in the F-model the internal time q(t)ln t. Indeed, in the F-model, the population size N(t) is asymptotically linear, N(t) t. Hence, when dq 1 ¼ , dt N ðtÞ then qðtÞ ¼

ðt

du  ln t: 0 N ðuÞ

Overall, both models show Darwinian “survival of the fittest” in the sense that asymptotically the distribution of the Malthusian parameter a over time becomes concentrated in an arbitrarily small vicinity of the maximal possible value of the parameter. The difference is that the rate of evolution in the F-model decreases dramatically compared to the rate of evolution of the density-dependent model. Practically, it means that frequency-dependent population is “much more polymorphic” than the density-dependent population of the same age; frequency-dependent population tends to a monomorphic state, but it takes an unrealistic amount time. The exclusion principle is valid for both of these models but only theoretically. Selection of the fittest species requires unrealistic population size for D-model and unrealistic time for F-model. Inhomogeneous logistic models, whether density-dependent or frequency-dependent, demonstrate non-Darwinian “survival of everybody.” The reason for this phenomenon is that the “internal time” q(t) for all these models does not increase indefinitely over time, unlike Malthusian inhomogeneous D- and F-models, but tends to a finite value q∗ that solves equaM ½q tion C0 ¼ 1. As a result, the limit distribution P(t ! ∞, a) for the considered logistic-like models with initial distribution P(0, a) coincides with the distribution of the inhomogeneous density-dependent Malthusian model with the same initial distribution P(0, a) taken at time t ¼ q∗. Let us emphasize that similarly to inhomogeneous Malthusian F-models, the evolution of the inhomogeneous frequency-dependent logistic model is much slower than the evolution of the density-dependent logistic model. In Fig. 7.8, one can see the evolution of the initial exponential truncated distribution in [0, 1] for the logistic D-model
 dlðt; aÞ N ¼ alðt; aÞ 1  dt C and for the logistic F-model
 dlðt; aÞ N ¼ aPðt; aÞ 1  : dt C

120

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

FIG. 7.8 Evolution of the distribution of Malthusian parameter in the (A) logistic density-dependent model and (B) logistic frequency-dependent model. The carrying capacity in both models is C ¼ 100. Initial distribution is truncated exponential in [0, 1] with s ¼ 2.

20 15 t 10 5 0 4

P

2 0 0.0 0.5 a

(A)

1.0

t

1000

500 0

4 P 2 0 0.0

(B)

0.5 a

1.0

7.8 Some notes on internal time and the competitive exclusion principle In this chapter, we considered several conceptual mathematical models of natural selection. In order for selection to operate, the community or population needs to be heterogeneous, and so the models appropriate for studying natural selection should be constructed within the frameworks of inhomogeneous population models. In particular, we considered inhomogeneous versions of classical Malthusian, Gompertzian, and different logistic-like models. The outcomes of natural selection in populations described by these models can be very different. Specifically, inhomogeneous Malthusian and birth-and-death models show Darwinian “survival of the fittest,” while inhomogeneous logistic-like and Gompertzian models show non-Darwinian “survival of everybody.” The dynamics of these models that initially are of large or even infinite dimensionality were investigated using the HKV method. To that end, we introduce an auxiliary keystone variable that can be interpreted as “internal time” of the population. All statistical characteristics of considered inhomogeneous models can be explicitly expressed with the help of this variable. The outcomes of natural selection in

7.8 Some notes on internal time and the competitive exclusion principle

121

inhomogeneous populations are determined by the asymptotic behavior of internal time q(t). Specifically: (1) If q(t) ! ∞ as t ! ∞, then we have Darwinian survival of the fittest. (2) If q(t) ! ∞ as !t∗ < ∞, then we have non-Darwinian survival of the common. (3) If q(t) ! q∗ < ∞ as t ! ∞, then we have non-Darwinian survival of everybody. There are no other options for population models that possess the internal time. In light of mathematical results described in this chapter, it is interesting to discuss the competitive exclusion principle, which is often considered to be a direct consequence of Darwinian “survival of the fittest.” A large body of literature is devoted to the discussion of this principle (known also as the Gause principle), which is one of the most important statements in ecology. In his 1934 book, G.F. Gause described his experiments, which he considered to be experimental proof of mathematical theory of struggle for existence developed mainly by Lotka and Volterra. Specifically, he considered it to be an empirical confirmation of an important theoretical result that followed from mathematical models, namely, that two species with similar ecology cannot coexist in the same space. Hardin (1968) reformulated the “exclusion principle” in a more aphoristic form: “complete competitors cannot coexist.” Later, this statement was generalized to the case of a community consisting of an arbitrary number of species: “no stable equilibrium is possible if some r species are limited by less than r resources” (Levin, 1970). It was assumed here that the growth rates of species depend linearly on the amount of available resources. Recently, it was shown (Szila´gyi et al., 2013) that the principle of competitive exclusion holds for template replicators if resources (nucleotides) affect growth linearly. It is important to notice that the assumption that the growth rates of species are linear functions of resources is crucial; if this assumption is relaxed, then coexistence of r species on k < r resources is possible, although perhaps not with constant densities (on this topic, see an important paper by Armstrong and McGehee (1980) and references within). Although this principle seems to follow directly from Darwinian natural selection, some contradictions have been identified, such as the “paradox of plankton.” Diversity of natural phytoplankton seems to contradict the competitive exclusion principle. Although most algae compete for the same inorganic nutrients, often more than 30 species coexist even in small parcels of water. Hutchinson (1961) posed his classic question: “How is it possible for a number of species to coexist in a relatively isotropic or unstructured environment, all competing for the same sorts of materials?” This problem was discussed in many papers; in particular, Wilson (1990) wrote that “the almost ubiquitous existence of multi-species communities is one of the few firm facts in ecology. How can alpha species diversity be as high as it is within most actual communities, in the face of the Principle of Gause that no two species can permanently occupy the same niche? Why does competitive exclusion not occur, leaving only one species - the one with the highest competitive ability?” Wilson also noticed that a similar question exists for tropical rain forest and coral reef communities. In this 1990 paper, he proposed 12 possible mechanisms to explain the paradox: 1. Niche diversification 2. Pest pressure

122 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

Equal chance Gradual climate change Intermediate-timescale disturbance Life history differences Initial patch composition Spatial mass effect Circular competitive networks Cyclic succession Aggregation Stabilizing coevolution

Several other hypotheses have additionally been proposed to address the problem of the exclusion principle, such as the following: (1) Resource models of coexistence (Levin, 1970): for maximal coexistence to occur, the M competitors must consume at least M resources in different proportions; this theory was refined by Armstrong and McGehee (1980). (2) Biodiversity under nonequilibrium conditions by species oscillations and chaos (Huisman and Weissing, 1999). (3) Neutral theory (Hubbell, 2001, 2006): evolution of ecological equivalence or niche convergence. Additionally, some of the modern theories suggest that the coexistence of species with similar competitive abilities can occur in nature as a result of a balancing act between fitness equalizing processes, such as trade-offs, and fitness stabilizing processes like the rare species advantage. One can reasonably ask: if there are so many exclusions from the exclusion principle, then maybe something is wrong either with the ways the principle was tested or with the principle itself? Why are there so many different deviations from this principle? Discussion of experimental and theoretical aspects of this problem is beyond the scope of this chapter, but we can provide insights into the exclusion principle from the point of view of conceptual mathematical models appropriate for modeling of outcomes of natural selection. An interesting and important point was suggested by Hardin (1968): “There are many who have supposed that the principle is one that can be proved or disproved by empirical facts, among them Gause himself. Nothing can be farther from the truth … The ‘truth’ of the principle is and can be established only by theory, not being subject to proof or disproof by facts … Indeed, let two non-interbreeding species that seem to have the same ecological characteristics be placed in the same location; if one of species extinguished the other, one says that the principle is proved. But if the species continue to coexist indefinitely, one may decide that there must be some subtle difference in ecology.” The “theory” here is based on the mathematical models of selection and struggle for existence, in accordance with Gause himself. The Gause principle can be seen as a special case (or consequence) of the Darwinian “survival of the fittest.” Conversely, a common opinion is that it is the struggle for existence that causes the Darwinian “survival of the fittest.” In this

Mathematical Appendix: the Newton diagram method

123

chapter, it was shown that the struggle for existence, if described by the logistic-like models that account both for exponential growth and limited resources, may not result in the Darwinian “survival of the fittest” after all. We showed that many mathematical models do not confirm the principle in its initial strong form as it was formulated by Gause and Hardin; in contrast, these models show coexistence of many species and even “survival of everybody.” So, instead of the statement “complete competitors cannot coexist,” we accept the statement “complete competitors may coexist,” at least in mathematical models. Moreover, even models that demonstrate Darwinian selection of the fittest may provide insights as to the clear deviation from the exclusion principle: For instance, if a population grows according to a frequency-dependent Malthusian model, then its rate of evolution is extremely slow, and thus it will take indefinite time for the fittest species to be selected. The selection process is occurring but it will never be over. In the next chapter, we will shift gears and dive into the discussion of the Principle of minimal information gain. We will show how with the HKV method it can be derived from system dynamics rather than postulated a priori, and discuss the implications of this result.

Mathematical Appendix (by F. Berezovskaya): The Newton diagram method and asymptotic behavior of q(t) for inhomogeneous birth-and-death equations A.1 Basic equation in the new form Function q(t) is defined by Eq. (7.31): n X dqðtÞ X ¼ pi exp ðbi t  di qðtÞÞ, pi > 0, pi ¼ 1, bi , di 2 R + : dt i¼1

(A.1)

The right-hand side is always positive; therefore q(t) increases monotonically. It is clear that it cannot tend to a constant, since in this case dq/dt would tend to zero; however, the righthand side of Eq. (A.1) increases indefinitely if q(t) tends to a constant. Hence limt!∞ q(t) ¼ ∞ . Denote v ¼ exp ðtÞ $ t ¼  log ðvÞ, u ¼ exp ðqÞ $ q ¼  log ðuÞ, u, v ! 0 as t ! ∞ because q(t) ! ∞ . In (u,v) coordinates, Eq. (A.1) becomes X X X b d d σb d σ du du dq dt u i pi v i u i u i pi u i v i u i pi u i v i ¼ ¼ ¼ ¼ : dv dq dt dv v vσ + 1 vσ + 1

(A.2)

(A.3)

Here σ ¼ maxi¼1,…,n {bi}, σ i ¼ σ  bi 0. Without loss of generality, we assume that b1  b2  …  bn and then 0 ¼ σ n  σ n1  …  σ 1 ¼ σ  b1  σ. The condition q(0) ¼ 0 implies that u(0) ¼ 1.

124

7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

Eq. (A.3) can now be written in the following form: dv  v0 ¼ vσ + 1 ¼ vPðv, uÞ, dτ X du  u0 ¼ u pi vσ i vdi ¼ uQðvuÞ, dτ i

(A.4)

where (0 ) means differentiation by some “dummy” positive variable τ. We have reduced the problem of asymptotic behavior of q(t) to the problem of analysis of the behavior of trajectories of system (A.4) in a positive neighborhood of the equilibrium point (v, u) ¼ (0, 0). The problem of asymptotic behavior of probabilities P(t; bj, dj) is reduced to computation of the following values:   pj ebj tdj qðtÞ pj vσ j udj ¼ lim v,u!0 Xn , j ¼ 1, …,n: (A.5) Pj ¼ lim t!∞ P t; bj , dj ¼ lim t!∞ Xn bj tdj qðtÞ σ i udi p e p v i i i¼1 i¼1 To solve this problem, we apply the method of Newton diagram, developed in Berezovskaya (2014), Bruno (2003) and described briefly in the next section.

A.2 The Newton diagram method Let us consider the Kolmogorov-type power vector field Z(x, y) ¼ {R(x, y), S(x, y)} given by the following system of differential equations: X x0 ¼ x rμν xμ yν ¼ xRðx, yÞ, μ,ν  0 μ , ν X (A.6) y0 ¼ y sμν xμ yν ¼ ySðx, yÞ: μ, ν Definition 1 A set M ¼ {(μ, ν) : jrμν j + j sμν j 6¼ 0} is the support of vector field (A.6); (rμν, sμν) is the vector-coefficient of the point (μ, ν) 2 M. Definition 2 Newton diagram (ND) Γ is the convex hull of {(μ, ν) + R 2+ } if (μ, ν) 2 M; Γ may consist of one vertex γ (0) or is a polygonal line, which consists of edges γ 2 Γ together with their vertices; see Fig. (A.1). s

Definition 3 (a). Index of the vertex (μ, ν) is the value β  βðμ, Þ ¼ rμνμν if rμν 6¼ 0 and β ¼ ∞ if rμν ¼ 0. 2 (b) Index of the edge γ is the value α  αðγ Þ ¼ μν12 μ ν1 > 0 if the points A1(μ1, ν1) , A2(μ2, ν2), and μ1 6¼ μ2, ν1 6¼ ν2 belong to the edge γ. Remark A.1 The edge index α(γ) is equal to the slope of the line l passing throw the points A1, A2 with negative direction of the ordinate axis. Let M be the support containing n 1 points and Γ be the Newton diagram of vector field (A.6). Denote Zγ (Rγ , Sγ } as the truncation of Eq. (A.6) to the edge γ; Zγ is the vector field of the form (A.6), where summation is performed over (μ, ν) 2 Mγ  M \ γ.

Mathematical Appendix: the Newton diagram method

125

Definition 4 Vector field Z(R, S) is nondegenerate if (1) index β of any vertex of Γ does not equal to indexes of edges adjacent to this vertex, (2) for any edge γ the functions Rγ (1, z), Sγ(1, z) have no common nonzero roots, (3) the function Fγ(z) ¼  αRγ (1, z) + Sγ (1, z), α ¼ α(γ) has no multiple nonzero roots. It is clear that nondegenerate Kolmogorov-type vector field Z has trivial orbits x ¼ 0 and y ¼ 0; hence the isolated singular point O is not monodromic (i.e., is not a center or a focus). The following theorem describes asymptotic behavior of orbits in a small neighborhood of singular point O (Berezovskaya 2014; Berezovskaya, 1979). Let us call to O-orbit any orbit (x(t), y(t)) such that (x(t), y(t)) ! (0, 0) as t !  ∞. Theorem A.1 Let Z be nondegenerate Kolmogorov-type vector field. Then all nontrivial O-orbits of Z have power asymptotics: y ¼ K∗ xρ ð1 + oð1ÞÞ, K∗ 6¼ 0, ρ > 0, x ! 0

(A.7)

where the power ρ is equal to the index β of a vertex 2 Γ or to the index α of an edge 2 Γ: _

_

(i) ρ ¼ β if and only if e α < β < α where e α, α are indexes of edges adjacent to the vertex (μ, ν) having _ index β; e α¼0 if vertex (μ, ν) belongs to ordinate axis, that is, μ ¼ 0, and α ¼ ∞ if vertex (μ, ν) ∗ belongs to abscise axis, that is, ν ¼ 0. The coefficient K in formula (A.7) is an arbitrary constant. (ii) ρ ¼ α ¼ α(γ) if and only if the function Fγ (z) has a root z ¼ K∗. Remark A.2 Several asymptotics defined by the theorem may exist simultaneously.

A.3 Asymptotics of orbits of system (A.4) Let us apply the Newton diagram (ND) method to analysis of the asymptotic behavior of system (A.4) X v0 ¼ vσ + 1 ¼ vPðv, uÞ, u0 ¼ u pi vσi udi ¼ uQðv, uÞ, (A.8) Pðv, uÞ ¼ vσ , Qðv, uÞ ¼

X

i

pi vσi udi and 0 ¼ σ n  σ n1  …  σ 1  σ  σ 0 :

i

The support M of systems (A.4) and (A.8) consists of n + 1 points: A0 ¼ A0(σ 0, 0) with vectorcoefficient (1,0), and Ai ¼ Ai(σ i, di) with vector-coefficients (0, pi), i ¼ 1, … , n. Let us construct the Newton diagram Γ of the system (see Fig. A.1). Between all the points on the ordinate axes d, let us choose the point D(0, min(dj)) ¼ D(0, d). It is clear that points A0, D are vertices of Γ. The line l that connects these two points has the slope αðlÞ ¼ σd0 with the negative direction of ordinate axis (see Remark A.1). The lines li i connecting the point A0(σ 0, 0) and the points Ai(σ i, di), i ¼ 1, … , n, have slopes αi ¼ σ0dσ . i σ0 σ0 i If αðlÞ ¼ d  α , i ¼ 1, …, n, then ND has unique edge γ 1, and α1  αðγ 1 Þ ¼ d is its index. Points i Ai(σ i,di) such that αi  σ0dσ ¼ σd0 ¼ α1 also belong to the edge γ 1 (and are enumerated A1i(σ 1i,d1i), i iEIγ 1 ), whereas other points of support M don’t belong to Γ (see Fig. A.1A).

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7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

d

d D

Ai

d g1

D

d

g2

A12 dc

d2 A11

d1

Ai

C A11

d1

g1 A0

A0 0

(A)

s2

s1

s0

s

0

sc

s1

s0 s

(B)

FIG. A.1 The Newton diagram is the convex polygonal line passing through the points A0(σ 0, 0), D(0, d) such that each Ai lies above or on it. (A) The diagram consists of one edge γ 1 whose index is α1; (B) the diagram consists of two edges γ 1, γ 2 whose indexes are α1 > α2.

If the minimal slope α1 ¼ min i αi is achieved by drawing a line connecting points A0 and C  C(σ c, dc), where σ c 6¼ 0, then the Newton diagram Γ has the edge γ 1 , whose vertices are c points A0(σ, 0) and C(σ c, dc) and α1  αðγ 1 Þ ¼ σσ dc . In this case, Γ has at least two edges. The case when ND of system (A.4) has exactly two edges, γ 1, γ 2, is presented in c Fig. A.1B. Here the edge γ 1 bounded by vertices A0, C has index α1 ¼ σ0dσ , and the edge c σc γ 2 bounded by vertices C and D has index α2 ¼ ddc < α1 . Generally the Newton diagram Γ can have some more edges. Thus we have proven the following statement. Proposition A.1 The Newton diagram Γ of system (A.4) always contains edge γ 1 with the vertex A0(σ 0, 0); it can also contain the edges γ 2,… γ r, r 2. All edges compose a polygonal line connecting the vertices A0(σ 0, 0) and D(0, d). Now we can apply Theorem A.1 to system (A.8) to find asymptotic behavior of orbits of Eq. (A.8). Proposition A.2 Orbits of system (A.8) have only one nontrivial asymptotics: u ¼ Kvρ ð1 + oð1ÞÞ, v ! 0

(A.9)

where K > 0, ρ ¼ α(γ), and γ is the edge of Newton diagram of Eq. (A.4) that has the vertex (σ 0, 0). Proof Due to Theorem A.1 a power ρ of asymptotics of system (A.4) can be equal only to a vertex or an edge index. The vertex (σ 0, 0) of Newton diagram has index β ¼ β0 ¼ 0. Index of any point Ai(σ i,di) 2 M, di > 0, and i ¼ 1 (including point D(0, d)) is equal to ∞ (according to Definition 3(a)). Thus system (A.8) has no orbits with power asymptotics (A.9) that correspond to vertices of Γ. Next, let us study the asymptotics (A.9) that correspond to edges of diagram Γ. Let us assume that the Newton diagram of system (A.4) has only one edge γP ¼ γ 1 with the vertices A0(σ, 0) and B(0, d) ; α ¼ σd0 is the edge index. Then Pγ (1, z) ¼ 1, Qγ (1, z) ¼ j2I(γ 1)pjudj. The function X σ0 Fγ ðzÞ ¼ αPγ ð1, zÞ + Qγ ð1, zÞ ¼  + pj udj d j2Iðγ Þ 1

127

Mathematical Appendix: the Newton diagram method

because pj 0 for j 2 I(γ 1), p0 6¼ 0 . This function has only one positive root, say z ¼ K∗, because P dj j2Iγ pjz monotonically increases with z. Applying Theorem A.1, we can state that the system has orbits with asymptotics (A.9) u ¼ Kvρ(1 + o(1)), where ρ ¼ α ¼ σd0 , K ¼ K∗ , and that there are no orbits with other positive asymptotics with ρ ¼ α. Now consider the case when the Newton diagram contains two edges, γ 1 and γ 2. Then the edge γ 1 has vertices A0(σ, 0) and C(σ c, dc) and the index α(γ 1) ¼ α1; the edge γ 2 has vertices C(σ c, dc), B(0, Pd), and d > dc and index α(γ 2) ¼ α2, α1 > α2. As in previous case the function Fγ 1(z) ¼ α1  j2I(γ 1)pjzdj has only one positive root. The truncation of systems (A.4) and P σ j dj (A.8) into edge γ is given by P (v, u)  0, Q (v, u) ¼ p v u , so the function 2 γ2 γ2 j2I(γ 2) j P Fγ 2(z) ¼ j2I(γ 2)pjzdj evidently has no positive roots (because all pj are positive). Due to Theorem A.1 the orbits of vector field (A.4) have no positive asymptotics (A.9) with ρ ¼P α2. Similarly, if the Newton diagram contains r > 2 edges, then the functions Fγ k(z) ¼  j2I(γ k)pjzdj and k ¼ 2, … , r have no positive roots, and so positive orbits of Eq. (A.4) have no asymptotics (A.9) with the powers ρ ¼ αk, k > 1. The proposition is proven. □ Notice now that, by definition,


 σ0  σi bi ¼ max i αð γ 1 Þ ¼ min i , di di   bi i ¼ max and therefore αð γ 1 Þ ¼ min i σ 0dσ i di . i Then, Propositions A.1 and A.2 imply the following theorem. Theorem A.2 Orbits of systems (A.4) and (A.8) that tend to (u ¼ 0, v ¼ 0)from  positive initial values have unique no-trivial asymptotics (A.9) u ¼ Kvρ(1 + o(1)). Here ρ ¼ max i P of the function Fγ (z) ¼  ρ + j2Iγpjzdj.

bi di

, and K is a positive root

A.4 Asymptotics of probabilities To compute the limiting probabilities Pj ¼ P(t; bj, dj) given by formula (A.5), we need the following estimations. Let X Φðx, yÞ ¼ fμ,ν xμ yν , fμ,ν 6¼ 0, Φð0, 0Þ ¼ 0 ðμ, νÞ2M M¼{(μ, v)} be the support of the function Φ and Γ be the corresponding Newton diagram, P that is, the convex hull of the points {(μ, ν) + R 2+} for (μ, ν) 2 M. Let Φγ (x, y) ¼ (μ,ν)2M\γ fμ,νxμyν be the truncation of Φ(x, y) to an edge γ with index α. Lemma A.1 There exist nonnegative constants λ, δ such that for arbitrary constant z,   Φðx, zxα Þ ¼ xλ Φγ ð1, zÞ + xδ φðx, zÞ , P where Φγ (1, z) ¼ (μ,ν)2M\γ fμ,νzν and φ(0, 0) ¼ 0.

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7. Inhomogeneous logistic equations and models for Darwinian and non-Darwinian evolution

Proof Let us write the function Φ(x, y) in the form X X e ðx, yÞ ¼ fμ,ν xμ yν + fμ,ν xμ yν : Φðx, yÞ ¼ Φγ ðx, yÞ + Φ ðμ, νÞ2M\γ ðμ, νÞ2Mn γ The edge γ belongs to the line μ + αν ¼ c ¼ const if (μ, ν) 2 M \ γ and μ + αν > c if (μ, ν) 62 M \ γ. Then X X fμ,ν xμ + αν zν ¼ xc fμ,ν zν , Φγ ðx, zxα Þ ¼ μ + αν¼c

e ðx, zx Þ ¼ Φ α

X

μ + αν>c

ν¼ðcμÞ=α

fμ,ν x

μ + αν ν

z ¼x

c

X μ, ν

fμ, ν xδμν zν

where δμν > 0. Thus, taking λ ¼ c and 0 < δ < min(μ,ν)2(MM\γ)(δμν), we finish the proof.

σ

p v ju

Asymptotic values of probabilities are defined by formulas (A.5):Pj ¼ lim v, u!0 Pnj

dj

p v σ i u di i¼1 i

,

j ¼ 1, …,n where σ i ¼ σ  bi 0, σ ¼ max P i¼1,…, n{bi}. Let us present the function G(u, v) ¼ ni¼1pivσiudi as the sum of two terms; the first one is the truncation of G(u, v) to the edge γ 1 having index ρ, and the second one contains all other summands: n X

X

pi vσi udi ¼

ðσ i , di Þ2M\γ 1

i¼1

n1 X

pi vσi udi +

pi vσi udi ¼

ðσ i , di Þ2Mn γ 1

X

pi vσ i udi +

i2Iðγ 1 Þ

pi vσi udi :

i62I ðγ 1 Þ

i ¼ ρ; hence σ i + ρdi ¼ σ Recall that, by definition, the index αð γ 1 Þ ¼ σ0dσ i σ 0 σ j we have dj < ρ; hence σ j + ρdj ¼ σ + δj, δj > 0 if j 2 I ðγ 1 Þ.




X

if i 2 I(γ 1); for any j 2
Iðγ 1 Þ,

Using power asymptotics (A.9) and Lemma A.1, we can write for j ¼ 1, … , n1: pj vσ j udj pj vσ j + ρdj K∗dj ð1 + oð1ÞÞ X X ¼ ¼ n X pi vσi + ρdi K∗di ð1 + oð1ÞÞ + vδ pj vσi + ρdi δ K∗di ð1 + oð1ÞÞ σ i di pi v u i2I ðγ 1 Þ

i¼1

¼

0 vσ @

i62I ðγ 1 Þ

vσ pj K∗dj ð1 + oð1ÞÞ X i2Iðγ 1 Þ

pi K∗di + vδ

X

1

pj K∗di vδj Að1 + oð1ÞÞ

pj K∗dj : ! X pi K∗di i2Iðγ 1 Þ

i62I ðγ 1 Þ

Therefore Pj ¼ lim v, u!0 Pn

X pj vσ j udj pj K∗dj X ¼ , j 2 I ð γ Þ and Pj ¼ 1: 1 n X pi K∗di j2I ðγ 1 Þ pi vσ i udi i¼1

i2I ðγ 1 Þ

As i¼1Pi ¼ 1, then Pj ¼ 0 for all j 2
Iðγ 1 Þ. We have proven the following theorem. □ n o   Theorem A.3 Let ρ ¼ max i dbii , S ¼ ðbi , di Þ : dbii ¼ ρ , and I ¼ {i : (bi, di)ES}; let K∗ be the single posP p K ∗di itive solution of the equation i2Ipizdi ¼ ρ. Then Pi ¼ P i ∗dj > 0 for i 2 I and Pj ¼ 0 for all j 2
Iðγ 1 Þ. pK j2I j

C H A P T E R

8 Replicator dynamics and the principle of minimal information gain Abstract In this chapter, we discuss a very general approach based on the principle of minimal information gain to infer unknown distributions subject to given testable information about the system. We show that the problem of rationale of this principle has a strong solution within the frameworks of selection systems and corresponding replicator equations, and prove that minimization of information gain is the underlying principle that follows and can be derived from the dynamics of selection systems instead of being postulated. These results are then applied to several specific selection systems, namely, the Malthusian inhomogeneous models, the model of global demography, models of tree stand self-thinning, inhomogeneous versions of classical logistic and Ricker models, and to the quasi-species theory. This Chapter is based on (Karev, 2010a,b).

8.1 Problem formulation Replicator equations describe dynamics of distributions in heterogeneous populations and communities under selective pressures. As was mentioned in previous chapters, very high or even infinite dimensionality is one of the most fundamental difficulties in studying the replicator equations. The HKV method helps to overcome this difficulty in many but not all cases. A very general approach to infer unknown distributions subject to some given testable information about the system can be based on the principle of minimal discrimination information, MinxEnt, or equivalently, the principle of maximum information entropy, MaxEnt. The MaxEnt principle stated most briefly posits, “when we make inferences based on incomplete information, we should draw them from that probability distribution that has the maximum entropy permitted by the information we do have” ( Jaynes, 1957). Here, “entropy” means the Shannon–Gibbs entropy of a discrete distribution {pi}, X pi log pi : (8.1) S½p ¼  i

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00008-4

129

# 2020 Elsevier Inc. All rights reserved.

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8. Replicator dynamics

Jaynes (1957, 2003) and his followers showed that essentially all known statistical mechanics can be derived from the MaxEnt principle. During last several decades, these methods have been successfully applied to analysis of a vast number of phenomena. Kullback (1997) formulated a similar (and, formally, a more general) principle using the KL divergence of distribution p from m (Kullback and Leibler, 1951) defined as ð I ½p : m  ¼

pðxÞ log A

  pð xÞ pð xÞ dx ¼ Ep log : mðxÞ mðxÞ

(8.2)

The value I[p : m] is also known as the cross entropy or information entropy and S[p : m] ¼  I[p : m] as the relative entropy. The KL divergence allows for unequal prior probabilities m and remains well-defined for continuous distributions, in contrast to the Shannon-Gibbs entropy, which can become undefined for nondiscrete probabilities. The KL divergence is always nonnegative but not symmetric; therefore it is not the true distance between distributions (see, e.g., Kapur, 1989; Dukkipati et al., 2006). Changes in the distribution can be measured with KL divergence between the initial and current distributions I[p : m]. The MinxEnt principle (Kullback and Leibler, 1951; Kullback, 1997) states that, given new facts, a new distribution p should be chosen, which is as hard to discriminate from the original distribution m as possible; so that the new data produce as small an information gain I[p : m] as possible. These facts, knowledge, experimental conditions or given physical (biological) constraints can typically be expressed as expected values over the unknown probability. The inference of p by minimizing I[p : m] is known as the principle of minimal discrimination information, MinxEnt, which is equivalent to the principle of maximum relative entropy, MaxEnt. In many biological applications the KL divergence I[p : m] can be interpreted as information gain; accordingly, MinxEnt can be reformulated as the principle of minimal information gain, and in what follows we will understand the principle in this sense. Shore and Johnson (1981) suggested an axiomatic explanation of the MaxEnt principle as a method for updating probabilities; this approach was developed further by several researchers (Skilling, 1988; Csisza´r, 1996). Nevertheless, the rationale of the MaxEnt method is substantially different from that of other statistical methods. According to Jaynes (1957), “the probability assignment which most honestly describes what we know should be the most conservative assignment in the sense that it does not permit one to draw any conclusions not warranted by the data.” A serious objection against this approach in natural sciences is that the principle of maximum entropy does not follow from the basic laws and fundamental theories and hence may or may be not postulated as an independent assertion. The problem was clearly formulated by Einstein in 1910: “… the statistics of a system should follow from its dynamics and, in principle, could not be postulated a priori” (Einstein, 1993). This statement points to a possible way to obtain a satisfactory rationale of the MaxEnt principle, but it still was not realized (and probably could not be realized) in statistical physics. There exists vast literature devoted to different aspects of both the rationale and applications of the principle of maximal (relative) entropy or minimal information gain; many interesting examples are given, for example, in Kapur (1989). Even a brief overview of the literature is out of the scope of this chapter. However, generally in applications to dynamical

8.2 MinxEnt algorithm and the Boltzmann distributions

131

systems the MinxEnt principle is used to estimate the unknown distribution p(x) at given constraints when the system is in equilibrium. Our aim here is to show that the problem of rationale of the principle of minimal information gain has a strong solution within the frameworks of selection systems and corresponding replicator equations. Later, we show that the MinxEnt principle is valid as an exact theorem for a wide class of selection systems not only in equilibrium states but also at every point along the system trajectory. More specifically, we prove the dynamical principle of minimal information gain in the following form: the solution to a replicator equation minimizes at every instant the KL divergence of the initial and current distributions at some time-dependent constraints; these constrains, in their turn, can be computed explicitly at every moment due to the dynamics of the selection system. The main conclusion follows from these results: the minimal KL divergence between initial and current distributions is an intrinsic property of selection system dynamics and of solutions of replicator equations; minimization of information gain is an underlying principle that follows and can be derived from the dynamics of selection systems instead of being postulated. These results are then applied to several specific selection systems some of which are familiar to the reader from Chapter 3, namely, the Malthusian inhomogeneous models, the model of global demography, models of tree stand self-thinning, inhomogeneous versions of classical logistic and Ricker models, and the quasi-species theory. We also give the “conjugate” description of the solution of RE based on the dynamical MinxEnt principle. We show that the KL divergence is the Legendre transform of the partition function of corresponding time-dependent Boltzmann distributions and that the solution to the escort system and the current mean values of the “traits” accounted for by the selection system are conjugate variables.

8.2 MinxEnt algorithm and the Boltzmann distributions MinxEnt is an inference algorithm that gives the least-biased probability distribution consistent with available information. Assume that a prior (or initial) pdf m is known; assume also that the expected values As of some variables φs, s ¼ 1, … , n over unknown pdf p are given: Ep[φs] ¼ As. We want to estimate the pdf p according to the principle of minimal information gain, MinxEnt. The solution to this problem, which actually is a standard problem on conditional extremum, is well known and can be obtained using the method of Lagrange multipliers (an introduction to this method can be found, e.g., in the textbook by H. Anton, Section 15.9 (Anton, 1999); systematic applications of this method to MinxEnt problems are given in Kapur (1989). To give an idea of how the pdf p can be computed based on MinxEnt, let us consider a P simple case of discrete distributions and a single constraint Ep[a] ¼ si¼1piai ¼ A. In this case, I ½p : m  ¼

s X i¼1

pi ln

pi , mi

where m ¼ {mi} is the prior probability; probability p ¼ {pi} should be estimated subject to prescribed mean value of a random variable a, such that Ep[a] ¼ A. Hence Pthe pdf p is the solution to a standard conditional extremum problem with the constraints si¼1pi ¼ 1, Ep[a] ¼ A.

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8. Replicator dynamics

According to the Lagrange method of undetermined multipliers, we need to construct the Lagrangian  s  X pi L¼ pi ln  ðλ0  1Þpi  λpi ai , mi i¼1 where λ0 and λ are the Lagrange multipliers. The extremum of L is given by the equation  s  X dL δL ¼ δpi ¼ 0: dpi i¼1 Since the values δpi are arbitrary, the only solution to this equation is dL ¼ 0 for all i: dpi Differentiation gives ln

pi  λ0  λai ¼ 0, mi

and therefore

Since

Ps

i¼1pi ¼ 1,

pi ¼ mi eλ0 + λai : it should be pi ¼

mi eλai , Z

where Z¼

s X

  mi eλai ¼ Em eλa :

i¼1 ln Z The Lagrange multiplier λ is defined by the constraint Ep[a] ¼ A; it is clear that Ep ½a ¼ ∂ ∂λ , ∂ ln Z so λ can be found from the equation ∂λ ¼ A. Similar results are valid in the general case. The “MinxEnt probability distribution” (MEPD for brevity) p that minimizes the discrimination information I[p : m] subject to the constraints Ep[φk(a)] ¼ Ak, k ¼ 1, …, n is ! n X 1 λs φs ðaÞ , (8.3) pðaÞ ¼ mðaÞexp Z s¼1

where

" ZðλÞ ¼ Em exp

n X k¼1

and the Lagrange multipliers λk solve the system

!# λk φk

(8.4)

8.2 MinxEnt algorithm and the Boltzmann distributions

133

∂ lnZ ¼ Ak : ∂λk

(8.5)

The minimized value of the discrimination information is I ½p : m ¼  ln ZðλÞ +

n X

λk A k :

(8.6)

k¼1

The MEPD distribution (8.3) belongs to the family of natural exponential distributions (see, e.g., Akin, 1982; Chentsov, 1982). Let us recall the following definition. A random n-dimension vector X ¼ (X1, … , Xn) has multivariate natural exponential distribution with parameters θ ¼ (θ1, … , θn) with respect to the positive measure ν on Rn if its joint density function is of the form f ðXÞ ¼ hðXÞeX

T

θsðθÞ

:

Here, s(θ) is the normalization function. So, the vector (φ1(a), … , φn(a)) has multivariate natural exponential distribution with parameters (λ1, … , λn) with respect to the measure m(a) and Z as the normalization function. Distribution of the form (8.3) is known in statistical thermodynamics P as generalized Maxwell-Boltzmann distribution. The expression eB, where B ¼ ns¼1λsφs(a) is the Boltzmann factor, and the normalizer Z as given by Eq. (8.4) is the generalized partition function. So, any MinxEnt distribution is the generalized Boltzmann distribution; conversely, any distribution of the form of Eq. (8.3) provides minimal I[p : m] under constraints Ep[φk] ¼ Ak, k ¼ 1, …, n. On the other hand, we have shown in Chapter 4 (see Proposition 4.2) that any generalized Boltzmann distribution solves a replicator equation. Conversely, if the distribution P(t, a) satisfies the replicator equation, then it is the generalized Boltzmann distribution. Overall, there exist close interconnections between three fundamental objects: MinxEnt principle, generalized Boltzmann (or exponential family of ) distributions, and replicator equations (or dynamics of selection system). Loosely speaking, they are in some sense equivalent. To be more precise, (A) the Kullback (or Jaynes’, for MaxEnt) theorem states that the MinxEnt distribution belongs to the exponential family, implying that MinxEnt ) generalized Boltzmann distribution (B) any generalized Boltzmann distribution (8.3) is the MinxEnt distribution with corresponding constraints and prior pdf and therefore satisfies the MinxEnt principle, suggesting the inverse implication: generalized Boltzmann distribution )MinxEnt (C) any time-dependent generalized Boltzmann distribution solves corresponding replicator equation implying that time-dependent Boltzmann distribution ) Replicator equation (D) inverse implication: Replicator equation (dynamics of selection system) ) time-dependent generalized Boltzmann distribution; it was proven in Chapter 4 (see Proposition 4.2) From (A), (B), (C), and (D), it follows that

134

8. Replicator dynamics

FIG. 8.1 The relationship between replicator equation, principle of minimum information gain, and subsequent connection to the generalized Boltzmann distribution.

(E) Replicator equation (dynamics of selection system) , Principle of minimal information gain. These implications are shown in Fig. 8.1. Let us consider aforementioned implication (E) in detail in the following section.

8.3 Selection systems and dynamical principle of minimal information gain In Chapter 4, we looked at the selection system of the form dlðt, aÞ ¼ lðt, aÞFðt, aÞ dt n X Fðt, aÞ ¼ ui ðt, GÞφi ðaÞ,

(8.7)

i¼1

where G(t) ¼ G1(t), … , Gm(t) is a set of regulators (see Chapter 4 for definitions) and ui(t, G) are ð t, a Þ continuous functions. The current system distribution Pðt,aÞ ¼ lN ðtÞ solves the replicator equation (see Proposition 4.1):  dPðt, aÞ ¼ Pðt, aÞ Fðt, aÞ  Et ½F dt

(8.8)

and is equal to Pðt,aÞ ¼

Pð0,aÞKt ðaÞ , E0 ½Kt 

(8.9)

P where Kt(a) ¼ exp( ni¼1qi(t)φi(a)). The keystone variables qi(t) solve the escort system (see Eq. 4.18): dqi ðtÞ ¼ ui ðt, G∗ ðtÞÞ, qi ð0Þ ¼ 0, i ¼ 1,…, n: dt

(8.10)

Eq. (8.9) shows that pdf P(t, a) is a generalized Boltzmann distribution with the Boltzmann factor Kt ¼ eB, where

8.3 Selection systems and dynamical principle of minimal information gain

BðqðtÞ; aÞ ¼

n X

135

qi ðtÞφi ðaÞ

i¼1

and the partition function is given by ZðqðtÞÞ ¼ E0 ½Kt :

(8.11)

Remark that in our case the partition function is completely known, given the initial pdf P(0, a) and the solution to the Cauchy problem (8.10). Within the frameworks of selection system (8.7), the partition function as a function of time has clear biological meaning: ZðqðtÞÞ ¼

N ðt Þ N ð 0Þ

is proportional to the current population size, which follows from formula (4.4) N(t) ¼ N(0) E0[Kt]. On the other hand, the partition function can be expressed using the mgf of the initial joint distribution of the r.v. φ1, … , φn. Indeed, let ! ð n X λi φi ðaÞ Pð0; aÞda: M0 ½λ1 , …, λn  ¼ exp A

i¼1

Then, ZðqðtÞÞ ¼ E0 ½Kt  ¼ M0 ½q1 ðtÞ,…, qn ðtÞ:

(8.12)

Comparing distribution (8.9) with the MinxEnt distribution (8.3), one can conclude that the solution to replicator equation (8.8) minimizes the KL divergence of the initial and current distributions under time-dependent constraints Ai(t) ¼ Et[φi] not only at equilibrium but also at each point of the system trajectory. These constrains in turn can be computed explicitly at every instant because the current pdf is known. Thus, we arrive at Theorem 8.1 (1) Let Pt ¼ P(t, a) be the solution (8.9) of replicator equation (8.8). Then, at every moment t, the distribution Pt provides minimum of I[Pt : P0] over all probability distributions compatible with the constraints Ai(t) ¼ Et[φi], i ¼ 1, ‥. , n. (2) The values of constraints evolve due to escort system (8.7) and at each time can be computed using equations Ai ðtÞ ¼

E 0 ½ φi K t  ∂ ¼ log M0 ½q1 ðtÞ, …, qn ðtÞ: E0 ½Kt  ∂qi

(8.13)

(3) Dynamics of the constraints are determined by the covariance equation dAi ðtÞ ¼ Covt ½F, φi : dt The first assertion of the theorem is already proved. Assertion 2 follows from Eq. (8.9) for the current pdf and from Eq. (8.12) for E0[Kt]. The last equation follows from the covariance Eq. (5.47) applied to Et[φi] ¼ Ai(t).

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8. Replicator dynamics

The following statement allows us to compute information gain given initial distribution and keystone variables. Proposition 8.1 For all t,

  I Pt : P0 ¼ Et ½B  ln M0 ðtÞ,   E0 ½BKt   ln E0 ½Kt  I Pt : P 0 ¼ 0 E ½Kt 

(8.14)

where B ¼ B(q(t); a). Indeed, by definition of information gain as given by Eq. (8.2),       t 0 Pt Kt ðaÞ t t I P : P ¼ E log 0 ¼ E log 0 ¼ Et ½B  log E0 ½Kt  ¼ Et ½B  log M0 ðtÞ: E ½Kt  P The second equation of (8.14) follows from the first one. Using Eq. (8.14) and taking into account that N(t) ¼ N(0)M0[t], we obtain the conservation law for selection systems:   I Pt : P0  Et ½B + lnN ðtÞ ¼ const, where the constant is equal to lnN(0). The following theorem is of interest from a theoretical point of view. Theorem 8.2 The information gain I[Pt : P0] solves the covariance equation   dI Pt : P0 ¼ Covt ½B, F dt

(8.15)

and can be computed as

Proof

n   X ∂ I P t : P0 ¼ qi ðtÞ log M0 ðq1 ðtÞ, …, qn ðtÞÞ  log M0 ðq1 ðtÞ, …, qn ðtÞÞ: ∂qi i¼1

(8.16)

Eq. (8.15) follows from the complete Price’ equation (Price, 1972; see also Page and Nowak, 2002; Rice, 2006):   dEt ½z dz ¼ Covt ½F,z + Et , (8.17) dt dt where z(t, a) is an arbitrary time-dependent trait. Applying the Price’ equation to Et[B] and taking into account that dB dt ¼ F (from definition of keystone variables), we get dEt ½B ¼ Covt ½F, B + Et ½F: dt According to Eq. (8.14),   dI Pt : P0 d t E0 ½FKt  E ½B  ln E0 ½Kt  ¼ Covt ½F, B + Et ½F  0 ¼ ¼ Covt ½F, B dt E ½K t  dt

(8.18)

8.4 MinxEnt principle and selection systems: Applications

137

because

due to Eq. (8.9). Next,

E0 ½FKt  ¼ Et ½F E0 ½Kt 

    X n   Pt Kt I Pt : P0 ¼ Et log 0 ¼ Et log 0 qi ðtÞEt ½φi ðaÞ  log E0 ½Kt  ¼ ¼ P E ½Kt  i¼1 ¼ from Eq:ð8:12Þ ¼ n X ∂ qi ðtÞ log M0 ðq1 ðtÞ, …, qn ðtÞÞ  log M0 ðq1 ðtÞ, …, qn ðtÞÞ: ¼ ∂qi i¼1

□

The last equation can be written in a shorter form. Let us recall the notion of the homogeneity operator (or Θ-operator): Θ¼

n X k¼1

xk

∂ : ∂xk

Then, using definition of partition function (8.12), Eq. (8.16) can be written as    I Pt : P0 ¼ Θ log ZðqðtÞ  log ZðqðtÞÞ:

(8.19)

In the following section, we demonstrate how these results can be applied to some classical population models within the frameworks of selection system (8.7). A similar theory can be developed for selection systems and replicator equations with discrete time (see Chapter 15).

8.4 MinxEnt principle and selection systems: Applications 8.4.1 Information gain in inhomogeneous Malthusian model Let us start from a simple selection system (8.7) with F ¼ φ(a); we can consider the value φ(a) ¼ a as the distributed parameter and study the simplest replicator equation  dPt ðaÞ ¼ P t ðaÞ a  E t ½ a : dt The corresponding inhomogeneous Malthusian model is dlðt, aÞ ¼ alðt, aÞ: dt

(8.20)

This model was studied in Chapter 2. For this equation, F ¼ a, and B ¼ at, and according to Theorem 8.2,   dI Pt : P0 ¼ Covt ½F, B ¼ tVart ½a, (8.21) dt so the information gain increases monotonically.

138

8. Replicator dynamics

For practical computations of I[Pt : P0] at different initial distributions with corresponding mgf M0, we can use Eq. (8.14); then,   (8.22) I Pt : P0 ¼ tEt ½a  ln M0 ½t: The current distribution of the inhomogeneous Malthusian model provides the minimal information gain subject to a single constraint, Et[a] ¼ A(t); this constraint changes with time due to model dynamics and can be computed as Et ½a ¼

d lnM0 ½t : dt

Let parameter a be normally distributed at the initial instant, so that M 0 ½ λ  ¼ eλ

σ =2 + λm

2 2

,

where m is the mean and σ is the variance. It was shown in Chapter 4, Proposition 4.3 that prameter distribution at any time t is also normal with the mean Et[a] ¼ m + tσ 2 and with the same variance σ 2. The information gain is equal to   I Pt : P0 ¼ tEt ½a  lnM0 ðtÞ ¼ t2 σ 2 =2, 2

and

  I Pt : P0 ! ∞ at t ! ∞:

Now, let parameter a be Gamma-distributed with coefficients k, s, η at the initial instant, such that   λ k λη for λ < s: M0 ½λ ¼ e 1  s Then, the parameter is also Gamma-distributed with coefficients k, s  t, η at any time t < s (see Proposition 2.4). The information gain is     kt t + k ln 1  , I Pt : P0 ¼ tEt ½a  ln M0 ½t ¼ st s and

Specifically, if k ¼ 1, mean s, then

  I Pt : P0 ! ∞ as t ! s < ∞: η ¼ 0, that is, the initial distribution is exponential with the   λ M0 ½λ ¼ 1  , s  t N ðtÞ ¼ N ð0Þ 1  , s

and

8.4 MinxEnt principle and selection systems: Applications

139

   t 0 t t + ln 1  : I P :P ¼ st s The latter example is applicable to conceptual models of global demography. It was shown in Chapter 3, Example 4 that inhomogeneous Malthusian model of the form (8.20) describes world population growth up to 1980 with high accuracy if the initial distribution of the growth rate is exponential with the mean 1/T, T ¼ 2025. The population grows in such a way that the distribution of the reproduction rate is expo1 nential at every instant t < T with the mean Et ½a ¼ Tt , providing minimum of the information t 0 gain I[P : P ] at each time moment with that mean reproduction rate. The “demographic explosion” in this model occurs at the moment t ¼ T when not only N(t) ¼ ∞ but also Et[a] ¼ ∞ , Vart[a] ¼ ∞ and I[Pt : P0] ¼ ∞. It is a corollary of the obviously unrealistic assumption that the exponentially distributed individual reproduction rate may take infinitely large values with nonzero probability. When the reproduction rate in the model is bounded, a 2 [0, b] (specifically, for real demography data b  0.114), and initial distribution is truncated exponential, then N(t) is finite for all t, even though it is increasing, and is very close to a hyperbola for a long time (up to 1990 at corresponding values of coefficients). Then the mean value of the reproduction rate Et ½ a  ¼

1 b + , b T  t 1  e ðTtÞ

and the truncated exponential distribution provides minimum information gain among all distributions in the interval a 2 [0, b] with that mean reproduction rate at each time moment. Let us emphasize again that for these models the MinxEnt is not an external assumption but a strong mathematical corollary resulting from model dynamics. It is interesting to compare these outcomes with those that follow from the standard application of the MaxEnt principle for estimation of an unknown distribution. It is known (see, e.g., Kapur, 1989, Section 3.2.1) that if the mean reproduction rate is the only quantity we can estimate from historical demographic data, then the most likely (the maximum entropy) distribution of the reproduction rate is exponential with an estimated mean. When the reproduction rate is bounded, a 2 (0, c), and the mean reproduction rate is again prescribed, then according to the MaxEnt principle, the most likely distribution is truncated exponential in that interval (Kapur, 1989, Section 3.3.1).

8.4.2 Information gain in the model of early biological evolution Nonhomogeneous Malthusian dynamics together with the principle of limiting factors were used in a model of early biological evolution (Zeldovich et al., 2007); see Chapter 5, t, a Þ mðaÞa0 Þb ¼ lðt, aÞð1a . Here, Example 5.3. The model can be formalized as the selection system dlðdt 0 a ¼ (a1, … , an), and m(a) ¼ min[a1, … , an]; b is the birth rate and a0 is the native state probability b ¼ 1. Considering ai as independent random variables with common of a protein; we let 1a 0 pdf f(a), the model can be reduced to the inhomogeneous Malthusian equation dlðt, mÞ ¼ lðt, mÞðm  a0 Þ dt

140

8. Replicator dynamics

with the initial pdf of m, gðmÞ ¼ nð1  GðmÞÞn1 f ðmÞ, where GðmÞ ¼

ðm

f ðaÞda:

0

This equation can be solved explicitly for given f(a) as described in Chapter 5. Let P(t, m) be the pdf of m at t moment. If f ðaÞ ¼

ea=T , 0 1 that contain less information than the sum of information for independent parts). In the earlier expression the term (1  q)Iq[S(1)]Iq[S(2)] may be considered an interaction term. With respect to prebiotic evolution, the “nonreductionist” character of parabolic replicator systems might reflect the importance of group selection and competition between groups of “selfish cooperators” (Koonin and Martin, 2005; Szathma´ry and Demeter, 1987). The use of q-entropy has to be justified by the properties of the system such as nonadditivity (Borland et al., 1998; Tsallis, 2009) when it is employed to derive an unknown probability distribution. However, when it is already known that the system is described by power law/Pareto distribution, it follows that these distributions can be obtained from maximization of the Tsallis q-entropy. It is well known (see theorems of Jaynes (1957) and Kullback (1997)) that maximization of the relative Boltzmann-Gibbs-Shannon entropy results in distributions that belong to the exponential family. In other words, the MaxEnt principle in this case is merely a restatement of the fact that the distribution belongs to the exponential family. Similarly, the principle of maximum of the relative q-entropy is merely a restatement of the fact that the given distribution belongs to the Pareto (or Tsallis) family. Hence q-entropy and the corresponding variational principle may be used in every case where Pareto distribution is observed; the applicability of these approaches does not depend on the assumption of nonadditivity of the system. It was emphasized in the literature (see, e.g., Plastino and Plastino, 1994) that nonextensive thermostatistics is based on the following two postulates: (1) The entropy of a system is given by the q-entropy, and (2) experimental measurement of an observable variable yields the q-expectation value. In practice, it is difficult to expect that these postulates can be verified directly for different complex systems of interest. In most cases the validity of the postulates should be decided exclusively based on the conclusions to which they lead and their comparison with experiment. The main point is that the variable of interest in the system follows the Pareto distribution, and this is the case for models of prebiotic evolution, where frequencies of species follow the Pareto distribution, and the growth rate is an observable variable. Moreover, frequencies of species have Pareto distribution (1 + ax) b at each time moment with parameter a proportional to time. Let us emphasize that we do not derive evolutionary laws from maximization of the Tsallis entropy. We move in the opposite direction: we prove that the distribution of clones in nonexponential population model is the Tsallis (or Paretolike) distribution. Hence, under appropriate constraints, system dynamics obey the principle

9.5 Discussion

167

of minimum Tsallis relative entropy independently of whether we accept (believe in) this principle or not and independently of any particular properties of the population. Nonadditivity of information gain is not a property of the system under consideration to be postulated a priori, but is the last element in the logical chain:

There exists a large body of literature devoted to derivation of particular (including experimental) distributions from variational principles. The MaxEnt principle and the Tsallis formalism have already been applied to many problems in widely different areas (physics: astrophysics, cosmology, and turbulence phenomena; mathematics: Le`vy flights, superdiffusion, and nonlinear Fokker-Planck equations; economics: analysis of market trends; biology and medicine; etc.; see some references at http://tsallis.cat.cbpf.br/biblio. htm). It seems that the only common property of all these systems is nonadditivity of the entropy functional; in fact, it is a formal mathematical assertion, which follows directly from the axioms of the Tsallis entropy (see, e.g., Beck, 2009 and references within for generalized Shannon-Khinchin axioms). With regards to biological interpretation, Tsallis q-entropy naturally applies to biologically realistic parabolic replicator systems unlike the Shannon-Boltzmann entropy, which only applies to idealized exponential systems. More generally, the results of this analysis indicate that the MaxEnt (MinxEnt) principle is a general optimization principle that governs the dynamics of evolving populations regardless of the specifics of the growth dynamics. Only the choice of the appropriate entropy (information) function depends on the growth law of a particular class of systems. In the next chapter, we will look in more detail at some models of inhomogeneous population extinction. This more philosophical chapter will tackle the topic of time perception and how it can change under various conditions, ranging from chemical inhibition, to disease, to what could be happening in a dying brain. We will dive deeper into the topic of “internal time” and show how some of these concepts can be formalized and maybe even understood within the frameworks of mathematical models of heterogeneous populations.

C H A P T E R

10 Modeling extinction of inhomogeneous populations Abstract In this chapter, we will look at two types of subexponential models of population extinction. Unlike the more traditional exponential models, the life span of populations described by subexponential models is finite. In the first model, the population is assumed to be composed of clones that are decreasing independently from each other. In the second model, we assume that the size of the population as a whole decreases according to the subexponential equation. We then investigate the “unobserved heterogeneity,” that is, the underlying inhomogeneous population model, and calculate the distribution of frequencies of clones for both models. We show that the dynamics of frequencies in the first model is governed by the principle of minimal Tsallis information loss. In the second model, the notion of “internal population time” is proposed; with respect to internal time, the dynamics of clone frequencies are governed by the principle of minimal Shannon information loss. The results of this analysis show that the principle of minimal information loss is the underlying law for the evolution of a broad class of models of population extinction. Finally, we propose a possible application of this modeling framework to mechanisms underlying time perception. This chapter is based on Karev and Kareva (2016).

10.1 Mathematical and nonmathematical motivations Questions of primary importance in modeling biological populations most frequently pertain to describing laws that govern population growth and development. Questions that focus on population decline and extinction as a primary focus of investigation appear to attract less attention despite their potential importance in the areas of ecology, paleontology, and environment preservation. Furthermore, one could ostensibly look at individual death from this point of view as well, where any individual and even any single organ can be viewed as a complex community of many different cell populations that undergo extinction with the death of the individual. For instance, human brain is one of the largest biological populations, consisting of approximately 1011 neurons and 1014 synapses. Looking at extinction of this

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00010-2

169

# 2020 Elsevier Inc. All rights reserved.

170

10. Modeling extinction of inhomogeneous equations

population from the point of view of the framework that will be presented here can provide unexpected and thought-provoking predictions. An investigation of the process of population extinction with the help of even seemingly uncomplicated conceptual models can thus be of general interest. Informal motivation for the presented investigation of nonlinear models of population extinction arose from a well-known short story “An Occurrence at Owl Creek Bridge” by Ambrose Gwinnett Bierce (born June 24, 1842, in 1913 Bierce disappeared without a trace while traveling with Mexican rebel troops). “An Occurrence at Owl Creek Bridge” is the story of Peyton Farquhar, a confederate sympathizer condemned to death by hanging from Owl Creek Bridge. During the execution, the rope breaks, and Farquhar falls into the river. Once he finally surfaces, he realizes his senses are superhuman. He can see the individual blades of grass and the colors of bugs on the tree leaves, despite the fact that he is whirling around in the river. Realizing that soldiers are still shooting at him, he escapes and makes it to dry land. He travels through the seemingly unending forest, attempting to reach his home. He begins experiencing strange physiological events, hearing unusual noises from the woods, and believes he has fallen asleep while walking. He wakes to see his perfectly preserved home, with his beautiful wife outside. As he runs towards her, he suddenly feels a searing pain in his neck; a white light flashes, and everything goes black. It is revealed that Farquhar never escaped at all; he imagined the story in the few seconds that passed between his execution and his death. A similar idea forms the basis of the plot of the novel “Pincher Martin: The Two Deaths of Christopher Martin” by Golding (2013). It is a story of psychophysical, spiritual and existential plight of a naval lieutenant who believes himself to be the sole survivor after a torpedo attack of his ship in the Atlantic Ocean during the Second World War. At the start of the novel, Martin is in the water fighting for his life. An important detail is highlighted: right away he kicks off his heavy seaboots. He is saved after being washed ashore a rocky mid-Atlantic islet, where he begins his struggle for survival. Martin uses his intelligence, education and training to find food, collect fresh water and alert any potential rescuers. As time goes by, a series of strange and terrifying events occur, which he at first dismisses as hallucinations that last for many days. At the end of the story, we find out that when his body was found, he didn’t even have time to kick off his seaboots. This means that his struggle for survival on the island never actually happened; he imagined the story in the few minutes that passed between the torpedo attack and his death. So a question can arise: What happens to time perception in a dying brain? Individual perception of time is closely tied to physiological processes in a complex way. It is possible that a dying person could be experiencing the world according to his or her own “internal” clock, and the time on this clock is determined by increasingly failing physiological processes as a result of death of neurons and disintegration of neural pathways. One can imagine that from the point of view of a dying brain, time could be dragging on increasingly slowly, while from the point of view of everybody else, the chronological time is in fact very brief. In this chapter, we will look at a possible way to distinguish between “internal” and “chronological” time and show that within certain contexts, “internal” time can indeed become increasingly long, while to the outside world it’s only moments.

10.3 Population of subexponentially decreasing clones

171

10.2 Problem formulation The simplest model of population decrease and extinction is exponential of the form where k is some positive constant and N(t) measures the current value of some property relevant to the process; typically, N(t) is the total population size. Radioactive decay can be described by this equation with high accuracy but in application to biological populations, the exponential model may be oversimplified. The model describes a population, which formally is immortal, and ignores an important fact that all real populations are polymorphic, and hence different parts of the population are likely to decrease at different rates. Nonhomogeneous models of population decline address this issue, since in these types of models, death rate of the entire population depends on the distribution of properties in the population, and so, different subpopulations decay at different rates. The problem of interest here is the evolution of the distribution of population composition and of related statistical characteristics, such as mean, variance and entropy for a given initial distribution. Important examples of exponential models of extinction come from forest ecology (Karev, 2003; see Example 7 in Chapter 3) and epidemic models (Dodson, 2010). Here, we consider nonexponential models of extinction of polymorphic populations, which show some new interesting phenomena compared to exponential models. p Nonexponential power models of population growth of the form dx dt ¼ kx , suggested by Szathma´ry and Smith (1997), were considered in Chapter 6. An important property of subexponential models of extinction is that the lifetime of corresponding populations is finite. Here, we investigate two types of inhomogeneous power models of population extinction. In the first model, we assume that the population is composed of independent clones, each of which decreases according to the subexponential equation; the problem of interest is the dynamics of total population size and distribution of densities of the clones. We show that the distribution dynamics follow the principle of minimal Tsallis information loss (Chapter 9). In the second model, we assume that the total size of the inhomogeneous population decreases according to the subexponential equation; the problem of interest is population composition and the dynamics of separate clones. We demonstrate within the frameworks of inhomogeneous population models the universal frequency-dependent representation of the power extinction models. We also show that clone distribution dynamics follow the principle of minimal Shannon information loss. Finally, we propose that the population as a whole can possess “internal time,” which tends to infinity as the “external,” or chronological time tends to a finite time moment of population extinction. We conclude with a discussion of a possible interpretation of the properties of internal time and how it can relate to time perception. dN dt ¼ kN,

10.3 Population of subexponentially decreasing clones In this section, we consider a model of a population composed of individuals xi, characterized by their own “extinction rate” ki; whose dynamics is given by the equation

172

10. Modeling extinction of inhomogeneous equations

dxi p ¼ ki xi : dt

(10.1)

As before, we call a set of all individuals of a certain type a clone. Let us emphasize that the dynamics of each clone in this model does not depend on the total population or on other clones. The solution to Eq. (10.1) is given by
 1 1p : xi ðtÞ ¼ xi ð0Þ 1  xi ð0Þp1 ki ð1  pÞt +

(10.2)

Here and henceforth the subscript “+” means a positive part of the corresponding expres, and its insion. Recall the definition of a q-exponential function, expq(x)  (1 + (1  q)x) 1/(1q) + 1q verse the q-logarithm function lnq x ¼ x 1q1; we used these functions in the previous chapter. Then the solution (10.2) can be conveniently written in the form
 (10.3) xi ðtÞ ¼ xi ð0Þ exp p xi ð0Þp1 ki t : According to Eq. (10.2), each clone has a finite lifetime. The ith clone becomes completely extinct at the moment Ti, that is, xi(Ti) ¼ 0, where extinction moment Ti is given by Ti ¼

xi ð0Þ1p : ki ð1  pÞ

(10.4)

Notice that for exponential and superexponential models (10.1) with p 1, every clone has an indefinite lifetime. It is of course an unrealistic property, which is why we will focus primarily on subexponential extinction models. Fig. 10.1 shows typical dependence of the extinction moment Ti, defined in Eq. (10.4), on the initial clone size and the value of the power p.

FIG. 10.1 Dependence of logarithm of the extinction time Ti, defined in Eq. (10.4), on the initial clone sizes xi ¼ e si, i ¼ 1, … , 100 and the power p; s ¼  0.1. Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

173

10.3 Population of subexponentially decreasing clones

Using the notion of extinction time Ti, we can now express the current density of the ith clone as ð1pÞ xi ðtÞ ¼ xi ð0Þð1  t=Ti Þ1= : +

(10.5)

The total population size is given by the formula 1
 p1 1  p N ðtÞ ¼ Σi xi ðtÞ ¼ Σi xi ð0Þ 1  xi ð0Þ ki ð1  pÞt +

(10.6)

ð1pÞ ¼ Nð0ÞΣi P0 ðiÞð1  t=Ti Þ1= : +

The population goes to extinction at the moment T 5 maxiTi. Typical dynamics of the total population size is shown in Fig. 10.2. The frequency of ith individual is given by
1=ð1pÞ p1 xi ðtÞ P0 ðiÞ 1  xi ð0Þ ki ð1  pÞt + ¼ , Pt ðiÞ ¼ 1 N ðt Þ
 Σj P0 ð jÞ 1  xj ð0Þp1 kj ð1  pÞt 1  p

(10.7)

+

which can also be written as Pt ðiÞ ¼ P0 ðiÞ

ð1pÞ ð1  t=Ti Þ1= +  1=ð1pÞ : Σj P0 ð jÞ 1  t=Tj +

The dynamics of clone frequencies are shown in Fig. 10.3.

FIG. 10.2 Typical dynamics over time of the total population size N(t) defined according to Eq. (10.6). Here P  si is the N(0) ¼ 1, P0(i) ¼ Ce si, i ¼ 1, … , 100; C ¼ 1/ 100 i¼1e normalization constant; p ¼ 0.9, s ¼ 0.1. Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

(10.8)

174

10. Modeling extinction of inhomogeneous equations

FIG. 10.3 Dynamics of clone frequencies Pt(i), defined in Eq. (10.7). Here P0(i) ¼ Ce si, i ¼ 1, … , 100; p ¼ 0.9, s ¼  1. Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

10.4 Dynamical principles of minimal Tsallis information loss Dynamical principle of minimal information gain was studied in the previous chapter. Again, here we submit that the information measure for dynamical models and systems should be chosen in accordance with system dynamics. In the case of subexponentially decreasing populations, the distribution of individual frequencies (10.7) is by definition the Tsallis distribution at each time moment, and accordingly, the Tsallis q-entropy is the appropriate information measure (for definitions, see Eqs. ( 10.9), (10.11) below). The Tsallis relative q-entropy of a discrete probability distribution {m(i)} given a reference distribution {r(i)} is defined as !     1 mðiÞ q1 mðiÞ Σi mðiÞ : (10.9)  1 ¼ Σi mðiÞlog q I q ½ m : r ¼ q1 rðiÞ rði Þ The divergence of the form (10.9) is known in statistics as “Cressie-Read divergence” (see Cressie and Read, 1984; Read and Cressie, 2012); a brief useful survey of this and other nonclassical entropies can be found in Gorban et al. (2010). The definitions, general properties, and theorems about the Tsallis entropy can be found, for example, in Borland et al. (1998) and Tsallis (2009). The main results and equations were given in Chapter 9. For convenience of readers, we repeat some definitions and equations below in the form appropriate for models of population extinction. The Tsallis relative q-entropy within the framework of a population extinction model can be interpreted as a measure of information loss. The inference of m by minimizing Iq[m : r] is known as the principle of minimum Tsallis relative q-entropy. The distribution that provides the minimum of Iq[m : r] with respect to the constraint

10.4 Dynamical principles of minimal Tsallis information loss

Σi uðiÞmðiÞq ¼ huiq

175 (10.10)

is the Tsallis distribution i1=ð1qÞ rðiÞ
 rðiÞ h mðiÞ ¼ ¼ 1  ð1  qÞrðiÞq1 βuðiÞ exp q rðiÞq1 uðiÞβ : + Z Z Here, Z is the normalization factor (the “q-partition function”): h i 1
 ZðβÞ ¼ Σi rðiÞ 1  ð1  qÞrðiÞq1 uðiÞβ 1q ¼ Σi rðiÞ exp q rðiÞq1 uðiÞβ :

(10.11)

(10.12)

The Lagrange multiplier β at a given constraint huiq can be calculated from the equation ∂ lnq Z ¼ huiq , ∂β

(10.13)

where ∂β∂ lnq Z ¼ Zq ∂β∂ Z, as follows from the definition of lnqx. One can then calculate the minimal information loss as Iq ½m : r ¼ lnq Z  βhuiq ¼ lnq Z + β

∂ lnq Z: ∂β

(10.14)

We can see that when r(i) ¼ P0(i), u(i) ¼ N(0)p1ki, q ¼ p, and β ¼ t, the distribution (10.11) exactly coincides with the distribution (10.7), which can be written in the form:
1=ð1pÞ
 P0 ðiÞ 1  P0 ðiÞp1 N ð0Þp1 ki ð1  pÞt P0 ðiÞ + exp p P0 ðiÞp1 N ð0Þp1 ki t : Pt ðiÞ ¼
1=ð1pÞ ¼ Z Σj P0 ð jÞ 1  P0 ð jÞp1 N ð0Þp1 kj ð1  pÞt +

(10.15) Notice that now the total population size can be written as N(t) ¼ N(0)Z(t) similar to Eq. (2.29) for Malthusian models of population growth. Let us reformulate these results. We do not seek an unknown distribution that would minimize the relative Tsallis entropy subject to a particular set of constraints. Instead, we have the solution (10.2) of model (10.1), which produces distribution (10.15) at each time moment. With this distribution, we can compute at each moment the p-mean of the death rate, P p t iuiPt(i)  huip. Importantly, we can compute this value knowing only the initial distribution P0(i) and initial population size N(0). (Technically, to compute huitp, one can use the formula huitp ¼ 

∂ ∂ lnp ZðtÞ ¼ Zp Z, ∂t ∂t

(10.16)

where Z(t) is defined by Eq. (10.12) with q ¼ p.) Considering the computed value huitp as a prescribed constraint, we can see that the distribution Pt (Eq. ( 10.15)) provides minimum for the Tsallis information loss, Ip[Pt : P0]. It means that within the frameworks of the model (10.1), the principle of minimal Tsallis information loss is not a hypothesis, but a strong mathematical assertion, which follows from system dynamics. In other words, the dynamics of extinction model (10.1) is governed by the principle of minimal Tsallis information loss. The following theorem holds, where we assume for

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10. Modeling extinction of inhomogeneous equations

FIG. 10.4

Dynamics of the Tsallis information loss Ip[Pt : P0] as defined in Eq. (10.17) when p ¼ 0.5 (dash) and p ¼ 0.7 (solid). Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

simplicity that N(0) ¼ 1. Theorem 10.1 Distribution of individuals in a population of subexponentially decreasing clones provides the minimal Tsallis information loss Ip[Pt : P0] at each time moment t among all probability distributions that have the p-mean of the population death rate equal to hkitp at time t. The information loss Ip[Pt : P0] at time t can be calculated according to its definition (10.9) or using the formula Ip ½Pt : P0  ¼  ln p ZðtÞ  thkitp ¼  ln p ZðtÞ + t

∂ ln p ZðtÞ: ∂t

(10.17)

Fig. 10.4 shows the dynamics of the Tsallis information loss with respect to different values of parameter p, when the initial distribution is geometric, P0(i) ¼ Ce i, i ¼ 0, … , 100, and C is the normalization constant. An important and perhaps unexpected conclusion can be drawn now. We considered a population composed of independent clones, each of which decreases according to the power equation with the same power but with its own specific death rate. The clones depend neither on each other nor on the population as a whole. Therefore it is difficult to expect the dynamics of such a population to admit some kind of “macro” description that ignores the “micro” dynamics of each independent clone. Nevertheless, such “macro” description does exist. We have shown that given the initial distribution of clone frequencies in the population, the knowledge of only the p-mean of the death rates at any time moment implies complete knowledge of the population distribution at that moment due to the principle of minimal Tsallis relative entropy (Tsallis information loss). The corresponding Tsallis distribution coincides with distribution (10.7), which follows from the “micro” dynamics of every clone. Let us emphasize that we do not claim a priori the principle of minimal Tsallis relative entropy; we proved that this principle was fulfilled for power model (10.1) due to the “micro” dynamics of the population.

177

10.5 Parametrically inhomogeneous models of population extinction

10.5 Parametrically inhomogeneous models of population extinction A general mathematical framework for investigation of inhomogeneous population models of Malthusian type was given in Chapter 2; see also Example 7 in Chapter 3. For clarity, the main results for the models of inhomogeneous population extinction are given below. Consider an inhomogeneous population composed of individuals with different death rates a. Let l(t, a) be the size of a-clone at the moment t. Dynamics of extinction of such a population can be described by the following model: dlðt, aÞ ¼ alðt, aÞgðN Þ, dt ð N ðtÞ ¼ lðt, aÞda,

(10.18) (10.19)

A

where g(N) is an appropriate function. The population size N(t) satisfies the equation dN ¼ NEt ½agðN Þ dt

(10.20)

ðt, aÞ and the pdf Pðt, aÞ ¼ lN ðtÞ solves the replicator equation of the form

  dPðt, aÞ ¼ Pðt, aÞ Et ½a  a gðN Þ: dt

(10.21)

dq ¼ gðN Þ, qð0Þ ¼ 0: dt

(10.22)

Ð Denote L0(λ) ¼ Ae λaP(0, a)da as the Laplace transform of the initial distribution. Let us formally define formally the auxiliary keystone variable q(t) as the solution to the Cauchy problem:

Then, lðt, aÞ ¼ lð0, aÞeaqðtÞ ¼ Nð0ÞPð0, aÞeaqðtÞ , ð N ðtÞ ¼ N ð0Þ eaqðtÞ Pð0, aÞda ¼ Nð0ÞL0 ðqðtÞÞ:

(10.23) (10.24)

A

Ð where L0(λ) ¼ Ae aλP(0, a)da is the Laplace transform of the initial distribution P(0, a). The equation for the variable q(t) can now be written in a closed form: dqðtÞ ¼ gðN ð0ÞL0 ðqðtÞÞÞ, qð0Þ ¼ 0: dt

(10.25)

With the solution to this equation, we can now completely solve the initial problem (10.18), (10.19). The current parameter distribution P(t, a) is determined by the formula Pðt, aÞ ¼

lðt, aÞ eqðtÞa : ¼ Pð0, aÞ L0 ðqðtÞÞ N ðtÞ

(10.26)

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10. Modeling extinction of inhomogeneous equations

The Laplace transform of the current distribution P(t, a) is given by the formula ð δaqðtÞa  e L0 ðδ + qðtÞÞ Pð0, aÞda ¼ : Lt ðδÞ ¼ Et eδa ¼ L0 ðqðtÞÞ A L0 ðqðtÞÞ The mean value of a over time is described by ð aeqðtÞa dlnL0 ðqÞ

t Pð0, aÞda ¼  E ½a ¼ q¼qðtÞ , L ð q ð t Þ Þ dq A 0

(10.27)

(10.28)

and dEt ½a ¼ gðNÞVart ½a: dt

(10.29)

10.6 Power extinction models and inhomogeneous F-models In this section, we investigate the subexponential inhomogeneous power models assuming that the total size of the inhomogeneous population decreases according to the equation dN ¼ kN p , p < 1: dt

(10.30)

Let us now construct the inhomogeneous model (10.18) and (10.19) in such a way that its total population size solves Eq. (10.30). Let g(N) ¼ kN1 and consider the inhomogeneous F-model dlðt, aÞ kalðt, aÞ ¼ ¼ kaPðt, aÞ: dt N ðt Þ

(10.31)

The total population size of F-model (10.31) solves the equation (see Eq. ( 10.20)) dN ¼ NEt ½agðN Þ ¼ kEt ½a: dt So, to obtain Eq. (10.30), we need to have Et[a] ¼ N(t)p for all t. 1 To this end, let us define the keystone variable through the equation dq or in dt ¼ gðN Þ ¼ kN a closed form (see Eq. ( 10.24)): dqðtÞ k , qð0Þ ¼ 0, ¼ dt N ð0ÞL0 ðqðtÞÞ where L0 is the Laplace transform of an unknown initial distribution. Then (see Eq. ( 10.29)) Et ½ a  ¼ 

dlnL0 ðqðtÞÞ : dq

Hence the equation Et[a] ¼ N(t)p ¼ N(0)pL0(q(t))p holds if 

1 dL0 ðqðtÞÞ ¼ Nð0Þp L0 ðqðtÞÞp , L0 ðqðtÞÞ dq

10.6 Power extinction models and inhomogeneous F-models

179

that is, if dL0 ðqÞ ¼ N ð0Þp L0 ðqÞp + 1 : dq Solving this equation given the initial condition L0(0) ¼ 1, we obtain L0 ðqÞ ¼ ð1 + pNð0Þp qÞ

1=p

:

(10.32)

It is known (and one can check with the help of computer packages, i.e. Mathematica) that the function L0(δ) ¼ (1 + βδ) ρ at β > 0 is the Laplace transform of the Gamma distribution 

a

aρ1 e β , a > 0: Pð a Þ ¼ ρ β Γ ð ρÞ

(10.33)

Hence the initial distribution of the parameter a should be the Gamma distribution (10.33) with ρ ¼ 1p and β ¼ pN(0)p. This distribution is completely characterized by its mean value N(0)p and variance pN(0)2p. Now, we can compute q(t) given the Laplace transform of the initial distribution (10.32) using the equation 1 dq ¼ kðN ð0ÞL0 ðqÞÞ1 ¼ kðN ð0ÞÞ1 ð1 + pNð0Þp qÞp : dt

The solution to this equation under initial condition q(0) ¼ 0 is

q ðt Þ ¼


 p 1p 1 + p 1  kN ð0Þ ð1  pÞt 1 pN ð0Þp

+

for p 6¼ 1

(10.34)

and qðtÞ ¼ eNð1 0Þ for p ¼ 1. The current pdf P(t, a) is again the Gamma distribution, as easily follows from formulas (10.27), (10.32), since kt

  1 p L0 ðδ + qðtÞÞ pN ð0Þp δ 1=p ¼ 1+ ¼ ð1 + pN ðtÞp δÞ Lt ð δ Þ ¼ p L0 ðqðtÞÞ 1 + pN ð0Þ qðtÞ

(10.35)

is the Laplace transform of the Gamma distribution (10.33) with parameters ρ ¼ 1p and β(t) ¼ pN(t)p. The current mean value of this Gamma distribution is Et ½a ¼ ρβðtÞ ¼ NðtÞp :

(10.36)

Hence the total population size of inhomogeneous population (10.31) with the initial pdf (10.33) solves the power equation (10.30), as desired. Next, it follows from Eq. (10.34) that q(t) increases monotonically in such a way that q(t) ! ∞ as t ! ∞ for p 1 and q(t) ! ∞ as t ! T for p < 1, where T¼

1 kN ð0Þ

1 + p

ð1  pÞ

< ∞:

(10.37)

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10. Modeling extinction of inhomogeneous equations

The population size for p 6¼ 1 is given by N ðtÞ ¼ N ð0ÞL0 ðqðtÞÞ ¼ Nð0Þð1 + pN ð0Þp qðtÞÞ

1 p


 1 1p ¼ N ð0Þ 1 + kN ð0Þ1 + p ðp  1Þt +

(10.38)

and N(t) ¼ N(0)e kt for p ¼ 1. Hence N(t) decreases monotonically and N(t) ! 0 as t ! ∞ for p 1 (exponential and superexponential models); N(t) ! 0 as t ! T < ∞ for p < 1 (subexponential model), where T is given by (10.37). We see that the lifetime of a population described by the exponential or superexponential equation is indefinite; in contrast, a subexponential population goes to extinction not asymptotically, as superexponential power models, but completely disappears at a certain finite time moment. This property makes subexponential power models of primary interest. Coming back to the general power model of extinction (10.30), we want to emphasize that we have found within the frameworks of inhomogeneous population models (10.18)–(10.19) a universal, that is, for all p > 0, frequency-dependent representation of power extinction models (F-models). Let us summarize our findings in the following theorem. Theorem 10.2 (1) Any power equation dN ¼ kN p , p > 0 dt

(10.39)

describes the dynamics of the total size of an inhomogeneous frequency-dependent model dlðt, aÞ kalðt, aÞ ¼ ¼ kaPðt, aÞ, dt N ðt Þ

(10.40)

where P(0, a), the initial distribution of the parameter a, is Gamma distribution (10.33) with ρ ¼ 1/p, β ¼ β(0)  pN(0)p, and Laplace transform L0(δ) ¼ (1 + β(0)δ)1/p. (2) The solution to the power equation exists for all t > 0 at p 1 and only up to the moment of population extinction, when N(T) ¼ 0: T¼

1 kN ð0Þ

p1

for p < 1:

ð 1  pÞ

(10.41)

(3) The solution to the inhomogeneous F-model (10.40) is lðtaÞ ¼ lð0aÞeaqðtÞ ,

(10.42)

where the auxiliary variable q(t) is given by

q ðt Þ ¼


 p 1p 1  kN ð0Þ1 + p ð1  pÞt 1

and by qðtÞ ¼ eNð1 0Þ when p ¼ 1. kt

pN ð0Þp

+

when p 6¼ 1

(10.43)

10.7 The “internal time” for F-models of extinction

181

(4) The total population size at the time t is
 1 1p N ðtÞ ¼ N ð0Þ 1 + kN ð0Þ1 + p ðp  1Þt +

(10.44)

for p 6¼ 1, and N ðtÞ ¼ N ð0Þekt for p ¼ 1. (5) The current distribution at any time moment t is Gamma distribution with the mean Et[a] ¼ N(t)p and the variance Vart[a] ¼ pN(t)2p.

10.7 The “internal time” for F-models of extinction We have shown that any power model of extinction (10.30) has a canonical representation in the form of the F-model (10.40). Now let us explore the following transformation. Define the following change in time in this model: dq ¼

k dt: N ðtÞ

(10.45)

Consider the equation dxðq, aÞ ¼ axðq, aÞ dq

(10.46)

lðt, aÞ ¼ xðqðtÞ, aÞ:

(10.47)

and let

Then, dl dx dq axðqðtÞ, aÞk aklðt, aÞ ¼ ¼ ¼ : dt dq dt N ðt Þ N ðt Þ Hence l(t, a), as defined by Eq. (10.47), solves Eq. (10.40). Noticeably, through change of time t ! q(t) as defined in Eq. (10.45), the F-model (10.40) becomes reduced to a simple Malthusian model of extinction (10.46). An explicit expression for q(t) is given by Eq. (10.43). The possibility of transformation of the initial power model (10.30) to the form (10.46) reveals some interesting properties of the power models of extinction. We can interpret the variable q(t) as “internal” time for F-models. In general, the internal time of a system can be defined as a sequence of internal events or states of the system (Meyen, 1984). Such interpretation becomes possible because each clone l(q, a) with respect to this time scale evolves as if it does not depend on other clones and on the population as a whole. The F-model with respect to the internal time q defined by Eq. (10.45) becomes identical to inhomogeneous Malthusian model with respect to the chronological or “external” time t, which describes free development of all clones in the population (see Fig. 10.5). Notice also that q(t) is the only time

182

10. Modeling extinction of inhomogeneous equations

FIG. 10.5 Plot of internal time q(t), defined in Eq. (10.43), for p < 1, in logarithmic scale, against real time. Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

scale for Eq. (10.40), which possesses this property. We already considered the keystone variable for Malthusian-type growth models as internal population time in Chapter 7, where this interpretation had more or less formal mathematical nature. Later in this chapter, we show that in the power models of extinction internal time might have real biological meaning. It follows from the explicit expression of the internal time (10.43) that if p 1, then q(t) is finite for all t and q(t) ! ∞ as t ! ∞. In contrast, if p < 1, then q(t) ! ∞ as t ! T < ∞, where T¼

Nð0Þ1p : k ð1  pÞ

(10.48)

This means that a finite “real-time” interval (0, T) corresponds to infinite duration of “internal” time. Every clone l(t, a) and the population as a whole tend to extinction simultaneously at time moment T. This is in contrast with the model considered in previous sections, where every clone has its own life duration. It is easy to show that the value of T at a given N(0) is minimal when p ¼ pðN ð0ÞÞ  popt ¼ 1 

1 : lnN ð0Þ

(10.49)


 N ð0Þ1p 1 dT 1 Indeed, dT dp ¼ kð1pÞ lnN ð0Þ + 1p , so dp ¼ 0 if p ¼ 1  lnNð0Þ : To guarantee that 0 < p < 1 , we should assume that lnN(0) > 1. The minimal value of T is attained when p ¼ popt and is equal to 1

lnN ð0ÞN ð0ÞlnNð0Þ e ¼ lnN ð0Þ, Tmin ¼ k k since 1

N lnN  e:

(10.50)

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10.8 Dynamical principles of minimal Shannon information loss

Summarizing the obtained results leads to the following theorem, where we assume that lnN(0) > 1. Theorem 10.3 The life duration of subexponential population (10.40) with any 0 < p < 1 is bounded from below by T  Tmin ¼ ke lnN ð0Þ. For a given N(0), this boundary is attained if p ¼ popt ¼ 1  lnN1ð0Þ. Conversely,
T ¼ Tmin at a given p if the initial population size is equal to 1 N ð0Þ ¼ exp 1p .

10.8 Dynamical principles of minimal Shannon information loss We already discussed the principle of minimal cross entropy (MinxEnt) in Chapter 8 (see also Kullback, 1997; Beck, 2009); the principle is based on the hypothesis that, subject to precisely stated prior data, the probability distribution that best represents the current state of knowledge is the distribution with the minimal cross entropy, known also as KL-divergence between the current distribution m and a reference distribution r. Recall the definition of KL-divergence, which is a generalization of the Boltzmann-Gibbs entropy or the Shannon information: ð mðxÞ dx: IKL ½m : r ¼ mðxÞ ln r ð xÞ A In our F-model of population extinction, the value of IKL[Pt : P0], where Pt ¼ P(t, a) is defined by Eq. (10.26), can be computed using the formula ð IKL ½Pt : P0  ¼ Pðt, aÞðaqðtÞ  lnL0 ðqðtÞÞÞda ¼ qðtÞEt ½a  lnL0 ðqðtÞÞ: A d The mean value is given by Et ½a ¼  dq lnL0 ðqðtÞÞ (see Eq. 10.28); therefore,

IKL ½Pt : P0  ¼ qðtÞ

d lnL0 ðqðtÞÞ  lnL0 ðqðtÞÞ: dq

(10.51)

With respect to the internal time of the model, frequencies of the clones (Eq. ( 10.26)) are given by Pðq, aÞ ¼ Pð0, aÞ

eaq L0 ðqÞ

(10.52)

and  d IKL Pq : P0 ¼ q lnL0 ðqÞ  lnL0 ðqÞ: dq Figs. 10.6 and 10.7 show a dramatic difference in behaviors of IKL[Pq : P0] versus the internal time q, and of IKL[Pt : P0] versus the “real,” or chronological time t; in both cases, P0(a) is the Gamma distribution given by Eq. (10.33) with ρ ¼ 1/p and β ¼ pN(0)p; p ¼ 0.96, N(0) ¼ 1011. An interesting property of the IKL[Pt : P0] dynamics is given by the following proposition.

184

10. Modeling extinction of inhomogeneous equations

FIG. 10.6 Plot of IKL[Pq : P0], defined in Eq. (10.51), against internal time q. Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

FIG. 10.7 Plots of IKL[Pt : P0], defined in Eq. (10.51), against “real,” or chronological time t. Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

Proposition 10.1

d qðtÞ IKL ½Pt : P0  ¼ Vart ½a: dt N ðt Þ

Indeed, according to Eq. (10.49),  dI KL ½Pt : P0  d  ¼ qðtÞEt ½a  lnL0 ðqðtÞÞ ¼ dt dt   dEt ½a dq d dq t ¼ qðtÞ  E ½a  lnL0 ðqðtÞÞ ¼ dt dt dq dt ¼

qðtÞVart ½t , N ðt Þ

(10.53)

10.8 Dynamical principles of minimal Shannon information loss

185

since d lnL0 ðqðtÞÞ ¼ Et ½a dq and dEt ½a Vart ½a ¼ dt N ðt Þ according to Eqs. (10.28) and (10.29). Therefore the KL-divergence increases monotonically, as we have seen in Figs. 10.6–10.7, and the rate of its increase is proportional both to the current variance of parameter a and the value of the internal time q(t). The importance of the KL-divergence IKL[Pq : P0] is twofold. Firstly, IKL[Pq : P0] is a general characteristic of the current state of a population that goes to extinction, which shows the “distance” between the current distribution Pq and the initial distribution P0. Secondly, the KL-divergence plays a central role in the Principle of minimal Boltzmann-Gibbs cross entropy, MinxEnt. According to known results (see, e.g., Borland et al., 1998; Dukkipati et al., 2006; Tsallis, 2008), the distribution m(x) that provides minimum for IKL[m : r] given the mean value of m equal to s, is the Boltzmann distribution: mðxÞ ¼

esx rðxÞ, Z

(10.54)

where Z is the normalization factor (the “partition function”). It is easy to see that distribution Pq ¼ P(q, a) (Eq. ( 10.52)), which is defined by the solution to the inhomogeneous model, coincides up to notation with the Boltzmann distribution (10.54) that provides the minimal value of KL-divergence IKL[Pq : P0]. As before, we submit that the information measure for dynamical models and systems should be chosen in accordance with the system dynamics. In the case of F-models of population decrease, the distribution of individual frequencies is the Boltzmann distribution at each time moment. Accordingly, the KL-divergence, which is a generalization of classical Shannon information, is the appropriate information measure for the model. The value of KL-divergence IKL[Pq : P0] can be interpreted as Shannon information loss during the internal time interval [0, q] in a decreasing population. Therefore the dynamics of F-model (10.40) is such that at each moment of the internal time, the Shannon information loss is minimal. It means that the Principle of minimal cross entropy, MinxEnt, in the form of the principle of minimal Shannon information loss for F-model, holds automatically because of model dynamics. Let us emphasize that we do not seek an unknown distribution that would minimize the KL-divergence IKL[Pq : P0] subject to a particular set of constraints, in contrast to the common approach. Instead, we have the solution (10.42) and (10.43) of F-model (10.40), which produces distribution (10.52) at each time moment. With this distribution, we can compute at each moment of internal time q the mean of the death rate, knowing only the initial distribution. Hence, given the initial distribution of clone frequencies in the population, the knowledge of only

186

10. Modeling extinction of inhomogeneous equations

the mean value of the death rates at any moment of internal time q yields complete knowledge of population distribution at that moment due to the principle of minimal KL-divergence. This means that the dynamics of F-models with respect to internal time are governed by the principle of minimal Shannon information loss.

10.9 Application of the model to time perception We may surmise that the aforementioned formally defined internal time, in application to one of the largest biological systems, the human brain, can be interpreted as a type of an internal clock model. In this section, we will first provide background on the currently accepted biological models of time perception and will then describe a possible application of one of the proposed extinction models to understanding alterations in time perception in a dying brain.

10.9.1 Some background information on time perception As can be seen through a survey of literature, subjective sense of time can be affected temporarily or permanently due to changes in synchronization between an “internal clock” and external “subjective” time. Arguably the most influential internal clock model, based on scalar expectancy theory (SET), was proposed by Gibbon (1977) and Gibbon et al. (1984). According to SET, temporal processing is regulated by a pacemaker, switch, and accumulator. The switch, controlled by attention, regulates the number of pacemaker pulses (clock ticks) that are collected into the accumulator. Time estimates depend on a number of “pulses” that have been accumulated between the switches: the more pulses have accumulated, the longer the perceived time interval. This model has since been modified and expanded, and the currently accepted most plausible model of time perception is the striatal beat frequency model (SBF), which has been proposed by Matell and Meck (2000, 2004). In the SBF, timing is based on the coincidental activation of medium spiny neurons by cortical neural oscillators in the basal ganglia in the midbrain. A broad array of studies showed that injections of substances that act as dopamine antagonists, such as cocaine and methamphetamine, cause shorter tasks to be perceived as long, slowing down the perception of time, while drugs that activate dopamine receptors, such as haloperidol and pimozide, speed up the perception of time (Meck, 1996, 2005; Coull et al., 2011). Moreover, patients with Parkinson’s disease, which involves degeneration of dopaminergic substances in the basal ganglia and specifically in substantia nigra par compacta (SNc), exhibit impaired timing perception (Rammsayer and Classen, 1997; Malapani et al., 1998). Similar problems with time perception have also been observed in patients with schizophrenia (Davalos et al., 2002; Penney et al., 2005) and attention deficit hyperactivity disorder (Barkley et al., 2001). These studies suggest that there exists an “optimal” level of dopamine that aligns one’s internal clock with external time. Noticeably, temporary changes to time perception can be affected by emotions (Droit-Volet and Meck, 2007; Droit-Volet, 2013) and reactions to threatening situations, where involvement of the amygdala in emotional memory contributes to the creation of secondary encoding of memories. This can create an erroneous sense of events spanning a greater period of time

10.9 Application of the model to time perception

187

than has actually passed and appears to be a function of recollection, not perception of time (Stetson et al., 2007). Another particularly interesting case of alterations in time perception could be occurring during near-death experiences (French, 2005). Anecdotal evidence suggests that time intervals during a near-death experience are perceived to be much longer than they actually are. Borjigin et al. (2013) reported a high frequency neurophysiological activity in the neardeath state of rats undergoing experimentally induced cardiac arrest. The levels of neurophysiological activity in the near-death state exceeded those found during conscious waking state, suggesting that the brain in fact might be highly active in the near-death state. These observations, coupled with the current state of knowledge about changes in time perception, allow formulation of the following hypothesis: during a near-death experience, the lack of oxygenation and nutrient access eventually causes death of all brain cells, including the dopaminergic substances in the basal ganglia, causing an experience of internal time to become increasingly longer compared with external time. This could be a possible mechanism to account for the perception of time that allows “life to flash before one’s eyes” in the moments preceding death.

10.9.2 Application of the proposed model to understanding time perception in a dying brain The second model of population extinction proposed in this paper allows decoupling of internal and external time. The proposed interpretation of obtained results allows simulating internal time tending to infinity in finite time interval, recapitulating qualitatively the possible experience of time perception in a dying brain. Recall that in this model, we assume that the population (e.g., of neural cells) is composed of clones l(t, a), which here stands for a set of all individuals in the population having the death rate (Malthusian parameter) equal to a. According to Theorem 10.2, the power equation (10.30) with p > 0 describes the dynamics of the total size of an inhomogeneous frequency-dependent extinction model dlðt, aÞ dt ¼ kaPðt, aÞ, where the initial distribution of the parameter a, P(0, a), is the Gamma distribution. Populations described by subexponential equation with p < 1 go to extinction at cer1p

0Þ tain finite time moment, that is, N(T) ¼ 0, when T ¼ ðNkðð1p ÞÞ. Within the context of our model, this suggests that the time to T depends on either the initial number of neurons or connections between them, both of which are logically consistent. Making a formal change of time t ! q through equation dq ¼ NkðtÞ dt allows rewriting the equation for the dynamics of cell clones l(t, a) with respect to this new independent variable q, aÞ ¼ alðq, aÞ, which becomes a standard Malthusian equation of population extincq as dlðdq tion. Here, we propose that q(t) is a possible representation of internal time. It is important that if 0 < p < 1, then q(t) ! ∞ as t ! T < ∞, where T is the time moment of population extinction. The internal time q(t) as a function of “real time” t is given by Eq. (10.43); the plot of internal time q(t) for p < 1 against real time is given in Fig. 10.5. There are three parameters that determine the dynamics of the internal time q(t): k, N(0), and p. As one can see, larger values of parameter k, which defines the death rate of the population as a whole, decrease extinction time T; increasing the initial value of N(0) predictably

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10. Modeling extinction of inhomogeneous equations

delays extinction time T; dependence of T on the value of p is nonmonotonic. N can be interpreted as either the number of neural cells or possibly the number of connections between cells. The latter interpretation might be preferable, since demise of connections between cells in a damaged or dying brain can lead to different parts of the brain living and dying as if separate from each other, which underlies the assertion that q(t) can in fact be representing internal time. An average human brain is estimated to range between 85 and 120 billion neurons (Herculano-Houzel, 2009), so we can estimate N(0) to be roughly 1010. The number of synapses (based on 1000 per neuron estimate) is about 1014, or 100 trillion. Furthermore, from Eq. (10.43), the expression kN(0)(p1) can be seen as a scaling quantity for “external” time. For the power p, we do not yet have a well-established biological interpretation, but we can identify certain properties. Specifically, if p > 1, the population defined in power equation (10.30) lives forever, but for p < 1, it will go extinct in finite time, defined by T. Hence the parameter p can be estimated based on the duration of the process of population extinction. Eq. (10.49) and Theorem 10.3 earlier give a simple estimation of p with respect to the initial population size N(0). The inhomogeneous F-model (10.40) noticeably does not depend on p; the rate of extinction of cells or synaptic connections between them is proportional to their frequency. However, the distribution of parameter a, which defines population clones, does depend on p. Specifically, the expected value and the variance of parameter a, which define completely the initial Gamma distribution, both depend on the value of p. Therefore theoretically the mean value of the parameter a can be used for estimation of the power p, as given by Et[a] ¼ N(t)p. According to Theorem 10.3, there exists a single value of p ¼ popt, given by equation, which minimizes T; for instance, popt ¼ 0.96 for N(0) ¼ 1011 and popt ¼ 0.97 for N(0) ¼ 1014; see Fig. 10.8.

FIG. 10.8 The plot of extinction moment T, defined in Eq. (10.41), plotted in logarithmic scale against the power p for N(0) ¼ 1011 (dotted) and N(0) ¼ 1014 (solid). Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

10.10 Discussion

189

FIG. 10.9 The plot of the optimal value popt given by Eq. (10.49), against N(0), plotted for 1010 < N(0) < 1015. Adapted from Karev, G.P., Kareva, I., 2016. Mathematical modeling of extinction of inhomogeneous populations. Bull. Math. Biol. 78, 834–858.

Furthermore, by plotting popt versus N(0) according to Eq. (10.49), we can see that for a very large magnitude of values of N(0), 1010 < N(0) < 1015, the values of popt, which provide minimum of T at given initial population size, are 0.957 < popt < 0.971, demonstrating that the “optimal” value of p belongs to a very narrow range (Fig. 10.9). The minimal value of T when p ¼ pmin is given by Theorem 10.3 and is proportional to lnN(0). The duration of death of the brain core in the state of clinical death is between 5 and 20 min (Safar, 1988). According to Eq. (10.50), T ¼ 300 s if lnNkð0Þ ¼ 300 e ffi 110, i.e. if, for instance, N(0) ¼ 1011 and k ¼ 0.23, or if N(0) ¼ 1014 and k ¼ 0.29. Then, within the framework of our model, if the process of death of the brain core can be p 11 14 described by power equation (10.30), dN dt ¼ kN , with 10 < N(0) < 10 , 0.96 < p < 0.97, and 0.23 < k < 0.29, then during T ¼ 5 min of “external,” or chronological time, the “internal” time of the underlying F-model can become indefinitely large and potentially infinite.

10.10 Discussion In this chapter we analyze two inhomogeneous models of population extinction. In the first model we assume that the population is composed of independent clones, each of which decreases according to the subexponential equation. In the second model we assume that the total size of the inhomogeneous population decreases according to the subexponential equation; we reveal and study the underlying inhomogeneous frequency-dependent population model. These conceptual models can be potentially applied for investigation of extinction processes in different ecological and biological systems. Furthermore, any individual and

190

10. Modeling extinction of inhomogeneous equations

even any single organ, for example, a human brain, can be viewed as a complex community of large populations of cells and connections between them that undergo extinction with the death of the individual, and could thus potentially be described using these models. We have shown that the dynamics of clone frequencies in both models follows the principle of minimal information loss, but the adequate information measures are different for these models. We submit that the information measure for dynamical models and systems should be chosen in accordance with the system dynamics. Then, the adequate information measure for the distribution of individual frequencies in the first model is Tsallis relative q-entropy, while the adequate information measure in the second model is the classical BoltzmannGibbs cross entropy (or Shannon information loss). The Tsallis entropy and distribution include the standard Shannon entropy and the Boltzmann-Gibbs distribution as a special case when q ! 1. A novel and important mathematical result that comes from this investigation is the following. We have already pointed out earlier that the principle of minimal relative entropy, MinxEnt, was successfully applied to various statistical, physical, and biological problems as a method for inference of unknown distribution subject to some given constraints (see Kapur, 1989; Kapur and Kesavan, 1992). What is important that in those models, this principle formally was a hypothesis and it was not clear why one can expect it to work as a description of nature. This problem has been intensively discussed in the literature over the last 50 years. In this chapter, we show that for the considered models of population extinction, the MinxEnt in the form of the principle of minimal information loss is neither an inference algorithm nor an external principle but a mathematical assertion that can be derived from system dynamics instead of being postulated. With the solution to the extinction models, we can compute the current mean values of the death rates at any instant. Then, treating these mean values as constraints, we can show that the Tsallis and Boltzmann distributions that minimize the Tsallis or Shannon information loss correspondingly coincide with the solutions to the extinction models, which were obtained independently of the MinxEnt algorithm. Remember that “information loss” here formally means the divergence in distributions of death rates between population at the initial time point and population at the current moment in time. Hence the principle of the minimal information loss can be considered as the variation principle that governs the dynamics of population extinction. In the previous chapter, we emphasized that a fundamental property of the Tsallis entropy is that it is nonadditive for independent subsystems. Thus, in the first model, for the system of subexponentially decreasing clones, the information about two exhaustive subsystems is insufficient to obtain information about the system as a whole. This “nonreductionist” character of the system might reflect the fact that different clones that make up the population go to extinction at different time moments. In contrast, in the second model, all clones and the population as a whole go to extinction simultaneously, at the same moment of real “chronological,” or “external” time. Additionally, there exists “internal time” in the population as a whole, such that with respect to this internal time, the system is described by the standard Malthusian model of extinction. An unexpected application of the proposed modeling framework lies in providing a possible explanation to the mechanisms underlying time perception. Biologically, perception of

10.10 Discussion

191

time and alignment of one’s “internal clock” with “external” chronological time appears to be primarily affected by the activity of the dopamine in basal ganglia in the midbrain (Matell and Meck, 2000, 2004). We hypothesize that when a brain is dying, such as after a cardiac arrest, the eventual cessation of activity of all neural cells, including dopamine system in the basal ganglia, can cause perception of time duration to increase dramatically compared with “external” chronological time. One can even further speculate that loss of neural synapses and neural cells during brain death, in the event that it occurs in accordance with the proposed model, would result in a minimization of loss of qualitative information, such as knowledge and memories, similar to the minimization of the Shannon information loss described above. This hypothesis would of course require further investigation. The proposed mathematical model allows replicating this effect of altered time perception in a dying brain through decoupling “internal” and “external” time using a parametrically heterogeneous subexponential power equation, where cells or possibly synapses N(t) in the population die at different rates. We divided the population of neural cells into clones, and death rate of each clone is proportional to its frequency in the population. We derived an equation for q(t), a variable that can be interpreted as describing “internal time,” because each clone, or subpopulation, with respect to this time scale evolves as if it does not depend on other clones or on the population as a whole. Within the context of a dying brain, this would describe a situation when the loss of connections between cells in the dying brain would indeed lead to subpopulations of cells dying independently of each other, which is logically consistent. Furthermore, from this equation we were able to also specify time of death T, where population of cells goes to extinction, that is, when N(T) ¼ 0. Conversely, knowing the specific time of death T, we can estimate other model parameters. One can also apply this model to explore hypotheses about what may be happening with time perception in the moments preceding death. Within the frameworks of this model, death of cells or synapses N(t) will result in internal time q(t) ! ∞ in a finite period of time, denoted by T. The details of what happens with time perception prior to brain death are not yet understood, including whether internal time indeed increases in the moments preceding death, allowing one to experience more than would normally be possible (Bierce, 2008). It is of course not known exactly what mechanisms governing perception of time in the human brain could be. Here, we proposed a possible mathematical formalization, which allows making logically consistent predictions. We have identified a small number of key parameters that could be involved in the decoupling of “internal” and “external” time in the human brain and proposed an explanation for mechanisms underlying time perception. We hope that this conceptual model can lay a foundation for further mathematical and theoretical exploration of this complex topic, which eventually might yield results to deepen the understanding of diseases, such as Parkinson’s, schizophrenia, and ADHD, where accurate time perception is compromised. In the next chapter, we step away from philosophy and turn our attention to the following question: so many models (logistic, Gompertz, etc.) describe curves of very similar shapes. Are there any insights that can be gained about the underlying population structure based on each model? We answer this question in the next chapter.

C H A P T E R

11 From experiment to theory: What can we learn from growth curves? Abstract In this chapter, we explore the connection between the shape of seemingly similar growth curves (such as logistic, Gompertz, or Verhulst) to pose the following question: what can we infer about intrinsic properties of a population (i.e., degree of heterogeneity or dependence on external resources) based on which growth function best fits its growth dynamics? We investigate several nonstandard classes of multiphase growth curves that capture different stages of population growth; these models include hyperbolic-exponential, exponentiallinear, and exponential-linear-saturation growth patterns. The constructed models account explicitly for the process of natural selection within inhomogeneous populations. Based on the underlying hypothesis for each of the models, we identify whether the population that is best fit by a particular curve is more likely to be homogeneous or heterogeneou, whether it may be growing in a density-dependent or frequency-dependent manner and whether it depends on external resources during any or all stages of its development. We apply these predictions to cancer cell growth and demographic data obtained from the literature. This chapter is based on Kareva and Karev (2018).

11.1 Problem formulation Finding a simple curve or a justified equation that fits experimental data well is a standard problem in population dynamics, as it allows making predictions about future dynamics of the population. Exponential and logistic curves for describing unrestrained and environmentally restrained population growth, respectively, are classical examples. The procedures for finding the best-fitting curves within a certain class of formulas (equations) are well developed, and the problem is often considered to be solved when the curve is found. For instance, tumor growth can be described by logistic or Gompertz curves, and there exists a relatively extensive debate on which curve provides a better fit; see, for instance, Benzekry et al. (2014). Finding the right curve to describe the trends observed in global demography is another example, where finding a correct equation influences dramatically predictions about future human population growth (see Chapter 3).

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00011-4

193

# 2020 Elsevier Inc. All rights reserved.

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In this chapter, we are interested in investigating a different angle of the data-equation relationship: if a data set is best fitted by a particular curve, what information about intrinsic population dynamics can be derived from this result? That is, we are interested not in making predictions about future population dynamics, but in inferring information about population structure and conditions under which the population has been growing based on the data that have been collected. Obtaining this information can become important in cases, when, for instance, intervention or population management strategies need to be devised. Using the theory developed in previous chapters, we investigate Gompertzian, inhomogeneous Malthusian, inhomogeneous logistic, linear-exponential, and three-stage models and identify whether the population that fits best is homogeneous or heterogeneous, whether it grows in a density-dependent or frequency-dependent manner and whether it depends on external resources during any or all stages of its development. Data that describe tumor growth dynamics, for instance, can be fit to various, often similarly shaped curves (e.g., logistic and Gompertz curves). This theory can provide an additional biomarker and a predictive tool to complement experimental research.

11.2 Approach and introductory example The capacity to grow exponentially under ideal environmental conditions is generally considered a common property of most populations, but in any realistic situation, eventually, limits to unrestrained growth are encountered. It generates the struggle for existence, leading to natural selection, which can operate only if the evolving population is nonhomogeneous. Mathematical frameworks for dynamics of inhomogeneous populations were developed in Chapters 2 and 4. We showed in Chapter 7, Section 4, that an inhomogeneous population composed of different exponentially growing clones can never demonstrate exponential growth, but instead always grows faster, “overexponentially.” The reason is that the population growth rate, which is equal for such a model to Et[a], is not a constant, but increases with time as long as Vart[a] > 0, that is, as long as the population remains inhomogeneous. We also showed (Theorem 7.4) that the Malthusian equation dN dt ¼ kN describes the dynamics of the total size of inhomogeneous frequency-dependent models (F-models for brevity): dlðt, aÞ ¼ kaPðt, aÞ dt

a  Pð0, aÞ ¼ N1ð0Þ e Nð0Þ ,

with exponentially distributed parameter a, where The solution to the F-model (11.1) is given by the formula lðt, aÞ ¼ e

a  N ð0Þekt

:

(11.1) 0  a < ∞. (11.2)

An important corollary follows from these results. The inhomogeneous population described by the F-model (11.1), which shows exponential growth of the total population size, consists of clones, each of which grows according the Gompertz curve that have the form G(t) ¼ r exp( ce kt) (see Chapter 7, Section 7.4 for a brief survey of the Gompertz curves). Similar statements can be proven about models that are more realistic than the exponential one, namely, logistic and generalized logistic equations.

195

11.3 Generalized logistic equation

11.3 Generalized logistic equation Let us consider the generalized logistic equation for a homogeneous population:
β !γ dN N α ¼ kN ðtÞ 1  , dt C

(11.3)

where C is the carrying capacity and k, α, β, γ are model parameters. Let us now consider an inhomogeneous population composed of different clones l(t, a), which now grow according to the generalized logistic equation: "
β #γ dlðtaÞ N α1 ¼ alðtaÞgðNÞ, where gðNÞ ¼ kN 1 (11.4) dt C (see Chapter 7, Section 3 for details). Eq. (11.4) describes a model of Malthusian type. According to results obtained in Chapter 2, the solution to this equation is given by equation lðt, aÞ ¼ lð0, aÞeaqðtÞ , where the keystone variable q(t) is defined by equation dq ¼ gðN Þ, qð0Þ ¼ 0: dt Here,

ð N ðt Þ ¼

lðtaÞda ¼ N ð0ÞM0 ½qðtÞ, A

where M0[λ] is the mgf of initial distribution of P(0, a). From Eq. (2.2) and Proposition 2.2, it follows that


β !γ dN N t t α ¼ E ½aNgðNÞ ¼ E ½aN ðtÞ 1  , dt C

(11.5)


β !γ dEt ½a N t t α1 ¼ Var ½agðN Þ ¼ Var ½aN ðtÞ 1  : dt C

(11.6) t

Eq. (11.5) coincides with initial Eq. (11.3) only if Et[a] is a constant, that is, if dEdt½a ¼ 0. But dEt ½a according to Eq. (11.6), dt > 0 as long the population remains inhomogeneous and until it reaches its carrying capacity. Hence an inhomogeneous population composed of different logistically growing clones (11.4) can never actually grow logistically according to Eq. (11.3) but instead always grows faster during an initial phase of growth according to Eq. (11.5). It was proven in Chapter 7 Section 7.3 that inhomogeneous population (11.4) remains inhomogeneous indefinitely and its composition as t ! ∞ is identical to the composition of an inhomogeneous Malthusian model at a certain time moment. Taking into account that all real populations are nonhomogeneous, the question arises on whether there exists a model of inhomogeneous population growth such that its total

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population size solves the generalized logistic equation (11.3). The answer is affirmative: it is possible to construct an inhomogeneous frequency-dependent model, whose total population size solves this equation. The approach is similar to the one we used for exponential equation in Theorem 7.4. Let us consider the inhomogeneous frequency-dependent model: "
 β #γ dlðt, aÞ N α2 ¼ aPðt, aÞgðN Þ ¼ alðt, aÞ kN 1 : (11.7) dt C The solution to this equation is given again by the formula l(t, a) ¼ l(0, a)eaq(t), where now the keystone variable q(t) is defined by equation "
 β #γ dq gðN Þ N α2 ¼ ¼ kN 1 , qð0Þ ¼ 0: (11.8) dt N C For this model,


β !γ dN N t t α1 ¼ E ½agðN Þ ¼ E ½aN ðtÞ 1  , dt C

(11.9)

Eq. (11.9) coincides with generalized logistic Eq. (11.3) if Et[a] ¼ N(t). According to Eq. (2.12), dM0 ½qðtÞ dlnM0 dq Et ½ a  ¼ ¼ ½qðtÞ: M0 ½qðtÞ dq Hence, taking into account Eq. (11.4), we obtain the equation for unknown mgf M0[λ]: dM0 ½qðtÞ dq ¼ N ðtÞ ¼ N ð0ÞM0 ½qðtÞ: M0 ½qðtÞ We can rewrite this equation as dM0 ½λ ¼ N ð0ÞM0 ½λ2 , M0 ½0 ¼ 1: dλ Its solution is M0[λ] ¼ (1  N(0)λ)1. It is the mgf of exponential distribution 

a

e N ð 0Þ , a > 0, Pð0, aÞ ¼ N ð 0Þ

(11.10)

with the mean value equal to N(0). We have proven the following proposition. Proposition 11.1 Total population size of the inhomogeneous frequency-dependent population (11.7) with initial exponential distribution (11.10) of the Malthusian parameter a solves the generalized logistic Eq. (11.3).

11.4 Gompertz versus Verhulst

197

11.4 Gompertz versus Verhulst Now, let us consider in detail the most important special case of the standard logistic equation
 dN N ¼ kN 1  : (11.11) dt C Taking α ¼ β ¼ γ ¼ 1 in all equations of the previous section, the equation for keystone variable q(t) becomes


 dq k N 1 1 1 1 ¼ 1   q ðt Þ : (11.12)  ¼k ¼k dt N C N ð0ÞM0 ½qðtÞ C N ð 0Þ C Its solution is given by


qðtÞ ¼

and hence lðt, aÞ ¼ Nð0ÞPð0, aÞe

aqðtÞ

  1 1   1  ekt N ð 0Þ C

(11.13)




 1 1 1 kt +  ¼ exp a e : C N ð 0Þ C

The mgf of initial exponential distribution is M0 ½λ ¼ 1N1ð0Þλ, and the total population size is N ðtÞ ¼ Nð0ÞM0 ½qðtÞ ¼

N ð 0Þ : 1  Nð0ÞqðtÞ

(11.14)

Substituting q(t) from Eq. (11.13) to Eq. (11.14), after some simple algebraic transformations, we obtain that C
: N ðtÞ ¼ C kt 1 1+e N ð 0Þ As one can see, this expression coincides with the formal solution to logistic Eq. (11.11). Next, let us show that parameter distribution at time t is exponential with the mean equal to N(t). For this, it suffices to show, according to Eq. (2.25), that the mgf Mt[λ] of the parameter distribution at time t is given by Mt ½λ ¼ 1N1ðtÞλ. Indeed, according to Eq. (2.10), Mt ½λ ¼

M0 ½λ + qðtÞ 1  N ð0ÞqðtÞ ¼ ¼ M0 ½qðtÞ 1  N ð0ÞqðtÞ  N ð0Þλ

1 1 : ¼ N ð0Þλ 1  N ðtÞλ 1 1  N ð0ÞqðtÞ

Let us collect the obtained results in the following theorem. Theorem 11.1 Logistic equation (11.11) describes the dynamics of the total size of an inhomogeneous F-model:
 dlðt, aÞ N ðtÞ ¼ kaPðt, aÞ 1  , dt C

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11. From experiment to theory

where initial distribution of parameter a is exponential with the mean N(0). The solution to the F-model is given by equation


 1 1 1 kt +  lðt, aÞ ¼ exp a e : (11.15) C N ð 0Þ C Total population size solves Eq. (11.11) and is given by the formula (11.14). The current distribution of the parameter at time t is exponential with the mean Et[a] ¼ N(t). Formula (11.15) shows that each clone l(t, a)   grows according to the Gompertz curve G(t) ¼ a

r exp( ce kt) with r ¼ e C and c ¼ a

1 1 N ð0Þ  C

. This means that, similarly to the exponential

population, a population that grows according to logistic curve (11.11) can be an inhomogeneous population composed of Gompertzian clones. Fig. 11.1 shows the dynamics of clones defined by Eq. (11.15) at different values of parameter a. The model described in Theorem 11.1 could be considered unrealistic for describing dynamics of real populations, since the population would contain clones with arbitrarily large values of parameter a. However, in this case the contribution of clones with large values of parameter a to the total population size is negligible. Indeed, let us denote ðB N ðt, BÞ ¼ lðt, aÞda, 0

so the total population size N(t) ¼ N(t, ∞). Then, C  N ðt, BÞ ¼ 1e kt 1 + e ðC=Nð0Þ  1Þ

Bekt ðC + N ð0Þðekt 1ÞÞ CN ð0Þ

! :

(11.16)

So, NNðtð,tBÞ Þ  1  eB=C for large t. Hence, N(t, B) gives a very good approximation of the exact solution to the logistic equation if CB  6  10, and therefore e B/C  0.0025  0.000045. The dynamics of N(t,B) over time for different value of B are shown in Fig. 11.2.

FIG. 11.1 Plots of l(t, a) as defined in Eq. (11.15), with N(0) ¼ 1, C ¼ 10, k ¼ 1. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

1 0.8

a = 10.0 a = 1.0 a = 0.1

I(t,a)

0.6 0.4 0.2 0 0

2

4

6 Time

8

10

11.4 Gompertz versus Verhulst

199

FIG. 11.2 Plots of N(t, B) as defined in Eq. (11.16) with B ¼ 10.30, 50; other parameters are fixed at N(0) ¼ 1, k ¼ 1, and C ¼ 10. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

The results of the last section can be summarized as follows. Both Malthusian and logistic equations describe the growth of total population size of specific inhomogeneous F-models. Recall that, by definition, F-models are the models, where the growth rate of each clone is proportional to its frequency in the total population. A possible derivation of frequency-dependent models is as follows: Assume that the growth rate of individuals within the population is controlled by a limiting external resource, which is divided uniformly between all individuals. Then the growth rate of a clone is proportional to its frequency. A monomorphic population in ideal conditions could grow exponentially; however, if the exponentially growing population is polymorphic, then one can hypothesize that (a) the population is composed of Gompertzian clones, each of which is described by a frequency-dependent model; (b) the growth of the population is not free but depends on an external resource, which is distributed uniformly between all individuals and hence proportionally to clone frequencies in the total population, thereby limiting growth rate of the population at each time moment. Importantly, similar conclusions are valid for the more realistic logistic equation as well. Real populations are polymorphic, and if a population grows logistically, then population growth most probably is controlled by an external dynamical resource at each time moment, not only close to saturation stage. The population may be also composed of Gompertzian clones with other parameters compared with the case of exponential growth. In general, if a population grows according to the solution to an F-model, then we can assume that at each time moment, population growth is controlled by an external dynamical resource, which is divided uniformly between all individuals in the population.

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11. From experiment to theory

Example 11.1 Logistic versus Gompertz curves Consider the following data set, which was extrapolated from Biebricher et al. (1985). As can be seen in Fig. 11.3, data set 1 (see supplementary excel file for this chapter for raw data) is fitted better by the logistic equation from which we can conclude that this population is more likely to be heterogeneous. Overall the results obtained in this section can have important implications. Assume there exist experimental data on population growth, but no a priori assumptions have been made about processes that govern it. If logistic curve fits the data better than the Gompertz curve, then the population is more likely to be polymorphic than monomorphic. Furthermore, assume one is able to determine if the population is indeed polymorphic. Then, one can conclude that population growth is controlled by an external (possibly dynamical) resource at all stages of population development, not only when the population approaches its carrying capacity. Finding such an external factor may be of great interest, especially for such populations as, for instance, cancer cells. This will be discussed in greater detail in the following sections.

11.5 Hyperbolic and hyperbolic-exponential growth Curves that have hyperbolic shape have been observed in a variety of circumstances, ranging from tumor growth curves (e.g., Almog et al., 2006; Naumov et al., 2006; Rogers et al., 2014) to global demography. Recall the following figure from Chapter 3 with some examples reported in the literature (Fig. 11.4). Data for global demography and corresponding models were considered in detail in Chapter 3. Tumor growth curves and corresponding models will be discussed below.

FIG. 11.3

Fitting of data reported in Fig. 3 in Biebricher et al. (1985). (A) Fitting with logistic curve, defined by

ð0Þ NðtÞ ¼ Nð0Þ + ðCN CN ð0ÞÞert , with C ¼ 80.64, N(0) ¼ 0.005838, and r ¼ 0.2376. (B) Fitting with the Gompertz model, N(t) ¼  kt

N(0)ec(1e ), with c ¼ 4.732, k ¼ 0.05165, and N(0) ¼ 0.7095. Parameter estimations were obtained using cftool in MATLAB. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

201

Tumor volume (mm3)

11.5 Hyperbolic and hyperbolic-exponential growth 1400 1200

3000

D

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b

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1000

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500

0

0

i

800 600

0

A

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mice

0 0

20 40 60 80 100 120 140 160

C

0

20 40 60 80 100 120 140 160

Time (days)

Time (days)

7,000,000,000 6,000,000,000 5,000,000,000 4,000,000,000 3,000,000,000 2,000,000,000 1,000,000,000

D

0 –10000

–6000

–8000

–4000

–2000

0

2015

Time (years)

FIG. 11.4 Examples of hyperbolic growth curves obtained from published literature. (A) Tumor growth curves reported Fig. 1D in Naumov et al. (2006). (B) and (C) Tumor growth curves reported in Fig. 2b,i in Rogers et al. (2014). (D) World population since 10,000 BCE, reported at https://ourworldindata.org/world-populationgrowth/. Data sources include History Database of the Global Environment (HYDE) before 1900, the UN publication “The World at Six Billion” for 1900–40, and the UN’s World Population Prospects: The 2015 Revision for 1950–2015. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

The simplest model that shows hyperbolic growth of population size is the inhomogeneous Malthusian model dlðdtt, aÞ ¼ alðt, aÞ with exponential or Gamma-distributed growth rate; this model was studied in detail in Chapter 2. If the initial distribution is the Gamma distribution (see Eq. ( 2.19)) with η ¼ 0, then its mgf is equal to M0 ½t ¼ ð1  stÞk , t < 1=s, k > 0, and the total population size N ðtÞ ¼ N ð0Þð1  stÞk shows hyperbolic growth. Notice that N(t) increases very slowly at the initial stage and then tends very quickly to infinity at the moment of “population explosion” T ¼ 1s ; see Fig. 11.4D. This unrealistic scenario becomes realized when the Malthusian parameter a can take arbitrarily large values, which can be avoided through truncating the initial distribution of parameter a to be restricted to a finite interval. Let the initial distribution be truncated exponential in the interval [0,b]:

202

11. From experiment to theory

Pð0,aÞ ¼ Vesa for 0  a  b, Pð0,aÞ ¼ 0 for all a62½0, b,

(11.17)

where V ¼ 1esBs is the normalization constant. Then the solution of the Malthusian model with this initial distribution is given by the following equation (see also Eq. ( 2.27)): N ðtÞ ¼ N ð0Þ

s 1  ebðstÞ s  t 1  ebs

(11.18)

Inhomogeneous Malthusian models with initial distribution concentrated in a bounded interval a 2 [0, b] possess some interesting shared properties independently of the specific initial distribution. According to Eq. (2.31), dN ¼ NEt ½a, dt population growth is superexponential, since population growth rate, which is equal to Et[a], increases over time as long as Vart[a] > 0. Hence Et[a] tends to the maximal possible value of a, Et[a] ! b. At this time, population growth becomes asymptotically exponential with the growth rate equal to b. All these phenomena were additionally illustrated in Chapter 3 with the example of global demography data and models.

11.6 Exponential-linear growth Another type of a model that can qualitatively replicate the dynamics observed in Fig. 11.4 is the exponential-linear model, constructed within the frameworks of F-models. Let us consider an inhomogeneous F-model (11.1): dlðt, aÞ ¼ kaPðt, aÞ: dt We have already shown that with exponentially distributed parameter a at the initial moment, the solution to this equation is given by the Gompertz curve (11.2) and the population size N(t) ¼ N(0)ekt. Now let us assume that the initial distribution of the Malthusian parameter is truncated exponential (11.17). Integrating Eq. (11.1) over a, we get dN ¼ kEt ½a: dt

(11.19)

It follows from Eq. (11.1) after simple algebra that dEt ½a Vart ½ak ¼ : dt N ðt Þ (Note that the last equation also follows from the covariance Eq. (5.47) applied to Eq. (11.1) in the form dlðdtt, aÞ ¼ lðt, aÞ NkaðtÞ.) Therefore Et[a] increases monotonically as long as Vart[a] > 0. So, Et[a] ! b, and the righthand side of Eq. (11.19) tends to kb ¼ const, implying asymptotic linear growth of N.

11.6 Exponential-linear growth

203

One may naturally assume that the initial stage of growth of a population described by Eq. (11.19) with initial exponential distribution and truncated exponential distribution are very similar if the value of b is large, and hence the initial dynamics of the model is close to exponential. Indeed, let us define the auxiliary variable by equation: dq k ¼ , qð0Þ ¼ 0: dt N dq Then, Eq. (11.1) reads dlðdtt, aÞ ¼ alðt, aÞ dt , and hence l(t, a) ¼ l(0, a)eaq(t). The mgf of truncated exponential distribution (11.17) is given by Eq. (2.27);

M0 ½λ ¼

s 1  eðbðλsÞÞ : s  λ 1  eðbsÞ

Then, NðtÞ ¼ N ð0ÞM0 ½qðtÞ ¼ Nð0Þ and hence

s 1  ebðsqðtÞÞ s  qðtÞ 1  ebs

dq k 1  ebs s  qðtÞ ¼ : dt N ð0Þ s 1  ebðsqðtÞÞ

(11.20)

(11.21)

With a solution to this equation, we can compute all statistical characteristics of interest: Pðt, aÞ ¼ ðb

eaqðtÞ 1  ebs eaqðtÞsa eaðqðtÞsÞ ðqðtÞ  sÞ Pð0, aÞ ¼ ¼ , M½qðtÞ s M½qðtÞ ðebðqðtÞsÞ  1Þ b

1 , s  qðtÞ 0   e2bq + e2bS  ebðq + SÞ 2 + b2 ðq  SÞ2 1 b2 ebðq + SÞ Vart ½a ¼ ¼  : 2 ðq  SÞ2 ðebq  ebS Þ2 ðebq  ebS Þ ðq  SÞ2 Et ½ a  ¼

aPðt, aÞda ¼

1  ebðsqðtÞÞ

(11.22)

+

(11.23)

For the purposes of analysis and computation, we can write this model in two different but equivalent forms: Version 1: dq k ¼ , dt N ðtÞ   (11.24) s 1  ebðqðtÞsÞ N ð t Þ ¼ N ð 0Þ : ð1  ebs Þðs  qðtÞÞ Version 2: dq k ¼ , dt NðtÞ
 (11.25) dN b 1 ¼ kEt ½a ¼ k + : dt 1  ebðsqðtÞ s  qðtÞ For example, let us consider the initial distribution (11.17) with s ¼ 1 and different values of the boundary b. As can be seen in Fig. 11.5, for large values of b, in the initial stages of growth

204

11. From experiment to theory

log(N(t))

104

102

B = 10 B = 100 B = 500

100 0

2

4

6

8

10

6

8

10

FIG. 11.5 Plots of N(t) as defined in Eq. (11.24) with different values of boundary b. (A) In the initial stages the population grows exponentially, as is confirmed by logarithmic transformation of the growth curve. (B) At later time points the population starts growing linearly. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

Time

(A)

N(t)

1500 B = 10 B = 100 B = 500

1000 500 0

(B)

0

2

4 Time

the population increases exponentially (i.e., logN(t) in Fig. 11.5A approximates well a linear function with respect to t). Then, after a transitional period, the shape of the curve changes and the population increases linearly (see Fig. 11.5B). Overall, we have proven the following theorem. Theorem 11.2 F-model (11.1) implies asymptotically linear growth of the total population size for any initial distribution of parameter a concentrated on a bounded interval. If initial distribution is truncated exponential, then the model shows exponential-linear growth of population size. Results obtained in this section can be summarized as follows: Nonstandard exponential-linear dynamics may be the evidence that a population (e.g., cancer cells) is inhomogeneous and is composed of clones such that distribution of their growth rates is close to truncated exponential distribution. Even more importantly the population growth is controlled by an external (possible dynamical) resource at all stages of population development, such that the resource is distributed uniformly between individuals in the population.

11.7 Virus-specific RNA replication and the three-stage model In their earlier work, Schuster (2011) proposed that in a closed system, where there exists no exchange of materials with the environment, the RNA replication process goes through three phases: exponential growth, linear growth, and saturation (Biebricher et al., 1985; Schuster, 2011). A schematic representation of the proposed kinetics of RNA replication in closed systems, as adapted from Schuster (2011), is shown in Fig. 11.6.

Concentration of RNA c(t)

11.7 Virus-specific RNA replication and the three-stage model

Exponential

Linear

Saturation or product inibition

exp(ft)

k.t

l-cxp(-kt)

205

Time t

FIG. 11.6

Schematic representation of three phases of replication process as proposed by Schuster (2011). The authors proposed that the time course of RNA replication shows three distinct growth phases: (i) an exponential phase, (ii) a linear phase, and (iii) a phase characterized by saturation through product inhibition. Adapted from Schuster, P., 2011. Mathematical modeling of evolution. Solved and open problems. Theory Biosci. 130, 71–89.

In their original work, Schuster and colleagues had the following hypothesis about the kinetics of RNA replication in closed systems. They proposed that the time course of RNA replication by Qβ-replicase shows three distinct growth phases: (i) an exponential phase, (ii) a linear phase, (iii) a phase characterized by saturation through product inhibition (Biebricher et al., 1983, 1985). The experiment [was] initiated by transfer of a very small sample of RNA suitable for replication into a medium containing Qβ-replicase and the activated monomers, ATP, UTP, GTP, and CTP in excess (consumed materials are not replenished in this experiment). In the phase of exponential growth, there [was] shortage of RNA templates, every free RNA molecule is instantaneously bound to an enzyme molecule and replicated, ft and the corresponding overall kinetics follows dx dt ¼ fx resulting in x(t) ¼ x(0)e . In the linear phase the concentration of template [was] exceeding that of enzyme, every enzyme molecule is engaged in replication, and overall kinetics is described by dx dt ¼ k0 e0 ðEÞ ¼ k, wherein e0(E) is the total enzyme concentration, and this yields after integration x(t) ¼ x(0) + kt.

The schematic given in Fig. 11.6 is somewhat exaggerated compared with the data that the authors cited, presumably to emphasize the transition from exponential to linear phase. In the reported data the transition is smoother. Notably the authors proposed describing the transition between three stages of growth using several separate models, which does not allow for understanding of how the transition between the three stages can occur naturally, as a result of system dynamics. Our numerical estimates of some of the reported data (such as curves reported in Biebricher et al. (1985)) are fitted well by a logistic model (see Fig. 11.3). However, a model that realizes all three regimes can be constructed in the following way. We have already shown that transition from exponential to linear growth can be described by an inhomogeneous F-model with a distributed Malthusian parameter, with initial

206

11. From experiment to theory

truncated exponential distribution. The transition to a saturation stage can be realized through addition of a logistic-like term. The resulting model is as follows. Consider an F-model:

 dlðt, aÞ N ðtÞ r ¼ kaPðt, aÞ 1  , (11.26) dt K where P(0, a) is the truncated exponential distribution (11.17). Define the auxiliary variable q(t) by the equation

N ðt Þ r k 1 dq K ¼ , qð0Þ ¼ 0: dt N ðt Þ

(11.27)

The solution to the F-model can be written again as l(t, a) ¼ l(0, a)eaq(t), so Eqs. (11.20), (11.22) and (11.23) apply also to the considered model. The difference is that the auxiliary variable q(t) is now defined not by Eq. (11.21), but by Eq. (11.27). Overall, similarly to the previous exponential-linear model, we can write our three-stage model in two different but equivalent forms: Version 1:

N ðt Þ r k 1 dq K ¼ , qð0Þ ¼ 0; dt N ðt Þ   s 1  ebðqðtÞsÞ : N ð t Þ ¼ N ð 0Þ ð1  ebs Þðs  qðtÞÞ Version 2:



N ðt Þ r k 1 dq K ¼ , qð0Þ ¼ 0; dt N ðtÞ
 dN b 1 t ¼ kE ½a ¼ k + : dt 1  ebðsqðtÞÞ s  qðtÞ

(11.28)

(11.29)

The three distinct stages of system (11.29) are illustrated in Fig. 11.7. The initial exponential stage is shown in Fig. 11.7A and is confirmed by logarithmic transformation of the same data in Fig. 11.7B. It is followed by the linear stage (Fig. 11.7C), which is followed by the saturation stage (Fig. 11.7D). The full curve is shown in Fig. 11.7E. Now let us compare the six types of models discussed up to this point: logistic, Gompertz, hyperbolic, hyperbolic-exponential, exponential-linear, and three-stage models. All six models can be used to describe population growth; however, populations described by these models have qualitatively different intrinsic properties. The results of these comparisons are summarized in Table 11.1.

11.7 Virus-specific RNA replication and the three-stage model

207

FIG. 11.7 Three-stage model, as described by system (11.29), with s ¼ 1, b ¼ 10, k ¼ 1, K ¼ 100, and r ¼ 4. (A) The exponential stage of the system growth, (B) confirmed by logarithmic transformation, followed by (C) the linear stage of population growth, followed by (D) the saturation stage of system growth. All three stages can be seen on in part (E). Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

Notably, for the logistic model, the initial distribution is exponential, while for three-stage model the initial distribution is truncated exponential. Hence solution to the three-stage model tends to the solution of the logistic model as the boundary b of truncated exponential distribution increases. In the next section, we will look at several examples specific for tumor growth and discuss their implications.

TABLE 11.1 Summary and implications of the proposed growth functions. Model

Likely monomorphic/ polymorphic

Figure

Logistic

100

Logistic growth

80 60 40 20 0

0

2

4

6

8

10

Time

Gompertz Gompertian growth curve

2.5

Other properties

Can be either monomorphic If it is polymorphic, then: or polymorphic but more – Frequency-dependent growth likely to be polymorphic – Population growth is controlled by an external resource uniformly distributed between individuals during all growth stages Can be either monomorphic – Likely composed of a single clone or polymorphic but more – Individuals may be either in likely to be monomorphic quiescent or actively growing state

´ 104

2 1.5 1 0.5 0 0

2

4

6

8

10

Time 12

Distributed malthusian model

Hyperbolic

Polymorphic

– Density-dependent growth of independent Malthusian clones – Distribution of clones close to exponential

Polymorphic

– Density-dependent growth of independent Malthusian clones – Distribution of clones likely close to truncated exponential – Population becomes nearly monomorphic when exponential stage supersedes the hyperbolic stage

Polymorphic

– Frequency-dependent growth – Distribution of clones likely close to truncated exponential – Population growth is controlled by an external dynamical resource during all stages of growth uniformly distributed between individuals during all stages of growth – Population becomes nearly monomorphic when linear stage supersedes the exponential stage

Polymorphic

– Frequency-dependent growth – Distribution of clones close to truncated exponential – Population growth is controlled by an external dynamical bounded resource uniformly distributed between individuals during all growth stages

10 8 6 4 2 0

0

20

40

60

80

100

Exponentiallinear

1030

1020

1010

100 0

1

2

3 Time

4

5

6

1500 Exponential-linear growth

Hyperbolicexponential

Hyperbolic-exponential growth (log scale)

Time

1000

500

0 0

2

4

6

8

10

Time

250 200 150 N(t)

Three-stage (exponential, linear, saturation)

100 50 0 0

20

40 Time

60

Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

11.8 Fitting experimental data to different models

209

11.8 Fitting experimental data to different models In this section, we take several data sets on tumor growth in various tissues that were published in the literature and fit them to the models described in the previous sections. Our goal is to try to infer, based on our theory summarized in Table 11.1, what possible mechanisms could have been underlying growth of these populations. We focused particularly on the following four models: logistic, Gompertzian, exponentiallinear, and three-stage. The data sets were extrapolated from the figures reported in the following publications: Naumov et al. (2006), Rogers et al. (2014), and Benzekry et al. (2014). Four data sets and corresponding fitted parameter values are given in Appendix to this chapter. Since the raw data were not published, we obtained the data sets from the figures using WebPlotDigitizer. We then performed parameter fitting using fmincon function in MATLAB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P and calculated standard mean square error MSE ¼ n1 ni¼1 ðxi  datai Þ2 and relative mean sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 Pn xi  datai 2 1 square deviation using the formula n i¼1 , where xi represents model predicdatai tions. The results are reported in the Appendix. The obtained goodness-of-fit estimations were consistent using both MSE and relative least squares.

11.8.1 Data set 1: Naumov et al. (2006) JNCI Consider the following data set, which was reported as one of the breast cancer growth curves in Naumov et al. (2006). In Fig. 11.8, one can see how the data fit logistic function (Fig. 11.8A), Gompertzian growth curve (Fig. 11.8B), exponential-linear (Fig. 11.8C), and the three-stage model (Fig. 11.8D). Data plotted with all four functions are shown in Fig. 11.8E, showing how the population would grow over time in each of the cases. As one can see, logistic function appears to provide a better fit, which is confirmed by low residual mean square error (see Appendix for values), suggesting that the population grows in a frequency-dependent manner and depends on a uniformly distributed external resource during all stages of growth.

11.8.2 Data set 2: Rogers et al. (2014) The following data set was obtained from Rogers et al. (2014), Fig. 2b. Similar to the previous case, the data are fitted to logistic (Fig. 11.9A), Gompertz (Fig. 11.9B), linear-exponential (Fig. 11.9C), and three-stage (Fig. 11.9D) functions. Neither function in this case provides a good fit, with logistic and three-stage models providing better fit than others. One can certainly see that the Gompertz function (Fig. 11.9B) provides a particularly poor fit, confirmed by RMSE and relative least-square deviation, at least suggesting that, according to Table 11.1, the population described here is highly unlikely to be monomorphic.

11.8.3 Data set 3: Rogers et al. (2014) The following data set was also obtained from Rogers et al. (2014), in this case from Fig. 2i. Here, both logistic (Fig. 11.10A) and Gompertz (Fig. 11.10B) functions provide a

210

11. From experiment to theory

FIG. 11.8 Data from Naumov et al. (2006), breast cancer in vivo. Data fitted to (A) logistic, (B) Gompertzian, (C) exponential-linear and (D) three-stage model. The data and all the growth functions are plotted in (E). Logistic function had the lowest mean square error (MSE). Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

relatively good fit, with the Gompertz function having a slightly smaller MSE (see Appendix for details). This suggests that in this case the population is more likely to be monomorphic.

11.8.4 Data set 4: Benzekry et al. (2014) Finally, consider the following data set, which was extrapolated from Benzekry et al. (2014), Fig. S1B, the lung cancer growth curve. Our calculations suggested that, while all

11.8 Fitting experimental data to different models

211

FIG. 11.9

Data extracted from Rogers et al. (2014), Fig. 2B. Data fitted to (A) logistic, (B) Gompertzian, (C) exponential-linear, and (D) three-stage model. The data and all the growth functions are plotted in (E). Gompertzian curve provides the worst fit (highest RMSE), allowing elimination of monomorphic population. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

the models provided a reasonable fit (Fig. 11.11), logistic model resulted in the lowest RMSE, suggesting that this population is inhomogeneous, grows in a frequency-dependent manner, and depends on a uniformly distributed external resource during all stages of growth. Notably, in their analysis, the authors predicted a better fit with a Gompertzian growth function;

212

11. From experiment to theory

Tumor volume (mm3)

3000

3000

Data set 5 Logistic

2000

2000

1000

1000

(A)

0 0

3000

50

100

(C)

0 0

3000

Data set 5 Gompertz

2000

2000

1000

1000

0 0

(B)

50 Time (days)

0 0

100

(D)

Data set 5 Lin.-exp.

50

100

Data set 5 3-Stage

50 Time (days)

100

14,000 12,000 10,000

Data Logistic Gompertz Lin.-exp. 3-Stage

8000 6000 4000 2000 0 0

(E) FIG. 11.10

100

200 Time (days)

300

Data extracted from Rogers et al. (2014), Fig. 2i. Data fitted to (A) logistic, (B) Gompertzian, (C) exponential-linear, and (D) three-stage model. The data and all the growth functions are plotted in (E). Gompertzian curve provides the best fit (lowest MSE), suggesting monomorphic population. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

213

11.8 Fitting experimental data to different models

3000

3000

Data set 5 Logistic

2000

2000 1000

Tumor volume (mm3)

1000

(A)

Data set 5 Exp.-lin.

0 0

50

3000

100

(C)

0 0

3000

Data set 5 Gompertz

2000

2000

1000

1000

0 0

50 Time (days)

(B)

0 0

100

(D)

50

100

Data set 5 3-Stage

50 Time (days)

100

18,000 16,000 14,000 12,000

Data set 5 Logistic Gompertz Exp.-lin. 3-Stage

10,000 8000 6000 4000 2000 0 0

(E)

50

100 Time (days)

150

FIG. 11.11 Data extracted from Benzekry et al. (2014), Fig. S1B, the lung cancer growth curve. Data fitted to (A) logistic, (B) Gompertzian, (C) exponential-linear, and (D) three-stage model. The data and all the growth functions are plotted in (E). Based on MSE values, logistic curve fits best, suggesting a polymorphic population that grows in a frequencydependent manner and depends on a uniformly distributed external resource during all stages of growth. Adapted from Kareva, I., Karev, G., 2018. From experiment to theory: what can we learn from growth curves? Bull. Math. Biol. 80, 151–174.

214

11. From experiment to theory

this discrepancy may be a result of variations either in parameter estimation methods or, which is more likely, in the fact that we had no access to original raw data and had to rely on the published figure to obtain the numbers for our analysis.

11.9 Simeoni model and exponential-linear growth One of the more widely used models of tumor growth that captures the exponential growth phase followed by a linear one was proposed by Simeoni et al. (2004), which assumes that there exists a “threshold,” determined by two model parameters, separating exponential and linear growth phases. However, this model, like many others, does not capture the fact that most tumors are genetically heterogeneous (Marusyk and Polyak, 2010; Burrell et al., 2013), a key property that largely underlies development of therapeutic resistance, since drugs typically can target only a subpopulation of cancer cells, effectively and inevitably leaving behind therapeutically resistant cell clones. The original Simeoni model has the following form: dxðtÞ λ0 xðtÞ ¼"
Ψ #1=Ψ , dt λ0 1+ ωðtÞ λ1

(11.30)

where x(t) are cancer cells, ω(t) is the total population size of cancer cells in a tumor population (in the absence of treatment ω(t) ¼ x(t)), λ0 is the parameter characterizing rate of exponential growth for tumor size ω < threshold tumor mass ωthreshold, λ1 is the parameter characterizing rate of linear growth for tumor size ω > ωthreshold, and Ψ is a fitting parameter. This model allows capturing exponential-linear dynamics that is frequently observed in mouse xenograft models (a common animal model for studying cancer growth, where human cancer cells are implanted into typically immunosuppressed mice to observe tumor growth in vivo). However, this model is empirical, and the transition between the two phases is achieved through variation of the two parameters λ0 and λ1 rather than arising from system dynamics. The frequency-dependent model described earlier might be able to capture the same dynamics, which in this case arises from intrinsic population heterogeneity, a known property of most tumors. Equations, parameter values, and their meaning for the Simeoni model are given in Table 11.2. As one can see in Fig. 11.12, both the Simeoni model (11.30) and the frequency-dependent model (11.1) fit the same data well (these data points were obtained from Fig. 7 in Simeoni et al. (2004)). Including population heterogeneity can be particularly important for treatment simulation, which is used to assess drug behavior in preclinical stages of drug development and is part of calculus for dose and schedule predictions for first-in-human doses. Cytotoxic (cell-killing) treatment is often simulated using a signal transduction model (Yang et al., 2010), where cytotoxic effect on the initial cancer cell population is spread over time as

11.9 Simeoni model and exponential-linear growth

215

TABLE 11.2 Equations, parameter values, and their meaning for the Simeoni et al. (2004) model of exponential-linear growth. Equations (Simeoni)

Meaning

x(t)

Population of untreated cancer cells

ω(t)

Total population size of cancer cells in a tumor population, in the absence of treatment ω(t) ¼ x(t)

dxðtÞ dt ¼

λ0 xðtÞ

Ψ 1=Ψ λ0 ωðtÞ 1+ λ1
 xðtÞ λ0 1 ωmax dxðtÞ Ψ 1=Ψ dt ¼
λ0 1+ ωðtÞ λ1

Exponential-linear growth of cancer cells over time (Simeoni et al., 2004)

Parameters

Meaning

λ0

Parameter characterizing rate of exponential growth for tumor size ω < threshold tumor mass ωthreshold

λ1

Parameter characterizing rate of linear growth for tumor size ω > threshold tumor mass ωthreshold

Ψ

Fixed parameter

ωmax

Carrying capacity




Exponential-linear growth of cancer cells over time, limited by carrying capacity (Haddish-Berhane et al., 2013)

FIG. 11.12 Simeoni model versus the F-model. Parameters used for Simeoni model: x1(0) ¼ 0.085, λ0 ¼ 0.23, λ1 ¼ 0.95, and Ψ ¼ 20. Parameters used for exponential-linear model: N(0) ¼ 0.085; k ¼ 0.42; and initial distribution is truncated exponential with a 2 [0, 4], s ¼ 20, where parameter s influences the duration of the exponential phase, while the interval of the initial distribution affects the slope of the linear phase. Data points were digitized from control in Fig. 7 of Simeoni et al. (2004).

216

11. From experiment to theory

the damaged cells go through progressive degrees of damage. This is captured through the following general model: dx1 ðtÞ ¼ Fðx1 ðtÞÞ  k2 cðtÞx1 ðtÞ, dt dx2 ðtÞ ¼ k2 cðtÞx1 ðtÞ  k1 x2 ðtÞ, dt dx3 ðtÞ ¼ k1 ðx2 ðtÞ  x3 ðtÞÞ, dt dx4 ðtÞ ¼ k1 ðx3 ðtÞ  x4 ðtÞÞ, dt

(11.31)

where F(x1(t)) is the growth rate of the cancer cell population, c(t) is the cytotoxic agent, k2 is the drug efficacy index, and k1 is the rate at which cells undergo additional damage, eventually dying at a rate  k1x4(t). Since cytotoxic chemotherapy primarily affects most rapidly proliferating cells, it is reasonable to assume that the death rate due to cytotoxic agent c(t) is directly proportional to the value of parameter a. To include this assumption in the model, we need to make the following transformation: Assume that
 dx1 ðt, aÞ k k ¼ ax1 ðt, aÞ  ak2 cðtÞx1 ðt, aÞ ¼ ax1 ðt, aÞ  k2 cðtÞ (11.32) dt N ðt Þ N ðt Þ Then the keystone variable q(t) becomes dqðtÞ k ¼  k2 cðtÞ: dt N ðt Þ

(11.33)

The resulting system of equations becomes dqðtÞ k ¼  k2 cðtÞ, dt N ðtÞ dx2 ðtÞ ¼ Et ½ak2 cðtÞN ðtÞ  k1 x2 ðtÞ dt dx3 ðtÞ ¼ k1 ðx2 ðtÞ  x3 ðtÞÞ dt

(11.34)

dx4 ðtÞ ¼ k1 ðx3 ðtÞ  x4 ðtÞÞ dt N ðtÞ ¼ N0 M0 ½qðtÞ Ntot ðtÞ ¼ N ðtÞ + x2 + x3 + x4 with the other statistical characteristics defined earlier. Note that in the equation for dxdt2 ðtÞ, in the first term, ax1(t) was replaced by Et[a]N(t), since we are now following the effect of the entire population of heterogeneous cells as it affects system dynamics over time. The effect of the cytotoxic agent is simulated through introducing equation C0 (t) ¼  k01C(t) as a simple

11.9 Simeoni model and exponential-linear growth

217

FIG. 11.13 Predicted treatment effects for a simulated drug for both models. Data for untreated and treated values were digitized from Fig. 8, experiment 1 in Simeoni et al. (2004). Parameter values used in simulations are given in Table 11.3. (A) Untreated population. (B) Dose of simulated drug necessary to achieve observed data using the Simeoni model is 160, given 3xQ4D. (C) Dose of simulated drug necessary to achieve observed data using the F-model is 2600, given 3xQ4D. The simulated drug preferentially affects rapidly growing clones in the exponential-linear model (system 11.34), resulting in smaller predicted effect compared with the classic Simeoni model, since the rapidly dividing clones are quickly eliminated from the population, while the slower growing ones remain.

case to illustrate the effects of drug administration on the simulated population. Parameters k1 ¼ 0.056 and k2 ¼ 0.0006 were taken from Simeoni et al. (2004). Parameter k01 was arbitrarily chosen to be 0.7. The dynamics of the simulated drug are identical for both cases, and all the parameters pertaining to cell damage and cytotoxic effects of the simulated drug were taken to be the same. In Fig. 11.13A the growth of an untreated tumor was reproduced using the two models; as one can see, both provide a good fit to the available data. In Fig. 11.13B the dose of the simulated drug is shown (160 units) that provides a fit between the data for a treated tumor and the Simeoni model. Finally, in Fig. 11.13C, a dose is found that results in the same tumor growth suppression (2600 units) by a simulated tumor described by the F-model. As one can see the difference is over 100-fold. Noticeably, 160 units of the simulated drug bring the predicted growth curve of the heterogeneous population close to the one described by the Simeoni model. This is consistent with the initial effects of a cytotoxic agent that eliminates cells that are sensitive to the drug, leaving behind a population of resistant cells that are remain unaffected; these resistant cells are responsible for the extremely high drug dose that is then required to achieve the effect observed in Fig. 11.13C. MATLAB code used to generate these figures is included in supplementary materials. Parameter values used in simulations are given in Table 11.3. Now, let us consider a case when all cells are equally susceptible to the drug: dx1 ðt, aÞ k ¼ ax1 ðt, aÞ  k2 cðtÞx1 ðt, aÞ dt N ðtÞ

218

11. From experiment to theory

TABLE 11.3 Parameters used in simulations. Parameter

Definition

Value

Units

References

Simeoni model λ0

Parameter characterizing rate of exponential growth of tumor size

0.27

day1

Simeoni et al. (2004), Table 3, exp. 1

λ1

Parameter characterizing rate of linear growth of tumor size

1.11

g day1

Simeoni et al. (2004), Table 3, exp. 1

Ψ

Fitting parameter

20

Simeoni et al. (2004)

Frequency-dependent growth model (F-model) k

Scaling parameter

0.42

n/a

n/a

s

Parameter of exponential distribution (for normal distribution, it would be mean and variance)

40

n/a

n/a

[B0, B1]

The interval of possible values of distributed parameter a

[0,5]

day1

n/a

Pharmacokinetic (PK) parameters k1

First-order rate constant of transit

0.056

day1

Simeoni et al. (2004), Table 3, exp. 1

k2

Measure of drug potency

20.2e-4

ng1 mL day1

Simeoni et al. (2004), Table 3, exp. 1

ke

Plasma clearance of the drug

0.7

L h1 kg1

Simeoni et al. (2004)

Initial conditions x(0)

Initial tumor size

0.037

g

Simeoni et al. (2004), Table 3, exp. 1

q(0)

Initial value of the keystone variable q(t)

0

n/a

n/a

c(0)

Initial concentration of the simulated drug

Varies

conc

n/a

Dosing interval

4

days

n/a

Introduce two keystone variables q1 and q2, such that dq1 ðtÞ k ¼ N ðtÞ dt dq2 ðtÞ ¼ k2 cðtÞ: dt Then, xðt, aÞ ¼ xð0, aÞeaq1 ðtÞ + q2 ðtÞ ¼ N ð0ÞPð0, aÞeaq1 ðtÞ + q2 ðtÞ

11.9 Simeoni model and exponential-linear growth

219

Then, total population size is ð ð N ðtÞ ¼ xðt, aÞda ¼ Nð0ÞPð0, aÞeaq1 ðtÞ eq2 ðtÞ da ¼ eq2 ðtÞ N ð0ÞM0 ½q1 ðtÞ 



and the distribution of cell clones in the population is given by Pðt, aÞ ¼

xðt, aÞ N ð0ÞPð0, aÞeaq1 ðtÞ + q2 ðtÞ eaq1 ðtÞ ¼ q ðtÞ ¼ Pð0, aÞ N ðt Þ M½q1 ðtÞ e 2 N ð0ÞM0 ½q1 ðtÞ

Formulas for mgf, expected value, and variance of the growth rate parameter a remain the same for both cases, with difference being in definition of keystone variable(s). The final system of equations for this model becomes dq1 ðtÞ k ¼ N ðt Þ dt dq2 ðtÞ ¼ k2 cðtÞ dt dx2 ðtÞ ¼ k2 cðtÞN ðtÞ  k1 x2 ðtÞ dt dx3 ðtÞ ¼ k1 ðx2 ðtÞ  x3 ðtÞÞ dt dx4 ðtÞ ¼ k1 ðx3 ðtÞ  x4 ðtÞÞ dt

(11.35)

Let us now compare the three cases: Simeoni model; F-model with more rapidly growing cells more susceptible to the drug; and F-model, where all cell clones are equally susceptible to the drug. We expect that predictions made by the Simeoni model and the F-model with all cells being equally susceptible to the drug give very similar predictions, but in fact, they are radically different, as one can see in Fig. 11.14.

FIG. 11.14 Comparison of the three models of tumor growth, where cells are killed by a cytotoxic drug according to the signal transduction model (11.31): Simeoni model (blue, solid), where F(x1) is given by Eq. (11.30); F-model with faster growing cells more susceptible to the drug (red, dash-dot) as per Eq. (11.34); and F-model, where all cells are equally susceptible to the drug (yellow, dash) as per Eq. (11.35).

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11. From experiment to theory

The reason for this counterintuitive result lies in how the fitness of an evolving population, as defined by difference between birth and death rate, changes over time in these models. In the Simeoni model the birth rate of all cancer cells remains constant, λ0 in the early stage of exponential growth, and λ1 in the later stage of linear growth. Therefore, when the drug is introduced, its effect needs to be greater than this constant growth rate. However, for the F-model, the growth rate increases over time as the population evolves, as reflected by the increasing value of Et[a], which we can compute explicitly; by contrast the value of Et[a] for the Simeoni model remains constant (see Fig. 11.15). The cell kill rate for F-model with equal sensitivity to treatment is k2, while the cell kill rate for F-model with differential sensitivity to the drug is k2  Et[a]. Since Et[a] increases over time, so does the total cell kill rate, making this population more responsive to treatment. Nevertheless, the rate of population evolution in this case outpaces drug efficacy, thereby making it less efficacious over time. This is consistent with many observations of cancer treatment, where one has to “run twice as fast just to stay in the same place” (Lewis Carroll, “Alice in Wonderland”), often requiring increased doses or alternative treatments just to keep up with the evolving tumor. Of course, these models are still simplifications of very complex processes of tumor growth, although they are based on some realistic assumptions. Experiments need to be run to determine when each model may be more applicable in different cases—for instance,

FIG. 11.15

Growth rates for the three models as they change over time, for untreated (left) and treated (rights) simulated tumors. According to the Simeoni model, the simulated tumor grows either with the rate λ0 (yellow) before the transitional threshold or at a rate λ1 (purple) after the threshold for linear growth. In contrast the growth rate of the F-models changes over time. With no treatment the two F-models are predictably growing at the same rate. With treatment, cells described by F-model are killed at a rate k2  Et[a], while cells with equal susceptibility to the drug are killed at a rate k2; therefore, for the population with equal sensitivity to the drug (red), the treatment is less effective, as reflected by the curves on the left.

11.10 Discussion

221

some cancer cell lines may be more homogeneous compared with others and thus could be described sufficiently well by the Simeoni model, which has been validated and used extensively in the literature. There also exist additional difficulties in estimating the initial distribution of cell clones in the population and identifying the ranges of possible parameter values. Finally, here, we used values of parameters of cell sensitivity to drug (k1 and k2) as reported in the literature. It is possible that methodology for estimating these parameter values for heterogeneous populations will be different, which will present an additional challenge. Here, we explored only the case of increased sensitivity to cytotoxic drug of rapidly dividing cells. After model validation, additional cases should be explored, including the effects of variations in immune cell sensitivity, which can be incorporated in these models as well. This is an exciting field that remains to be validated and explored.

11.10 Discussion Finding an appropriate function that best fits the data may have not only predictive value. It may provide insights into the nature of the population that is growing according to one or another growth law and the conditions under which this growth has occurred. Finding justifications for making these distinctions has been the focus of this work. We have shown here that an inhomogeneous (polymorphic) population composed of different exponentially growing clones can never demonstrate either exponential or logistic growth during the entire time scale, but instead grows faster, “overexponentially.” A homogeneous (monomorphic) population can grow exponentially in the absence of competition; it can grow logistically if there exists a limitation on external resources. If the population is polymorphic and consists of exponential or logistic clones, then the total population size grows faster than exponentially or logistically. Nevertheless, polymorphic population can show exponential or logistic growth if the population consists of clones that grow according to the Gompertz curve. Inhomogeneous populations can demonstrate exponential and logistic growth if the total population size is described by specific inhomogeneous frequency-dependent models (or F-models, where the growth rate of a clone is proportional to its frequency in the total population). If a population grows in accordance with the solution to an F-model, then we can assume that the population growth depends on an external (perhaps, dynamical) resource, which is divided uniformly between all individuals in the population. If the logistic curve fits the data better than Gompertz curve, the population is more likely polymorphic than monomorphic. If one is able to determine if the population is indeed polymorphic, then one can conclude that the population grows dependently on an external (possibly dynamical) resource at all stages of population development, not only when the population size becomes large. Determination of this external factor may be of crucial interest, especially for such populations as tumor cells. In a number of mouse xenograft models, tumor growth curves were reported, which exhibit extended period of near-negligible growth, followed by a sharp exponential-like growth phase (see Fig. 11.4). Such behavior can be captured by inhomogeneous logistic or, more likely, exponential-linear models.

222

11. From experiment to theory

Exponential-linear dynamics may be the evidence that the tumor is inhomogeneous, and the distribution of the clones’ growth rates is close to truncated Malthusian distribution. Even more importantly, this can mean that the population growth depends on an external resource at all stages of population development, such that the resource is distributed uniformly between individuals in the population. One may expect that at this stage the system still has enough external resource for growth and has not yet reached the saturation stage. Adding a saturation stage to the exponential-linear dynamics allows reproducing threestage dynamics, including linear, exponential, and saturation stages, which was observed in viral RNA replication models (Biebricher et al., 1985; Schuster, 2011). A key difference of the proposed model from the models proposed by Schuster et al. (Schuster, 2011) is that this model allows replicating all three dynamical regimes with just one model. The constructed three-stage model is perhaps the simplest one, which allows us to explain the transition from one stage of development to another due solely to internal system dynamics. Hence, if one observes the three-stage growth curve, then one can assume that the system is inhomogeneous and the growth rates of different clones follow the truncated Malthusian distribution. Moreover the population in this case is once again likely to depend on an external, possibly dynamical, resource at all stages of population development, not only during the saturation stage.

11.11 Applications and implications Here, we have extracted several data sets from published literature and compared the data with our models (see Figs. 11.9–11.12). We observed that depending on the data set, different functions fit it better, with logistic model providing better fits in the majority of cases, implying (according to our theory) that the population described is heterogeneous, grows in a frequency-dependent manner, and depends on a uniformly distributed external resource during all stages of growth. In one of the cases, where neither model provided good fit to the data (Fig. 11.9), we were nevertheless able to eliminate Gompertzian growth, suggesting at least that the population is not monomorphic. Analysis and predictions made in this section are based on our theory, summarized in Table 11.1, but they of course require experimental verification. Nevertheless, should this theory prove correct, it can provide invaluable tools for inferring information about the nature of the population, that is, whether it is monomorphic or polymorphic, and the conditions under which the population is evolving, that is, whether it depends on an external resource at all stages of growth. An example of such a population would be hormone-dependent tumors, such as some breast and prostate cancers, among others (Wirapati et al., 2008; Jozwik and Carroll, 2012; Brisken, 2013; Spring et al., 2016). Other examples could be nutrient-related, such as phosphorus (Elser et al., 2007; Kareva, 2013) or glucose and glutamine (Kareva and Hahnfeldt, 2013; Chang et al., 2015; Gillies and Gatenby, 2015; Kareva, 2015). Identification of such resources for each tumor might provide crucial guidance into effective therapeutic avenues, such as with estrogen-dependent breast cancers (Wirapati et al., 2008; Spring et al., 2016).

Appendix: Supplementary material

223

Our theory, if confirmed, can also allow making better predictions about further population growth, since, even if the initial stages of growth look similar, over time the shape of the growth curves varies depending on the model. Such analytical insights can provide an additional biomarker and a predictive tool to complement experimental research.

Appendix: Supplementary material Supplementary material related to this chapter can be found on the accompanying CD or online at https://doi.org/10.1016/B978-0-12-814368-1.00011-4

C H A P T E R

12 Traveling through phase-parameter portrait Abstract There exist numerous parametrically homogeneous models that have been fully analyzed using welldeveloped mathematical tools, such methods as bifurcation theory. While in a parametrically homogeneous system, the behavior of the population is restricted to the specific domain determined by the constant value of the bifurcation parameter, in parametrically heterogeneous systems, where the expected value of a parameter changes over time, we can see it “travel” through the phase parameter portrait. Using the HKV method we can build upon previously analyzed parametrically homogeneous systems to reveal complex, rich, and sometimes unexpected behaviors that can help answer many interesting and important new questions. In this Chapter, we will demonstrate this “travel” through the bifurcation diagram using three examples: consumer-resource models of the tragedy of the commons; resource distribution strategies; and an example of the effect of oncolytic virus therapy on a population of cancer cells.

Numerous parametrically homogeneous models exist that have been fully analyzed using well-developed mathematical tools such as bifurcation theory. In a parametrically homogeneous system, the behavior of a population is restricted to the specific domain defined by the constant value of the bifurcation parameter. However, in a parametrically heterogeneous systems, the expected value of the parameter changes over time, enabling it to “travel” through the phase parameter portrait. Using the HKV method, we can build upon previously analyzed parametrically homogeneous systems to reveal complex, rich, and sometimes unexpected behaviors that can help answer interesting and important new questions. In this Chapter, we will demonstrate precisely how parameters “travel” through the bifurcation diagram using three examples: a model of resource (over)consumption; a model that evaluates resource distribution strategies; and a model that describes the effects of oncolytic virus therapy on a population of cancer cells.

12.1 Introduction Ecological systems are complex and adaptive, composed of diverse agents that are interconnected and interdependent. Heterogeneity within these systems is necessary for evolution and adaptability of system components, thus enabling them to withstand and recover

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00012-6

225

# 2020 Elsevier Inc. All rights reserved.

226

12. Traveling through phase-parameter portrait

from environmental perturbations. However, heterogeneity also makes the appearance and short-term prosperity of overconsumers possible, which can in turn lead to exhaustion of the common resources, known as the tragedy of the commons (Hardin, 1968). If the population’s survival depends on this resource, then the short-term adaptations to increase resource consumption can eventually lead to collapse of the entire population, also known as evolutionary suicide (Parvinen, 2005). Elinor Ostrom has focused on the question of avoiding the tragedy of the commons from the point of view of collective decision making (Ostrom, 2015). She observed that mutually satisfactory and functioning institutions of collective action could be developed in small communities where possible resource overconsumption by individuals was immediately noticed and punished. For example, members of a community living on Spanish huertas enacted a system to ration fresh water, where each user is able to closely watch and punish other members of the community for stealing scarce water through a combination of monetary fines and public humiliation. This practice of inflicting punishment/tax for overconsumption on transgressing members of a community provides a guide to maintain scare resources by effectively selecting against overconsumers. Another example Ostrom cites involves a successful adaptive governance system of several rural villages in Japan. These communities have been successfully managing about 3 million hectares of forests and mountain meadows for centuries in large part because “accounts were kept about who contributed to what to make sure no household evaded its responsibilities unnoticed. Only illness, family tragedy, or the absence of able-bodied adults… were recognized as excuses for getting out of collective labor… But if there was no acceptable excuse, punishment was in order.” Another approach communities could employ to avoid resource overconsumption involves relying on carrot rather than the stick, that is, through bestowing reward/subsidy to underconsumers. In these cultures, the community introduces some kind of “social currency” that rewards cooperating individuals with social status (Vollan and Ostrom, 2010). Yet in some, and perhaps most, cases communities cannot self-regulate, requiring government intervention. An example occurred recently, which require inordinately large amounts of water, despite a state-wide drought, following the reasoning of “if I don’t use it, my competitor will” (it is also a classic case of a game theoretical problem of selection of cooperation, which will be covered in Chapters 13 and 14). Only government regulations were able to begin addressing the increasing water crisis (Frank, 2015; Niles and Wagner, 2017). However, sometimes the need to implement interventions is unclear until it is too late. This can be addressed if transitional regimes preceding a critical situation can be identified. In the next example, we describe a mathematical model of consumer-resource interactions that demonstrate how classical tools of dynamical system analysis, combined with the HKV method can be used to identify transitional regimes precluding system collapse, and help evaluate intervention strategies.

Example 12.1. Sustainability: Using a parametrically heterogeneous model to investigate resource depletion, transitional regimes, and intervention strategies In this first example, we focus on a consumer-resource model, where the resource is critical for survival of the population of consumers. Let us start with a model of a homogeneous

Example 12.1. Sustainability: Using a parametrically heterogeneous model to investigate resource depletion

227

population of consumers that was initially proposed in Krakauer et al. (2008) in the context of niche construction, and was later expanded in Kareva et al. (2012). This model contains two coupled differential equations written as follows: 1 0 C B C B C B B dN ðtÞ N ðtÞ C C B ¼ rNðtÞ B c  C |{z} C B |ffl ffl {zffl ffl } dt kz ð t Þ |fflffl{zfflffl} Bconsumption |ffl{zffl} C C population B consumers dynamic A growth rate @ carrying capacity

dzðtÞ ¼ dt |ffl{zffl} shared resource

γ  δzðtÞ |fflfflfflfflffl{zfflfflfflfflffl} natural resource turonver

N ðt Þð1  c Þ : + e zðtÞ + N ðtÞ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

(12.1)

change in resource caused by consumers ðdepletion if c>1, restoration if c 1 results in resource depletion, while c < 1 results in its restoration. Resource z(t) also has a natural turnover rate, which can allow for sustainable coexistence of consumers with the resource. However, since increase in proliferation rate of consumers with a larger value of c results in increased consumption of the resource, it will lead to depletion of shared resources and eventual collapse of the population that depends on it. Several questions can be asked of this model, such as: (1) How will such a system behave when the number of overconsumers in it changes? What are the possible dynamical regimes that such a system can realize as it approaches resource exhaustion and eventual population collapse? (2) Can we identify transitional regimes that can serve as warning signals of approaching collapse? (3) What, if any, intervention measures can be implemented to prevent the tragedy of the commons? Answering these questions requires a combination of both classical bifurcation analysis, and of the HKV method, which allows visualizing a system’s evolutionary trajectories.

Question 1. How will such a system behave when the number of overconsumers in it changes? This question can be answered by conducting stability and bifurcation analysis, as was done in Kareva et al. (2012). In this work, the authors progressively increased the value of parameter c and observed a series of dynamical regimes ranging from sustainable coexistence with the common resource (with ever-decreasing domain of attraction as the value of c

228

12. Traveling through phase-parameter portrait

FIG. 12.1 Bifurcation diagram of System (12.1) in the (γ, c) and (N, z) phase-parameter spaces for fixed positive parameters e and δ. The nontrivial equilibrium point A is globally stable in Domain 1; it shares basins of attraction with equilibrium O in Domains 2 and 3. The separatrix of O and the unstable limit cycle that contains point A, serve, correspondingly, as the boundaries of the basins of attraction. Only equilibrium O is globally stable in Domains 4, which p also ffiffiffiffiffi contains an unstable nontrivial A, and 5, which contains the elliptic sector. Domain 6 exists only for δ > 5 + 24, where the stable limit cycle that is in turn contained inside an unstable limit cycle, shares basins of attraction with equilibrium O. Boundaries between Domains K, S, H, Nul, C correspond to appearance of an attractive sector in a neighborhood of O, appearance of unstable limit cycle containing A, change of stability of equilibrium A via Hopf bifurcations, disappearance of positive A, and a saddle-node bifurcation of limit cycles, respectively. Adapted from Fig. 4 in Kareva, I., Berezovskaya, F., Castillo-Chavez, C., 2012. Transitional regimes as early warning signals in resource dependent competition models. Math. Biosci. 240, 114–123.

increased), to sustained oscillatory regimes, to population collapse due to depletion of the common resource. The results of this analysis are summarized in Fig. 12.1. In Domain 1, when the value of c is small, the shared carrying capacity remains large, allowing stable long-term coexistence of the population with its resource. In Domain 2, as the value of c is increased, a parabolic sector appears near the origin, which results in the decrease of the domain of attraction of the nontrivial equilibrium point A. The population can still sustainably coexist with the resource even with moderate levels of overconsumption, but the range of initial conditions where such coexistence is possible, decreases. As the value of c is further increased, an unstable limit cycle appears around point A through a catastrophic Hopf bifurcation in Domain 3, and via “generalized” Hopf bifurcation in Domain 6, further shrinking the domain where sustainable coexistence of resource and consumers is possible. Finally, in Domains 4 and 5, population extinction is inevitable due to extremely high overconsumption rates that the resource cannot support and from which it cannot recover.

Question 2. Can we identify transitional regimes that can serve as warning signals of approaching collapse? Answering this question requires the ability to visualize evolutionary trajectories as the population evolves toward higher levels of resource (over)consumption. This can be achieved

Example 12.1. Sustainability: Using a parametrically heterogeneous model to investigate resource depletion

229

using the HKV method. In the parametrically homogeneous case analyzed in Kareva et al. (2012), parameter value c is always a constant, and so the population dynamics are always restricted to a single domain of Fig. 12.1 that corresponds to that fixed value of parameter c. To introduce population heterogeneity, we assume that each consumer is characterized by their own value of parameter c (rather than all individuals having the same average value of c). Each consumer is now denoted as x(c, t), and so System (12.1) can now be rewritten as follows: 1 0 C B C B C B B dxðc, tÞ N ðt Þ C C B ¼ rxðc, tÞ B c  C |{z} C B |fflfflffl{zfflfflffl} dt kz ð t Þ |fflfflffl{zfflfflffl} Bconsumption |ffl{zffl} C C population B consumers dynamic A growth rate @ carrying capacity

dzðtÞ ¼ dt |ffl{zffl} shared resource

γ  δzðtÞ |fflfflfflfflffl{zfflfflfflfflffl} natural resource turonver

N ðtÞð1  cÞ + e , zðtÞ + N ðtÞ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

(12.2)

change in resource caused by consumers ðdepletion if c>1, restoration if c1, restoration if c > > < dt ¼ rxðα, tÞ α c1  kzðtÞ + ð1  αÞ NðtÞ + zðtÞ  ϕ , (12.8)     N ðt Þ > > > dz ¼ γ  δzðtÞ + e Et ½αð1  c1 Þ + 1  Et ½α ð1  c2 Þ : : dt NðtÞ + zðtÞ The last equation in System (12.8) can be deduced from the last equation of System (12.7) by integrating over α; here we denote to Et[α] as the mean value of α over the current distribution of clones, Pðα, tÞ ¼ xNðαð,tÞtÞ. If the selective pressures acting upon the population are sufficiently strong, then we should expect to see the distribution of clones P(α, t) change over time, and the question of what the final distribution will be, or what the transitional regimes that can be observed as the system stabilizes, do not have an intuitive and predictable answer. This is due to the fact that the state of the environment in which the population evolves is determined not only by the amount of resources that the individuals have to compete for, but also by the population composition and individuals themselves. Consequently, different types of clones can impose different selective pressures on each other, further affecting overall system dynamics over time. These effects of intrapopulation selective pressures on system demonstrate why dynamics cannot be captured without taking into account population heterogeneity. The proportion of each type of clone within the population can be tracked through the expected value of the

238

12. Traveling through phase-parameter portrait

parameter α, which in the homogeneous system was just a constant, but in a parametrically heterogeneous system is a function of time. Introduce auxiliary variables q(t) and g(t) such that 8 dqðtÞ zðtÞ > > > < dt ¼ zðtÞ + N ðtÞ , (12.9) > dg bN ðtÞ > > ¼ : : dt kzðtÞ Then the equation for the rate of change of the frequency of each clone can be written as      dxðα, tÞ dgðtÞ dgðtÞ ¼ rxðα, tÞ α c1  ϕ : (12.10) + ð1  α Þ c 2 dt dt dt The solution to Eq. (12.10) is xðα, tÞ ¼ xðα, 0Þexp ½rððαc1  ð1  αÞϕÞt + ð1  αÞc2 qðtÞ  αgðtÞÞ:

(12.11)

Then the total population size N(t) becomes: ð1

ð1

0

0

N ðtÞ ¼ xðα, tÞdα ¼ N ð0Þ erðc2 qðtÞϕtÞ  erαððc1 + ϕÞtc2 qðtÞgðtÞÞ Pð0, αÞdα

(12.12)

¼ N ð0Þerðc2 qðtÞϕtÞ  M0 ½rððc1 + ϕÞt  c2 qðtÞ  gðtÞÞ, where N(0) is initial population size, and M0 is the mgf of the initial distribution P(0, α). The distribution of clones over time is given by Pðt, αÞ ¼

xðα, tÞ eαΩðtÞ ¼ , N ðtÞ M0 ½ΩðtÞ

(12.13)

where Ω ¼ r((c1 + ϕ)t  c2q(t)  g(t)). The mean value of α at time t is ð1

ð1

E ½α ¼ αPðt, αÞdα ¼ Pð0, αÞ t

0

0

αeαΩ dM0 ðΩÞ =M0 ðΩÞ: dα ¼ dt M0 ðΩÞ

(12.14)

Putting together all these expressions, we obtain the following system of equations: 8  t    N ðt Þ dz t > > > dt ¼ p  dzðtÞ + re E ½αð1  c1 Þ + 1  E ½α ð1  c2 Þ zðtÞ + N ðtÞ , > > > > < dq zðtÞ ¼ , (12.15) > dt zðtÞ + N ðtÞ > > > > > dg bN ðtÞ > : ¼ , dt kzðtÞ

Example 12.2. Natural selection in resource allocation strategies

239

where N(t) is defined in Eq. (12.12) and Et[α] is the mean value of the parameter α, which can be calculated from Eq. (12.14). The auxiliary variables q(t) and g(t), which were not present in the original model, are the “keystone” quantities that govern the system dynamics and determine all of its statistical characteristics. The resulting parametrically heterogeneous system allows us to explore the changes in predicted evolutionary trajectories depending on the initial composition of the population with respect to different strategies. The results of these simulations reveal that the direction of population evolution in this system is extremely sensitive to initial population composition (see Fig. 12.6). This suggests that even though one strategy might be preferable for a parametrically homogeneous population, in a parametrically heterogeneous case the direction of the evolutionary trajectory is determined primarily by the initial distribution of clones with the population. This can be interpreted as “founder effect,” when the initial composition of the small population determines the subsequent evolutionary trajectory of the population over time (Lambrinos, 2004). Furthermore, the mean value of α does not always reach an equilibrium value, i.e., the population does not always tend toward some fixed strategy, whether pure or mixed. Starting in Domain 6, we observe that not only do the population size and amount of extrinsic resources begin oscillating (which corresponds to the system ‘entering’ the domain of attraction of the stable limit cycle) but so does the Et[α]. In this case, as the system evolves, no final distribution of clones becomes fixed over time (Fig. 12.7). This suggests that the standard approach using a fixed value of the parameter α (or its mean value for a heterogeneous population) can yield incorrect predictions within the domain of model parameters and, hence, is not justified in the general case.

FIG. 12.6 The effects of difference in the initial composition of the population with respect to different strategies. Different initial distributions were chosen to be (A) uniform initial distribution (B) truncated exponential initial distribution with parameter μ ¼ 1.1 (note: population crashes after time t ¼ 32) and (C) truncated exponential initial distribution with parameter μ ¼ 10.1. Initial conditions are such as to fall within Domain 1. All parameters held constant at r ¼ 1, e ¼ 1, b ¼ 1, k ¼ 1, N0 ¼ 0.1, c2 ¼ 0.2, c1 ¼ 0.6, d ¼ 1, p ¼ 1, ϕ ¼ 0.14. One can see that the initial composition of the population can have dramatic effects on the direction in which the population will evolve over time. The values of μ were chosen arbitrarily for illustrative purposes. Adapted from Fig. 6 in Kareva, I., Berezovkaya, F., Karev, G., 2013a. Mixed strategies and natural selection in resource allocation. Math. Biosci. Eng. 10, 1561–1586.

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FIG. 12.7 Trajectories and distribution of clones throughout the population. In this case, one not only observes stable oscillatory behavior in the amount of resource and total population size but also a shift between the two strategies. That is, the population evolves not toward eventual dominance of just one pure strategy but shifts between two strategies. Initial distribution is uniform. Initial conditions are such as to fall within Domain 6. Parameters are r ¼ 1, e ¼ 1, b ¼ 0.9, k ¼ 1, ϕ ¼ 1.2, N0 ¼ 0.1, c2 ¼ 8.75, c1 ¼ 9, d ¼ 24, p ¼ 7.72. Adapted from Fig. 4 in Kareva, I., Berezovkaya, F., Karev, G., 2013a. Mixed strategies and natural selection in resource allocation. Math. Biosci. Eng. 10, 1561–1586.

This model demonstrates clearly that there is in fact no “optimal” resource allocation strategy even for a specific resource. Starting with different initial distributions even within the same domain on the phase-parameter space can lead to different system behaviors and different strategies being favored by natural selection in the long run. We could observe it by calculating numerical solutions of the system, with uniform initial distribution of strategies within the population, and truncated exponential initial distribution (parameter α bounded on the interval [0,1]) with different parameters of the distribution. As one can see in Figs. 12.6 and 12.7, even when everything else is equal, the direction in which the system evolves depends greatly on the initial distribution. Therefore, when one is trying to predict the direction in which a system will evolve, simply knowing the rules that govern its dynamics might not be enough to make an accurate prediction. Rather, one also needs to know the composition of the population that is playing by these rules. This is true even in the case of perfect information, i.e., when every individual in the population knows the rules and plays to maximize his or her own fitness.

Example 12.3 Cancer and oncolytic viruses Here we discuss a perfect example of an organism engaging in evolutionary suicide— cancer: by overconsuming the resources that are shared with normal cells, a tumor kills its host, thereby eventually killing itself. Cancer cells are genetically heterogeneous, and the

241

Example 12.3 Cancer and oncolytic viruses

dynamics of the tumor-host (consumer-resource) system, which entails coevolution under the selective pressure that is imposed by host environment is highly complex and nonlinear. Thus, to precisely define the conditions for successful therapy, mathematical models are needed. In this final example, we look at a model proposed by (Karev et al., 2006; Novozhilov et al., 2006), which describes the dynamics of cancer cells that can be infected by an oncolytic virus, i.e., a virus that can specifically infect and kill cancer cells but leave normal cells unharmed (Bartlett et al., 2013; Chiocca and Rabkin, 2014; Kaufman et al., 2015). The proposed model considers two types of cancer cells, infected and uninfected, growing in a logistic fashion. The system is described by the following two equations:   dX X+Y βXY ¼ r1 X 1   dt K X +Y |{z} |fflffl{zffl ffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} uninfected cancer cells

logistic growth to shared

rate of virus

carrying capacity K

transmission

dY dt |{z}

  X+Y ¼ r2 Y 1  + K |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

βXY X +Y |fflffl{zffl ffl}

infected cancer cells

logistic growth to shared carrying capacity K

rate of virus transmission



δY , |{z}

(12.16)

death of infected cancer cells

where X is the size of the uninfected cancer cell population; Y is the size of the infected cancer cell population; r1 and r2 are the maximum per capita growth rates of uninfected and infected cells, respectively; K is the carrying capacity; β is the transmission coefficient, and δ is the rate of virally induced death rate of infected cells. All the parameters of the model are assumed to be non-negative. Model (12.16) is subject to initial conditions X (0) ¼ X0 > 0 and Y(0) ¼ Y0 > 0. The concentration of viral particles is not explicitly included; it is assumed that virus abundance is proportional to infected cell abundance (Nowak and May, 2000). The following questions can be asked and answered by this model: (1) What are the transitional regimes that occur as the cancer cell population gains resistance to the virus? a. Can we use the model to infer dynamics of evolution of resistance? (2) Why are cytotoxic therapies effective in some patients and not others? Similar to the previous cases, both classic bifurcation analysis and HKV method will be necessary to answer these questions.

Question 1. What are the transitional regimes that occur as the cancer cell population gains resistance to the virus? Can we use the model to infer dynamics of evolution of resistance? Once again, in order to answer this question, bifurcation analysis needs to be performed. A full bifurcation diagram can give a sense of what transitional regimes a population goes

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through as it moves from the area of phase-parameter space of tumor elimination to that of tumor growth. Rescaling model (12.16) by letting t ! r1 t,XðtÞ ! XKðtÞ , YðtÞ ! YKðtÞ , leads to the system dX βXY ¼ Xð1  ðX + YÞÞ  , dt X+Y dY βXY ¼ γYð1  ðX + YÞÞ +  δY, dt X+Y

(12.17)

where β ¼ b/r1, γ ¼ r2/r1, and δ ¼ a/r1. In model (12.17) nondimensional parameters and scaled sizes of cell populations are used, so that X + Y  1 for any t. The complete phase-parameter portrait of System (12.17), constructed in Novozhilov et al. (2006) is shown in Fig. 12.8. The phase-parameter portrait of System (12.17) exhibits all possible outcomes of oncolytic virus infection, i.e., no effect on the tumor (domains I and II in Fig. 12.8A), stabilization or reduction of the tumor load (domains IV and V), and complete elimination of the tumor (domain VIII). Moreover, there are two domains (domains III and VII), where the final outcome crucially depends on the initial conditions and can result either in failure of the virus therapy, or in stabilization (domain III) or even elimination (domain VII) of the tumor. The full bifurcation diagram answers the first part of the first question by providing a full description of the transitional regimes that are possible in this model. However, like most population models, System (12.16) assumes population homogeneity, i.e., that all call have identical attributes, which is not biologically accurate. Two parameters of particular interest in this model that can help answer questions about evolution of resistance are transmission coefficient β, and parameter of cytotoxicity δ.

FIG. 12.8 Phase-parameter portrait of System (12.17) given as a cut of the positive parameter space (γ, β, δ) for an arbitrary fixed value of 0 < γ < 1 (A) and 1 < γ (B). The boundaries of the domains are α1 ¼ (δ, β, γ) : δ  β ¼ 0, α2 ¼ (δ, β, γ) : γ  δ ¼ 0, α3 ¼ (δ, β, γ) : γβ  δ ¼ 0, and α4 ¼ (δ, β, γ) : γ  δ  1 + β ¼ 0. Adapted from Novozhilov, A.S., Berezovskaya, F.S., Koonin, E.V., Karev, G.P., 2006. Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models. Biol. Direct 1, 6.

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Example 12.3 Cancer and oncolytic viruses

Question 2. Why are cytotoxic therapies effective in some patients and not others?

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The answer to this question came from simulations, which showed that initial composition of a population may be one of the factors underlying emergence of resistance in some tumors. Specifically, Figs. 12.9–12.13 show that differences in variance of the initial distribution of cell clones within the population can lead to qualitatively different final outcomes of oncolytic virus therapy. These results may shed some light on the question of variability in therapeutic successes for other interventions, a topic that is of vital importance.

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FIG. 12.9 Solutions of System (12.20)–(12.21) with Gamma-distributed parameter β on [1.5, ∞]. Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green, and black, respectively. The initial conditions are X(0) ¼ 0.5, Y(0) ¼ 0.1, parameter values γ ¼ 1, δ ¼ 2. The initial mean of the parameter is E0[β ¼ 2.5], and the initial variances are 0.06 (A), 0.1 (B), 0.3 (C), 0.4 (D). Simulations were conducted by Artem Novozhilov. Adapted from Karev, G.P., Novozhilov, A.S., Koonin, E.V., 2006. Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics. Biol. Direct 1, 30.

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FIG. 12.10 Et(β) versus time. The initial conditions are X(0) ¼ 0.5, Y(0) ¼ 0.1, parameter values γ ¼ 1, δ ¼ 2. The initial mean of the parameter is E0[β ¼ 2.5], and the initial variances are 0.06 (top), 0.1 (second from top), 0.3 (third from top), 0.4 (bottom). Simulations were conducted by Artem Novozhilov.

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Solutions of System (12.20)–(12.21) with Gamma distributed parameter β on [0.1, ∞]. Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green, and black, respectively. The initial conditions are X(0) ¼ 0.5, Y(0) ¼ 0.1, parameter values γ ¼ 0.5, δ ¼ 0.3. The initial mean is E0[β] ¼ 3.5, the initial variances are 1.5 (A) and 0.5 (B). Simulations were conducted by Artem Novozhilov. Adapted from Karev, G.P., Novozhilov, A.S., Koonin, E.V., 2006. Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics. Biol. Direct 1, 30.

FIG. 12.11

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Example 12.3 Cancer and oncolytic viruses 11 1 10 0.9 9 0.8 8 7 0.6

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FIG. 12.12 Solutions of System (12.24)–(12.25) with Gamma-distributed parameters β on [1, ∞) and δ on [1, ∞). Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green and black, respectively. The initial means of the distributions are E0[β] ¼ 11 and E0[δ] ¼ 10, initial variances are σ β 2 ð0Þ ¼ 0 and σ δ 2 ð0Þ ¼ 0:5. The initial conditions are X(0) ¼ 0.5, Y(0) ¼ 0.1, and γ ¼ 1. (A) Change in tumor load over time (B) Change in mean values of parameters β and δ with respect to each other (compare with Fig. 12.8). Simulations were conducted by Artem Novozhilov. Adapted from Karev, G.P., Novozhilov, A.S., Koonin, E.V., 2006. Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics. Biol. Direct 1, 30. 11

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Solutions of System (12.24)–(12.25) with Gamma-distributed parameters β on [2, ∞) and δ on [0.9, ∞). Uninfected cells, X(t), infected cells, Y(t), and the total tumor load, X(t) + Y(t), are shown in blue, green, and black, respectively. The initial means of distributions are E0[β] ¼ 11 and E0[δ] ¼ 10, initial variances are σ β 2 ð0Þ ¼ 8 and σ δ 2 ð0Þ ¼ 0:5. The initial conditions X(0) ¼ 0.5, Y(0) ¼ 0.1, γ ¼ 0.7. (A) The change in tumor load over time. (B) Change in mean values of parameters β and δ with respect to each other. Simulations were conducted by Artem Novozhilov. Adapted from Karev, G.P., Novozhilov, A.S., Koonin, E.V., 2006. Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics. Biol. Direct 1, 30.

FIG. 12.13

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12. Traveling through phase-parameter portrait

12.2 Distributed susceptibility Assume that the transmission coefficient β is distributed within the population of uninfected tumor cells. Assume also that uninfected cells with a particular value of β beget daughter cells with the same parameter value. Denote B to be the set of all possible values of β, and x(t, Ð β) to be the density of uninfected cells that have a given value of β at moment t. Let X(t) ¼ Βx(t, β)dβ be the size of the uninfected cancer cell population, and p(t, β) ¼ x(t, β)/X(t) be the pdf of the distribution of β at time t. Then the inhomogeneous version of model (12.17) with distributed parameter β becomes dxðt, βÞ βxðt, βÞYðtÞ ¼ xðt, βÞ½1  ðXðtÞ + YðtÞÞ  , dt XðtÞ + YðtÞ dYðtÞ Et ðβÞXðtÞYðtÞ ¼ γYðtÞ½1  ðXðtÞ + YðtÞÞ +  δYðtÞ, dt XðtÞ + YðtÞ

(12.18)

where Et[β] is the mean value of β at time t. The initial conditions are xð0, βÞ ¼ X0 pð0, βÞ, Yð0Þ ¼ Y0

(12.19)

Integrating the first equation in (12.18) over β results in the following system dX XY ¼ X ð 1  ð X + Y Þ Þ  Et ð β Þ , dt X+Y dY XY ¼ γYð1  ðX + YÞÞ + Et ðβÞ  δY: dt X+Y

(12.20)

In order to solve this system, it is necessary to know the explicit expression for Et[β]. Introduce an auxiliary keystone variable where q(t): dq Y ¼ , qð0Þ ¼ 0: dt X+Y

(12.21)

Let Mβ[λ] be the mgf of the initial distribution p(0, β). Then

1 dMβ ½λ

: Et ðβÞ ¼ Mβ ½qðtÞ dλ λ¼qðtÞ Notice that System (12.20) differs from (12.17) in that the fixed value of parameter β is replaced by its current mean value Et[β] in the distributed model, which depends on system dynamics and the population sizes (see Eq. (12.21) and earlier). According to the Price equation, applied to the first equation of System (12.18),   dEt ðβÞ βYðtÞ YðtÞ t ¼ Cov β, 1  ðXðtÞ + YðtÞÞ  0, and the population of uninfected cells is inhomogeneous, i.e., Vart[β] > 0. The inequality shows the direction of selection: the cells with lower parameter β values are selected for. The bifurcation diagram of the homogeneous model (12.16) (Fig. 12.8) allows one to identify transitional behavior of the solutions of Eq. (12.20) as Et[β] changes over time. Next, it was assumed that β is Gamma-distributed on [η, ∞) with positive parameters k, s, η > 0. The choice of the Gamma distribution can be justified because it suitably approximates most unimodal distributions concentrated on the positive half-line. It can also be shown that at any time Et ½β ¼ η + ðsqkðtÞÞ, so Et[β] decreases monotonically over time and Et[β] ! η as t ! ∞ (because q(t) !  ∞, see Eq. (12.21)). Hence, as the system evolves, over time the parametric point with coordinates (γ, δ, Et[β]) travels in the parameter space of model (12.17) along the line connecting points (γ, δ, Et[β]) and (γ, δ, η). Results of several numerical simulations of System (12.20)–(12.21) are shown in Fig. 12.9. Parameter values were chosen to start in domain VIII (Fig. 12.8A) (eradication of the tumor), cross domain VII (bistable situation), and end up in domain I (no effect of virus therapy). The solutions shown in Fig. 12.9 show that the degree of heterogeneity plays an important role in the model dynamics. Parameter values and initial conditions are the same for all four simulations; the difference comes from different initial variances of β: the greater the initial variance, the faster the unfavorable domain I is reached. Conversely, the initial variance of the distribution can be small enough, so the time during which the size of the tumor remains negligible (X + Y is close to zero) is greater than the patient’s life span. These results emphasize the need to know not only the final value of Et[β] but also the transient behaviors. Fig. 12.9 shows that in a heterogeneous population one can recapitulate the phenomenon of tumor recurrence after a relatively long period of what appears to be controlled tumor burden, a behavior that could not be observed in the original homogeneous model (12.17). This phenomenon can be interpreted as tumor dormancy, a topic that was briefly discussed in Chapter 2. In this model, tumor reappearance is likely the outcome of the following mechanism: after the virus kills off the susceptible tumor cells and is cleared (given that it cannot infect healthy tissues), resistant cancer cells can develop into a new tumor This example shows that heterogeneous model (12.18), has both inherited dynamical regimes from model (12.17) and revealed new regimes that are driven by population heterogeneity. The change of the mean parameter value for the cases presented in Fig. 12.9 is shown in Fig. 12.10. Some additional trajectories that can correspond to the phenomenon of tumor dormancy driven by population heterogeneity can be seen in Fig. 12.11. Other possible cases can be similarly analyzed; for example, if parameters are chosen such that the starting point belongs to domain IV (Fig. 12.8A) and the final point belongs to domain II, it is not difficult to predict that, first, there will be a short period of time when the tumor load remains constant, followed by a period of linear growth (when Et[β] belongs to domain V), until finally the tumor grows logistically in domain II; a detailed discussion of the possible mechanisms underlying various growth laws, including logistic and linear, can be found in Chapter 11.

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12. Traveling through phase-parameter portrait

12.3 Distributed susceptibility and distributed cytotoxicity Together with differential susceptibility considered in the previous section, parameter δ (the rate of killing of infected cells) can also be assumed vary between individual infected cells. Such heterogeneity can be attributed either to higher intrinsic death rate of cells that are susceptible to infection by the virus, or perhaps it is virus strains that vary in their ability to kill cells. The latter assumption of viral heterogeneity can transform model equations, which, assuming that δ takes values from set Δ, and y(t, δ) is the density of infected cells with parameter value δ, take the form: ∂xðt, βÞ βxðt, βÞYðtÞ ¼ xðt, βÞ½1  ðXðtÞ + YðtÞÞ  , ∂t XðtÞ + YðtÞ ∂yðt, δÞ Et ðβÞXðtÞyðt, δÞ ¼ γyðt, δÞ½1  ðXðtÞ + YðtÞÞ +  δyðt, δÞ, ∂t XðtÞ + YðtÞ

(12.22)

where the initial conditions are xð0, βÞ ¼ x0 ðβÞ ¼ X0 p1 ð0, βÞ, yð0, δÞ ¼ yðδÞ ¼ Y0 p2 ð0, δÞ:

(12.23)

Integrating System (12.22)–(12.23) over parameters β, δ we get the following system of ODEs: dX XY ¼ Xð1  ðX + YÞÞ  Et ðβÞ , dt X+Y dY XY ¼ γYð1  ðX + YÞÞ + Et ðβÞ  Et ðδÞY, dt X+Y Xð0Þ ¼ X0 , Yð0Þ ¼ Y0

(12.24)

where the mean parameter values are



1 dMβ ½λ

1 dMδ ½λ

t E ½β  ¼ , E ðδ Þ ¼ , Mβ ½q1 ðtÞ dλ λ¼q1 ðtÞ Mδ ½t dλ λ¼t t

for the given mgf of p1(0, β) and p2(0, δ), and the auxiliary variable becomes dq1 Y , q1 ð0Þ ¼ 0 ¼ X+Y dt

(12.25)

The transitional behavior of cell populations (prior to reaching the steady state—if it is ever reached) can be quite complex. This can be seen in Fig. 12.11, which describes solutions of Eqs. (12.24)–(12.25) in Fig. 12.12A, and the path of the mean parameter values in Fig. 12.12B. Due to the structure of the bifurcation diagram of System (12.17), the mean parameter values can pass through domains of qualitatively different dynamical behavior many times, thus resulting in a complex and unpredictable trajectory. Moreover, even if the initial and final parameter points belong to the same domain, during transitional phases the mean parameter values can “visit” other domains (Fig. 12.13).

12.4 Conclusions

249

In Fig. 12.13, the initial and final parameter values belong to domain VIII (Fig. 12.8), where complete eradication of the tumor cells occurs. On its way to extinction, however, the tumor load behaves in an irregular, complex way. This example shows the possibility that complex and erratic observational data can be explained not only by random effects and noise but also by the innate heterogeneity of the cell and virus populations. In conclusion, the model in this example demonstrated that qualitatively distinct behaviors that are absent in the homogeneous model can appear as natural consequences of tumor and oncolytic virus heterogeneity. For example, the model shows that time to tumor recurrence can depend on the heterogeneity of the transmission coefficient (Fig. 12.9), and that the effects of tumor cell heterogeneity are not limited to trivial resistance of a subpopulation of cells to the virus, i.e., recurrence might ensue even when all cells have a nonzero probability of infection. Thus, interpretation of the results of anticancer therapy should take into account the possibility that irregular, quasichaotic behavior can be caused not only by random fluctuations but also by the heterogeneity of the tumor and the virus (Fig. 12.13). Additional results can be found in Karev et al. (2006).

12.4 Conclusions Classic techniques for analysis of dynamical systems can provide critical insights into the possible dynamical regimes that a system can realize. Unfortunately, doing full bifurcation analysis is labor intensive and is not always possible due to increasing complexities of many models. However, there already exists a very rich body of literature of fully analyzed parametrically homogeneous models in many fields, including ecology (Bazykin, 1998; Berezovskaya et al., 2005), epidemiology (Brauer et al., 2001; Novozhilov, 2008, 2012), among others. As the examples presented here demonstrate, even relatively simple two-dimensional systems can reveal rich, unexpected, and meaningful behaviors. Application of the HKV method to introduce population heterogeneity in a meaningful way, and utilizing previously performed analysis can reveal a new layer of understanding of many existing models that was not accessible before. This of course is possible only if we ask the right questions. In the next two chapters, we will dive in to the topic of game theory, including a discussion of evolution of cooperation both within heterogeneous populations, and within games, and how new insights on evolution of cooperation can be obtained using the HKV method.

C H A P T E R

13 Evolutionary games: Natural selection of strategies Abstract In this chapter, we model and analyze the process of natural selection between all possible mixed strategies in classical two-player two-strategy games. We derive and solve an equation that is a natural generalization of the Taylor-Jonker replicator equation that describes dynamics of pure strategy frequencies. We then investigate the evolution of not only frequencies of pure strategies but also total distribution of mixed strategies. We show that the process of natural selection of strategies for all games obeys the dynamical principle of minimal information gain (see Chapter 8). We also show a principal difference between mixed-strategy hawk-dove (HD) game and all other 2
2 matrix games (prisoner’s dilemma, harmony and stag hunt games). Mathematically, for HD game the limit distribution of strategies is nonsingular, and the information gain tends to a finite value, in contrast to all other games. Biologically the process of natural selection in the HD game follows nonDarwinian “selection of everybody,” while for all other games we observe Darwinian “selection of the fittest.”

13.1 Problem formulation Mathematical game theory was initially developed for economic and social problems to make predictions about how different behavioral strategies can affect individuals’ behavior through affecting the predicted “payoff” that corresponds to each strategy. Evolutionary game theory (EGT) originated as its application to biological problems, since, per Dobzhansky’s famous dictum, “Nothing in biology makes sense except in the light of evolution” (Dobzhansky, 1973); see also an interesting discussion of this dictum in (Griffiths, 2009). EGT looks at strategies that species or individuals within the species can employ to increase their payoff, which in the case of EGT is their fitness. The first explicit applications of mathematical game theory to evolutionary biology were given in Lewontin (1961) and Maynard Smith and Price (1973) and were later formalized in a seminal monograph by Maynard Smith (1982). EGT further found interesting applications in social sciences, where evolution is not biological but economical or cultural; EGT provides relevant assumptions and mathematical tools for investigation of corresponding problems. Within the framework of EGT, strategies are seen as inherited programs that control the

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00013-8

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13. Strategy selection in evolutionary games

individuals’ behavior. “…The word strategy could be replaced by the word phenotype; for example, a strategy could be the growth form of a plant, or the age at first reproduction, or the relative numbers of sons and daughters produced by a parent” (Maynard Smith, 1982, Chapter 1). Understanding mechanisms underlying strategy selection can allow making predictions about evolutionary trajectories, deepening our understanding of the underlying biology. Let us consider a specific example. Assume a large population of individuals (of the same species) that can adopt two different strategies, which we denote as 0 and 1. A strategy i player receives payoff Gij when playing against a strategy j player; G ¼ (Gij), i, j ¼ 0, 1 is a matrix of expected payoffs. Within the frameworks of EGT, the payoff is considered as an individual’s fitness. In the succeeding text, we will call the 0-strategy players “cooperators” and 1-strategy players “defectors,” keeping in line with the classical “prisoner’s dilemma” and “hawk-dove” games (see Hofbauer and Sigmund, 1998; Broom and Rychta´r, 2013 for details). These games will be discussed in greater detail in the following sections. The central concept suggested in Maynard Smith and Price (1973) for game analysis was the concept of an evolutionarily stable, or uninvadable, strategy (ESS). An ESS is a strategy such that if all members of a population adopt it, then no mutant strategy could invade the population through natural selection. Later a dynamical approach for game analysis known as replicator dynamics was offered by Taylor and Jonker (1978); for details and further development of replicator dynamics, see, for example, Hofbauer and Sigmund (1998) and Webb (2007). In the simplest problem formulation, it is assumed that every individual can adopt only a single pure strategy; the population is divided into two subpopulations with frequencies x(t) and 1  x(t) such that all individuals in the first subpopulation adopt strategy 1 (defectors) and in the second subpopulation they adopt strategy 0 (cooperators). Then the dynamics of the frequency x(t) are described by the Taylor-Jonker replicator equation dx ¼ xð1  xÞððA + BÞx  AÞ, dt

(13.1)

A ¼ G00  G10 , B ¼ G11  G01 :

(13.2)

where

This way the dynamics of strategy frequencies depend on only two quantities, A and B, instead of the original four elements of the payoff matrix. The derivation of the Taylor-Jonker equation can be found in the original paper by Taylor and Jonker (1978) or in the textbook by Hofbauer and Sigmund (1998); we do not reproduce it here because later we will derive a more general equation. Eq. (13.1) always has equilibria x ¼ 0 and x ¼ 1 and may have an additional equilibrium x∗ ¼ A A+ B if 0 < A A+ B < 1. Replicator equation by Taylor-Jonker describes four well-known classes of games depending on the signs of A and B: (1) If A < 0 and B > 0, then x ¼ 1 is a stable equilibrium; the corresponding game is prisoner’s dilemma (PD), where defectors dominate over cooperators. (2) If A > 0 and B < 0, then x ¼ 0 is a stable equilibrium; the corresponding game is harmony (H), where cooperators dominate over defectors. In both cases, there are no other stable states. These two games are identical up to reordering of the strategies.

13.2 The model and the main equations

253

(3) If both A and B are positive, then the game becomes stag hunt (SH), or a coordination game. It is characterized by bistability, where both equilibria x ¼ 0 and x ¼ 1 are stable, and their areas of attraction are divided by the unstable equilibrium x∗ ¼ A A+ B. (4) Finally, if both A and B are negative, then there exists internal (polymorphic) stable equilibrium x∗ ¼ A A+ B, and equilibria x ¼ 0 and x ¼ 1 are unstable; this is the hawk-dove (HD) game, also known as snowdrift or the game of chicken. The main results on stability properties of replicator dynamics can be found in Hofbauer and Sigmund (1998) and Cressman (2013). In addition to the two described classical approaches to analyzing evolutionary games, namely, studying evolutionarily stable strategies and modeling the dynamics of strategy frequencies, we develop here a third approach, which generalizes the second one. Specifically, we model and study the process of natural selection between all possible mixed strategies in the population. This was achieved by deriving and solving an equation that describes the dynamics of the frequencies of mixed strategies and is a natural generalization of the TaylorJonker replicator equation that describes dynamics of only pure strategy frequencies. An investigation of the evolution of total distribution of mixed strategies revealed a key difference between the mixed-strategy hawk-dove (HD) game and other classical games (prisoner’s dilemma, harmony, and stag hunt games). Mathematically, for HD game, the limit distribution of strategies is nonsingular, and the information gain tends to a finite value, in contrast to all other games. Biologically the process of natural selection in HD game follows non-Darwinian “survival of everybody”; for all other games, we observe Darwinian “survival of the fittest” (Karev, 2018). These results were obtained with the help of the HKV method, which was developed in Chapters 2 and 4.

13.2 The model and the main equations In what follows, we assume that every individual not only can adhere to one “pure” strategy but also can adopt each strategy with its own (hereditary) probabilities, that is, keep a mixed strategy. Let l(t, α) be the set of all individuals that adopt strategy 1 (defection) with probability α and strategy 0 (cooperation) with probability 1  α; we refer to the subpopulation l(t, α) as α-clone. Then l(t, 0) consists only of “cooperators,” and l(t, 1) consists only of “defectors”; in the simplest case, the total population consists of these two clones only. In our case, there exists an indefinite number of mixed strategies, which can be adopted by individuals from the population, and each mixed strategy is characterized by its own value of the parameter α, 0  α  1. In what follows, we will denote the density of α-clone also by l(t, α). Our aim is to trace the natural selection of mixed strategies in a process of “honest competition”; this process is described by evolution of distribution of parameter α. To formulate a mathematical model, let us introduce the necessary notation and exact definitions. Let P(0, α) be the pdf of the initial distribution of parameter α. Let S0 be the support of P(0, α), that is, S0 ¼ {α : P(0, α) > 0}. For example, if S0 consists of only two points, S0 ¼ {0, 1}, then the population is composed of only two clones, one of which consists of cooperators and the

254

13. Strategy selection in evolutionary games

other of defectors. If all mixed strategies are possible in the population, then S0 coincides with the unit interval, S0 ¼ [0, 1]; this is the case that is of primary interest to us. In what follows, we always assume that the points α ¼ 0 and α ¼ 1 belong to S0; it means that individuals that adopt pure strategies are present in the population atÐ the start. The total population size at time t is N(t) ¼ 10l(t, α)dα. The distribution of parameter α at time t is defined as Pðt, αÞ ¼

lðt, αÞ : N ðtÞ

The population at time t is composed of defectors D(t) and cooperators C(t), where the numbers of defectors and cooperators are given by ð1 DðtÞ  αlðt, αÞdα, 0

CðtÞ 

ð1

ð1  αÞlðt, αÞdα ¼ N ðtÞ  DðtÞ:

0 DðtÞ Let us denote xðtÞ ¼ N ðtÞ to be the frequency of defectors in the total population and yðtÞ ¼ C ð tÞ to be the frequency of cooperators in the population, so that x + y ¼ 1. Then the frequency N ð tÞ of defectors is determined by the following equation: ð1 DðtÞ αlðt, αÞ ¼ dα ¼ Et ½α, (13.3) xðtÞ ¼ N ðt Þ N ð t Þ 0

where Et[α] is the mean value of parameter α at time t over the probability P(t, α). Following standard assumptions of two-strategy evolutionary game theory, we assume that cooperators have an average number of offspring per individual during the time interval (t, t + Δt) equal to ðG00 yðtÞ + G01 xðtÞÞΔt, and defectors have an average number of offspring equal to ðG10 yðtÞ + G11 xðtÞÞΔt: Consequently, we arrive at the master model dlðt, αÞ ¼ lðt, αÞ½ð1  αÞðG00 yðtÞ + G01 xðtÞÞ + αðG10 yðtÞ + G11 xðtÞÞ: dt

(13.4)

This equation describes the dynamics of clones in a “mixed strategy game”: the outcome of this game is determined by the payoff matrix, initial distribution of parameter α, and by Eq. (13.4). Furthermore, we can assume that each individual in the population has initialfitness F0; a standard assumption is that F0 corresponds to logistic growth, that is, F0 ¼ r 1  N K , where r, K are positive constants. Then dlðt, αÞ ¼ lðt, αÞ½ð1  αÞðG00 yðtÞ + G01 xðtÞÞ + αðG10 yðtÞ + G11 xðtÞÞ + F0 : dt

13.2 The model and the main equations

255

Notice that the initial fitness of F0 affects the dynamics of clone sizes and total population size but does not change the dynamics of frequencies of defectors and cooperators or the distribution of parameter α. Because of this, we focus only on Eq. (13.4). Taking into account that y(t) ¼ 1  x(t), let us rewrite Eq. (13.4) as follows: dlðt, αÞ ¼ lðt, αÞ½αððG10  G00 Þ + ðG11  G01  G10 + G00 ÞxðtÞÞ + G00 + ðG01  G00 ÞxðtÞ dt ¼ lðt, αÞ½αððA + BÞxðtÞ  AÞ + G00 + ðG01  G00 ÞxðtÞ,

(13.5)

where A ¼ G00  G10 and B ¼ G11  G01 (see Eq. 13.2). Denote Fðα, tÞ ¼ αððA + BÞxðtÞ  AÞ + G00 + ðG01  G00 ÞxðtÞ:

(13.6)

dlðt,αÞ ¼ lðt, αÞFðα, tÞ, dt

(13.7)

Then

so F(α, t) is the fitness of individuals of the clone l(t, α). According to the Price covariance equation (discussed in Chapter 5, Example 5.9) applied to Eq. (13.7), dEt ½α ¼ Covt ½α, Fðα, tÞ ¼ Vart ½αððA + BÞxðtÞ  AÞ: dt Taking into account that x(t) ¼ Et[α] (see Eq. 13.3), we proved the following proposition. Proposition 13.1

dxðtÞ ¼ Vart ½αððA + BÞxðtÞ  AÞ: dt

(13.8)

Due to the importance of this proposition, in Section 13.3 we provide a direct proof of Eq. (13.8). It is clear that replicator equation (13.1) is a special case of Eq. (13.8) when Vart[α] ¼ x(1  x), that is, when parameter α is distributed according to the Bernoulli distribution. This means that α can take only two values, 0 and 1, with probabilities P(α ¼ 1) ¼ x and P(α ¼ 0) ¼ 1  x. In this case the total population is composed of only two clones, one of which is characterized by the 1 strategy and has size l(1, t) ¼ x(t)N(t) and another of which is characterized by the 0 strategy and has size l(0, t) ¼ (1  x(t))N(t). Despite Eq. (13.1) being a special case of the replicator equation (13.8), there exists a key difference between these equations. Eq. (13.1) can be solved explicitly using standard methods, while Eq. (13.8) cannot because the current variance Vart[α] is unknown in the general case. One can recognize here a particular case of the well-known problem of “dynamical insufficiency” of the Price equation: it cannot be used alone to determine dynamics of the model over time because to compute the mean value of a trait one needs to know its variance, and to compute the variance one needs to know the third-order moment, etc. (see, e.g., Barton and Turelli, 1987; Frank, 1997). A way to overcome the dynamical insufficiency of the Price equation was suggested in Chapter 5. We use this approach in the next section.

256

13. Strategy selection in evolutionary games

13.3 Solution to the replicator equation The only way to overcome the problem of dynamical insufficiency of Eq. (13.8) is to compute Et[α] independently; in our case, we need to compute l(t, α) as defined by Eq. (13.5). To solve Eq. (13.5), we apply the HKV method described in Chapter 4. Let us formally introduce the auxiliary keystone variable as described by the following equation: dq ¼ xðtÞ, qð0Þ ¼ 0: dt

(13.9)

Then, denoting for brevity zðtÞ ¼ G00 t  ðG01  G00 ÞqðtÞ, wðtÞ ¼ ðA + BÞqðtÞ  At,

(13.10)

we can write the solution to Eq. (13.5) as lðt, αÞ ¼ lð0, αÞeαwðtÞ + zðtÞ :

(13.11)

Let M0 ½δ ¼

ð1

eδα Pð0,αÞdα

0

be the mgf of the initial distribution P(0, α) of parameter α; we assume that this function is known. Then ð1 NðtÞ ¼ lðt, αÞdα ¼ Nð0ÞezðtÞ M0 ½wðtÞ: (13.12) 0

The current distribution of the parameter α is then given by Pðt, αÞ ¼

lðt, αÞ eαwðtÞ ¼ Pð0, αÞ : N ðtÞ M0 ½wðtÞ

(13.13)

Notice, that P(t, α) depends only on w(t) and does not depend on z(t). The current mean value of parameter α is given by Et ðαÞ ¼

ð1 0

αPðt, αÞdα ¼

ð1

αeαwðtÞ dlnM0 ½w ðwðtÞÞ: Pð0, αÞdα ¼ dw M ½ w ð t Þ  0 0

(13.14)

DðtÞ t Taking into account Eq. (13.3), xðtÞ ¼ N ðtÞ ¼ E ðαÞ, we can now write Eq. (13.9) for the auxiliary variable q(t) in a closed form   dq d ¼ lnM0 ðwÞ , qð0Þ ¼ 0: (13.15) dt dw w¼ðA + BÞqðtÞAt

257

13.4 Equilibria of frequencies

Notice that due to Eq. (13.13), the current distribution of α depends only on the variable w(t). According to Eqs. (13.10) and (13.15), dw dq d ¼ ðA + BÞ  A ¼ ðA + BÞ lnM0 ðwÞ  A, wð0Þ ¼ 0: dt dt dw

(13.16)

Eq. (13.16) is the main step in our approach to solving replicator equation (13.8). With its solution, we are able to compute the distribution of mixed strategies at every time moment using an explicit formula (13.13); we can now also compute other characteristics of interest, in particular, the frequencies of cooperators and defectors, using Eq. (13.9). In what follows, it would be useful to notice that, according to Eqs. (13.9) and (13.10), dw ¼ ðA + BÞxðtÞ  A: dt

(13.17)

To avoid trivial cases and technical details, we will assume as a rule that the support of the initial distribution is the entire interval [0, 1], that is, P0(α) > 0 for all 0  α  1. In this case the support of the current distribution P(t, α), according to Eq. (13.13), is also an entire interval [0, 1]; therefore Vart[α] > 0 for all t.

13.4 Equilibria of frequencies The main equation for the dynamics of the frequency x(t) in model (13.4) is Eq. (13.8). Now we are able to derive this equation directly. To this end, let us prove that dxðtÞ dwðtÞ ¼ Vart ½α : dt dt

(13.18)

Indeed, according to Eq. (13.16),

  dxðtÞ d d d d dw ¼ lnM0 ½wðtÞ ¼ lnM0 ½wðtÞ : dt dt dw dw dw dt

Next, ð1

2 ðwðtÞαÞ

  α e Pð0, αÞdαÞ d d lnM0 ½wðtÞ ¼ 0  dw dw M0 ½wðtÞ   2 ¼ Et α2  ðEt ðαÞÞ ¼ Vart ½α:

ð 1

Since dw ¼ ðA + BÞxðtÞ  A, dt we have dx ¼ Vart ½αððA + BÞx  AÞ: dt

ðwðtÞαÞ

αe

2 Pð0, αÞdα

0

M0 ½wðtÞ2

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13. Strategy selection in evolutionary games

dw Corollary 13.1 Eq. (13.18) shows that dx dt and dt are of the same sign, so x(t) monotonically increases or decreases together with w(t). Each equilibrium w∗ of Eq. (13.16) or (13.17) corresponds to the equilibrium value of the frequency x∗. Replicator equation (13.1) allows one to find all equilibrium values of frequencies in twostrategy two-player games; the results are well known and are briefly described in the introduction to this chapter. Eq. (13.8) allows us to prove similar results for model (13.4). Firstly, let us note that Eq. (13.8) always has equilibria x ¼ 0 and x ¼ 1 because in these cases corresponding distributions are concentrated in points α ¼ 0 or α ¼ 1, respectively, and therefore Var[α] ¼ 0. Stability of these equilibria can be checked through standard analysis. Assume that the initial distribution of parameter α is not concentrated in a single point. Then Vart[α] > 0 for all t, and so existence and stability of internal equilibrium 0 < x∗ < 1 are defined by equation (A + B)x∗  A ¼ 0. Standard analysis implies the following:

Theorem 13.1 (1) The frequency x(t) has an equilibrium value 0 < x∗ < 1 if and only if the values A ¼ G00  G10 and B ¼ G11  G01 are nonzero and of the same sign. The equilibrium x∗ ¼ A A+ B is stable if these values are negative and is unstable otherwise. (2) The points x ¼ 0 and x ¼ 1 are always equilibrium values of x(t); x ¼ 0 is a stable state if A > 0 and unstable if A < 0; x ¼ 1 is a stable state if B > 0 and unstable if B < 0. (3) If A ¼ 0, then x(t) monotonically increases if B > 0 and monotonically decreases if B < 0. (4) If B ¼ 0, then x(t) monotonically increases if A < 0 and monotonically decreases if A > 0. (5) If A ¼ B ¼ 0, then x(t) ¼ const for all t and P(t, α) ¼ P0(α) for all t. Overall, all equilibria of general equation (13.8) are identical to corresponding equilibria of the standard equation (13.1), but the dynamics of the frequency x(t) may be different. An example is given below (see Fig. 13.5). Now we are able to study not only the dynamics of frequency x(t) ¼ Et(α) but also the dynamics of the distribution of parameter α. For any specific game, these dynamics are completely determined by the values of A and B and the mgf M0 of the initial distribution of parameter α. Let us now summarize the main formulas, which will be used later to compute the distribution dynamics for different games: Pðt, αÞ ¼ Pð0, αÞ xð t Þ ¼

eαwðtÞ , M0 ½wðtÞ

dlnM0 ½w ðwðtÞÞ, dw

(13.19) (13.20)

where w(t) is the solution to the Cauchy problem dw d ¼ ðA + BÞ lnM0 ðwðtÞÞ  A, wð0Þ ¼ 0: dt dw

(13.21)

Theorem 13.1 together with formula (13.19) allows us to make qualitative conclusions about asymptotical behavior of the distribution of parameter α. Indeed, according to Theorem 13.1, three cases are possible: (1) x(t) ! 0, (2) x(t) ! 1, and (3) x(t) ! x∗, 0 < x∗ < 1 as t ! ∞. In the first two cases, the limiting distributions P(t, α) as t ! ∞ are singular and concentrated in the

13.5 Dynamics of the distribution of strategies

259

points α ¼ 0 and α ¼ 1, respectively, independently of the initial distribution. In the last case, x∗ ¼ A A+ B, and hence there exists a limiting stationary solution w∗ as determined by Eq. (13.17), ∞ < w∗ < ∞. This solution determines the limiting stationary distribution P∗(α) through Eq. (13.19): P∗ ðαÞ ¼ Pð0, αÞ

eαw∗ , M0 ðw∗ Þ

and this limiting distribution crucially depends on the initial distribution P(0, α).

13.5 Dynamics of the distribution of strategies 13.5.1 Replicator equation The process of natural selection of strategies in model (13.4) is captured by the evolution of the distribution of parameter α: Pðt, αÞ ¼ where

lðt, αÞ , N ðt Þ

ð N ðt Þ ¼

lðt, αÞdα A

and (see Eq. 13.7) dlðt, αÞ ¼ lðt, αÞFðt, αÞ: dt It is well known (see, e.g., Hofbauer and Sigmund, 1998; see also Chapter 4, Proposition 4.1) that the pdf P(t, a) solves the replicator equation of the form   dPðt, αÞ ¼ Pðt, αÞ Fðt, αÞ  Et ½Fðt, αÞ : dt According to definition (13.6) of the function F(t, α),   Fðt, αÞ  Et ½Fðt, αÞ ¼ ððA + BÞxðtÞ  AÞ α  Et ½α : Hence, as x(t) ¼ Et[α] (see Eq. 13.3), dPðt,αÞ ¼ Pðt, αÞððA + BÞxðtÞ  AÞðα  xðtÞÞ: dt

(13.22)

The solution to this equation is given by formulas (13.19)–(13.21). Now, to investigate the process of natural selection of strategies, we take the initial distribution of the parameter to be uniform in the interval [0, 1]; in this case, ð1 ðeδ  1Þ : (13.23) M0 ½δ ¼ eδα dα ¼ δ 0

260

13. Strategy selection in evolutionary games

A more general case is the truncated exponential distribution on the interval [0, 1] of the form Pð0, αÞ ¼

s esα , 0  α  1 1  es

(13.24)

1 with the mean value E½α ¼ 1s + 1e s and mgf

  s  eδs  1 M0 ½δ ¼ : 1  es δs

(13.25)

d 1 1 lnM0 ½w ¼ : + sw dw 1e sw

(13.26)



Then

The uniform distribution is a limit case of the truncated exponential distribution as s ! 0. One more representative case is the normal distribution truncated on the interval [0, 1]. The pdf of this distribution depends on two parameters, m and s, and has the form ðαmÞ2

Pð0, αÞ ¼ C e s , 0  α  1 hpffiffiffiffiffi     i mffi pffi p with normalization constant C ¼ 2= πs Erf 1m + Erf . s s The mgf of this distribution is given by the formula 



 sw2 2 + 2m + sw 2m + sw mw + 4 p ffiffi p ffiffi e Erf + Erf 2 s 2 s



 M0 ½w ¼ : 1m m Erf pffiffi + Erf pffiffi s s

(13.27)

(13.28)

Eqs. (13.19) and (13.21) allow us to compute the current distribution of parameter α given any initial distribution and its mgf. In the following sections, we investigate the dynamics of parameter α that describes the process of natural selection of a mixed strategy in different games and for different initial distributions of α. Initial distributions of interest are truncated exponential given by Eqs. (13.24) and (13.25) and truncated normal given by Eqs. (13.27) and (13.28); other distributions can be used in the same way (see Chapter 17, Math Appendix for a list of some of the more commonly used distributions). A good introduction and description of the considered games can be found, for example, in Broom and Rychta´r (2013, Chapter 4) and Gintis (2009, Chapter 3).

13.5.2 Prisoner’s dilemma A large body of literature is devoted to the discussion and analysis of different versions of the game known as the prisoner’s dilemma (PD). In this game the defection strategy gives the player a higher payoff than the cooperation strategy irrespective of which strategy is used by the second player; therefore a rational individual should defect. However, both players would get a higher payoff if they cooperated, which makes this a dilemma.

13.5 Dynamics of the distribution of strategies

261

The PD game not only is widely applied to many social and behavioral problems but also has (perhaps limited) applications to some biological situations (see e.g., Turner and Chao, 1999; Broom and Rychta´r, 2013). An extreme case of the PD game is the well-known “tragedy of the commons,” a situation when the strategy of overexploitation of a shared resource on which the population depends (defecting) rather than preserving it (cooperating) can lead to resource destruction and consequent population collapse (Hardin, 1968); it was covered in more detail in Chapter 12. It is well known that the frequency of defectors tends to one for a two-person two-strategy PD game. This statement is also true for a mixed-strategy PD game. The elements of the payoff matrix of such a PD game satisfy inequalities: G10 > G00 > G11 > G01 , so that A ¼ G00  G10 < 0, B ¼ G11  G01 > 0: Then, according to assertions (1) and (2) of Theorem 13.1, lim t!∞ xðtÞ ¼ 1, and therefore the final distribution of parameter α is concentrated in the point α ¼ 1 independently of the initial distribution. Additionally, we can trace the process of strategy selection described by the evolution of the distribution of parameter α. A standard example of the PD game is defined by the payoff matrix   b  c c GPD ¼ , b > c > 0: b 0 Then A ¼  c, B ¼ c, and dw dt ¼ c (see Eq. 13.21); therefore w(t) ¼ ct, and Pðt, αÞ ¼ Pð0, αÞ

eαct : M0 ðctÞ

We can see that the dynamics of the distribution of parameter α is identical to the dynamics of the distribution of the Malthusian parameter in the simplest inhomogeneous Malthusian t, αÞ model given by equation dlðdt ¼ αclðt, αÞ; see Chapter 2. Consider a numerical example of the mixed-strategy PD game with the payoff matrix   1 1 ; GPD ¼ 2 0 then A ¼ 1, B ¼ 1, w ¼ t, Pðt, αÞ ¼

Pð0, αÞeαt : M 0 ðt Þ

The dynamics of the distribution of parameter α is shown in Fig. 13.1. Fig. 13.2 shows the dynamics of the population with an initial normal truncated distribution of parameter α as per Eq. (13.27) with the mgf given by Eq. (13.28).

13.5.3 Coordination game or SH game The coordination game, also known as the stag hunt (SH) game, is characterized by bistability, when both equilibria x ¼ 0 and x ¼ 1 are stable and there exists an unstable equilibrium 0 < x∗ < 1 that divides the domains of attraction of the stable equilibria. This game is defined by the payoff matrix, whose elements satisfy the condition

262

13. Strategy selection in evolutionary games

40

t

30

20

10 0

30 20

P(t,a) 10

1.0

0.5

0.0

0

a

FIG. 13.1 The dynamics of the distribution of parameter α for the PD game. The initial distribution (blue) is truncated exponential (Eq. 13.24 with s ¼ 10); the final distribution (red) is concentrated at the point α ¼ 1. 60 40

t 20 0 20 15 10 5 1.0

P(t,a)

0

0.5 a 0.0

FIG. 13.2

Dynamics of the population for the PD game with initial truncated normal distribution (blue) on the interval [0,1] (Eq. 13.27) with m ¼ 0.5, s ¼ 0.01, to the final distribution (red), concentrated at the point α ¼ 1.

G00 > G10 , G11 > G01 : In this case, A ¼ G00  G10 > 0, B ¼ G11  G01 > 0, and x∗ ¼

A : A+B

Evolution of the distribution of parameter α crucially depends on the initial mean value E0[α]; if E0[α] < x∗, then the limit distribution is concentrated in the point α ¼ 0, and if E0[α] > x∗, then the limit distribution is concentrated in the point α ¼ 1. Consider the following numerical example. Let A ¼ B ¼ 1; then x∗ ¼ 12. Now assume the initial distribution is truncated exponential on the interval [0,1]. If s ¼ 0.01, then E0[α] ¼ 0.4992, and the distribution of α is concentrated in the point α ¼ 0; if s ¼  0.01, then E0[α] ¼ 0.5008, and the distribution of α is concentrated in the point α ¼ 1 (see Fig. 13.3). As we can see, in this

263

13.5 Dynamics of the distribution of strategies

FIG. 13.3

Evolution of the population for SH game with an initial truncated exponential distribution given by Eq. (13.24). Here the initial conditions are (A) E0[α] ¼ 0.4992 and (B) E0[α] ¼ 0.5008.

100

t 50

0 60 40 P(t,a) 20 0 1.0 0.5 a 0.0

(A) 100 t 50

0 60 40 P(t,a) 20 0 1.0 0.5 a

(B)

0.0

264

13. Strategy selection in evolutionary games

game, a very small difference in the initial distribution can lead to dramatic differences in the outcome of natural selection of strategies. Now let the initial distribution be truncated normal on the interval [0,1]. Let us once again consider the SH game with A ¼ B ¼ 1. The dynamics of the population are shown on Fig. 13.4. The SH game is an example of a situation when the selection of one or another pure strategy crucially depends on the initial composition of the population, and so any random perturbation can have dramatic effects on the predicted outcome. FIG. 13.4 Evolution of the population for SH game with an initial truncated normal distribution on the interval [0,1] given by Eq. (13.27) with m ¼ 0.5. Initial conditions are (A) E0[α] ¼ 0.499 and (B) E0[α] ¼ 0.501. 15

P(t,a)

10

600

5 400

0 0.0

t 200 0.5

a

(A)

0

1.0

15 10

P(t,a)

600

5 400

0 0.0

t 200 0.5

a

(B)

1.0

0

13.6 Natural selection of strategies in a “hawk-dove” game

265

13.6 Natural selection of strategies in a “hawk-dove” game The “hawk-dove” (HD) game was introduced by Maynard Smith and Price (1973) and can be described as follows. Consider individuals within a population competing for resources V. Doves avoid confrontation, while hawks provoke fights. Thus, on average, two doves will share the resources, and the average increase in fitness for doves is V/2. A dove that encounters a hawk will leave all the resources to the hawk. If a hawk meets a hawk, they escalate until one of the two gets knocked out. The winner’s fitness is increased by V, while the loser’s fitness is reduced by C, so that the average increase in fitness is (V  C)/2, which is negative if the cost of the injury exceeds the gain from winning the fight. The importance of the HD game was clearly explained by Webb (2007), Section 8.4: “The biological significance of the hawk-dove game is that it provides an alternative to groupselectionist arguments for the persistence of species whose members have potentially lethal attributes (teeth, horns, etc.). The question to be answered is the following. Because it is obviously advantageous to fight for a resource (having it all is better than sharing), why don’t animals always end up killing (or at least seriously maiming) each other? The groupselectionist answer is that any species following this strategy would die out pretty quickly, so animals hold back from all out contests ‘for the good of the species.’ The ‘problem’ with this is that it seems to require more than just individual-based Natural Selection to be driving Evolution. So, if group selection is the only possible answer, then that would be a very important result. However, the hawk-dove game shows that there is an alternative—one that is based fairly and squarely on the action of Natural Selection on individuals. So, applying Occam’s Razor, there is no need to invoke group selection.” Qualitative behavior of strategies in this game is characterized by the existence of a stable polymorphic state 0 < x∗ < 1. This state exists if the following inequalities hold: G00 < G10 and G11 < G01 ; then A ¼ G00  G10 < 0 and B ¼ G11  G01 < 0: It follows from Proposition 13.1 that in this case, x∗ ¼ A A+ B is the equilibrium frequency of defectors (hawks). This result is well known for the two-strategy HD game. However, the approach developed in previous sections allows us to expand it by tracing the process of natural selection of strategies for a mixed-strategy HD game to obtain some new and interesting results. Notice that in the cases of the PD and SH mixed-strategy games, the final distribution of strategies is always concentrated at the points α ¼ 0 or α ¼ 1, and there are no other stationary distributions; only one pure strategy can be selected over time. In contrast, the set of possible stationary distributions for the HD game is extremely rich. The distribution P(t, α) is stationary, by definition, if dPðdtt, αÞ ¼ 0 for all α. The dynamics of current distribution P(t, α) is defined by the replicator equation (13.22) dPðt, αÞ ¼ Pðt, αÞððA + BÞxðtÞ  AÞðα  xðtÞÞ: dt

266

13. Strategy selection in evolutionary games

Hence, if the frequency of defectors is given by x ¼ A A+ B, then the distribution P(t, α) is stationary. Recall that x(t) ¼ Et[α] (see Eq. 13.3). We have proven the following proposition. Proposition 13.2 Let 0 < A A+ B < 1 and P(α) be any distribution of α 2 [0, 1] such that E½α ¼ A A+ B. Then P(α) is a stationary distribution. The current distribution of parameter α given the initial distribution P(0, α) is described by equation (13.19). Then, if 0 < A A+ B < 1, a nonsingular limit stationary distribution of α exists and has the form P∗ ðαÞ ¼ where w∗ solves the equation

dw dt ¼ 0,

Pð0αÞeαw∗ , M0 ðw∗ Þ

(13.29)

that is (see Eq. 13.21),

d A lnM0 ðw∗ Þ ¼ E∗ ½α ¼ : dw A+B Here we denote E∗[α] to be the mean value of α with respect to distribution P∗(α). It follows from Eq. (13.29) that all mixed strategies that were present in the population initially do not disappear over time but remain in the population indefinitely. A classic example of a hawk-dove game is defined by the following payoff matrix: 0 1 Coop Def B 0 C, C > V: (13.30) G ¼ @ Coop V=2 A V C Def V 2 C Here A ¼  V2 , B ¼ VC 2 < 0, A + B ¼  2 . Consequently, Eq. (13.8) becomes   dxðtÞ V C ¼ Vart ½αððA + BÞxðtÞ  AÞ ¼ Vart ½α  xð t Þ dt 2 2

(13.31)

and the equilibrium frequency of defectors is x∗ ¼ VC. In addition to these known results, Eqs. (13.13) and (13.16) allow us to study the dynamics of the distribution of parameter α that describes the process of natural selection of strategies. Eq. (13.16) now becomes dw V C d ¼  lnM0 ðwðtÞÞ: dt 2 2 dw This equation has a unique stable equilibrium w∗, which solves the equation d V lnM0 ðwÞ ¼ : dw C If the initial distribution of α is truncated exponential as per Eq. (13.24), then d 1 1 lnM0 ðwÞ ¼ , + dw 1  esw s  w

267

13.6 Natural selection of strategies in a “hawk-dove” game

so

  dw V C 1 1 ¼  + , dt 2 2 1  esw s  w

(13.32)

and w∗ solves equation 1 1 V + ¼ : ∗ sw ∗ 1e sw C Now that we have a solution to Eq. (13.32), we can compute the pdf P(t, α) according to Eq. (13.14) for all t and therefore can describe all the statistical characteristics of this game. Consider the following numerical example, where V ¼ 2, C ¼ 6. The dynamics of the frequency x(t) of defectors depends on the initial distribution of α and in particular on the parameter s of the truncated exponential distribution (13.24). However, the final value x∗ does not depend on the initial distribution, as can be seen in Figs. 13.5 and 13.6. Complete information about the process of natural selection of strategies can be obtained from the dynamics of the distribution of parameter α. Let the initial distribution be truncated exponential on the interval [0,1]. Then the current distribution at time t is also a truncated exponential with pdf Pðt, αÞ ¼

eαðwðtÞsÞ ðwðtÞ + sÞ : 1  ewðtÞs

(13.33)

x

0.8

0.6

0.4 10

20

30

40

50

60

70

t

0.2

FIG. 13.5 Dynamics of the defector frequency in the hawk-dove game as described by Eq. (13.33). The initial distribution of α is truncated exponential with parameter s, as defined in Eq. (13.24); s ¼ 10 (red, bottom) and s ¼ 10 (blue, top); the limit frequency of defectors is x∗ ¼ VC ¼ 13.

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13. Strategy selection in evolutionary games

x

0.30

0.25

0.20

0.15

(A)

10

20

30

40

50

60

70

t

x 0.9 0.8 0.7 0.6 0.5 0.4

(B)

10

20

30

40

50

60

70

t

FIG. 13.6 The hawk-dove game. Comparison of the dynamics of the frequency of defectors defined by replicator equation (13.1) (green) and by Eq. (13.31) (red), with A ¼  1, B ¼  2. The initial distribution of α is truncated exponential in [0,1] with parameter s, as defined in Eq. (13.24). (A) x(0) ¼ E0[α] ¼ 0.1, s ¼ 10; (B) x(0) ¼ E0[α] ¼ 0.9, s ¼  10.

13.6 Natural selection of strategies in a “hawk-dove” game

269

The limit distribution is P∗ ðαÞ ¼

eαðw∗ sÞ ðw∗ + sÞ , 1  ew∗ s

where w∗ is the root of equation es + ew ð1  s + wÞ ¼ V=C: ðes  ew Þðs  wÞ

(13.34)

The dynamics of the distribution of parameter α given by Eq. (13.33) is shown in Fig. 13.7. We have seen that the total frequency of defectors, which is equal to Et[α], tends to VC ¼ 1=3 in agreement with known results. A new piece of information is that the distribution of mixed strategies characterized by parameter α tends over time to truncated exponential distribution, whose mean value is equal to the limit frequency of defectors. Notice that the support of all current and limit distributions coincides with the support of the initial distribution regardless of what the initial distribution was. This means that all mixed strategies that are present initially will also be present in the final state of the system. Let us now assume that the initial distribution is normal, truncated in the interval [0,1] as defined by Eqs. (13.27) and (13.28). Then, according to Eqs. (13.19) and (13.21), the current pdf is given by the formula 2

2eðα + m + sw=2Þ =s 



 , Pðt, αÞ ¼ pffiffiffipffiffi 1 + m + sw=2 m + sw=2 pffiffi pffiffi π s Erf + Erf s s where w(t) is the solution to equation 0

1  p ffiffi B C e e1=s  e2m=s + w s dw 



 C ¼ A + ðA + BÞB m + sw=2 + : @ pffiffiffi 1 + m + sw=2 m + sw=2 A dt pffiffi pffiffi π Erf + Erf s s ð1 + ðm + sw=2Þ2 Þ=s



100

t 50 0 2.0 P(t,a)

1.5 1.0 0.5 0.0 0.5 a 1.0

FIG. 13.7 The dynamics of a population with an initial truncated exponential distribution of α in the hawk-dove game with s ¼  1 (blue).

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13. Strategy selection in evolutionary games

200

P(t,a)

150

4

100

2 0 0.0

t

50 0.5

a

1.0

0

FIG. 13.8 Evolution of a population with an initial truncated normal distribution (blue) in the hawk-dove game with A ¼  1, B ¼  2. Parameters of the initial distribution are m ¼ 12 , s ¼ 0:01; the final distribution (red) is truncated normal with m ¼ 13 , s ¼ 0:01.

A typical evolutionary trajectory of truncated normal distribution is shown on Fig. 13.8. The “game of chicken” is just a version of the HD game and differs only in interpretation. A standard example of this game is given by the following payoff matrix:   b  c b  2c G¼ , b > 0, c > 0, b  2c > 0: b 0 For this game, A ¼ c, B ¼ 2c  b < 0, so condition (1) of Theorem 13.1 holds and 0 < x∗ ¼

A c ¼ b > c, then B > 0 , and the “game of chicken” becomes a prisoner’s dilemma, where x∗ ¼ 1 is a single equilibrium frequency of defectors. If 2c ¼ b, then x∗ ¼ 1 is also a single limit value of the frequency. Fig. 13.9 illustrates the dynamics of the frequency of defectors in the “game of chicken” and its transition to a prisoner’s dilemma when the value of b decreases. Blue and green curves correspond to the case b > 2c (game of chicken), and the red curve corresponds to the case b < 2c (PD game). We are also able to trace the dynamics of the distribution of α given by Eqs. (13.19) and (13.21), where now

271

13.7 Natural selection of strategies and the principle of minimum information gain

x 1.0 0.9 0.8 0.7 0.6 0.5

20

40

60

80

100

t

0.4 0.3

FIG. 13.9 Dynamics of the defector frequency for the “game of chicken.” The initial distribution of α is a truncated exponential in [0,1] with parameter s ¼  1, as defined in Eq. (13.24). Parameters for the blue curve (bottom) are b ¼ 8, c ¼ 2 and for the green curve (middle) are b ¼ 5, c ¼ 2. With change of parameter b, the “game of chicken” becomes PD game, as in the red curve (top) with b ¼ 3, c ¼ 2.

dw d ¼ c + ðc  bÞ lnM0 ðwÞ: dt dw If the initial distribution of α is a truncated exponential as per Eq. (13.24), then this last equation becomes dw es + ew ð1  s + wÞ ¼ c + ðc  b Þ : dt ðes  ew Þðs  wÞ

(13.35)

The dynamics of the current distribution of α is shown in Figs. 13.5 and 13.6. If b ¼ 8, c ¼ 2, then we have an HD game (or “game of chicken”), and the final distribution is nonsingular (see Fig. 13.10A) with 0 < E[α] < 1. If, however, b ¼ 3, c ¼ 2, then we have the PD game, and the final distribution is concentrated in the point α ¼ 1; see Fig. 13.10B.

13.7 Natural selection of strategies and the principle of minimum information gain In this section, we show that the dynamics of the distribution of strategies as it was described by Eqs. (13.19)–(13.21) obey the principle of minimal information gain for all considered games. The principle was discussed in Chapter 8; recall that it was based on the hypothesis that given a prior probability distribution m and precisely defined prior data, the probability distribution p that best represents the current state of knowledge is the distribution that minimizes KL divergence between p and m. The KL divergence I[p : m] is a measure of

272

13. Strategy selection in evolutionary games

FIG. 13.10

2.0

P(t,a) 1.5

100

1.0 0.5 0.0

50

a

t

Dynamics of a population with an initial truncated exponential distribution of α with s ¼ 1; the initial distribution is shown in blue, and the final distribution is shown in red. (A) Top panel with b ¼ 8, c ¼ 2 depicts the HD game with A ¼  2, B ¼  6; the final distribution is a truncated exponential. (B) Bottom panel with b ¼ 3, c ¼ 2 depicts the PD game with A ¼  2, B ¼ 1, and the final distribution is concentrated in the point α ¼ 1.

0.5 1.0

(A) 20

15

0

t 10

5

0 20 15

P(t,a)

10 5 0 1.0 0.5

(B)

0.0

a

information gain in moving from a prior distribution m to a posterior distribution p; it is defined by the formula ð h pðαÞ pi dα ¼ Ep ln : I ½p : m ¼ pðαÞ ln (13.36) mðαÞ m A I[p : m] is also known in natural sciences as relative or cross entropy; therefore the principle of minimal information gain is mathematically equivalent to the principle of minimum of relative or cross entropy, known also as MinxEnt. It was shown in Chapter 8 that within the frameworks of the mathematical theory of selection, MinxEnt is not an external independent hypothesis but a strong mathematical assertion, which follows from the dynamics of corresponding models. In what follows, we give a direct proof of this statement in application to our models of strategy selection.

13.7 Natural selection of strategies and the principle of minimum information gain

273

A general theory for application of the principle of minimal information gain was developed by Kullback (1997). The probability density function (pdf ) m(α), α 2 A in Eq. (13.36) is assumed to be given. It is also assumed that posterior data in the form of the expected value of some variable, T(α), is given, Ep[T] ¼ C ¼ const. Then the distribution that provides the minimum of I[p : m] subject to the constraint Ep[T] ¼ C is given by the following equation (see Kullback, 1997; Theorem 2.1): pðαÞ ¼ mðαÞ Here M(w) ¼

Ð

wT(α) m(α)dα, Ae

ewTðαÞ : M ðw Þ

(13.37)

and the multiplier w is the solution to the equation d log ðMðwÞÞ ¼ C: dw

(13.38)

If the pdf p is defined this way, then I ½p : m ¼ Cw  log ðMðwÞÞ:

(13.39)

Now let us apply Kullback’s theorem to our problem of strategy selection using “inverse logic.” That is, we do not seek an unknown distribution Pt that would minimize the KL divergence I[Pt : P0] subject to the mean value Et[α] ¼ x(t). Instead, we already know the distribution of strategies given by Eq. (13.13) and the mean value Et[α] given by Eq. (13.14) at each time point. Let us plug into formulas (13.36)–(13.39) the following quantities: A ¼ [0, 1], T(α) ¼ α, m ¼ P0, and p ¼ Pt. Then it can be easily seen that distribution (13.13) coincides with the distribution (13.37) that minimizes the information gain I[Pt : P0] subject to the given mean value Et[α] ¼ x(t). Therefore the following theorem holds. Theorem 13.2 The current strategy distribution (13.13) provides minimum of information gain I[Pt : P0] over all probability distributions Pt ¼ P(t, α) with a given value of Et(α) at every time t; the value of I[Pt : P0] can be computed as I ½Pt : P0  ¼ wðtÞxðtÞ  ln M0 ½wðtÞ ¼ wðtÞ

dlnM0 ½wðtÞ  lnM0 ½wðtÞ: dw

(13.40)

Let us emphasize that Theorem 13.2 is valid for any mixed-strategy game described by Eq. (13.4). Theorem 13.2 implies an interesting corollary, Corollary 13.2

I ½Pt : P0  ¼ lnPðt, xðtÞÞ  ln ðPð0, xðtÞÞ:

(13.41)

Indeed, according to Eqs. (13.40) and (13.19), exp ð I ½Pt : P0 Þ ¼

ewðtÞxðtÞ Pðt, xðtÞÞ : ¼ M0 ½wðtÞ Pð0, xðtÞÞ

(13.42)

Proposition 13.3 Let the pdf of the initial distribution satisfy 0 < P(0, α) < ∞ for all α 2 [0, 1]. Then information gain I[Pt : P0] tends to a finite value for the HD game and I[Pt : P0] ! ∞ for the PD, H, and SH games as t ! ∞.

274

13. Strategy selection in evolutionary games

Indeed, it was proven in Theorem 13.1 that lim t!∞ xðtÞ ¼ x∗ as t ! ∞, 0 < x∗ ¼ A A+ B < 1, and A < 0, B < 0 for the HD game. Then Eq. (13.17) has a finite stable equilibrium w∗. Hence wðtÞ ! w∗ as t ! ∞, and so Pðt, αÞ ! δð1Þ, I ½Pt : P0  ! w∗ x∗  lnM0 ðw∗ Þ: The second assertion of the proposition follows from Eq. (13.41). Let, for example, lim t!∞ xðtÞ ¼ 1 as in the PD game. Then the distribution of parameter α over time becomes concentrated at the point α ¼ 1. This means that the density of this distribution satisfies Pðt, αÞ ! δð1Þ, where δ(α) is the Dirac δ-function. Then Pðt, xðtÞÞ ! ∞ as t ! ∞: Therefore Pðt, xðtÞÞ ! ∞ as lim Pð0, xðtÞÞ ¼ Pð0, 1Þ < ∞: t!∞ Pð0, xðtÞÞ Then it follows from Eq. (13.42) that lim I ½Pt : P0  ¼ ∞:

t!∞

The case lim x(t) ! 0 as t ! ∞ can be considered in the same way. The following figures show the dynamics of KL divergence I[Pt : P0] between the initial and current strategy distributions for different games considered in Sections 13.3–13.5. In all cases the initial distribution was taken as truncated exponential according to Eq. (13.24) with parameter s. Fig. 13.11 shows I[Pt : P0] for the prisoner’s dilemma and harmony games. Recall that for these games the distribution P(t, α) becomes concentrated over time at the points α ¼ 1 and α ¼ 0, respectively, independently of the initial distribution. In both cases, I[Pt : P0] ! ∞ as t ! ∞. Fig. 13.12 shows the dynamics of the KL divergence for the stag hunt game; in this case the distribution P(t, α) becomes concentrated over time at either α ¼ 1 or α ¼ 0, depending on the initial distribution. Fig. 13.13 shows the dynamics of KL divergence for the hawk-dove game; in this case the limit distribution P(t, α) as t ! ∞ is nonsingular, and its support coincides with the support of the initial distribution. The limit value of I[Pt : P0] as t ! ∞ is finite, in contrast to the other games. Overall, within the framework of the considered model of strategy selection, the dynamical version of the principle of minimal information gain can be derived from the model’s dynamics instead of being postulated a priori. Minimization of information gain at any time point is an intrinsic property of the process of natural selection of strategies for all games due to the

275

13.7 Natural selection of strategies and the principle of minimum information gain

FIG. 13.11 Dynamics of KL divergence as defined in Eq. (13.36). (A) Prisoner’s dilemma, A ¼  1, B ¼ 1, s ¼ 10. (B) Harmony game, A ¼ 1, B ¼  1, s ¼ 10.

KL

10 8 6 4 2

20

(A)

40

60

80

100

t

KL

3.0 2.5 2.0 1.5 1.0 0.5

(B)

20

40

60

80

100

t

game dynamics. The initial distribution of strategies and the current mean value, which is equal to the current frequency of one of the pure strategies, completely define the current distribution of strategies due to MinxEnt; no other details about the selection process or the game itself are necessary. The principle of minimum information gain is not an external hypothesis but a mathematical theorem within the frameworks of the model of strategy selection. It can therefore be considered to be an underlying variational principle, which governs the selection process for all games at every time moment.

KL 3.0 2.5 2.0 1.5 1.0 0.5 20

40

60

80

100

t

FIG. 13.12 Dynamics of KL divergence for the stag hunt game: A ¼ B ¼ 1 and s ¼  0.01. Due to symmetry, dynamics of information gain is identical in both cases. KL

FIG. 13.13 Dynamics of KL divergence for hawk-dove game. (A) A ¼  1, B ¼ 1, s ¼ 10. (B) A ¼  1, B ¼ 1, s ¼  10.

1.2 1.0 0.8 0.6 0.4 0.2

(A)

20

40

60

80

100

t

KL

4

3

2

1

(B)

20

40

60

80

100

t

13.8 Discussion

277

13.8 Discussion In this chapter, we studied the process of natural selection of mixed strategies in classical 2x2 matrix games. The current state of the strategy selection process is described by the probability distribution of parameter α, which represents the (heritable) probability of an individual playing one of two strategies. This distribution can be considered as a distribution of mixed strategies in the game, and its dynamics are the main problem of interest. The problem was completely solved using the hidden keystone variable (HKV) method; the distribution of parameter α at any time is defined by Eqs. (13.19)–(13.21). The dynamics of frequencies of pure strategies is described by Eq. (13.8), which generalizes a well-known replicator equation by Taylor and Jonker (1978). These general results were applied to known 2
2 matrix games. We were able to show that the dynamics of strategy distributions in the prisoner’s dilemma (PD) and harmony (H) games essentially depend on the initial distribution of mixed strategies. In both cases the limit distribution is singular, that is, only a pure strategy can be selected over time from all possible mixed strategies, in agreement with known results. Similarly, in the stag hunt (SH) game, only one pure strategy can be selected over time, but the initial population composition, and, more specifically, its mean value, has a critical impact on what strategy will finally be selected. In all of these cases, only a single pure strategy can be selected. In contrast, in the hawk-dove (HD) game, not only the overall dynamics but also the shape of the final distribution of mixed strategies depends on the initial distribution. The final distribution is not singular, and any mixed strategy that was initially present in the population will be present in the final distribution. Another principal difference of the HD game from all other considered games is that KL divergence between current and initial distributions of strategies (information gain) tends to a finite value, while it tends to infinity for other games. We would like to emphasize that the HD game has clear biological interpretation and may be used to explain the persistence of species, whose members have potentially lethal attributes. Interestingly the process of natural selection of strategies for all considered games obeys the dynamical principle of minimal information gain. It means that given the initial distribution and the value of the frequency of one of pure strategies, at any time, the current distribution of mixed strategies provides minimal information gain over all probability distributions. Formally, we can postulate this principle (as Kullback and Jaynes did in statistics and statistical physics accordingly) and then construct a solution to the model by solving a corresponding variational problem. What is important is that now we know for certain that this way we obtain the distribution that exactly coincides with the solution to the model. Hence the principle of minimal information gain is the underlying optimization principle, whose “invisible hand” governs the process of natural selection of strategies in these games. In the next chapter, we describe a mathematical framework for analyzing natural selection not just between strategies, but between games. We show that the distribution of games changes over time due to natural selection. We also investigate the question of mutual invasibility of games with respect to different strategies and different initial population composition. Finally, we discuss the applicability of the developed framework to understanding games that cancers play.

C H A P T E R

14 Natural selection between two games with applications to game theoretical models of cancer Abstract In the previous chapter, we modeled and studied the process of natural selection between all possible mixed strategies in classical two-player two-strategy games. In this chapter, we describe a mathematical framework for analyzing natural selection not only between strategies but also between games. We provide theoretical analysis of natural selection between the games of prisoner’s dilemma (PD) and Hawk-Dove (HD) and demonstrate that while the dynamics of cooperators and defectors within their respective games are as expected, the distribution of games changes over time due to natural selection. We also investigate the question of mutual invasibility of games with respect to different strategies and different initial population compositions. We conclude with a discussion of how the proposed approach can be applied to other games in cancer, such as motility versus stability strategies that underlie the process of metastatic invasion. This Chapter is based on (Kareva and Karev, 2019).

14.1 Model description Consider an (infinitely) large community composed of individuals that can play in one of two games with hereditary probability; we will refer to these games as G1 and G0 (notably, G1 can be interpreted as Prisoner’s dilemma (PD) and G0 can be interpreted as Hawk-Dove (HD); however, other interpretations and game choices are possible as well). Let us assume that every individual can play the G1 game with hereditary probability β and the G0 game with hereditary probability (1
β). Assume also that every individual can adopt only one of two different pure strategies, s1 or s0; for interpretation, we refer to s1 strategy as defecting and to s0 strategy as cooperating. It means that the community is divided into two populations: individuals from the s1 population use the s1 strategy, and individuals from the s0 population use the s0 strategy. The kth game has a matrix of expected payoffs Gk ¼ (Gkij), k ¼ 0, 1; i, j ¼ 0, 1.

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00014-X

279

# 2020 Elsevier Inc. All rights reserved.

280

14. Natural selection between games

What we want to find out is which game or a mixture of games, and which strategy, whether pure or mixed, will dominate over time as a result of natural selection.

14.1.1 The model Let us refer to the set of all individuals that play the G1 game with hereditary probability β as a β-clone; denote L(t; β) to be the total size of β-clone in t moment. Let l(t; s1, β) and l(t; s0, β) be the number of individuals in β-clone that adopt s1 or s0 strategy accordingly, so Lðt; βÞ ¼ lðt; s1 , βÞ + lðt; s0 , βÞ:

(14.1)

In what follows, we consider only the case when β is equal to either 0 or 1. The total number of individuals that use the s1 strategy (defectors) becomes DðtÞ ¼ lðt; s1 , 0Þ + lðt; s1 , 1Þ

(14.2)

and the total number of individuals that use the s0 strategy (cooperators) at moment t becomes CðtÞ ¼ lðt; s0 , 0Þ + lðt; s0 , 1Þ:

(14.3)

The total community size at time moment t is given by NðtÞ ¼ Lðt, 0Þ + Lðt, 1Þ or N ðtÞ ¼ DðtÞ + CðtÞ:

(14.4)

Let us denote the frequencies of G0 or G1 games in the community at t moment to be Pðt; βÞ ¼

Lðt; βÞ , β ¼ 0,1: N ðt Þ

(14.5)

Then the frequencies of defectors s1 and cooperators s0 in game β at time moment t are given by lðt; s1 , βÞ DðtÞ lðt; s0 , βÞ : pðt; s0 , βÞ ¼ CðtÞ

(14.6)

DðtÞ CðtÞ pðt; s1 , βÞ + pðt; s0 , βÞ: N ðt Þ N ðt Þ

(14.7)

pðt; s1 , βÞ ¼

Then, Pðt; βÞ ¼

Denote also the frequencies of strategies within β players: p1 ðt; βÞ ¼

lðt; s1 , βÞ Lðt, βÞ

p0 ðt; βÞ ¼

lðt; s0 , βÞ : Lðt, βÞ

(14.8)

281

14.1 Model description

We are interested in the dynamics of the frequencies P(t; β) of players that play G0 or G1 games and in the dynamics of the total numbers of defectors D(t) and cooperators C(t) over time. Let us denote for brevity the payoff matrix as
 Gβ ¼ G0 + β G1
G0 , β ¼ 0, 1: Its elements are

  Gβij ¼ G0ij + β G1ij
G0ij :

(14.9)

Finally, let us denote the frequency of s1 players in the total community as xð t Þ ¼

DðtÞ : N ðtÞ

(14.10)

Then, (1
x(t)) is the frequency of s0 players in the community. The payoff of s0 players in β game (interpreted as the average number of offspring per individual) during the time interval (β) (t, t + Δt) is equal to (G(β) 00 (1
x(t)) + G01 x(t))Δt  F0(β, x(t))Δt, and s1 players in the β game have an average number of offspring:   ðβ Þ ðβÞ G10 ð1
xðtÞÞ + G11 xðtÞ Δt  F1 ðβ, xðtÞÞΔt: We now finally arrive at equations that constitute our master model, with β ¼ 0 corresponding to G0 (HD) game and β ¼ 1 corresponding to G1 (PD) game:    dlðt; s1 , βÞ ðβ Þ ðβÞ ðβÞ ¼ lðt; s1 , βÞF1 ðβ, xðtÞÞ ¼ lðt; s1 , βÞ G10 + xðtÞ G11
G10 dt (14.11)    dlðt; s0 , βÞ ðβ Þ ðβÞ ðβÞ ¼ lðt; s0 , βÞF0 ðβ, xðtÞÞ ¼ lðt; s0 , βÞ G00 + xðtÞ G01
G00 : dt

14.1.2 Solution to the model To solve the model using the HKV method, let us introduce the keystone variable q(t) through the following escort equation: dqðtÞ ¼ xðtÞ, qð0Þ ¼ 0: dt

(14.12)

Then the solutions to Eq. (14.11) are given by the formulas lðt; s0 , βÞ ¼ lð0; s0 , βÞeðG00 t + qðtÞðG01
G00 ÞÞ ðβ Þ

ðβ Þ

ðβ Þ

ðβ Þ ðβ Þ ðβ Þ lðt; s1 , βÞ ¼ lð0; s1 , βÞeðG10 t + qðtÞðG11
G10 ÞÞ :

Eq. (14.12) for keystone variable q(t) now reads

(14.13)

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14. Natural selection between games

dqðtÞ DðtÞ ¼ dt N ðt Þ

ð 0Þ ð 0Þ ð 0Þ ð 1Þ ð 1Þ ð 1Þ lð0; s1 , 0ÞetG 10 + qðtÞðG11
G10 Þ + lð0; s1 , 1ÞetG 10 + qðtÞðG11
G10 Þ ¼ ð 0Þ ð 0Þ ð 0Þ ð 1Þ ð 1Þ ð 1Þ ð 0Þ ð 0Þ ð 0Þ lð0; s0 , 0ÞetG 00 + qðtÞðG01
G00 Þ + lð0; s0 , 1ÞetG 00 + qðtÞðG01
G10 Þ + lð0; s1 , 0ÞetG 10 + qðtÞðG11
G10 Þ ð1Þ ð 1Þ ð 1Þ + lð0; s , 1ÞetG 10 + qðtÞðG11
G10 Þ : (14.14)

1

With a solution to this equation, we can compute clone sizes, total size of the population and the number of s0 and s1 strategists, that is, the numbers of defectors and cooperators and their frequencies. We can also compute all other statistical characteristics of the system.

14.2 Natural selection between games: Hawk-Dove versus Prisoner’s dilemma Prisoner’s dilemma and Hawk-Dove games are “opposite” games in some sense, with one selecting for a pure strategy, while the other selects for a mixed one with respect to cooperation versus defection. Therefore it can be of great interest to trace natural selection between these two games within a population, where individuals can play in any of these games. Traditional form of the payoff matrix for prisoner’s dilemma is   b
c
c Gð1Þ ¼ , 0 < c < b: (14.15) b 0 The matrix that corresponds to the HD game looks as follows:   b
c c Gð0Þ ¼ : b 0

(14.16)

Let us study natural selection between the PD and HD games corresponding to these payoff matrices. The keystone equation (14.14) (after some algebra) now reads dqðtÞ ect ðlð0; s1 , 0Þ + lð0; s1 , 1ÞÞ ¼ 2cqðtÞ : dt e lð0; s0 , 0Þ + lð0; s0 , 1Þ + ect ðlð0; s1 , 0Þ + lð0; s1 , 1ÞÞ

(14.17)

We can see that this equation does not depend on parameter b, and parameter c is just a time-scaling parameter, so we can let c ¼ 1. Let us assume for now that initial values lð0; s0 , 0Þ ¼ lð0; s1 , 0Þ ¼ lð0; s0 , 1Þ ¼ lð0; s1 , 1Þ ¼ 1, that is, assume that the two games and all the strategies are equally represented at the initial time moment. Then, dqðtÞ 2et ¼ : 2q dt 1 + e ðtÞ + 2et

14.2 Natural selection between games: Hawk-Dove versus Prisoner’s dilemma

283

FIG. 14.1 Dynamics of the frequency of the defector strategy x(t) as given by Eq. (14.19) during natural selection process between a population of PD players and a population of HD players. All games and strategies are equally represented at t ¼ 0. As one can see, after initial increase in the number of defectors, the population equilibrated at half the population (in both games) pursuing the defector strategy, that is, x(0) ¼ x(∞) ¼ 1/2.

The solution to this equation is

 pffiffiffipffiffiffiffiffiffiffiffiffiffiffi qðtÞ ¼ ln
1 + 2 1 + et :

(14.18)

Now, we can compute all the statistical characteristics of the model. DðtÞ Fig. 14.1 shows dynamics of frequency of defectors xðtÞ ¼ N ðtÞ in the total population. The value of x(t) can be computed using the formula xð t Þ ¼

dqðtÞ et ¼ pffiffiffipffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffiffiffiffiffiffiffiffi : dt 2 1 + et
1 + 2 1 + et

(14.19)

Next, let us look not at the frequency of strategies but the frequency of individuals that are playing one or the other game. The frequency of G1 players is equal, by definition, to Pðt; 1Þ ¼

lðt; s0 , 1Þ + lðt; s1 , 1Þ : N ðt Þ

(14.20)

In this example, where all the games and all the strategies are equally represented at the initial time moment and q(t) is given by Eq. (14.18), the frequency of G1 players (PD) is given by Pðt; 1Þ ¼

1 + et pffiffiffipffiffiffiffiffiffiffiffiffiffiffi : 4 + 4et
2 2 1 + et

(14.21)

The plot of P(t; 1) is shown in Fig. 14.2. Notice that P(t; 1) ! 0.25 as t ! ∞, that is, only 1/4 of players will play in the first (PD) game in the end, regardless of which strategy they were

284

14. Natural selection between games

FIG. 14.2 Dynamics of individuals that keep playing the PD game (G1 players) during natural selection process between a population of PD players and a population of HD players. The frequency of G1 players as given by Eq. (14.21). As one can see, over time, only ¼ of players will continue playing PD game (regardless of strategy), even though they started at ½.

using initially. This suggests that HD game that selects for a combination of cooperators and defectors can persist and even largely “outcompete” the defector-dominated PD game. Next, let us study the frequencies of strategies within each game. By definition, the frequency of defectors and cooperators within a game is given correspondingly by lðt; s1 , βÞ lðt; s0 , βÞ + lðt; s1 , βÞ lðt; s0 , βÞ p0 ðt; βÞ ¼ , β ¼ 0, 1: lðt; s0 , βÞ + lðt; s1 , βÞ

p1 ðt; βÞ ¼

(14.22)

As we can see in Fig. 14.3A, the frequency of defectors within the PD game p1(t, 1), as defined by Eq. (14.22) with β ¼ 1, tends to 1, as is expected. In Fig. 14.3B, we can see the frequency of defectors p1(t, 0) within the HD game, as defined in Eq. (14.22) with β ¼ 0. Note that over

FIG. 14.3 Change over time in strategy frequency within each game, with all the games and strategies initially equally represented within the population. (A) Frequency p1(t, 1) of s1 strategy (defectors) within the 1st (PD) game as given by Eq. (14.22), β ¼ 1; as we can see, within PD, only defectors remain over time despite initially constituting half of the population. (B) Frequency p1(t, 0) of s1 strategy (defectors) within the 0th (HD) game, as given by Eq. (14.22), β ¼ 0. As one can see, the final frequency of cooperators is 1/3, despite initially constituting half of the population.

14.3 Mutual invasibility of both strategies and games

285

time, it tends to 1/3, while the limit stable frequency of defectors in HD game with payoff matrix (14.16) is 1/2. To summarize, we observe the following behaviors with regard to games and strategies during interactions of two populations, where PD and HD games are equally represented in the beginning and when defectors and cooperators are equally represented as well. Over time, (1) After initial increase in the number of defectors, the population equilibrates at half the population (in both games) pursuing the defector strategy. (2) Only ¼ of players will continue playing PD game, regardless of strategy, with the remaining individuals playing the more “cooperative” HD game. (3) Within the PD game, defector strategy dominates (as is expected). (4) Within the HD game, 1/3 of the population keeps the defector strategy, while 2/3 maintains the cooperator strategy (as is expected). Overall, the dynamics of cooperators and defectors within their respective games is as expected, but the distribution of games changes over time, with ¼ of the population playing the PD game and ¾ playing HD game over time.

14.3 Mutual invasibility of both strategies and games Finally, let us consider another example, which shows how a small number of HD gamers can invade a population that initially was composed mainly of PD gamers. Assume the initial composition of the population is given by lð0; s0 , 0Þ ¼ 0:01, lð0; s1 , 0Þ ¼ 0:01, lð0; s0 , 1Þ ¼ 10, lð0; s1 , 1Þ ¼ 1:

(14.23)

Hence the initial proportion of HD players is l(0; s0, 0) + l(0; s1, 0) ¼ 0.02, and the initial proportion of PD players is l(0; s0, 1) + l(0; s1, 1) ¼ 11. Then the keystone variable that solves Eq. (14.17) with initial conditions (14.23) becomes h  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii (14.24) qðtÞ ¼ ln 50
11 + 121:2202 + 0:02ð10 + 1:01ð
1 + 2et ÞÞ : Fig. 14.4 shows dynamics of frequency of defectors x(t) in total population as computed by formula xðtÞ ¼

dqðtÞ 0:00184et pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ dt 11:034 + 0:0037et
1: 121:4 + 0:0404et

(14.25)

We can see that the initially a very small proportion of defectors (equal to ðlð0; s1 , 0NÞ +ð0lÞð0; s1 , 1ÞÞ  0:1Þ tends over time to ½. Next, let us consider a situation, when the population of PD gamers is invaded by a small population of HD gamers. The frequency of G1 (PD) gamers with payoff matrices (14.15) and (14.16) can be computed using the formula: Pðt, 1Þ ¼

lðt; s0 , 1Þ + lðt; s1 , 1Þ lð0; s0 , 1Þ + et lð0; s1 , 1Þ ¼ 2qðtÞ : N ðtÞ e lð0; s0 , 0Þ + lð0; s0 , 1Þ + et ðlð0; s1 , 0Þ + lð0; s1 , 1ÞÞ

(14.26)

286

14. Natural selection between games

FIG. 14.4 Dynamics of the frequency of defectors x(t) in total population given by Eq. (14.25). The initial composition of the population consisted of a very small number of defectors (x(0) ¼ 0.092), with the PD and HD games equally represented in the initial time moment. Over time, x(∞) ¼ 0.5, showing how a very small initial number of defectors can invade, but only half of the entire population.

In the considered example, Pðt, 1Þ ¼

10 + et  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 : 10 + 1:01et + 25
11 + 121:4 + 0:0404et

(14.27)

The frequency of G0 (HD) gamers is calculated as P(t, 0) ¼ 1
P(t, 1). Fig. 14.5 shows the dynamics of frequencies of PD gamers and HD gamers over time. We can see that a very small initial proportion of HD gamers (equal to 0.002) tends over time to 0.5, with HD gamers eventually taking over half of the entire population. Finally, let us consider the distribution of games within defectors and cooperators. By definition, the frequency of G1 and G0 gamers among all s1 strategists (defectors) is given by the formula

FIG. 14.5

The dynamics of games, where a population of PD gamers is invaded by a small number of HD gamers. (A) Dynamics of frequency of PD-game players, as described by Eq. (14.27), showing how the population of PD gamers decreases to half the entire population. (B) Dynamics of frequency of HD gamers, where the small number of HD gamers can over time come to constitute half of the entire population.

14.3 Mutual invasibility of both strategies and games

f ðt; s1 , βÞ ¼

lðt; s1 , βÞ , β ¼ 0, 1, lðt; s1 , 1Þ + lðt; s1 , 0Þ

287 (14.28)

and the frequency of G1 gamers and G0 gamers among all s0 strategists (cooperators) is given by the formula f ðt; s0 , βÞ ¼

lðt; s0 , βÞ , β ¼ 0, 1: lðt; s0 , 1Þ + lðt; s0 , 0Þ

(14.29)

It is easy to show that in the example with payoff matrices (14.15) and (14.16), the proportions of HD gamers and PD gamers among all s1 strategists (defectors) are constants that are equal to lð0; s1 , 0Þ ðlð0; s1 , 0Þ + lð0; s1 , 1ÞÞ and lð0; s1 , 1Þ , ðlð0; s1 , 0Þ + lð0; s1 , 1ÞÞ respectively. In contrast, proportions of HD gamers and PD gamers among all s0 strategists (cooperators as given by Eq. 14.29), can be calculated using equation f ðt; s0 , βÞ ¼

lð0; s0 , βÞe2qðtÞ , β ¼ 0, 1: ðlð0; s0 , 0Þe2qðtÞ + lð0; s0 , 1ÞÞ

(14.30)

Then, the frequency of cooperators tends to 1, and the frequency of defectors tends to 0; Fig. 14.6 shows dynamics of the frequencies with initial conditions defined in Eq. (14.23). To summarize, we have shown here that games can be mutually invasible: a population of PD players can be invaded by a small number of HD players. At the same time, a population

FIG. 14.6

Change over time in the distribution of cooperators and defectors between the games in the population. (A) Frequency f(t; s0, 1) of PD gamers among all s0 strategists (cooperators) tends to 0 over time as given by Eq. (14.30), β ¼ 1, while (B) the frequency f(t; s0, 0) of HD gamers between all s0 strategists (cooperators) tends to 1 as calculated using Eq. (14.30), β ¼ 0. These results illustrate that even though there may have been cooperators present initially in the PD game, over time, all the cooperators can be found only among HD players.

288

14. Natural selection between games

of cooperators can be easily invaded by defectors. However, an important observation is that while defectors can invade a population of cooperators in a situation, where two games are being played at the same time, they do not take over but constitute only half of the population. An optimistic way to look at this result is that even when defectors predominate in the population in the beginning, eventually, cooperators establish their place as well and cannot be fully outcompeted regardless of the initial composition of the population. In the next section, we will look at several examples of different games as they are applied to cancer biology and dynamics.

14.4 Games tumors play Most tumors are characterized by high genotypic and phenotypic heterogeneity (Alizadeh et al., 2015; Fidler, 1978; Marusyk and Polyak, 2010; Merlo et al., 2006), with different cell phenotypes interacting with normal structures and malignant ones. This suggests that tumor dynamics is likely to be subject not only to natural selection but also to various phenotypic strategies that cells within a tumor could utilize to increase their fitness. This hypothesis, coupled with developments in evolutionary game theory (EGT), led to applications of EGT to trying to understand cancer and its various aspects, including interactions between the cells themselves (Dingli et al., 2009; Hummert et al., 2014; Tomlinson, 1997), strategies in metabolism (Kareva, 2011; Kianercy et al., 2014), secretion of public goods and toxins or “public bads” (Archetti, 2013), and allocation of resources for evolution of resistance (Orlando et al., 2012). The goal of application of EGT to cancer is to enable understanding and prediction of evolutionary trajectories of the tumor to devise optimal interference strategies, suggesting a “game” between cancer and treatment (Orlando et al., 2012). An important distinction that needs to be made in trying to apply EGT to understanding tumor progression is that between evolutionary game theory and natural selection. Specifically, in game theory, the behavior or “strategy” of the player, such as a cancer cell, can change depending on the behavior of its neighbors within the cell’s lifetime. Natural selection, however, assumes a fixed “strategy,” which may or may not get passed onto the next generation. For instance, the p53 mutation that results in the loss of capacity for programmed cell death (Liu et al., 2010) is a trait that is subject to natural selection but is not a game. However, resource allocation towards aerobic or glycolytic metabolism is a strategy that can change over a cell’s lifetime depending on the behavior of other cells in its environment and thus can be analyzed as a game. In what follows, we will briefly discuss two games in the context of cancer and look at how these games can be better analyzed within the frameworks of the methods, described earlier.

Game 1: Metabolism and resource allocation In the model proposed in Kianercy et al. (2014), the authors looked at glucose and lactate metabolism within the frameworks of game theory. The authors analyzed the metabolic strategy of hypoxic and oxygenated cells, with the latter being represented by a payoff matrix of a Hawk-Dove game. The drawback of the approach lied largely in the fact that the elements of

14.4 Games tumors play

289

the payoff matrix were stationary, and within the frameworks of this particular problem formulation, oxygenated cells’ dynamics were independent of hypoxic cells’ behavior, which is probably not biologically realistic. In Kareva (2011), cancer metabolism and the associated resource allocation payoffs were formulated as a different game. Specifically, it was suggested that aerobic and anaerobic (glycolytic) cells are in fact playing the game of prisoner’s dilemma: aerobic cells require less glucose to yield a certain number of ATPs (energy units for all cells); glycolytic cells require significantly more raw resources to obtain the same number of ATPs but the by-product of glycolysis, lactic acid, gives them a competitive advantage against aerobic cells when it is accumulated in large enough quantities to increase aerobic cells’ mortality. Therefore, to gain competitive advantage, glycolytic cells need to cooperate to produce a sufficient amount of “public goods” (Driscoll and Pepper, 2010; Pepper, 2012), but for each individual cell at each time moment, it is energetically unprofitable to cooperate. Therefore, normal state of aerobic metabolism is the “defector strategy,” which is the evolutionarily stable strategy in a game of prisoner’s dilemma. Introducing the “cooperator” strategy of glycolytic cells within this interpretation requires modification of the environment, which reduces the costs associated with the “cooperator” strategy, thereby allowing it to invade the population. This can occur either through the inflow of additional resources or more likely due to the tumor outgrowing its blood supply, causing the cells to revert to glycolysis as a survival strategy. The process of lactic acid accumulation that starts as a “passenger process” of the tumor ecology may eventually become the “driver process” (Kareva, 2015) through sufficiently changing the environment and the population composition to allow previously impossible persistence of the “cooperator” strategy.

Game 2: Motility versus stability Another possible game in cancer dynamics has to do with ability of the cells to become mobile, which is necessary to initiate the metastatic process. Acquisition of a mobile phenotype requires epithelial cells to undergo an epithelial to mesenchymal transition (EMT). It involves the loss of cell-cell adhesion and apicobasal polarity and a significant decrease in proliferative capacity. These costs are offset by an acquisition of the ability to migrate individually and invade base membrane and blood vessels (Barriere et al., 2015; Brabletz et al., 2018; Jolly et al., 2015; Micalizzi et al., 2010; Revenu and Gilmour, 2009). The plasticity in the ability of a cell to become motile or remain stationary within its lifetime makes it possible to consider the “motility versus stability” transition as two strategies within a game. Notably, EMT appears to not be a binary process (Kalluri and Weinberg, 2009; Nieto, 2013). Some cells can attain a hybrid epithelial/mesenchymal (E/M), or “intermediate” phenotype (Arnoux et al., 2005; Fustaino et al., 2017; Jolly et al., 2015; Micalizzi et al., 2010; Revenu and Gilmour, 2009), maintaining some characteristics of both motile and stationary cells. This can result in the formation of motile cell clusters, composed of several motile, stationary, and intermediate phenotypes, a fascinating topic that warrants further analysis within the context of game theory. A major ecological trade-off that most populations experience, including those of cells within a tumor, is that of food versus safety (Fedriani and Manzaneda, 2005; Kotler, 2016;

290

14. Natural selection between games

Nonacs and Dill, 1990; Pettersson and Br€ onmark, 1993). A cancer cell that becomes outcompeted for resources in the crowded microenvironment of a primary tumor may become motile to escape a nutrient-deprived environment (Amend and Pienta, 2015). Additionally the presence of immune cells, the cancer cells’ natural predator, and the associated immune-related cytokines may also select for a motile strategy as a way to escape predation. Acquisition of motility has numerous disadvantages, which include increased nutritional demands, since motile cells appear to use fast but energetically inefficient aerobic glycolysis (Shiraishi et al., 2015), and the loss of much of the cells’ proliferative capacity and thus decreased fitness. The motility versus stability game, therefore, is likely to select for some mixed strategy, where, if a sufficient number of cells has left the primary tumor, subsequent reduced depletion of the local resources would allow stationary cells to proliferate until conditions once again begin to favor migration. The environment that these cells inhabit remains dynamic, and therefore each cell’s strategy will depend both on the behavior of other cells around it and on the overall ecology of the tumor. These dynamic changes could keep shifting tumor composition with regard to each strategy depending on its ecology. The hypothesis that motility versus stability game is a variation of a game with a stable mixed strategy, such as Hawk-Dove, remains to be explored further. However, if this is the case, it is possible and even likely that cancer cells are playing at least two games at the same time but using the same pool of resources and under the same set of food and safety constraints. And therefore, the payoffs for resource allocation would be determined along the stable-pure-strategy PD dimension with regard to creation of “public bads” and along the stable-mixed-strategy HD dimension, allocating the resources to stability or motility.

14.5 Some conclusions Evolutionary game theory has been applied extensively to understanding various aspects of cancer biology. What yet remains to be investigated is that different cells at the same time could be playing several games. This will influence how cells will behave over time under the varying conditions of tumor ecology, depending on how much a strategy costs in one game and how much it takes away from being able to invest in a strategy in another game. A broader view of several mutually influential games that tumors may be playing at the same time might provide deeper insights into the whole that is bigger than the sum of its parts, driving the development of smarter therapeutic interventions. In the next chapter, we will look at the application of the HKV method to discrete systems and show that the key theoretical results can translate to maps, revealing new dynamics even in well-known discrete systems.

C H A P T E R

15 Discrete-time selection systems Abstract In this chapter, we develop a theory of inhomogeneous maps or selection systems with discrete time, and explore their evolution. The theory is similar to that developed for selection systems with continuous time but we tried to give the reader a possibility to read this chapter independently of previous chapters. To this end, we give here all necessary definitions, although some of them repeat the definitions given in the previous chapters. We show here that knowing the initial distribution of a selection system allows us to determine system distribution explicitly on the entire time interval. All statistical characteristics of interest, such as mean values of the fitness or any trait, can be predicted effectively for indefinite time, and these predictions dramatically depend on the initial distribution. We show that the problem of dynamic insufficiency, characteristic of both the Price equation and the FTNS, can be resolved within the framework of selection systems with discrete time. We derive formulas for solutions of the Price equation and the FTNS. We also show applications of the developed theory to several other problems of mathematical biology such as dynamics of inhomogeneous logistic and Ricker models, and selection in rotifer populations. Complex behavior of the total population size, the mean fitness and other traits can all be observed in inhomogeneous populations with density-dependent fitness. The HKV method allows investigating the temporary dynamics of these quantities.

15.1 Main definitions Let us assume that a population consists of individuals, each of which is characterized by its own value of n parameters (a1, … , an) ¼ a. Here, we do not specify the vector parameter a, whose components may be arbitrary traits; for example, ai could be the number of alleles of ith gene, as in simple genetic models; we can also think of a as the entire genome. For the general case, we will denote A to be the domain of the vector parameter a. Let l(t, a) be the population density at time moment t. Informally, l(t, a) is the number of all individuals with a given vector parameter a; the subpopulation of all these individuals is referred to as the a-clone. In general, fitness of an individual depends on the individual’s vector parameter and on the “environment,” which in turn may depend on time. Then, in the next time instant, lðt + 1, aÞ ¼ wt ðaÞlðt, aÞ,

(15.1)

where the reproduction rate wt(a) (fitness, by definition) is a nonnegative function. The initial density l(0, a) is assumed to be given.

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00015-1

291

# 2020 Elsevier Inc. All rights reserved.

292 Let N(t) ¼

15. Discrete-time selection systems

Ð

Al(t, a)da

be the total population size; define Pt ð aÞ ¼

lðt, aÞ N ðt Þ

(15.2)

to be the current probability distribution. If l(0, a) is given, then the initial distribution P0(a) is also given. It is important to note that if Pt(a∗) ¼ 0 for a particular a∗ at some instant t, then Pt0 (a∗) ¼ 0 for all t∗ > t. Hence selection system (15.1) describes evolution of a distribution with a support that does not increase with time; it may thus be interpreted as the process of selection (see survey by Gorban, 2007). We will show that for a large class of models (15.1)–(15.2), the current distribution Pt(a) of parameter a can be computed if we assume that the initial distribution P0(a) is known. This class of models is defined by a certain condition on the reproduction rate wt(a). We assume that wt(a) > 0, and hence we can write the reproduction rate in the form wt(a) ¼ eBt(a), where Bt(a) ¼ ln(wt(a)). The exponential form of the reproduction coefficient is more appropriate in many cases for systems with discrete time. Indeed, let lðt + Δt, aÞ ¼ lðt, aÞð1 + Bt ðaÞΔtÞ for a small time interval Δt. Then, dlðt, aÞ ¼ lðt, aÞBt ðaÞ: dt The difference analogue of this equation is not lðt + 1, aÞ ¼ lðt, aÞBt ðaÞ but rather lðt + 1aÞ ¼ lðtaÞeBt ðaÞ because lðt + ΔtaÞ ¼ lðtaÞð1 + Bt ðaÞΔtÞ  lðtaÞeBt ðaÞΔt as Δt ! 0; this equation coincides with the previous one if Δt is taken as the time unit. In general, there is no one-to-one correspondence between difference and differential equations that describe the same system. Taking into account thatP any smooth function of two variables t and a can be approximated by a finite sum of the form ni¼1φi(a)gi(t), where φi depends on a only and gi depends on t only, we will assume further that the fitness is of the form ! n X φi ðaÞgi ðtÞ : (15.3) wt ðaÞ ¼ exp i¼1

For example, if we think of a as the entire genome, then Eq. (15.3) defines the map from the set of all possible genotypes {a} ¼ A to the set of corresponding fitnesses. Generally speaking, identification of this map is one of the central problems in biology. Within the framework of the master model (15.1)–(15.3), we take an individual’s fitness to depend on a given finite set

15.2 Malthusian inhomogeneous maps

293

of traits labelled by i ¼ 1, … , n. Function φi(a) describes the quantitative contribution of a particular ith trait (or gene) to the total fitness, and gi(t) describes a possible variation of this contribution with time depending on environment, population size, etc. Let us emphasize that we do not assume that contributions of different traits are independent of one another; on the contrary, the evolution of this dependence is one of the central problems explored here.

15.2 Malthusian inhomogeneous maps To clarify how exactly this approach applies to investigation of inhomogeneous maps, let us consider the simplest but important example of the Malthusian version of inhomogeneous model (15.1). The model of population growth in the absence of regulation of density (or size) of a population in a stable environment is of the form lðt + 1, aÞ ¼ wt ðaÞlðt, aÞ:

(15.4)

Let us collect here the main assertions about the discrete inhomogeneous Malthusian model, which follow from Theorem 15.1 given below in s. 15.3. For any measurable function φt(a) defined on the probabilistic space (A, Pt) (which can be considered to be a random variable on this space), we will denote ð E t ½ φt  ¼

A

φt ðaÞPt ðaÞda:

Let P0(a) be the initial distribution of the vector parameter a for inhomogeneous map (15.4). Then, (1) The population size Nt satisfies the recurrence equation Nt + 1 ¼ Nt Et ½w: (2) The current mean value of the fitness can be computed by the formula   E0 wt + 1 Et ½w ¼ : E0 ½wt 

(15.5)

(15.6)

(3) The current distribution Pt(a) is given by the formula Pt ð a Þ ¼

P0 ðaÞwt ðaÞ : E0 ½wt 

(15.7)

The evolution of the mean fitness, population size, and density dramatically depends on the initial distribution of fitnesses even for this simplest model. Let us assume for simplicity that the fitness itself is a distributed parameter, and consider the evolution of current distribution of w for a different initial pdf.

294

15. Discrete-time selection systems

(1) Let P0(a) be Γ-distribution with pdf (probability density function) given by P0 ðwÞ ¼

sk wk1 esw Γ ðkÞ

for x 0, where s, k > 0. Then, Pt ðwÞ ¼

sk + t wk + t1 esw Γ ðk + t Þ

is again Γ-distribution with parameters s, k + t; its mean is Et ½w ¼

ðk + t Þ s

and variance is given by Vart ½w ¼

ðk + t Þ : s2

Next, Nt + 1 ¼ Et ½wNt ¼

k+t Nt s

and hence Nt ¼ N0

Γðk + tÞ : st Γ ðk Þ

Indeed,   E0 w t ¼

ð∞

wt P0 ðwÞdw ¼

ð∞

0

0

sk wk1 + t esw Γðk + tÞ dw ¼ t : s ΓðkÞ Γ ðkÞ

According to Eq. (15.7), Pt ð w ¼ x Þ ¼

P0 ðw ¼ xÞxt sk xk1 + t esx ΓðkÞst sk + t xk1 + t esx ¼ ¼ E0 ½wt  ΓðkÞ Γðk + tÞ Γðk + tÞ

is the pdf of the Γ-distribution with parameters s, k + t; its mean is E t ½w  ¼

ðk + t Þ , s

so Nt + 1 ¼

ðk + tÞ Nt s

due to formula (15.5). So, in this case the mean fitness increases linearly with time, and the k + tÞ! population size increases extremely quickly, Nt ¼ ððk1 Þ!st N0 . All other examples below can be proven in the same way.

15.2 Malthusian inhomogeneous maps

295

(2) Let P0(w) be the log-normal distribution 2

P0 ðwÞ ¼

ð ln wmÞ 1 pffiffiffiffiffi e 2σ 2 , w > 0: wσ 2π

Then the moments of log-normal distribution are given by the formula of    t σ 2 t2 σ2 E0 wt + 1 2 + mt σ2 t + 2 + m E0 w ¼ e 2 ¼ e , and Et ½w ¼  eσ t : t E0 ½w  The current distribution of the fitness is

! xt1 ð ln x  mÞ2 t2 σ 2  tm :  Pt ðw ¼ xÞ ¼ pffiffiffiffiffi exp  2σ 2 2 σ 2π

Next, Nt + 1 ¼ Nt Et ½w ¼ eσ so Nt ¼ N0 exp

Xt1  k¼0

2

1 t + 2σ 2 + m Nt ,

   σ 2 k + σ 2 =2 + m ¼ N0 exp σ 2 t2 =2 + mt

We see that in this case, the mean fitness Et ½w  eσ t increases exponentially over time, 2 2 while the population grows faster, proportionally to N0 eσ t =2 + mt . 2

(3) Let P0(w) be the Beta distribution with parameters (α, β) in the interval [0, b], P0 ðwÞ ¼

Γðα + βÞ wα1 ðb  wÞβ1 ; ΓðαÞΓðβÞ bα + β1

the moments of Beta distribution are   bk Γðα + βÞΓðα + tÞ : E0 w t ¼ ΓðαÞΓðα + β + kÞ Hence

  E 0 wt + 1 bðα + tÞ Et ½w ¼  b: ¼ t ðα + β + t Þ E0 ½ w 

Next, Nt + 1 ¼

b ðα + t Þ Nt , ðα + t + β Þ

so Nt ¼ N0 bt

Γ ð α + β Þ Γ ð α + t  1Þ Γðα + βÞ t β  N0 bt : ΓðαÞ Γðα + β + t  1Þ ΓðαÞ

296

15. Discrete-time selection systems

We used here the main asymptotic property of Γ-functions: Γ(t + c)/Γ(t)  tc at large t for any constant c. Hence the fate of a population dramatically depends on the value of b: if b  1, the population goes to extinction; if b > 1, the size of the population increases indefinitely. In the case when b ¼ 1, the mean fitness tends to 1. One could expect that the total population size would tend to a stable nonzero value over time but in fact the population goes to extinction at a power rate, Nt  t β. (4) Let P0(w) be the uniform distribution in the interval [0, b]. Then,   E0 wt ¼ bt =ðt + 1Þ; hence Et ½w ¼

b ð t + 1Þ  b, ð t + 2Þ

and Nt ¼

N 0 bt : t+1

The fate of a population in this case also depends on the value of b: if b > 1, the size of the population increases indefinitely; if b  1, the population becomes extinct. (5) Fitness may be a function of a single selective trait w ¼ w(a). Selection with strict truncation (Crow and Kimura, 1979; Shnol and Kondrashov, 1994) corresponds to fitness defined as follows: wðaÞ ¼ C, if a  b ¼ const,

wðaÞ ¼ 0 if a > b:

Let pb ¼ P0 ða  bÞ, Then,

  E0 wt ¼ Ct pb :

Hence, Pt ðaÞ ¼

P0 ð a Þ for a  b pb

and Pt ðaÞ ¼ 0 for a > b: This means that Pt(a) is equal to the conditional probability P0(a) when a  b: Pt ð a Þ ¼

P0 ð a Þ χ fa  bg: P0 ða  bÞ

15.3 Evolution of the main statistical characteristics of inhomogeneous maps

297

Hence, probability Pt(a) does not change after the first selection step, Pt(a) ¼ P1(a) for all t > 1. Then, ΔE0 ½w ¼ E0 ½wja  bE0 ½w at the first selection step and ΔEt ½w ¼ 0 for any t > 0. Next,

  E0 wt + 1 Et ½ w  ¼ ¼C E0 ½wt 

and hence Nt + 1 ¼ CN t : Thus the total size of the population increases (decreases) exponentially, Nt ¼ N0Ct, unless C ¼ 1. A more realistic model should take into account some form of regulation of population size; these types of models will be studied later. We have seen that the mean fitness can increase indefinitely as a linear, exponential, power, etc. function of time depending on the initial distribution if its support is unbounded. Then the total population size also increases indefinitely. If fitness was initially distributed on a finite interval [0, b], then its mean tends over time to the maximal possible value b. The fate of the entire population depends on the particular value of b: if b > 1, asymptotically, the population increases exponentially; if b < 1, the population goes to extinction (if b ¼ 1, population behavior may very depending on the initial distribution). The exact values of the mean fitness and any other trait can be computed according to formulas (15.6) and (15.7) at any time if the initial distribution of the trait is known. Notice that all results of this section can be extended to the models with “factorized” fitness of the form w(a) ¼ f(a)g(Nt). More general results are given in the next section.

15.3 Evolution of the main statistical characteristics of inhomogeneous maps Let us return to the master model of inhomogeneous maps lðt + 1, aÞ ¼ wt ðaÞlðt, aÞ wt ðaÞ ¼ exp

n X

!

φi ðaÞgi ðtÞ :

(15.8)

i¼1

Theorem 15.1 shows that the model can be reduced to a nonautonomous map on I  R1 and completely analyzed. Let us denote ! t n Y X wk ðaÞ ¼ exp φi ðaÞGi ðtÞ , (15.9) Kt ðaÞ ¼ k¼0

i¼1

298

15. Discrete-time selection systems

where Gi(t) ¼

Pt

s¼0gi(s).

It is easy to see that wt Kt1 ¼ Kt

and lðt + 1, aÞ ¼ Kt ðaÞlð0, aÞ:

(15.10)

We can think of the function Kt(a) as the reproduction coefficient for the [0, t]-period or, for brevity, t-fitness. Let us note that sometimes functions gi(t) and hence Gi(t) can be well defined not for all 0 < t < ∞, but only for 0 < t < T, where T is a certain finite time moment. Accordingly, all assertions below are valid only for t < T. Later, we do not specify this condition if it is not necessary. Let us denote φ ¼ (φ1, … , φn) and let p(t; φ) be the pdf of the random vector φ at time t, that is, p(t; x1, … , xn) ¼ Pt(φ1 ¼ x1, … , φn ¼ xn). The master model defines a complex transformation of the distribution Pt(a) or, equivalently, the pdf p(t; φ) over time. Let λ ¼ (λ1, … , λn); denote ! ! ð ð n n X X λi φi ðaÞ Pðt, aÞda ¼ exp λi xi pðt; x1 , …, xn Þdx1 ,…, dxn (15.11) Mt ðλÞ ¼ exp A

R

i¼1

i¼1

to be the moment generation function (mgf ) of the pdf p(t; φ) of the random vector φ. The mgf of the initial distribution, M0(λ), is crucially important for the theory developed below. For example, E0[Kt] can be easily computed with the help of M0(λ): E0 ½Kt  ¼ M0 ðGðtÞÞ, where G(t) ¼ (G1(t), … , Gn(t)). Theorem 15.1 Let P0(a) be the initial distribution of the vector parameter a for inhomogeneous map (15.3). Then, (1) The population size Nt satisfies the recurrence equation Nt + 1 ¼ Nt Et ½wt  and can be computed as Nt ¼ N0 E0 ½Kt1  ¼ N0 M0 ½Gðt  1Þ: (2) The current distribution Pt(a) satisfies the recurrence equation Pt + 1 ð a Þ ¼

Pt ðaÞwt Et ½wt 

and can be computed as P t ð aÞ ¼

P0 ðaÞKt1 ðaÞ ; E0 ½Kt1 

15.3 Evolution of the main statistical characteristics of inhomogeneous maps

(3) Let ψ(a) be a random variable on the space (A, Pt). Then, Et ½ψ  ¼

E0 ½ψKt1  : E0 ½Kt1 

Proof Rewriting Eqs. (15.1) and (15.3) as

! n X lðt + 1, aÞ ¼ exp φi ðaÞgi ðtÞ , lðt, aÞ i¼1

we see that n X

lðt, aÞ ¼ lð0, aÞ exp

φi ð aÞ

! gi ðsÞ ¼ lð0, aÞKt1 ðaÞ:

s¼0

i¼1

Then,

t1 X

ð Nt ¼

lðt, aÞda ¼ N0 E0 ½Kt1  A

and Pt ðaÞ ¼

lðt, aÞ P0 ðaÞKt1 ðaÞ : ¼ Nt E0 ½Kt1 

Next, integrating the equality l(t + 1, a) ¼ wt(a)Pt(a)Nt over a, we obtain Nt + 1 ¼ Nt Et ½wi : The mean value of r.v. ψ(a) at time t is

ð

ð Et ½ ψ  ¼

ψ ðaÞPt ðaÞda ¼

ψ ðaÞKt1 ðaÞP0 ðaÞda A

E0 ½Kt1 

A

¼

E0 ½ψKt1  : E0 ½Kt1 

Specifically, Et ½wt  ¼ So,

E0 ½wt Kt1  E0 ½Kt  ¼ : E0 ½Kt1  E0 ½Kt1 

  Pt + 1 ð aÞ K t ð aÞ E0 ½Kt  wt ¼ : ¼ Pt ð aÞ Kt1 ðaÞ E0 ½Kt1  Et ½wt 

□ Denote

" kðaÞ ¼ lim t!∞ 1=t

t n X X s¼0

i¼1

!# φi ðaÞgi ðsÞ

299

300

15. Discrete-time selection systems

to be the average reproduction rate of an a-clone. Then, ! t X n X Kt ða1 Þ ¼ exp ðφi ða1 Þ  φi ða2 ÞÞgi ðsÞ  etðkða1 Þkða2 ÞÞ : Kt ða2 Þ s¼0 i¼1 The following corollary helps to understand the evolution of the distribution, and explains the Haldane principle within the framework of inhomogeneous maps. Let P0(a2) > 0. Then,  Pt ða1 Þ P0 ða1 Þ Kt1 ða1 Þ P0 ða1 Þ tðkða1 Þkða2 ÞÞ ¼ e : (15.12)  Pt ða2 Þ P0 ða2 Þ Kt1 ða2 Þ P 0 ð a2 Þ Hence the evolution of a heterogeneous population leads to a (exponentially fast) replacement of individuals with smaller values of k(a1) by those with larger values of k(a2), even though the fraction of the latter in the initial distribution was arbitrarily small. Let a∗ be a point of global maximum of k(a), and let P0(a∗) > 0; then, k(a) < k(a∗) implies Pt(a) ! 0. Therefore, any stationary or limit distribution (if it exists) should be concentrated in the set of points of global maximum of the average reproduction rate k(a) on the support of the initial distribution. This version of the Haldane principle was established in Semevsky and Semenov (1982). The dynamics of the distribution on bounded time intervals is also of interest and, perhaps, is even more important for applications than detailed mathematical description of limit distributions (which may not exist even for plain population models). Within the framework of the master model, the general case of the complete description of Pt(a) is given by Theorem 15.1. Dynamics and change over time of the initial distributions is of interest for many practical problems. The following lemma is critical for understanding the evolution of distributions. Lemma 15.1 Mt ½λ ¼ Proof

M 0 ½ λ + G ð t  1Þ  : M 0 ½ G ð t  1Þ 

! ! ð n n X X Kt1 ðaÞ P0 ðaÞda λi φi ðaÞ Pt ðaÞda ¼ exp λ i φ i ð aÞ Mt ½λ ¼ exp E 0 ½Kt1  A A i¼1 i¼1 ! ð n X 1 M0 ½λ + Gðt  1Þ : exp ðλi + Gi ðt  1ÞÞ φi ðaÞP0 ðaÞda ¼ ¼ E0 ½Kt1  A M0 ½Gðt  1Þ i¼1 ð

□

Let us start from the simplest but important case, when the random variables φi are independent at the initial time instant.

Theorem 15.2 Let random variables {φi, i ¼ 1,...... , n} be independent at the initial time instant; assume the initial pdf is pi0(xi) with the mgf-s Mi0(λi), so that p0 ðx1 , …, xn Þ ¼

n Y i¼1

pi0 ðxi Þ

15.3 Evolution of the main statistical characteristics of inhomogeneous maps

and M0 ðλ1 , …, λn Þ ¼

n Y

301

Mi0 ðλi Þ:

i¼1

Then, for any t 2 [0, T], their distributions have the mgf Mit ðλi Þ ¼

Mi0 ðλi + Gi ðtÞÞ Mi0 ðGi ðtÞÞ

and Mt ðλ1 , …, λn Þ ¼

n Y Mi ðλi + Gi ðtÞÞ 0

i¼1

Mi0 ðGi ðtÞÞ

:

Proof According to Lemma 15.1, M t ðλ Þ ¼

n M 0 ð λ + G ð t  1Þ Þ Y Mi0 ðλi + Gi ðtÞÞ ¼ : M 0 ð G ð t  1Þ Þ Mi0 ðGi ðtÞÞ i¼1

□

The evolution of the pdf of a random vector φ ¼ (φ1, … , φn) in a general case of correlated random variables {φi, i ¼ 1, … , n} is of great practical interest because it helps to explore the dynamics of an inhomogeneous population as it depends on the correlations between random variables φi(a). Let us call a class S of probability distributions of the random vector φ ¼ (φ1, … , φn) invariant with respect to the model, Eq. (15.3), if pð0, φÞ 2 S ) pðt, φÞ 2 S for all t: Let MS be the set of moment-generating functions for distributions from the class S. The following criterion of invariance immediately follows from Lemma 15.1. Criterion. A class S of pdfs is invariant with respect to models (15.1) and (15.3) if and only if M0 ðλÞ 2 MS )

M0 ðλ + GðtÞÞ 2 MS for all t: M0 ðGðtÞÞ

We can prove with the help of this criterion that many important distributions are invariant with respect to models (15.1) and (15.3). Let us recall some definitions (see Kotz et al., 2004). A random vector X ¼ ðX1 , …, Xn Þ has a multivariate normal distribution with the mean EX ¼ m ¼ ðm1 , …, mn Þ

302

15. Discrete-time selection systems

and a covariance matrix

if its mgf is

   C ¼ cij , cij ¼ cov Xi , Xj

    1 MðλÞ ¼ E exp λT X ¼ exp λT m + λT Cλ : 2

A random vector X ¼ (X1, … , Xn) has a multivariate polynomial distribution with parameters (k; p1, … , pn), if PðX1 ¼ p1 , …, Xn ¼ pn Þ ¼ P for ni¼1mi ¼ k. The mgf of the polynomial distribution is MðλÞ ¼

n X

k! n pm1 …pm n m1 !…mn ! 1

!k pi e

λi

:

i¼1

A general class of multivariate natural exponential distributions is very important for applications; this class includes multivariate polynomial, normal, and Wishart distributions as special cases. A random n-vector X ¼ (X1, … , Xn) has multivariate natural exponential distribution (NED) with parameters θ ¼ (θ1, … , θn) with respect to the positive measure ν on Rn if its joint density function is of the form fθ ðXÞ ¼ hðXÞeX

T

θsðθÞ

,

(15.13)

where s(θ) is a function of parameters, and the generating measure μðdXÞ ¼ hðXÞνðdXÞ is not concentrated on any affine hyperplane of Rn. The mgf of NED (15.13) is    MðλÞ ¼ Ev exp λT X ¼ exp ðsðθ + λÞ  sðθÞÞ: For any NED, ∂ sðθÞ ¼ mi , ∂θi and   ∂2 sðθÞ ¼ cov Xi , Xj : ∂θi ∂θj Theorem 15.3 Let us assume that in the initial time moment, the random vector φ ¼ (φ1, , φn) has (i) multivariate normal distribution with the mean vector m(0) and covariance matrix C ¼ (cij). Then at any time t the vector φ also has the multivariate normal P distribution with the same covariance matrix C and the mean vector m(t), mi(t) ¼ mi(0) + 1/2 nk¼1(cik + cki)Gk(t  1);

303

15.3 Evolution of the main statistical characteristics of inhomogeneous maps

(ii) multivariate polynomial distribution. Then at any moment t the vector φ has the multivariate p eGi ðt1Þ polynomial distribution with parameters (k; p1(t), , pn(t)), where pi ðtÞ ¼ Pni Gj ðt1Þ ; pe j¼1 j

(iii) multivariate natural exponential distribution on Rn (15.13) with parameters θ ¼ (θ1,......... , θn). Then at any moment t the vector φ also has the multivariate NED with parameters θ + G(t  1) and the moment-generating function Mt ðλÞ ¼ exp ðsðθ + λ + Gðt  1ÞÞ  sðθ + Gðt  1ÞÞ: Proof (i) The mgf of the initial distribution of the vector φ is M0 ½λ ¼ eλ

T

mð0Þ + 1=2λT Cλ

;

due to Lemma 15.1, Mt ðλÞ  ¼ M0 ðλ + Gðt  1ÞÞ=M0 ðGðt  1ÞÞ ¼

¼ exp ðλ + Gðt  1ÞÞT mð0Þ + 1=2ðλ + Gðt  1ÞÞT Cðλ + Gðt  1ÞÞ  Gðt  1ÞT mð0Þ + 1=2Gðt  1ÞT CGðt  1Þ ¼ 0 1 ! n n n X X X ¼ exp @ λi mi ð0Þ + 1=2 ðcik + cki ÞGk ðt  1Þ + 1=2 λi cik λk A i¼1 k¼1 i, k¼1

This is the mgf of the desired multivariate normal distribution. (ii) If vector φ has the polynomial distribution at the initial time moment, then M0 ½λ ¼

n X

!k pi eλi

:

i¼1

Due to Lemma 15.1, n X

Mt ðλÞ ¼

!k pi exp ðλi + Gi ðt  1ÞÞ

i¼1 n X

!k pi exp ðGi ðt  1ÞÞ

¼

n X

!k pi ðtÞe

λi

i¼1

i¼1

with p eGi ðt1Þ pi ðtÞ ¼ Pn i Gk ðt1Þ , where Mt(λ) is the mgf of desired multivariate polynomial distribution. pe k¼1 k

(iii) If vector φ has at the initial time the multivariate natural exponential distribution with the mgf M0 ðλÞ ¼ esðθ + λÞsðθÞ ,

304

15. Discrete-time selection systems

then Mt ðλÞ ¼ □

esðθ + λ + Gðt1ÞÞsðθÞ ¼ esðθ + λ + Gðt1ÞÞsðθ + Gðt1ÞÞ : esðθ + Gðt1ÞÞsðθÞ

Theorem 15.3 states that multivariate normal, polynomial, Wishart, and natural exponential distributions are invariant with respect to the master model. By contrast, the multivariate uniform distribution is an example of a distribution that is not invariant. Definition Let S be a bounded Borel set in Rn and mesS be its Lebesgue measure. A random vector X ¼ (X1, … , Xn) has a multivariate uniform distribution in S if mesS > 0; its pdf P(X) ¼ 1/mesS if X 2 S and is 0 otherwise. The moment-generating function of the uniform distribution in the case of rectangle S ¼ {bi  ai  ci, i ¼ 1, … , n} is n  Q

e λi c i  e λi b i

Mðλ1 , …, λn Þ ¼ i¼1n Q

ðci  bi Þλi

 :

i¼1

Assume that vector φ ¼ (φ1, … , φn) has at the initial moment a multivariate uniform distribution in a rectangle. Then Lemma 15.1 implies that n Q

ð exp ððλi + Gi ðt  1ÞÞci Þ  exp ððλi + Gi ðt  1ÞÞbi ÞÞ

M½λ1 , …, λn  ¼  ni¼1 Q

:

ð1 + λi =Gi ðt  1ÞÞexp ðGi ðt  1Þci Þ  exp ðGi ðt  1Þbi Þ

i¼1

So at any time t > 0, vector φ has not the uniform but the multivariate truncated exponential distribution in this rectangle.

15.4 Self-regulated inhomogeneous maps The theory developed in Section 15.3 for population models with the fitness of the form of Eq. (15.3) can be applied only if the time-dependent components gi(t) are known explicitly. As a rule, this is not the case for most interesting and realistic models, where the time-dependent component should be computed according to the current population characteristics. For example, the logistic-type model that describes deterioration of the environment due to population increase corresponds to the function g(Nt) ¼ 1  Nt/C, where C is the carrying capacity; as another example, the Ricker model corresponds to the function g(Nt) ¼ λe βNt. In general, population regulation (change of environment due to population growth) may be defined not only by the total population size but also by the so-called regulators, which are averages over population density ð Si ðtÞ ¼ si ðaÞlðt, aÞda (15.14) A

305

15.4 Self-regulated inhomogeneous maps

or population frequency

ð Hi ðtÞ ¼

hi ðaÞPðt, aÞda,

(15.15)

A

where si(a), hi(a) are appropriate functions. For example, if si(a) ¼ hi(a) is the biomass of an individual with parameter a, then H(t) is the average biomass of individual, S(t) is the total population biomass, and the population growth rate may depend on S(t) ¼ NtH(t). Total population size Nt is also a regulator with s(a) ¼ 1. Note that there exists a simple relationship between regulators (15.14) and (15.15): ð ð sðaÞlðt, aÞda5Nt sðaÞPðt, aÞda: A

A

Nevertheless, it may be useful in some situations to distinguish between the “densitydependent” (15.14) and “frequency-dependent” (15.15) regulators. So, let us specify the theory developed in Section 15.3 to the model lðt + 1, aÞ ¼ lðt, aÞwt ðaÞ 0 1 n m X X   wt ðaÞ ¼ exp @ ui ðSi ðtÞÞϕi ðaÞ + vj H j ð t Þ ψ j ð aÞ A , i¼1

(15.16)

j¼1

which we will refer to as the self-regulated inhomogeneous population model. Here the individual fitness wt(a) may depend on some density-dependent and frequency-dependent regulators, Si(t) and Hj(t). The initial distribution l(0, a) of individuals over the vector parameter a is assumed to be given. The main new problem is that the values of regulators are not given but should be computed at each point in time. Now, t-fitness given by Eq. (15.9) is equal to 0 0 11 t t n m X Y X X   @ Kt ðaÞ ¼ wk ðaÞ ¼ exp @ ui ðSi ðkÞÞφi ðaÞ + vj Hj ðkÞ ψ j ðaÞAA: k¼0

k¼0

i¼1

j¼1

Let f(a) be a (measurable) function on A and λ ¼ (λ1, … , λn), δ ¼ (δ1, … , δn). For a given initial distribution P0(a), introduce the functional Φ(f; λ, δ) such that 0 1 ð n m X X Φð f ; λ, δÞ ¼ f ðaÞexp @ λi φi ðaÞ + δj ψ j ðaÞAP0 ðaÞda: (15.17) A

i¼1

j¼1

This functional, if known, can help compute the values of all regulators. (Notice that this functional is a generalization of the moment-generating function M0 introduced earlier, M0[λ, δ] ¼ Φ(1; λ, δ).) Denote t X ðu1 ðS1 ðkÞÞ, …, un ðSn ðkÞÞÞ, SðtÞ ¼ k¼0

HðtÞ ¼

t X k¼0

ðv1 ðH1 ðkÞÞ, …, vm ðHm ðkÞÞÞ:

306

15. Discrete-time selection systems

Then the following useful formula is valid:   E0 fKt ¼ Φð f ; SðtÞ, HðtÞÞ:

(15.18)

Theorem 15.4 Let P0(a) be the initial distribution for inhomogeneous self-regulated model (15.16) and Φ(f; λ, δ) be the corresponding functional (15.17). Then the total population size and the regulators can be computed recurrently using the following equations: Nt ¼ N0 E0 ½Kt1  ¼ N0 Φð1; Sðt  1Þ, Hðt  1ÞÞ; Si ðtÞ ¼ N0 E0 ½si Kt1  ¼ N0 Φðsi ; Sðt  1Þ, Hðt  1ÞÞ;     E0 hj Kt1 Φ hj ; Sðt  1Þ, Hðt  1Þ ¼ Hj ðtÞ ¼ : E0 ½Kt1  Φð1; Sðt  1Þ, Hðt  1ÞÞ

(15.19)

(15.20)

Proof Let N∗t , S∗i (t), H∗j (t) solve the recurrent systems (15.19)–(15.20) for given N0, l(0, a), P0(a), and henceforth be known as S(0) and H(0). Define 0 1 n m 

X X   w∗t ðaÞ ¼ exp @ ui S∗i ðtÞ ϕi ðaÞ + vj Hj∗ ðtÞ ψ j ðaÞA i¼1

Kt ðaÞ ¼

t Y

j¼1

w∗k ðaÞ

k¼0

  ∗ ðaÞ=E K ∗ P∗t ðaÞ ¼ P0 ðaÞKt1 0 t1 l∗ ðt, aÞ ¼ P∗t ðaÞNt∗ Let us prove that Nt∗ , S∗i ðtÞ, Hj∗ ðtÞ, P∗t ðaÞ, l∗ ðt, aÞ satisfy Systems (15.14)–(15.16). This assertion is clearly valid at t ¼ 0; assume that it is valid at t  1. Then, w∗t1 ðaÞ ¼ wt1 ðaÞ and ∗ ðaÞ ¼ K Kt1 t1 ðaÞ:

Next, ∗ w∗ ðaÞ ¼ lðt, aÞ ¼ lðt  1, aÞwt1 ðaÞ ¼ l∗ ðt  1, aÞw∗t1 ðaÞ ¼ P∗t1 ðaÞNt1 t1 ∗ ðaÞ ¼ lð0, aÞK ∗ ðaÞ ¼ l∗ ðt, aÞ; ¼ P0 ðaÞN0 Kt1 t1 ð ð ð   ∗ ðaÞda ¼ N E K ∗ ∗ Nt ¼ lðt, aÞda ¼ lð0, aÞKt1 ðaÞda ¼ lð0, aÞKt1 0 0 t1 ¼ Nt A A A ð ð   ∗ ðaÞda ¼ N E sK ∗ ∗ SðtÞ ¼ sðaÞlðt, aÞda ¼ sðaÞlð0, aÞKt1 0 0 t1 ¼ S ðtÞ A A ð ð    ∗  ∗ ðaÞda=N ∗ ¼ E hK ∗ ∗ HðtÞ ¼ hðaÞPt ðaÞda ¼ hðaÞlð0, aÞKt1 0 t t1 =E0 Kt1 ¼ H ðtÞ A

□

A

15.5 The Price equation and the Fisher fundamental theorem for maps

307

15.5 The Price equation and the Fisher fundamental theorem for maps We discussed the Price equation and the Fisher fundamental theorem in Chapter 5 within the frameworks of selection systems with continuous time. Initially, however, the Price equation was formulated for systems with discrete time. For the sake of completeness, let us derive the Price and Fisher equations within the framework of general discrete-time model (15.2). The simplest version of the FTNS in terms of population genetics is as follows. Let N be the population size, ni be the number of alleles Ai, pi ¼ ni/N be the frequency, and wi be the “fitness” of the ith allele. Then, n0i ¼ wi ni ,

(15.21)

where primes denote onePtime step into the future, and where wi is the fitness of alleles Ai. The mean fitness is E[w] ¼ ipiwi, and its variance is Var[w] ¼ E[w2]  E2[w]. It follows from Eq. (15.21) that p0i ¼ wi pi =E½w and ΔE½w

X

 p0i wi  pi wi ¼ Var½w=E½w:

(15.22)

i

So, within the framework of model (15.21), the FTNS (15.22) is a very simple mathematical assertion, which is not specific to genetic selection. Notice that model (15.21) actually describes the dynamics of a subdivided (inhomogeneous) population with fitness distributed over subpopulations; the subpopulations are composed of individuals that have the same fitness. Then assertion (15.22) is exactly Li’s theorem (Li, 1967): in a subdivided population, the rate of change of the overall growth rate is proportional to the variance in the growth rates of the subpopulations. Now, let us derive the Price equation for inhomogeneous maps (15.1)–(15.2). Let zt(a) be a character of an individual with the given vector parameter a, which can vary with time. Then, Et + 1 ½ z t  ¼

Et ½zt wt  Et ½wt Δzt  , Et + 1 ½Δzt  ¼ Et ½wt  E t ½w t 

and Et ½wt ΔEt ½zt  ¼ Et ½wt ðEt + 1 ½Δzt  + Et + 1 ½zt   Et ½zt Þ ¼ Et ½wt Δzt  + Et ½zt wt   Et ½zt Et ½wt  : ¼ Et ½wt Δzt  + Covt ½zt wt  We have now obtained the second, or complete, Price equation: ΔEt ½zt  ¼

Covt ½zt wt  + Et ½wt Δzt  : Et ½wt 

(15.23)

If the character z does not depend on t, that is, Δzt ¼ 0, then the second Price equation implies the first Price equation, also known as the covariance equation (Li, 1967; Robertson, 1968; Price, 1970): Covt ½zwt  ΔEt ½z ¼ : Et ½wt 

308

15. Discrete-time selection systems

The difference between mean values of the trait after and before selection (at t time moment) is ΔEt ½zt  ¼ Et + 1 ½zt + 1   Et ½zt ; it is known as the selection differential and is an important characteristic of selection process. The covariance equation and the Price equation show the relationship between selection differential and fitness. If zt ¼ wt, then Vart ½wt  + Et ½wt Δwt  , (15.24) ΔEt ½wt  ¼ Et ½wt  which is the FTNS for time-dependent fitness. If the fitness does not depend on time, that is, if Δwt ¼ 0, then ΔEt ½wt  ¼

Vart ½wt  , Et ½wt 

(15.25)

which is the standard form of FTNS. Price (1972) claimed that his equation is the exact, complete description of evolutionary change under all conditions, in contrast to the FTNS and covariance equation, where the “environment” is fixed. It is worth noting that the Price approach did not consider the effects of mutations; indeed, it follows from equation ni ðtÞ ¼ wi ðt  1Þni ðt  1Þ ¼ ni ð0Þ

t1 Y

w i ðsÞ

s¼0

that the subpopulation ni(t) is composed of individuals of type i that are derived from type i individuals at time 0. A more general “replicator-mutator” version of the Price equation was derived in Page and Nowak (2002). The Price equation was applied not only to biological problems, such as evolutionary genetics, sex ratio, and kin selection (see Rice, 2004, Chapter 6; Crow and Nagylaki, 1976, etc.), but also to social evolution (Frank, 1998), evolutionary economics (Knudsen, 2004), etc. We discussed in Chapter 5 the problem of dynamical insufficiency of the Price equation: to calculate the dynamics of a mean trait with the help of the Price equation alone, we need to solve the equation for covariance, which in turn includes moments of higher order. In general, this is impossible unless higher moments are expressed in terms of lower moments (see, e.g., Barton and Turelli, 1987; Frank, 1997). The Price equation does not allow one to predict changes in the mean of a trait beyond immediate response if one knows only the value of covariance of the trait and fitness at this moment. To make this equation a useful tool in the field of mathematical biology, one needs to overcome the problem of its dynamic insufficiency. In practice, it means that some quantities in the Price equation should be calculated independently of others. We have already shown how the problem can be resolved within the framework of selection systems with continuous time. The theory developed in Sections 15.3 and 15.4 allows us to resolve the problem of dynamical insufficiency of the Price equation and the FTNS for discrete-time systems. For master model (15.1) and for the self-regulated model (15.16), all

309

15.5 The Price equation and the Fisher fundamental theorem for maps

statistical characteristics of interest can be computed effectively given the initial distribution. Specifically, we can compute the mean value of any trait and in this sense solve the Price equation, the covariance equation, and the equation of the FTNS. Proposition 15.1 (On the complete Price equation) (i) For master models (15.1) and (15.3) with known initial distribution, the solution to the Price equation (15.23) is given by the formula Et ½zt  ¼ E0 ½zt Kt1 =E0 ½Kt1 :

(15.26)

(ii) For self-regulated model (15.16), the solution of the Price equation is given by the formula Et ½zt  ¼ Φðzt ; Sðt  1Þ, Hðt  1ÞÞ=Φð1; Sðt  1Þ, Hðt  1ÞÞ:

(15.27)

Current values of regulators S(t), H(t) can be computed recursively with the help of Theorem 15.4, using Eqs. (15.19)–(15.20). Indeed, equality (15.26) was proven earlier. Applying Eq. (15.18), we obtain Eq. (15.27). We assumed in Proposition 15.1 that all terms in the right-hand sides of (15.26) and (15.27) are well defined and finite. Under this assumption, all terms of the Price equation can be computed explicitly: (i) For models (15.1)–(15.3), Covt ½wt zt  E0 ½zt Kt  E0 ½zt Kt1  ¼  Et ½ w t  E0 ½Kt  E0 ½Kt1  Et ½wt Δzt  E0 ½Δzt Kt  ¼ : Et ½wt  E0 ½Kt  (ii) For model (15.16), Covt ½wt zt  Φðzt ; SðtÞ, HðtÞÞ Φðzt ; Sðt  1Þ, Hðt  1ÞÞ  ¼ Et ½wt  Φð1; SðtÞ, HðtÞÞ Φð1; Sðt  1Þ, Hðt  1ÞÞ Et ½wt Δzt  ΦðΔzt ; SðtÞ, HðtÞÞ : ¼ Et ½wt  Φð1; SðtÞ, HðtÞÞ The FTNS is a special case of the Price equation when zt ¼ wt. Proposition 15.2 (On the Fisher FTNS) (i) For master models (15.1)–(15.3) with a known initial distribution, the solution of the FTNS equation (15.24) is given by the formula Et ½ w t  ¼

E0 ½Kt  : E0 ½Kt1 

If M0[λ] is the mgf of the initial distribution of φ ¼ (φ1, … , φn), then (in notation of Section 15.4)

310

15. Discrete-time selection systems

E t ½w t  ¼

M0 ½GðtÞ : M0 ½Gðt  1Þ

(ii) For self-regulated model (15.16), the solution of the FTNS equation is Et ½wt  ¼

Φð1;SðtÞ, HðtÞÞ : Φð1;Sðt  1Þ;Hðt  1ÞÞ

(15.28)

The current values of regulators S(t), H(t) can be computed recursively using Theorem 15.4. All the terms of the FTNS equation can also be computed explicitly. Let us stress that Eqs. (15.27) and (15.28) define the solutions of the Price and FTNS equations for self-regulated selection systems by explicit recursive procedures; these procedures can be easily realized computationally and sometimes can even yield solutions in analytical form. Let us summarize the discussion of the Price and Fisher equations. • The FTNS in the form “the rate of increase in fitness of any organism at any time is equal to its total variance in fitness at that time” is valid as a mathematical assertion for a broad class of models of inhomogeneous populations, where fitness does not change over time. • The FTNS in the form that allows dependence of fitness on time is a special case of the full Price equation, which is valid under quite general conditions. • The Price equation is not dynamically sufficient; it is a mathematical identity within the framework of a corresponding model and hence cannot be “solved”; it cannot predict population dynamics beyond the immediate response without additional assumptions. • The developed theory of selection systems with discrete time allows us to compute effectively the distribution of the system and all statistical characteristics of interest at any time. In particular the problem of dynamic insufficiency for the Price equations can be resolved if the initial distribution of the parameters is known.

15.6 Applications and examples It is well known that nonlinear population models with discrete time (maps) can realize very complex and even counterintuitive behaviors depending on the values of model parameters. The main dynamical regimes of the corresponding inhomogeneous models are determined by the behaviors of the original homogeneous models but have some essentially new interesting peculiarities due to “inner bifurcations,” that is, changes that arise because of internal dynamics of the system.

15.6.1 Ricker’ model with discrete time The classical Ricker model Nt+1 ¼ Ntλ exp( bNt), where λ and b are positive parameters, takes into account population dependence of the reproduction rate on the population size. Let us consider the inhomogeneous Ricker model with a single distributed parameter a:

15.6 Applications and examples

lðt + 1, aÞ ¼ lðt, aÞλ0 aebNt ,

311 (15.29)

where the fitness is w(a, Nt) ¼ λ0ae bNt, λ0 is the scaling multiplier and b > 0 is a constant. According to Eqs. (15.5)–(15.6), the inhomogeneous Ricker model takes the form Nt + 1 ¼ Et ½wNt   λ0 E0 at + 1 bNt e E t ½w  ¼ : E0 ½ a t 

(15.30)

Let the initial distribution of a be Γ-distribution with parameters (s, k). Then, as it was proven in Section 15.3, Pt(a) is again Γ-distribution with parameters (s, k + t), and  k + t bNt e Et ½w ¼ λ0 s k + t bNt N t + 1 ¼ N t λ0 : e s   These formulas show that the coefficient λ0 k +s t of the inhomogeneous Ricker model, which determines the dynamics of the model, increases indefinitely with time. As a result, according to the theory of the plain homogeneous Ricker model, after a period of monotonic increase, cycles of period 2, then cycles of period 4, and then almost all cycles of Feigenbaum’s cascade (see, e.g., Devaney, 2018; Wiggins, 2003, pp. 384–386) appear and realize as parts of a single trajectory; see Fig. 15.1. Hence, after a time, we observe that the population size begins to oscillate with increasing amplitude. Practically it means that the population goes to extinction over time because there exist points in time when its size happens to be ≪ 1. The reason for the appearance of these remarkable nonclassical trajectories shown in Fig. 15.1 is as follows. If parameter λ0 is small and/or s is large, then the sequence   k + t λ0 s , t ¼ 0, 1, … takes values close to all bifurcation values of the coefficient of the Ricker model. It follows that a notable phenomenon, the “almost complete” sequence (with the step λ0/s) of all possible bifurcations of the homogeneous Ricker model is realized within the framework of a unique inhomogeneous Ricker model. The trajectory {Nt}∞ 0 in some sense mimics the bifurcation diagram of the plain Ricker model; see Fig. 15.1. We would like to emphasize that Fig. 15.1 shows a trajectory of model (15.30) such that to each value of t corresponds a single value of Nt; for clarification, the enlarged section of the graph is given in Fig. 15.2. The process of evolution of the population, described by the model, goes through different stages with the speed that depends on s. Let us explore the evolution of the mean reproduction rate (mean fitness) of the inhomogeneous Ricker model given by Eq. (15.30). Since fitness depends explicitly on the total population size, it may not increase monotonically over time (in contrast to the plain FTNS and in accordance with the full Price equation). Indeed, this is the case for model (15.29). The dynamics of mean fitness (15.30) together with the total population size for the inhomogeneous Ricker model with initial Γ-distribution of the parameter is shown on Fig. 15.1. We can see that after periods of increase and stable behavior, the mean fitness starts to oscillate with increasing amplitude, as does the total population size.

312

15. Discrete-time selection systems

FIG. 15.1 The trajectory of total population size (top) and mean fitness (bottom) for the inhomogeneous Ricker model with Γ-distributed parameters a (λ0 ¼ 1, E0[a] ¼ 3, Var0[a] ¼ 0.1). Adapted from Karev, G.P., 2008. Inhomogeneous maps and mathematical theory of selection. J. Differ. Equ. Appl. 14, 31–58.

FIG. 15.2 An enlarged section of the model trajectory given in the upper panel of Fig. 15.1. Adapted from Karev, G.P., 2008. Inhomogeneous maps and mathematical theory of selection. J. Differ. Equ. Appl. 14, 31–58.

15.6 Applications and examples

313

A similar phenomenon is observed for any initial distribution of the parameter with unbounded support, for example, for log-normal distribution. Another type of behavior is observed if the initial distribution has bounded support, that is, the parameter can take any value from a bounded set A. Let P0(a) be the Beta distribution in [0,1] with parameters α, β. Then Pt(a) is again the Beta distribution with parameters α + t, β. Hence,  t+α Et ½w ¼ λ0 ebNt t+ α + β k+α Nt + 1 ¼ Nt λ0 ebNt : t+α+β Choosing an appropriate value of λ0, we can observe any possible behavior of the model as its final dynamical behavior. Fig. 15.3 illustrates this assertion.

FIG. 15.3 The trajectory of total population size and mean fitness for the inhomogeneous Ricker model with Beta distributed parameter (λ0 ¼ 14.5, E0[a] ¼ 0.1, Var0[a] ¼ 0.02). The final dynamics of the model is eight-cycle. Adapted from Karev, G.P., 2008. Inhomogeneous maps and mathematical theory of selection. J. Differ. Equ. Appl. 14, 31–58.

314

15. Discrete-time selection systems

15.6.2 Inhomogeneous logistic map A well-known logistic map is of the form Nt+1 ¼ λaNt(1  Nt), 0 < λ < 4, and 0  Nt  1. Consider the inhomogeneous logistic model lðt + 1,aÞ ¼ λalðt, aÞð1  Nt Þ, where λ ¼ const and a is the distributed parameter; then, wt(a) ¼ λa(1  Nt). The inhomogeneous logistic model takes the form Nt + 1 ¼ Et ½wNt   λð1  Nt ÞE0 at + 1 : Et ½w ¼ E 0 ½at  λE0 ½at + 1  The model makes sense only if 0 < E0 ½at  < 4. Let P0(a) be the Beta distribution in [0,1] with parameters α, β. Then,  t+α ð1  Nt Þ: Et ½w ¼ λ t+α+β Choosing an appropriate value of 0 < λ < 4, we can observe (as t ! ∞) any possible behavior of the plain logistic model as final dynamical behaviors of the inhomogeneous logistic model. Specifically, at λ ¼ 4, almost all cycles of Feigenbaum’s cascade appear over time and become realized as parts of a single trajectory, as a result of “inner bifurcations” of the inhomogeneous logistic model. Fig. 15.4 illustrates this assertion and also showcases the complex behavior of the mean fitness (which is very different from the plain FTNS).

15.6.3 Inhomogeneous Ricker map with two distributed parameters The Ricker model takes into account the regulation of the reproduction rate of a population in a way that is more appropriate compared with the logistic map. Consider the inhomogeneous version of the Ricker model with discrete time and two distributed parameters a ¼ lnλ and b: lðt + 1; a, bÞ ¼ lðt; a, b,Þwða, b, Nt Þ, where wða, b, Nt Þ ¼ eabNt :

Then, Kt ðaÞ ¼ and

t Y k¼0

wk ðaÞ ¼ exp ðt + 1Þa  b

t Y

! Nt

k¼0

lðt + 1; a, bÞ ¼ lð0; a, bÞexp ðt + 1Þa  b

t X k¼0

Compared with Eq. (15.16), for this example we should set sðaÞ ¼ 1, u1 ðxÞ ¼ 1, u2 ðxÞ ¼ x,

! Nt :

315

15.6 Applications and examples

FIG. 15.4 The trajectory of total population size and mean fitness for inhomogeneous logistic model with Betadistributed parameter a (λ ¼ 4; E0[a] ¼ 0.1, Var0[a] ¼ 0.005). Adapted from Karev, G.P., 2008. Inhomogeneous maps and mathematical theory of selection. J. Differ. Equ. Appl. 14, 31–58.

so that SðtÞ ¼ Nt , u1 ðSðtÞÞ ¼ 1, u2 ðSðtÞÞ ¼ Nt : Let

ð

eλ1 a + λ2 b P0 ða, bÞdadb

M 0 ½λ1 , λ2  ¼ A

be the mgf of the initial joint distribution of parameters a and b. Then, ! t X E0 ½Kt  ¼ M0 t + 1,  Nk : k¼0

Applying Theorem 15.1, we obtain Nt ¼ N0 M0 t + 1, 

t X k¼0

! Nk ,

(15.31)

316

15. Discrete-time selection systems

Pt ða, bÞ ¼ P0 ða, bÞ exp at  b

t1 X

!, M0 t, 

Nk

k¼0

t1 X

! Nk :

k¼0

These formulas completely solve the inhomogeneous Ricker model. The selection differential for the model is ΔEt ½wt  ¼

E0 ½Kt + 1  E0 ½Kt   , E0 ½Kt  E0 ½Kt1 

where E0[Kt] for given initial distributions can be computed recurrently by Eq. (15.31).

15.6.4 Selection in natural rotifer community A mathematical model of zooplankton populations was suggested in Snell and Serra (1998) and studied systematically in Berezovskaya et al. (2005). The model depends on parameters a, which characterizes quality of the environment, and γ, which is a species-specific parameter. The model takes the form Nt + 1 ¼ Nt wðNt , a, γ Þ,  wðNt , a, γ Þ ¼ exp a + 1=Nt  γ=Nt2 : Let us consider now a model of a community that consists of different rotifer populations; individuals in these populations may have different reproduction capacities subject to constant toxin exposure in their environment. The model takes the form lðt + 1, a, γ Þ ¼ lðt, a, γ ÞwðNt , a, γ Þ, and Kt ða, γ Þ ¼ exp ðt + 1Þa + 1=

t X

Nk  γ=

k¼0

In this example, we set sð1, γ Þ ¼ 1; φ1 ða, γ Þ ¼ a, u1 ðxÞ ¼ 1, φ2 ða, γ Þ ¼ γ, u2 ðxÞ ¼ 1=x2 so that S1 ðtÞ ¼ S2 ðtÞ ¼ Nt , u1 ðS1 Þ ¼ 1, u2 ðS2 ðtÞÞ ¼ 1=Nt2 :

t X k¼0

! N2 k :

317

15.6 Applications and examples

Let M0(λ1, λ2) be the mgf of the initial distribution of a and γ. Then, ! ! t t X X 2 E0 ½Kt  ¼ exp 1= Nk M0 ðt + 1Þ  1= Nk k¼0

Nt ¼ N0 exp 1=

t1 X

k¼0

! Nk M0 t  1=

k¼0

t1 X

!

Nk2

k¼0

! ! t1 t1 X X 2 2 Pt ða, γ Þ ¼ P0 ða, γ Þ exp ta  γ= Nk =M0 t  1= Nk : k¼0

k¼0

These equations completely solve the inhomogeneous model of rotifer community. A simpler case when only parameter a is distributed was studied in Karev et al. (2008). Even in this case the trajectory NtEt[a] has a very complex transitional regime as the population evolves from initial to final state (see Fig. 15.5). The outcome of population dynamics depends dramatically on the initial variance of parameter a. The trajectory Nt has a very complex transitional regime from initial to final behavior; see Fig. 15.5 (independent parameters a, γ are both Γ-distributed).

FIG. 15.5

The trajectory of the total population size and mean fitness for inhomogeneous rotifer model with Γ-distributed parameter a; E0[a] ¼ 5.2; the value of parameter γ ¼ 0.044 is fixed. (A) and (C) the initial variance is Var0[a] ¼ 0.035 and the population reaches a stable asymptotical state; (B) and (D) the initial variance is Var0[a] ¼ 0.026 and the population becomes extinct as it is being trapped in a domain of attraction of 0. Adapted from Karev, G.P., 2008. Inhomogeneous maps and mathematical theory of selection. J. Differ. Equ. Appl. 14, 31–58.

318

15. Discrete-time selection systems

15.7 Discussion In this chapter, we contribute to the general theory of selection systems with discrete time and develop methods for studying evolution of such systems in detail. The Fisher fundamental theorem of natural selection, the Price equations and the Haldane principle are well-known general results of mathematical selection theory. We explored these assertions within the framework of a general class of selection systems with discrete time; we obtained the main results as consequences of the explicitly derived equation for the distribution of system parameters. In 1970s Price tried to find a general formula that could be applied to any (not necessarily biological) problem of selection with the ultimate goal of developing a formal theory. The Price equation was an outstanding contribution to the future theory but its “dynamical insufficiency” does not allow one to predict changes in the mean of a trait beyond the immediate response. However, both the Price equation and FTNS are mathematical identities, and therefore their “solutions” cannot be approached in the same way as the solutions of typical equations. The only way to predict the dynamics of a trait over a long period of time using the Price equation is to compute all the values on the right-hand side of the equation independently of the left-hand side for all time moments of interest. In general, it can be done only if the entire distribution during the total time interval is known or can be computed (and then the Price equation is not necessary). Examples considered in this chapter show the differences in the global dynamics of a selection system as impacted by the initial distribution. Note that the Price equation does not depend on the initial distribution but only on the mean and covariance of the trait and fitness at a given instant. Hence the Price equation describes a general instant property of any selection system, which does not depend on the global dynamics of the system. This independence is the reason for the theoretical universality and restricted practical utility of the Price equation taken alone. The Haldane optimal principle can be considered as one of the first general assertions about selection systems. This principle describes asymptotical behavior of population composition; a version of this principle is given here in Section 15.3. The composition of a population in a stable equilibrium is such that the population is concentrated at the points of global maximum of the mean reproduction coefficient. The system “forgets” all peculiarities of its previous dynamics. The asymptotical composition of general “systems with inheritance” driven by selection was explored in mathematical detail in Gorban (2007). So the theory in its most recent formulation allows one to predict behavior of selection systems only at the first time step and “at infinity” (if the limit distribution exists and is stable). Let us emphasize that the current dynamics of the population and its distribution during protracted but finite time intervals is also of interest and, perhaps, is of primary importance in applications. In this chapter, we developed methods that allow us to determine the current distribution of a selection system with discrete time. Specifically, formulas for computation of the mean of any trait at any time moment were derived for a given initial distribution; these mean values, of course, satisfy the Price equation and in this sense give its “solution.” This approach provides a way to resolve the problem of dynamical insufficiency of the Price equation and of the FTNS. We then applied the developed approach to several examples, which showcased complex behavior of the total population size and the mean fitness for inhomogeneous populations with size-dependent fitness.

15.7 Discussion

319

The developed theory can be applied to a wide class of inhomogeneous population models with discrete time and distributed parameters. Model behavior may be different and even counterintuitive even for simplest linear maps depending on the initial distribution. Nonlinear inhomogeneous models of self-regulated populations show complex dynamical behavior; their trajectories can mimic parts of the model bifurcation diagram. These models can have extremely complex transitional regimes as the populations they describe evolve from initial to final behavior. For instance, all cycles of the Feigenbaum’s cascade and even chaotic behavior can appear over time and become realized as part of a single trajectory due to “inner bifurcations” of the inhomogeneous model. Additionally, any possible behavior of the corresponding “homogeneous” model can be observed as final behavior (at t ! ∞) with appropriate initial distribution. The obtained results may be useful for understanding the dynamic peculiarities of inhomogeneous maps and the crucial role of initial distributions; we hope that theorems and methods presented here can help investigations (both analytical and numerical) of inhomogeneous self-regulated population models with discrete time, which appear in different areas of mathematical biology.

C H A P T E R

16 Conclusions Abstract In this book, we discussed and applied a powerful method for solving a wide class of selection systems and replicator equations called the hidden keystone variables, or HKV method. It allows reducing complex inhomogeneous models to “escort” systems of ODEs for auxiliary “keystone” variables that in many cases can be investigated analytically. Noticeably, even if an analytical solution to the escort system is not available, numerically solving the resulting ODE system is much simpler than studying the initial system.

In this book, we showed applications of the method to a variety of problems, ranging from global demography, to evolution of altruism, to time perception in a dying brain, to the tragedy of the commons, to oncolytic virus therapy for heterogeneous populations of cancer cells. We dove into discussion of the Principle of minimal information gain, using HKV method to show that it is the underlying variational principle that governs replicator dynamics. We showed application of the method to game theory, including strategy selection both within and between games. Finally, we showed that the method can be applied to both continuous and discrete systems, and demonstrated how incredibly rich the dynamics can become for inhomogeneous maps. The HKV method allows computing, often explicitly, all statistical characteristics of interest (such as mean or variance of a parameter that represents an evolving trait of interest), making it possible to visualize evolutionary trajectories of a heterogeneous population under selection over time. When combined with classical tools of dynamical system analysis, such as bifurcation theory, this “travel” through the bifurcation diagram can reveal new, rich, complex and unexpected behaviors that can help answer new evolutionary questions in biology. It is our hope that we have provided here both a strong theoretical foundation and an abundant number of examples to both learn from and take forward in future research. “Nothing in biology makes sense except in light of evolution,” and the HKV method allows shedding a little more light into the study of this complicated and exciting world.

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00016-3

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# 2020 Elsevier Inc. All rights reserved.

C H A P T E R

17. Math Appendix Moment-generating functions for various initial distributions Abstract This is supplementary material, which summarizes formulas for moment-generating functions for various initial distributions that may be useful for calculations using the HKV method.

17.1 Distributions on the entire line or half line Exponential distribution pðxÞ ¼ sesx , 0  x < ∞, M½λ ¼

s , λ < s, sλ

1 1 m ¼ , σ2 ¼ 2 : s s Normal distribution

! 1 ðx  m Þ2 pðxÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp  , ∞ < x < ∞, 2σ 2 2πσ 2   λ2 σ 2 mλ + 2 : M½λ ¼ e

Standard Gamma distribution pð xÞ ¼

Modeling Evolution of Heterogeneous Populations https://doi.org/10.1016/B978-0-12-814368-1.00017-5

sα α1 sx x e , 0  x < ∞, α > 0, ΓðαÞ

323

# 2020 Elsevier Inc. All rights reserved.

324

17. Math Appendix. Moment-generating functions for various initial distributions

M½λ ¼

1 , λ < s, ð1  λ=sÞα

α α m ¼ , σ2 ¼ 2 : s s Gamma distribution pð xÞ ¼

sα ðx  ηÞxα1 eðxηÞ=s , x > η > 0,α > 0, ΓðαÞ M½λ ¼

eλη , λ < s, ð1  λ=sÞα

α α m ¼ η + , σ2 ¼ 2 : s s Poisson distribution pð xÞ ¼

ec cx , x ¼ 0,1, …, c > 0: x!

MðλÞ ¼ ecð exp ðλÞ1Þ :

17.2 Distributions on a finite interval Uniform distribution pð xÞ ¼ M½λ ¼

1 , a  x  b, ba

eλb  eλa , λ 6¼ 0; Mð0Þ ¼ 1, λðb  aÞ

m¼

a+b ðb  aÞ2 , σ2 ¼ : 2 12

Truncated exponential distribution   pðxÞ ¼ Cesx , a  x  b, C ¼ s= eas  ebs : M ½λ ¼

s ebλ + as  eaλ + bs , ðeas  ebs Þ sλ 1 beas  aebs m ¼ + as , s e  ebs

17.3 Many-dimensional distributions

σ2 ¼

325

1 eða + bÞs ðb  aÞ2  : 2 s2 ðebs  eas Þ

Truncated normal distribution pðxÞ ¼ Ce

hpffiffiffiffiffi     i ðxmÞ2 ma pffi pffi s , a  x  b with normalization constant C ¼ 2= πs Erf bm + Erf : s s sλ2 emλ + 4

M½λ ¼







 2b  2m + sλ 2a + 2m + sλ pffiffi pffiffi Erf + Erf 2 s 2 s



: bm ma p ffiffi p ffiffi Erf + Erf s s

Beta distribution with parameters (α, β) in the interval [0, b] P0 ðwÞ ¼

Γðα + βÞ wα1 ðb  wÞβ1 , ΓðαÞΓðβÞ bα + β1

the moments of Beta distribution are given by  bk Γðα + βÞΓðα + tÞ : E0 wt ¼ ΓðαÞΓðα + β + kÞ Symmetric triangular (or Simpson) distribution pð xÞ ¼

2 2  ja + b  2xj, a  x  b, b  a ðb  aÞ2 

2ðeλb=2  eλa=2 M½λ ¼ λðb  aÞ

2 :

Asymmetric triangular distribution

a  x  b, 8 2 xa > < for a  x  θ b  am  a , pð xÞ ¼ > : 2 b  x for θ  x  b b  ab  m  aλ   2 be  aebλ + ða  bÞeθ + θ ebλ  eaλ M½λ ¼ : λ2 ðb  aÞðθ  aÞðb  θÞ

17.3 Many-dimensional distributions Multivariate normal distribution A random vector X ¼ (X1, … , Xn) has a multivariate normal distribution with the mean EX ¼ m ¼ ðm1 , …, mn Þ, and a covariance matrix

326

if its mgf is

17. Math Appendix. Moment-generating functions for various initial distributions

   C ¼ cij , cij ¼ cov Xi Xj      1 MðλÞ ¼ E exp λT X ¼ exp λT m + λT Cλ : 2

Multivariate polynomial distribution A random vector X ¼ (X1, … , Xn) has a multivariate polynomial distribution with parameters (k; p1, … , pn), if PðX1 ¼ p1 , …, Xn ¼ pn Þ ¼ P for ni¼1 mi ¼ k: The mgf of the polynomial distribution is MðλÞ ¼

k! p1 m1 …pn mn , m1 !…mn !

n X

!k pi e

λi

:

i¼1

Multivariate natural exponential distributions This class of distributions includes multivariate polynomial, normal, and Wishart distributions as special cases. A random n-vector X ¼ (X1, … , Xn) has multivariate natural exponential distribution (NED) with parameters θ ¼ (θ1, … , θn) with respect to the positive measure ν on Rn if its joint density function is of the form PðXÞ ¼ hðXÞeX

T

θsðθÞ

,

where s(θ) is a function on parameters and the generating measure μðdXÞ ¼ hðXÞνðdXÞ is not concentrated on any affine hyperplane of Rn. The mgf of NED is h T i MðλÞ ¼ Eν eλ X ¼ esðθ + λÞsðθÞ : For any NED, ∂ s ð θÞ ¼ m i ∂θi and   ∂2 sðθÞ ¼ cov Xi , Xj : ∂θi ∂θj

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Index Note: Page numbers followed by f indicate figures and t indicate tables.

A

D

Allee-type models distributed, three, 22–23, 22f inhomogeneous, 19–23 Allometric principle, 80 Allometric scaling, 81 Allometry, 81 Altruism, evolution of, 60–61 Antimicrobial agents, on heterogeneous microbial populations, 33–36 Asymptotics of orbits of system, 125–127 Asymptotics of probabilities, 127–128 Attention deficit hyperactivity disorder, 186

Darwinism, 2, 83 “Selection of everybody”, 253 “Selection of the fittest”, 119–120, 122, 281 Darwin’s theory, 74 Death rate, birth-and-death equation with, 59–60 Density-dependent Malthusian model, 117 Discrete-time Ricker’ model with, 310–313 selection system, 291–293 Distributed birth rate, birth-and-death equation with, 59–60 Distributed carrying capacity, logistic equation with, 108–109 Distributed cytotoxicity, 249–250 Distributed Malthusian parameter, logistic equation with, 16–19 Distributed parameters, logistic equation with two, 110–115 Distributed susceptibility, 242–250 Distributions dynamics of different initial, 11–13 dynamics of specific, 44–48 D-models for equation vs. F-models for equation, 95–96, 95–96t, 97f Dobzhansky’s thesis, 1 Dynamical principles of minimal information gain, 158–160 of minimal Shannon information loss, 183–186 of minimal Tsallis information loss, 174–176, 176f Dynamic carrying capacity, 227, 235 Dynamics of distribution of strategy coordination game or SH game, 261–264 Prisoner’s dilemma, 260–261 replicator equation, 259–260 Dynamics of specific distributions, 44–48

B Beta distribution, 325 Bifurcation analysis, 2, 4 Bifurcation theory, 4, 225, 321 Birth-and-death equation, 59–60, 111 inhomogeneous, 123–128 Birth-and-death models, 119–120 information gain in inhomogeneous, 143–146 Boltzmann distribution, 131–134, 158 Boltzmann-Gibbs entropy, 158, 183, 185, 190 Boltzmann-Gibbs-Shannon entropy, 166 Brain death, time perception in, 187–189

C Cancer game theoretical models of, 279–282 and oncolytic viruses, 240–242 Canonical form of power model, 85–90 Carrying capacity, 110–115 Competitive exclusion principle, 119–123 Consumer-resource model, 226–227 Coordination game, 261–264 Cressie-Read divergence, 174 Cross entropy, 130, 155–156, 158, 272 Cytotoxicity, distributed, 249–250 Cytotoxic therapy, 242, 243–247f

E Ecological systems, 7, 225–226 EGT. See Evolutionary game theory (EGT) Eigen quasi-species theory, 154 Epithelial to mesenchymal transition (EMT), 289

337

338

Index

Equation-based models, 2 Equilibria of frequencies, 257–259 Evolutionarily stable, or uninvadable, strategy (ESS), 252 Evolutionary game theory (EGT), 154, 251–252, 254, 288, 290 Evolutionary suicide, 225–226, 240 Exponential distribution, 323–326 truncated, 324–325 Exponential equation, inhomogeneous model for, 90–93 Exponential-linear dynamics, 222 Exponential-linear growth, 202–204, 214–221, 215t Extinction F-models of, 181–183 subexponential models of, 171 Extinction models, power, 178–181

F Fisher-Haldane-Wright equation (FHWe), 70–73 Fisher’s fundamental theorem, 74–78, 91, 307–310 Fisher’s fundamental theorem of natural selection (FTNS), 74–75 F-models for brevity, 85, 194 cell kill rate for, 220 for equation vs. D-models for equation, 95–96, 95–96t, 97f of extinction, 181–183 inhomogeneous, 178–181, 188 vs. Simeoni model, 215f Forest stand self-thinning models, 36–40 Frequency-dependent model, 85, 100–101 inhomogeneous model, 93, 101, 194, 196 Malthusian model, 122 FTNS. See Fisher’s fundamental theorem of natural selection (FTNS)

G Game theoretical models of cancer, 279–282 Game theory evolutionary, 251–252, 254 mathematical, 251–252 Gamma distribution, 12–13, 323–324 Gause’s exclusion principle, 120–122 Global demography, conceptual models of, 25–32 Gompertz curves, 193–194 vs. logistic curves, 200 Gompertz equation, inhomogeneous, 106–107 Gompertz function, 92 Gompertz model, 107, 119–120 inhomogeneous, 107 vs. Verhulst model, 197–199 Group selection model, 60–61

Growth curves breast cancer, 209 of heterogeneous population, 217 hyperbolic, 201f logarithmic transformation of, 204f lung cancer, 210–214, 213f tumor, 200, 201f, 221

H Haldane principle, 318 for selection systems, 73–74 Hawk-Dove (HD) game, 277 natural selection of strategies in, 265–271, 267–271f vs. prisoner’s dilemma (PD) game, 282–285, 283–284f, 286–287f Heterogeneous Malthusian growth model, 15f Heterogeneous model, parametrically, 226–234 Hidden keystone variable (HKV) method, 2, 3t, 4–5, 7–11, 36, 57–59, 154, 225–226, 231, 234, 261, 321 Homogeneous (monomorphic) population, 221 Homogeneous model, 225, 235 Homogeneous system, 225, 230f parametrically, 2–4 Hyperbolic-exponential growth, 200–202, 201f Hyperbolic law, 25

I Information entropy, 130 Information gain in inhomogeneous birth-and-death models, 143–146 Ricker’ model, 146–147 Information gain in models of early biological evolution, 139–140 of inhomogeneous communities, 149–152 of tree stand self-thinning, 140–141 Inhomogeneous Allee-type models, 19–23 distribution of carrying capacity b, 20–21 distribution of growth parameter a, 20 distribution of parameter m, 21–22 three distributed Allee models, 22–23, 22f Inhomogeneous birth-and-death equation, 101, 123–128 Inhomogeneous birth-and-death models, information gain in, 143–146 Inhomogeneous communities, information gain in models of, 149–152 Inhomogeneous density-dependent models, 100–101 Inhomogeneous extinction model, 37 Inhomogeneous F-model, 178–181, 188 Inhomogeneous frequency-dependent model, 101, 194, 196 Inhomogeneous Gompertz equation, 106–107 Inhomogeneous Gompertz model, 107 Inhomogeneous logistic equation, 16–17, 101–103, 106f

Index

Inhomogeneous logistic map, 314 Inhomogeneous logistic model, 8, 31, 103, 119 Inhomogeneous Malthusian equation, 62, 139 Inhomogeneous Malthusian model, 7–11, 13–17, 27–29, 30f, 103 extinction model, 140–141 F-models, 119 information gain in, 137–139 Inhomogeneous maps evolution of statistical characteristics of, 297–304 Malthusian, 293–297 self-regulated, 304–306 Inhomogeneous model, 141 of communities, reduction theorem, 53–55 dynamics of distributions in, 115–119 for exponential equation, 90–93 Malthusian, 131 of populations, reduction theorem, 50–53 Inhomogeneous populations, 67–70, 221 Inhomogeneous power models, of population extinction, 171 Inhomogeneous prey-predator Volterra model, 65–67 Inhomogeneous Ricker equation, 63–64 Inhomogeneous Ricker map, with two distributed parameters, 314–316 Inhomogeneous Ricker’ model, information gain in, 146–147 Internal population time, 119–123 Internal time, 171 for F-models of extinction, 181–183

K Keystone equation, 13 KL divergence, 130–131 Kolmogorov-type equations, 4 Kullback’s theorem, 273

L Lagrange method, 132 Laplace transform, 16, 39 Limiting factors, principle of, 61–63 Linear systems, quasi-species equation and, 141–143 Logistic curves, 193 vs. Gompertz curves, 200 Logistic equation, 195–196 with distributed carrying capacity, 108–109 with distributed Malthusian parameter, 16–19 generalized, 195–196 inhomogeneous, 16–17, 101–103, 106f with two distributed parameters, 110–115 Logistic growth model, parametrically heterogeneous, 18f

339

Logistic inhomogeneous model, 31, 103 with distributed Malthusian parameter, 104–106 Logistic-type equations, 110

M Malthusian dynamics, nonhomogeneous, 139 Malthusian equation, 79 inhomogeneous, 139 Malthusian extinction model, inhomogeneous, 140–141 Malthusian growth rate, 99 Malthusian inhomogeneous maps, 293–297 Malthusian inhomogeneous models, 131 Malthusian model, 83 frequency-dependent, 122 growth model, 29, 30f inhomogeneous, 7–11, 13–16, 27–28, 137–139 of population growth, 25 Malthusian parameter, 13, 101–103, 107, 110–115 dynamics of distributions of, 118f logistic equation with distributed, 16–19 logistic inhomogeneous models with distributed, 104–106 Many-dimensional distributions, 325–326 Mathematical game theory, 251–252 Maximum entropy (MaxEnt) principle, 129–130, 158 Maximum information entropy principle, 129, 152–153 Maxwell-Boltzmann distribution, 133 Metabolism and resource allocation, 288–289 mgf. See Moment-generating function (mgf) Minimal discrimination information principle, 130 Minimal information gain, dynamical principle of, 134–137 Minimal relative entropy (MinxEnt) principle, 130–134, 155–156, 190 and selection systems, 137–147 Minimal Shannon information loss, dynamical principles of, 183–186 Minimum information gain principle, 134f, 160–165 natural selection of strategies and, 271–276 Minimum of nonextensive information gain principle, 165 Minimum Tsallis relative q-entropy principle, 174–175 MinxEnt principle. See Minimal relative entropy (MinxEnt) principle MinxEnt probability distribution (MPED), 132–133 Moment-generating function (mgf), 8–9, 326 distributions on entire line/half line, 323–324 distributions on finite interval, 324–325 many-dimensional distributions, 325–326 natural exponential distribution, 326 polynomial distribution, 326

340

Index

Motility acquisition of, 290 vs. stability, 289–290 Motivation informal, 170 philosophical, 169–170 MPED. See MinxEnt probability distribution (MPED) Multivariate natural exponential distributions, 326 Mutual invasibility, of strategies and games, 285–288

N Natural exponential distributions, multivariate, 326 Natural rotifer community selection, 316–317 Natural selection, 115–119, 121 between games, 282–285 in resource allocation strategies, 234–240 of strategies and principle of minimum information gain, 271–276 of strategies in Hawk-dove (HD) game, 265–271, 267–271f tragedy of the commons, 236–237, 236f Newton diagram method, 124–125, 125f Non-Darwinian selection, 97, 253 “Survival of everybody”, 82, 97, 100–101, 103, 107, 119–120, 122 “Survival of the common”, 82 “Survival of the fittest”, 82, 97, 99–101, 103, 109–111, 114, 119–122 Nonhomogeneous Malthusian dynamics, 139 Nonhomogeneous models, of population decline, 171 Nonlinear models of population extinction, 170 Nonlinear power equations, for replicator dynamics, 155 Nonlinear quadratic law, 26 Normal distribution, 11, 323, 325

O

Parametrically inhomogeneous models, of population extinction, 177–178 Parkinson’s disease, 186 Payoff matrix, 254, 261–262 PD game. See Prisoner’s dilemma (PD) game Phase-parameter portrait, 231f, 242, 242f Philosophical motivation, 169–170 Poisson distribution, 12, 38, 141 Population decline, nonhomogeneous models of, 171 Population dynamics, 1–2, 193 Population extinction inhomogeneous power models of, 171 nonlinear models of, 170 parametrically inhomogeneous models of, 177–178 Population growth, Malthusian model of, 25 Population heterogeneity, 1–2, 4 as reason of power growth, 83–85 Population of subexponentially decreasing clones, 171–173, 172–174f Power equations, subexponential, 155–156 Power extinction models, 171, 178–181 Power growth, population heterogeneity as reason of, 83–85 Power law, 79, 82, 87 Power model, canonical form of, 85–90 Power replicator equations, 79–83 Prey-predator Volterra model, inhomogeneous, 65–67 Price covariance equation, 255 Price equation, 74–78, 307–310, 318 Prisoner’s dilemma (PD) game, 260–261, 262–264f, 277 vs. Hawk-Dove (HD) game, 282–285, 283–284f, 286–287f Probability density function (pdf), 8 Punishment/generous reward approach, 233

Q

ODEs escort system of, 50–51, 58–59, 65 infinitely dimensional system of, 7 Oncolytic viruses cancer and, 240–242 therapy, 225, 242 Orbits of system, asymptotics of, 125–127

q-entropy, 166 Tsallis relative, 174–175, 190 Quadratic law interpretation of, 26 nonlinear, 26 Quasi-species equation, 141 and linear systems, 141–143 Quasi-species theory, 141

P

R

Parabolic community, SG-model of, 163–164 Parabolic replicators population of, 160–165 population of freely growing, 156–157 Parameter distribution technique, 2 Parametrically heterogeneous model, 18f, 226–234 Parametrically homogeneous systems, 2–4

Radioactive decay, 171 RE. See Replicator equations (RE) Reduction theorem, 2, 4–5 for inhomogeneous models of communities, 53–55 for inhomogeneous models of populations, 50–53 Relative entropy, 130, 158, 272 Boltzmann-Gibbs, 190

Index

Replicator dynamics, 252 nonlinear power equations for, 155 Replicator equations (RE), 2, 7, 129, 131, 133–134, 134f, 137, 148, 153, 256–257, 259–260 power, 79–83 Ricker equation, inhomogeneous, 63–64 Ricker’ model with discrete time, 310–313 information gain in inhomogeneous, 146–147 Robertson covariance equation, 75

S

SBF. See Striatal beat frequency model (SBF) Scalar expectancy theory (SET), 186 Schmalhausen formula, 80–82 “Selection of everybody”, Darwinian, 253 “Selection of the fittest”, Darwinian, 119–120, 122, 281 Selection system dynamics, conjugative approach to, 147–149 Selection systems, 3t, 41–44, 57–59, 134–137 Haldane principle for, 73–74 MinxEnt principle and, 137–147 with self-regulations, 48–50 Self-regulated inhomogeneous maps, 304–306 Self-regulations, selection systems with, 48–50 SET. See Scalar expectancy theory (SET) SG parabolic community, 163–164 Shannon–Gibbs entropy, 129 Shannon information loss, 190 dynamical principles of minimal, 183–186 SH game. See Stag hunt (SH) game Simeoni model, 214–221, 220f vs. F-model, 215f Simulations, parameters used in, 218t Social currency, 226 Stability vs. motility, 289–290 Stag hunt (SH) game, 261–264, 263–264f Striatal beat frequency model (SBF), 186 Subexponential models, 86–87

341

of extinction, 171 growth models, 80 Subexponential power equations, 155–156 Superexponential models, 93–94 “Survival of everybody”, non-Darwinian, 82, 97, 100–101, 103, 107, 119–120, 122 “Survival of the common”, non-Darwinian, 82 “Survival of the fittest”, Darwinian, 82, 97, 99–101, 103, 109–111, 114, 119–122 Susceptibility, distributed, 242–250 Sustainability, 226–234 bifurcation analysis, 227–228, 228f tragedy of the commons, 232–234 Szathmary and Smith (SS-model), 156, 163–164

T Taylor-Jonker replicator equation, 252–253 Time perception application of the model to, 186–189 background information on, 186–187 in brain death, 187–189 Tragedy of the commons, 225–226, 232–234, 236–237 Truncated exponential distribution, 324–325 Truncated Poisson distribution, 38 Tsallis entropy, 158, 165–167, 190 Tsallis formalism, 167 Tsallis information gain, 160, 160f dynamics of, 160, 160f Tsallis information loss, dynamical principles of minimal, 174–176, 176f Tsallis relative entropy, 156, 166–167 q-entropy, 174–175, 190 Tumor, 288–290 growth models, 219f

V

Verhulst model vs. Gompertz model, 197–199 Virus-specific RNA replication and three-stage model, 204–208, 205f, 207f, 208t Volterra model, inhomogeneous prey-predator, 65–67