Current Trends in Dynamical Systems in Biology and Natural Sciences [1 ed.] 3030411192, 9783030411190

Table of contents :
Preface
Contents
About the Editors
Modelling Ecological Systems from a Niche Theory to Lotka-Volterra Equations
1 Introduction
2 Definition of Lotka-Volterra Dynamical Model for an Ecological Niche
3 Lotka-Volterra Dynamical Model for Interacting Niches in a Ecosystem
4 The Master Equation and the Fluctuations in the Populations
5 Numerical Simulations
6 Conclusions
References
Accurate Recognition of Spatial Patterns Arising in Spatio-Temporal Dynamics of Invasive Species
1 Introduction
2 Model and Method
3 Topological Characteristics of Spatial Pattern
3.1 The Number of Objects
3.2 The Fragmentation Rate and the Density of Objects
4 Sensitivity of Spatial Pattern to the Cut-Off Parameter
5 Reconstruction of Spatial Patterns from Sparse Data
6 Conclusions
References
Collocation Techniques for Structured Populations Modeled by Delay Equations
1 Introduction
2 Collocation to Improve Continuation
2.1 Prototype Problem
2.1.1 State-Dependency
3 Collocation to Compute Periodic Solutions
3.1 Experimental Results
4 Conclusions and Future Work
References
Herding Induced by Encounter Rate, with Predator Pressure Influencing Prey Response
1 Introduction
2 Intraspecific Interaction
3 Modification of Well-Mixedness as a Result of Herding
4 Interspecific Interactions
5 The Specialist Predator Model
5.1 Boundedness
5.2 The No-Herding Counterpart of (13)
6 The Generalist Predator Model
6.1 Boundedness
6.2 The No-Herding Counterpart of (32)
7 Numerical Simulations
7.1 The Specialist Predator Model
7.2 The Generalist Predator Model
8 Conclusions
References
Harvesting Policies with Stepwise Effort and Logistic Growth in a Random Environment
1 Introduction
2 Variable Effort Optimal Policy
3 Constant Effort Optimal Policy
4 Comparison of Policies
5 Conclusions
References
Mathematical Modeling of the Population Dynamics of Long-Lived Raptor Species: Application to Eurasian Black Vulture Colonies
1 Introduction
2 The Model
3 Estimation of p
4 Application to Eurasian Black Vulture Colonies
5 Concluding Remarks
References
On the Role of Inhibition Processes in Modeling Control Strategies for Composting Plants
1 Introduction
2 The Aerobic Biodegradation Model
3 Optimal Aeration Control Problem
4 Modeling Oxygen Uptake
5 Numerical Investigation of the Optimal Control Problem
5.1 Optimal Time-Profiles
5.2 Performance Indices
6 The Bioeconomic Model
7 Numerical Investigations of the Bioeconomic Problem
8 Concluding Remarks
References
Optimal Control of Invasive Species with Budget Constraint: Qualitative Analysis and Numerical Approximation
1 Introduction
2 An Optimal Control Model for Invasive Species
2.1 Model Formulation
2.2 First Order Necessary Conditions for Optimality
3 The State-Control Optimality System
3.1 Phase-Space Analysis in the State-Control Plane
3.2 Analysis of the Optimal Solution Through the Phase-Space Analysis
4 Numerical Approximation of the Temporal Dynamics
4.1 Numerical Results
5 Conclusions
References
A Shape Optimization Problem Concerning the Regional Control of a Class of Spatially Structured Epidemics: Sufficiency Conditions
1 Introduction
2 Preliminaries
2.1 A Spatially Structured Man-Environment-Man Epidemic Model
2.2 A Regional Optimal Control Problem
3 Dual Dynamic Programming Method for Problem (P)
4 Sufficient Optimality Conditions
5 Sufficient Conditions for -Optimality
5.1 -Optimality
5.2 Computational Algorithm
6 Conclusions
References
The Interplay Between Voluntary Vaccination and Reduction of Risky Behavior: A General Behavior-Implicit SIR Model for Vaccine Preventable Infections
1 Introduction
2 The SIR Model: Mandatory vs. Voluntary Vaccination
2.1 The Case of Constant Vaccination Coverage
2.2 Voluntary Vaccination: A Phenomenological Model
2.3 Modeling the Information Index M(t)
3 Behavior–Modulated Contact Rate
3.1 Extending the Capasso–Serio Behavioral Model
4 A General SIR Model Embedding Behavioral Feedbacks on Both Vaccination Propensity and the Contact Rate
4.1 Modelling Human Behavior and Its Implications for Infection Control
4.2 Behavioural Responses and Infection Dynamics
5 Concluding Remarks
References
PC-Based Sensitivity Analysis of the Basic Reproduction Number of Population and Epidemic Models
1 Introduction
2 The Basic Reproduction Number
2.1 Definition
2.2 Approximation
3 PC-Based Global Sensitivity Analysis
3.1 PCEs
3.2 Sobol' Decomposition and PC-Based Sobol' Indices
4 Examples
4.1 Influenza Epidemic Model with Vaccination [27, 28]
4.2 Age-Structured Epidemic Model [21]
5 Conclusions
References
Linear Dynamics of mRNA Expression and Hormone Concentration Levels in Primary Cultures of Bovine Granulosa Cells
1 Introduction
2 Experimental Materials and Methods
2.1 Datasets
3 Regulatory Function Approach: Gene and Hormone Regulatory Matrices (GHRMs)—Effect of Treatment on the Strength of the Interactions
3.1 GHRM: Numerical Example
4 Correlation Matrices
5 Comparison Between GHRMs for Different Treatments
5.1 Comparison Between GHRMs for Different Treatments for the Secondary Follicles
5.2 Comparison Between GHRMs for Different Treatments for the Dominant Follicles
5.3 Comparison Between Secondary and Dominant Follicles for Different Treatments Using GHRMs
6 Conclusions
References

Citation preview

21 Maira Aguiar · Carlos Braumann Bob W. Kooi · Andrea Pugliese Nico Stollenwerk · Ezio Venturino Editors

Current Trends in Dynamical Systems in Biology and Natural Sciences

Se MA

SEMA SIMAI Springer Series Volume 21

Editors-in-Chief Luca Formaggia, MOX-Department of Mathematics, Politecnico di Milano, Milano, Italy Pablo Pedregal, ETSI Industriales, University of Castilla-La Mancha, Ciudad Real, Spain Series Editors Mats G. Larson, Department of Mathematics, Umeå University, Umeå, Sweden Tere Martínez-Seara Alonso, Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain Carlos Parés, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain Lorenzo Pareschi, Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Ferrara, Italy Andrea Tosin, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Torino, Italy Elena Vázquez-Cendón, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, A Coruña, Spain Jorge P. Zubelli, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brazil Paolo Zunino, Dipartimento di Matemática, Politecnico di Milano, Milano, Italy

As of 2013, the SIMAI Springer Series opens to SEMA in order to publish a joint series aiming to publish advanced textbooks, research-level monographs and collected works that focus on applications of mathematics to social and industrial problems, including biology, medicine, engineering, environment and finance. Mathematical and numerical modeling is playing a crucial role in the solution of the complex and interrelated problems faced nowadays not only by researchers operating in the field of basic sciences, but also in more directly applied and industrial sectors. This series is meant to host selected contributions focusing on the relevance of mathematics in real life applications and to provide useful reference material to students, academic and industrial researchers at an international level. Interdisciplinary contributions, showing a fruitful collaboration of mathematicians with researchers of other fields to address complex applications, are welcomed in this series. THE SERIES IS INDEXED IN SCOPUS

More information about this series at http://www.springer.com/series/10532

Maira Aguiar • Carlos Braumann • Bob W. Kooi • Andrea Pugliese • Nico Stollenwerk • Ezio Venturino Editors

Current Trends in Dynamical Systems in Biology and Natural Sciences

Editors Maira Aguiar Department of Mathematics University of Trento Trento, Italy Bob W. Kooi Faculty of Science VU University Amsterdam Amsterdam, The Netherlands Nico Stollenwerk Department of Mathematics “CMAF-CIO” University of Lisbon Lisbon, Portugal

Carlos Braumann Department of Mathematics University of Évora Évora, Portugal Andrea Pugliese Department of Mathematics University of Trento Trento, Italy Ezio Venturino Department of Mathematics “Giuseppe Peano” University of Turin Torino, Italy

ISSN 2199-3041 ISSN 2199-305X (electronic) SEMA SIMAI Springer Series ISBN 978-3-030-41119-0 ISBN 978-3-030-41120-6 (eBook) https://doi.org/10.1007/978-3-030-41120-6 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The Ninth Edition of the International Workshop “Dynamical Systems Applied to Biology and Natural Sciences (DSABNS)” was held at the Department of Mathematics of the University of Torino, Italy, from February 7th to 9th, 2018. The workshop program included both theoretical methods and practical applications, covering research topics in population dynamics, epidemiology of infectious diseases, eco-epidemiology, molecular and antigenic evolution, and methodological topics in the natural sciences and mathematics. Since 2010, the DSABNS workshop, which was upgraded to a conference in 2019, has been organized by the Mathematical Biology Group of the Center for Mathematics, Fundamental Applications and Operations Research (CMAF-cIO) of Lisbon University, in collaboration with professors and researchers from Portugal, Italy, and the Netherlands. From 2010 to 2015, the event was held at Lisbon University, during which time it acquired a broad organizational structure and attracted an increasing number of participants. From 2016 to 2017, the workshop was held in Évora (Portugal) and it then moved to Italy, being held in Torino in 2018 and Naples in 2019. As a traditional “no registration fee” scientific event, the DSABNS attracts researchers and students from different countries around the world who draw on their own funding to attend and present their recent scientific results. A book of abstracts (with ISBN number) is also published at the end of each event. The Ninth DSABNS 2018 in Torino attracted the participation of 133 delegates from 30 countries. There were 13 plenary talks, 10 invited talks, 58 contributed talks, and a poster session. In this book, we have collected papers based on the research topics presented during DSABNS 2018, centering mainly on topics involving ecology and epidemiology but even touching on waste recycling and a genetic application. Some contributions also involve the application of numerical techniques to problems of structured populations. In ecology, the contributions range from a theoretical investigation aimed at reconstructing the interactions of populations from a niche theory to other issues as the study of suitable techniques for the assessment of the patterns generated by invasive species in the spatiotemporal domain. v

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In the former case, chapter “Modelling Ecological Systems from a Niche Theory to Lotka-Volterra Equations”, the concept of fitness landscape allows a stochastic description of species dynamics and the introduction of the notion of fitness potential for the evolution of a mutual ecosystem. Feasibility of its thermodynamic equilibrium, whose distribution is a multinomial negative distribution, is provided by the study of a master equation. In chapter “Accurate Recognition of Spatial Patterns Arising in Spatio-Temporal Dynamics of Invasive Species”, it is remarked that being able to distinguish between the patchy spatial density patterns and continuous front spatial density patterns is essential for the implementation of control measures against invasive species. A model consisting of two integrodifference equations is proposed to investigate various spatial density distributions. With it, several topological characteristics are generated, among which it is found that the number of objects in the visual image of a spatial distribution offers the most reliable conclusion for distinguishing between continuous and patchy spatial structures. The two most relevant features of the monitoring protocol are found, namely the threshold density value and the number of sampling locations. More abstract problems related to population theory are studied in the next two chapters. In chapter “Collocation Techniques for Structured Populations Modeled by Delay Equations”, an improved numerical scheme is proposed based on piecewise polynomial collocation to reduce delay systems to systems of ordinary differential equations or to approximate a periodic solution. For realistic models of structured populations, the proposed method substantially improves performances in comparison with the existing ones that rely on an external ordinary differential equations solver. Its adaptability for the computation of periodic solutions is demonstrated. A view differing from the classical predator-prey models is taken in chapter “Herding Induced by Encounter Rate, with Predator Pressure Influencing Prey Response”, where the effects of herding are investigated, observing that populations living together have less than well-mixed interactions. A range of models is thus obtained for a single population, specifically hyperbolic models which exhibit intermediate growths between the exponential and the logistic ones. In the context of Lotka-Volterra intermingling populations, this formulation stabilizes coexistence. For predators, predation pressure is reduced, as well as access to resources. The latter is modeled via a reduction in carrying capacity with increasing predator pressure, while predator escape is formulated in terms of the degree of herding. The latter is the stronger, the larger the predator pressure becomes. Hopf bifurcations are possible, leading to stable limit cycles for specialist predators and unstable ones when generalist predators are considered. Still in the context of ecology, in chapter “Harvesting Policies with Stepwise Effort and Logistic Growth in a Random Environment” constant and variable effort harvesting policies to maximize the expected total discounted profit are investigated over a finite horizon in the presence of stochastic fluctuations naturally occurring in real-life ecological situations. Due to the inapplicability and other shortcomings of the optimal variable effort policy, constant effort policies were considered. They are easy to implement, have no such shortcomings, and surprisingly provide a profit

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that is only slightly lower. The paper then studies variable effort stepwise strategies, where the effort is kept constant over one or two years and then updated. These stepwise policies are easy to implement at the cost of reducing the already low profit advantage of the optimal variable effort strategy. In chapter “Mathematical Modeling of the Population Dynamics of Long-Lived Raptor Species: Application to Eurasian Black Vulture Colonies”, a stochastic approach is also employed for the investigation of the population dynamics of raptor species. The long-lived Eurasian black vulture colonies are examined via discrete-time branching models, identified by time rather than by generation. A distinguishing feature in the population is the consideration of the coexistence of individuals from different generations. The most informative reproductive parameters are estimated in a non-parametric statistical setting using a Bayesian estimation procedure. Real data coming from the region of Extremadura (Spain) are used in the simulations. Specifically, the colonies used for the sampling represent two of the largest breeding colonies worldwide. They are located in the National Park of Monfragüe and in the Sierra San Pedro. Control theory is also employed for waste recycling in chapter “On the Role of Inhibition Processes in Modeling Control Strategies for Composting Plants”, in particular for the composting process of biocells. It allows optimization of the ways to provide air when inhibition due to a high concentration of oxygen occurs, thereby guaranteeing that the aerobic biodegradation process proceeds smoothly. Special attention is devoted to the assessment of the minimal cost of the control policy thus devised. A further application of control is presented in chapter “Optimal Control of Invasive Species with Budget Constraint: Qualitative Analysis and Numerical Approximation”. It concerns the optimal removal of invasive species, addressing the best temporal resource allocation strategy to achieve it. The optimality system in the state and control variables is derived, and phase-space analysis is used to provide qualitative insights about the behavior of the optimal solution. In particular, a practical situation involving plants is considered. The problem is reduced to a boundary-valued nearly-Hamiltonian system which is solved by suitable exponential Lawson symplectic approximations. An application to a real plant ground-reclaiming case is finally provided. Control theory also represents the link with the second part of the contributions, describing investigations performed in the domain of epidemiology. A stabilization problem for an epidemic model, described by a reaction-diffusion system with feedback, is considered in chapter “A Shape Optimization Problem Concerning the Regional Control of a Class of Spatially Structured Epidemics: Sufficiency Conditions”, where sanitation measures are envisaged. The main aim is the assessment of control programs administered only in a given subdomain of the region of interest that induce an effective disease eradication in the whole habitat. The sufficient optimality conditions are obtained and an approximate conceptual algorithm is discussed. Vaccination, as an explicit disease control measure accounting for people’s behavior, is considered in chapter “The Interplay Between Voluntary Vaccination

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and Reduction of Risky Behavior: A General Behavior-Implicit SIR Model for Vaccine Preventable Infections”. Two broad classes of behavior-implicit SIR models are reviewed: prevalence-dependent vaccination and prevalence-dependent contact rate. Then behavior-dependent and nonlinear and linear forces of infection are set in a general framework that also encompasses epidemic memory. These two different issues are here combined in a single unified approach that allows an assessment of the complicated interplay between the different behavioral responses due to various epidemiological parameters. As a result, sustained oscillations of vaccine coverage, risky behavior, and infection prevalence are obtained. In epidemiology, a fundamental concept is the disease basic reproduction number R0 . In the presence of parameter uncertainties, the sensitivity estimation of the stochastic model is allowed by suitable numerical methods using polynomial chaos expansions. Evaluation of Sobol indices by polynomial chaos-based methods are presented in chapter “PC-Based Sensitivity Analysis of the Basic Reproduction Number of Population and Epidemic Models”, showing how R0 is affected by varying the input parameters. The newly developed computational model for R0 introduced here allows for the efficient and versatile treatment of rather complex epidemic models. Finally, an application to genetics is presented in chapter “Linear Dynamics of mRNA Expression and Hormone Concentration Levels in Primary Cultures of Bovine Granulosa Cells”. The Gene Regulatory Matrices technique is here generalized to encompass also hormones, specifically estradiol (E2) and progesterone (P4), by constructing a directed weighted graph to model the interactions of several mRNA encoding enzymes. This allows the calculation of hormone concentration from the concentration of mRNA. This approach had previously been attempted only via differential equations, which are, however, limited by the need for accurate knowledge of the decay rates of hormones and mRNA. The novel technique with Gene and Hormone Regulatory Matrices allows estimation of the concentration on the whole network by using only a subset of its nodes. The models are constructed from data obtained in experiments providing gene expression and hormone concentration levels for primary bovine granulosa cells. The collection of selected papers presented in this SEMA SIMAI Springer Series followed the traditionally rigorous reviewing standards of journals that are traditional to this series. The authors are indebted and express their thanks to Luca Formaggia and SIMAI for the kind invitation to contribute to this series. Trento, Italy Évora, Portugal Amsterdam, The Netherlands Trento, Italy Lisbon, Portugal Torino, Italy May 2019

Maira Aguiar Carlos Braumann Bob W. Kooi Andrea Pugliese Nico Stollenwerk Ezio Venturino

Contents

Modelling Ecological Systems from a Niche Theory to Lotka-Volterra Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Paolo Freguglia, Eleonora Andreotti, and Armando Bazzani

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Accurate Recognition of Spatial Patterns Arising in Spatio-Temporal Dynamics of Invasive Species. . . . . . . .. . . . . . . . . . . . . . . . . . . . Natalia Petrovskaya and Wenxin Zhang

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Collocation Techniques for Structured Populations Modeled by Delay Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Alessia Andò and Dimitri Breda

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Herding Induced by Encounter Rate, with Predator Pressure Influencing Prey Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Henri Laurie, Ezio Venturino, and Iulia Martina Bulai

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Harvesting Policies with Stepwise Effort and Logistic Growth in a Random Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Nuno M. Brites and Carlos A. Braumann

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Mathematical Modeling of the Population Dynamics of Long-Lived Raptor Species: Application to Eurasian Black Vulture Colonies . . . . . . . . . . 111 Casimiro Corbacho, Manuel Molina, and Manuel Mota On the Role of Inhibition Processes in Modeling Control Strategies for Composting Plants .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125 Giorgio Martalò, Cesidio Bianchi, Bruno Buonomo, Massimo Chiappini, and Vincenzo Vespri Optimal Control of Invasive Species with Budget Constraint: Qualitative Analysis and Numerical Approximation.. . . .. . . . . . . . . . . . . . . . . . . . 147 Angela Martiradonna, Fasma Diele, and Carmela Marangi

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A Shape Optimization Problem Concerning the Regional Control of a Class of Spatially Structured Epidemics: Sufficiency Conditions . . . . . 165 Sebastian Ani¸ta, Vincenzo Capasso, Marta Lipnicka, and Andrzej Nowakowski The Interplay Between Voluntary Vaccination and Reduction of Risky Behavior: A General Behavior-Implicit SIR Model for Vaccine Preventable Infections . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 185 Alberto d’Onofrio and Piero Manfredi PC-Based Sensitivity Analysis of the Basic Reproduction Number of Population and Epidemic Models . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 205 Francesco Florian and Rossana Vermiglio Linear Dynamics of mRNA Expression and Hormone Concentration Levels in Primary Cultures of Bovine Granulosa Cells . . . . 223 Malgorzata J. Wieteska, John A. Hession, Katarzyna K. Piotrowska-Tomala, Agnieszka Jonczyk, Pawel Kordowitzki, Karolina Lukasik, Dariusz J. Skarzynski, and Leo Creedon

About the Editors

Maira Aguiar is a biologist who has also trained in the mathematical modeling of biological systems, with emphasis on nonlinear dynamics, bifurcation analysis, and biostatistics. Her research investigates problems in public health epidemiology, focusing on the dynamics of vector-borne diseases. She has authored more than 35 papers and is frequently invited as a plenary speaker at international scientific meetings. Since 2018, she has been Vice President of the European Society for Mathematical and Theoretical Biology. Carlos A. Braumann is Emeritus Professor in the Department of Mathematics and a member of the Research Centre in Mathematics and Applications, Universidade de Évora, Portugal, working on stochastic differential equations and biological applications. He has been an elected member of the International Statistical Institute since 1992 and has been President of both the European Society for Mathematical and Theoretical Biology (2009–2012) and the Portuguese Statistical Society (2006– 2012). Bob W. Kooi’s main research interests concern interacting populations in Life Sciences—Ecology, Evolution, Epidemiology, and Biochemistry—using mathematical models based on physical/chemical processes at different organizational levels: at the individual level, the Dynamic Energy Budget model, and at higher levels, unstructured/physiologically structured populations and community and ecosystem models. The emphasis is on sensitivity, perturbation, bifurcation, and nonlinear dynamics analysis techniques. Andrea Pugliese is Professor of Mathematical Analysis and Mathematical Biology at the University of Trento. He obtained a Master’s in Mathematics at the University La Sapienza of Rome and a PhD in Ecology and Evolution at the State University of New York at Stony Brook. He is the author of more than 90 scientific publications, mainly in the areas of mathematical epidemiology and ecology. He is an editorial board member for the Journal of Mathematical Biology and the Journal of Biological Dynamics. xi

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About the Editors

Nico Stollenwerk received his PhD in theoretical physics from the University of Clausthal. He previously worked at the Research Center Jülich, Germany and is currently the Principal Investigator of the Mathematical Biology group at CMAF, Lisbon University, where he has designed a Biomathematics PhD course. He is a coauthor of the book “Population Biology and Criticality” and has also coauthored many articles in international journals and more than 40 book chapters, as well as refereeing contributions in international congresses. Ezio Venturino received his PhD in Applied Mathematics from SUNY at Stony Brook in 1984. He is currently Professor of Mathematics in the Department of Mathematics, University of Turin, Italy. He has visited a number of international institutions worldwide and has a wide scientific collaboration network. His earlier research focused on numerical analysis, mainly methods for integral equations, and he is currently engaged in research on nonlinear models for biological and ecological applications.

Modelling Ecological Systems from a Niche Theory to Lotka-Volterra Equations Paolo Freguglia, Eleonora Andreotti, and Armando Bazzani

Abstract This paper is an attempt to analyze the notion of ecological niche as a community of different species and of ecosystem as a set of niches in order to formulate a dynamical model for an ecosystem. Our assumption is that the concept of fitness landscape allows to model the phenotype dynamics of an ensemble of species as a stochastic process. To take into account the interaction structure of different communities in the niches and the environment we introduce an ecological fitness potential to formulate a Lotka-Volterra system which describes the evolution of a mutual ecosystem in presence of finite resources. To explicitly consider the effect of fluctuations in the numerousness of the species, we associate a master equation to the average Lotka-Volterra system and we study the conditions of existence of a detailed balance equilibrium (i.e. a thermodynamic equilibrium) for the ecosystem. The explicit solution for the equilibrium probability distribution is a multinomial negative distribution and we discuss the relation between the detailed balance condition and relative species abundance distribution in the framework of Hubbell’s neutral theory. Moreover the theoretical distribution implies the existence of a correlation among the relative species distribution associated to the different communities. We use numerical simulations to illustrate the results on simple models.

P. Freguglia () Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy E. Andreotti Department of Mechanics and Maritime Sciences, Chalmers University of Technology, Gothenburg, Italy e-mail: [email protected] A. Bazzani Department of Physics and Astronomy, University of Bologna, Bologna, Italy INFN Bologna, Bologna, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_1

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Keywords Hubbell’s neutral theory · Relative species abundance · Lotka-Volterra equation · Master equation

1 Introduction An ecological system (or ecosystem) is composed by different interacting communities (as a set of trophic equivalent species) which live together in different niches in a given environment (see for instance [1–3]). The environment provides the necessary resources to sustain the ecosystem, whose number of individuals is necessary finite, but the interaction structure among the species of different niches may change the environment itself modifying the resources at disposal to the different species. The dynamics (evolution) of an ecosystem could be represented by ecological transformations of the phenotype characters of representative individuals that perform a migration into the different niches. According to this point of view the ecological niches become attraction basins of the ecological transformations in a phenotype space and one could define a geometrical structure through the concepts of fitness landscape and fitness potential [4] where the niches are modeled by the potential wells associated to the local minima. An ecological system consists on an ensemble of different niches associated to specific environmental conditions where different communities live together performing the different type of biological interactions (from predator-prey interactions to mutual interactions). According to A.T. Peterson [5]: “The type and number of variables comprising the dimensions of an environmental niche vary from one species to another [and] the relative importance of particular environmental variables for a species may vary according to the geographic and biotic contexts”. As G.E. Hutchinson [6] says, in order to analyze the dynamics and the ecological transformations into a niche, one uses mathematics and statistics, and one tries to explain how species coexist within a given community. In defining a niche model one has also to consider the effect of an alteration of an ecological niche by its inhabitants that is part of the mechanism of niche construction (see A.T.Peterson [7]). That is we have to consider the mutual influence between the environment and the organisms or individuals of different species which live in the environment. Even if Richard Lewontin [8] says “the organisms propose, but the environment decides”, actually also the individuals modify the environment (see i.e. exploitive system, Conrad H. Waddington (1959) and the action of human people). Examples of environment modifications from the organisms are i.e. aeries, nests, dens or barriers as the beaver constructions. The aim of this paper is to propose a mathematical description of the ecological niche in a ecosystem context and to derive a stochastic dynamics that could describe the evolution of an ecosystem [9, 10]. The concept of niche has been extensively considered to understand the structure of ecological systems [11]. One of the main challenge is to understand the universal properties observed in the relative species abundance (RSA) distribution and in the species abundance distribution (SAD) in an ecosystem [12–14]. The Hubbell’s neutral theory [15–18] assumes that the

Modelling Ecological Systems from a Niche Theory to Lotka-Volterra Equations

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Number of species 50 40 30 20 10 0 1

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16 32 64 128 256 512 1024 2048 4096 8192 Number of individuals per species

Fig. 1 Lognormal abundance distribution for moth samples; the continuous curve is an interpolation together with a log-normal distribution (the picture is downloaded from http://web2.uwindsor. ca/courses/biology/macisaac/55-437/lecture9.htm)

differences among species of an ecological niche are “neutral”, i.e. irrelevant to their success. In many ecological systems the RSA distribution is well interpolated by a log-normal distribution or a log-series distribution: in the Fig. 1 we plot an example of RSA distribution together with a log-normal interpolation (Fig. 1). Previous works have proposed an explanation of the diversity and relative abundance of species in ecological communities, assuming that the differences between trophically similar species of a community of species are irrelevant to their success (neutral theory). Recently in the paper [19] the RSA distribution is derived from a single master equation of a birth death process [20] in the framework of neutral theory. However the niche theory of biodiversity remarks the relevance of the interactions among the different species, so that there has been a debate on the problem of developing an unified approach to model an ecological system [18, 21, 22]. A fundamental remark is that our observations on ecosystems reflect the property of a stationary state that is an equilibrium condition among all the different populations where the effect of interactions can be hidden. Despite of the several models that have been proposed to describe the SAD, there is an intrinsic difficulty to make any predictions beyond the hollow-curve SAD itself [23]. The resilience concept applied to ecosystems is essentially related to the stability properties of the equilibrium solution: i.e. the capability of the system to recover its equilibrium in presence of the environmental perturbations. Several studies have been performed to understand the effect of species interactions on the resilience of the ecosystem [24–26]; the methodology mainly consists in computing the spectral properties of random matrices, whose entries represent the interaction typologies among the species and in considering the stability of linear dynamical systems associated to these matrices, which represent the dynamics of an ecosystem near its equilibrium solution. Theoretical and numerical results show as the introduction of interactions among species in a random way may deteriorate the stability of the

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equilibrium solution (particularly in the thermodynamics limit). We remark that the previous approaches aim to relate the structure of interaction network to the stability properties of the equilibrium solution in order to understand the selection processes underlying the structure of an ecosystem. Here we adopt a different point of view: we assume that the equilibrium state of an ecosystem and its stability properties are given and we consider the effect of species interactions on global properties of the system, like the species abundance distribution and the correlation among these distributions when one considers the presence of different ecological niches. Therefore in our approach the interactions do not change the equilibrium of the system and its linear stability, but they affect the correlation among the fluctuations of the different populations the global non linear behavior of the dynamics. We address this problem from a dynamical point of view studying the stochastic dynamics of a hypothetical ecological described by a Lotka-Volterra like system [27], which describes the mutual interactions of species into different ecological niches. The fluctuations effect of the population abundances is introduced by a master equation where the evolution of discrete populations depends on the birth and death rates [28]. In the thermodynamics limit the average dynamics of the master equation reduces the Lotka-Volterra system, but for finite size populations it allows to define a stationary population distribution that could be related to RSA distribution. Using the correspondence between Statistical Mechanics and the master equation [22] we analyze the possibility of a detailed balance (DB) condition for the stationary state of mutual interacting communities. The DB condition characterizes the thermodynamic equilibrium of the system where each community is an equilibrium with any other community separately. According to the (DB) the statistical properties of the communities at the equilibrium are equivalent to the corresponding properties of independent communities. This condition suggests as in the equilibrium state the niche theory and the neutral theory could predict the same RSA distribution [29–31]. We explicitly compute the stationary distribution in the DA condition which results to be a multinomial negative distribution, whose marginal distributions are negative binomial distributions. We recall that the negative binomial distribution has been proposed as a model for the RSA distribution according to the Hubbell‘s neutral theory [19]. The existence of a dynamical model underlying the stationary state gives the possibility of performing predictions on the ecosystem evolution when perturbations or environmental changes are present once the interaction structure of an ecosystem is known. The theoretical results point out that a mutual interaction among the communities implies the existence of a correlation among the corresponding RSA distributions. This correlation effect could be used in analyzing empirical data to measure the interaction among different communities even in the framework of neutral theory. However both the problems of measuring the interactions among species in a stationary state [32] or of modeling of ecological interaction networks are currently been studied [33, 34]. The paper is organized as follows: in the second section we discuss the main features of a dynamical model for an ecosystem starting from the concept of the fitness landscape and the existence of ecological niches; in the third section we introduce a Lotka-Volterra system that may describe the evolution of a mutual

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ecosystem composed by different ecological niches in presence of finite environmental resources; in the forth section we study the solution of a master equation for a one step stochastic process whose average dynamics recovers the Lotka-Volterra system and we point out the condition for the existence of a DB equilibrium state, whose probability distribution is a negative multinomial distribution; finally in the fifth section we show some results of numerical simulations on a model with two communities and the relation of our results with a model for the RSA distribution.

2 Definition of Lotka-Volterra Dynamical Model for an Ecological Niche An ecological niche indicates the fit of different species living in a given environment and it can be associated to a subset Bi of the phenotype space. From a statistical physics point of view, each niche i contains a community of species that reach a stationary distribution characterized by an average value n∗i for the population, that measures the success of the niche itself. Moreover, it is natural to relate the success of a niche to the exponential of a energy (the fitness of the phenotypes in the niche) Vi according to the Maxwell-Boltzmann distribution n∗i ∝ exp (Vi )

(1)

We recall that the distribution (1) maximizes the Entropy of the system with the constraint in the average fitness. In the Eq. (1) there is the implicit assumption that higher is the fitness Vi , greater is the attractiveness of the corresponding phenotypes. From a mathematical point of view, this assumption would mean that in a randomly chosen regular potential it is expected with high probability a positive correlation between the deepness of a well and its extension: this has been observed in numerical studies of potentials in condensed matter physics [35]. The correspondence (1) is also consistent with experimental observations [36] and it suggests a possibility to measure the fitness potential. The fitness associated to a phenotype Xi characteristic of a species i could be measured as the logarithm of the number of individuals ni that express the phenotype Xi : Vi = ln(ni ). The idea of a fitness potential (or fitness landscape) has been introduced to represent the evolution process of the phenotype characters under the influence of an external environment. The fitness of the population increases during time toward the local maxima of the fitness landscape following the local gradient of the landscape. To derive a population dynamics in presence of different ecological niches, we represent the fitness landscape by a potential U (n), assuming a functional dependence between the numerousness of a species ni and its phenotype characters. Then we get the dynamical system d n˙ i ∂U ln(ni ) = =− (n) dt ni ∂ni

(2)

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that simulates as the evolution of the average population in each niche tends to increase moving along the gradient of the fitness potential of the whole ecosystem. Each component of the gradient of the fitness potential represents the reproduction rate (i.e. the success) of the individuals in the niche. In the case of a quadratic potential U (n) = −



gi ni +

i

1 ni gij nj 2 ij

the Eq. (2) takes the form of a Lotka-Volterra like system [27] ⎛ n˙ i = ⎝gi −



⎞ gij nj ⎠ ni

(3)

j

The non trivial equilibrium solutions are determined by the equation ∂U ∗ (n ) = 0 ∂ni and their stability requires that the eigenvalues of the symmetric matrix ∂ 2U (n∗ ) ∂ni ∂nj are all positive (i.e. the critical point of the fitness potential is a minimum). In generic situation one could have different equilibria taking into account the possibility of extinction of a population in a niche. In the next section we introduce a possible Lotka-Volterra model for an ecosystem which is consistent with the niche theory approach.

3 Lotka-Volterra Dynamical Model for Interacting Niches in a Ecosystem We specify the system (3) in the case of an ecological system composed by different niches that perform a mutual interaction. Our first assumption is that there exists a stable equilibrium stationary solution for the average population dynamics in the different niches due to the limited resources of the environment. Let n∗i the equilibrium value for the i niche, we introduce a logistic equation for each niche to model the relaxation process toward the equilibrium state   ni n˙ i = gi 1 − ∗ ni ni

(4)

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where ni is the average numerousness of the i community and gi > 0 is the generation rate. The stability of the equilibrium solution depends on the eigenvalue of the linearized system −gi < 0. The logistic equation (4) describes an ensemble of non-interacting communities, which have a stable equilibrium state n∗ defining their carrying capacity. In such a case, the carrying capacity is the result of the fitness of the single community in the given environment, that is unaffected by the presence of the other communities. This picture is consistent with a neutral assumption to explain the RSA distribution in each niche, since the success of a given species is due to random effects. The extinction of rare species due to random fluctuations in the population numerousness is an independent event in the different niches, so that the stability of the ecological system is the stability of the single niche. However the evolution process has pointed out as the mutual cooperation could be one of the selected features [37] and food-web structures could influence extinction process in ecosystems [38]. We assume the existence of a mutual interaction among the communities that can be associated to a fitness measure of the whole ecosystem. The dynamical model introduces a coupling term in the Eq. (4) in a similar manner as in the quasi-species model, without changing the equilibrium solution and its stability properties. Let φj > 0 the relative fitness contribution of the community j to the ecological system, we introduce a measure of the mutual fitness as Φ(n) =

 φj j

n∗j

nj

(5)

with the normalizing condition (this is a technical condition to introduce a parameter which modulates the interaction strength) 

φj = 1

j

The mutual fitness potential (5) can be the consequence of the modification of the environment by the different species or of the production of nutrients by the ecosystem itself(food wed structure). The mutual fitness is a global properties of the ecosystem and the coefficient φj /n∗j measures the impact of an individual j community on the ecosystem fitness. We modify the logistic equation by increasing the reproduction rate of each community as a function of the mutual fitness (5): each community takes advantage from the presence of the other communities in the ecosystem. This effect may simulate the influence on the environment of the communities or the existence of food webs. The logistic equations (4) become a Lotka-Volterra system ⎛ ⎞⎤    nj ni n˙ i = ⎣gi 1 − ∗ + a ⎝ φj ∗ − 1⎠⎦ ni ni nj ⎡

j

(6)

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where the parameter a > 0 modulates the mutual interaction effect. We remark that the mutual fitness contributes in the same way to the evolution of each community to stress the fact that the fitness is a global property of the ecosystem. By comparing Eq. (6) with Eq. (3), we see that the existence of a fitness potential U (n) requires the condition  φj ∂U =a nj ∂ni n∗j

∀i

j

which is satisfied only if φi φk = ∗ n∗i nk

∀ i, k

We will see as the existence of the fitness potential in the average dynamics (6) corresponds to the DB condition for the associated master equation, that describes the fluctuations effect. According to Eq. (6) the contribution of the mutual fitness is small if nj  n∗j (i.e. when the numerousness of a community is much less than the equilibrium value) and it increases with nj until the effects of the limited resources become relevant. We also remark that the parameter a changes the reproduction rate of each community, which is gi − a in the system (6). For seek of simplicity we set gi = g for all the communities and we study the stability of the equilibrium solution using the linearized system ⎡ δ n˙ i = − ⎣gδni − a

 j

⎤ δn j n∗i φj ∗ ⎦ nj

δni = ni − n∗i

(7)

where we recognize the matrix Lij = gδij − aφj

n∗i n∗j

The following result can be easily proved Theorem 1 If the inequality g > a is satisfied, the equilibrium solution of the system (6) ni = n∗i is stable. The matrix Lij is similar to the matrix gδij − a1i φj whose characteristic equation reads det|(λ − g)δij a1i φj | = 0

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Since 1i φj is a stochastic matrix (indeed a singular stochastic matrix), its spectrum has an eigenvalue equal to 1 whereas all the other eigenvalues vanishes (all the columns are equal). Then the matrix Lij has an eigenvalue λ = g − a whereas all the other eigenvalues are λ = g. The stability of the linear system (7) follows if all the eigenvalues have a positive real part: i.e. g − a > 0. The relaxation time scale toward the equilibrium solution is (g − a)−1 , therefore for a fixed value of the parameter g (the reproduction rate in the absence of interaction), the stability (or the resilience) of the equilibrium solution decreases when the interaction strength a increases (cfr. [24]). If we keep constant the different g − a when varying a, the stability of the equilibrium solution is not changed. The solution ni = 0 ∀ i is clearly unstable and the system (6) have other equilibrium solutions characterized by the extinction of some communities.

4 The Master Equation and the Fluctuations in the Populations The Lokta-Volterra equation (6) describes the average dynamics of the ecological system, but the finite size of the populations in the communities requires to take into account the fluctuation effect to compare the model simulations with the empirical observations. The master equation [20] has been extensively used to model population dynamics. Let P (n, t) the probability to observe the numerousness n in the ecosystem at time t, we assume that all the populations change by a single representative’ individual each unit time (Δni = ±1) and we interpret the birth rate and the death rate as the probability transition rates among the states of the system that differ for a single individual (one step process). The corresponding master equation reads

  ni − 1) g (ni − 1)P (n, t) n∗i i ⎛ ⎞ ⎤  nj − Ei− ⎝(g − a) + a φj ∗ ⎠ ni P (n, t)⎦ nj

P˙ (n, t) =



(Ei+

(8)

j

where the one-step character for the stochastic process means that the probability of population changes by more than one individual in a time interval Δt is o(Δt). We remark that the physical space ni > 0 ∀i is invariant in the Eq. (8) and we neglect the possibility of extinction since we are interested to study the ecosystem near the equilibrium solution. The Van Kampen operators [20] Ei± are defined by the formula Ei± P (n) = P (n ± eˆi )

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where eˆi is the canonical base of the Euclidean space. A straightforward calculation shows that the mean values  < n˙ i >= ni P˙ (n, t) n

satisfy the average dynamics (6) in the thermodynamics limit ni  1. The equilibrium condition corresponds to the stationary distribution function Ps (n) that is the solution of ⎡ ⎛ ⎞ ⎤     nj ni (ni − 1) − Ei− ⎝(g − a) + a (Ei+ − 1) ⎣g φj ∗ ⎠ ni ⎦ P (n) = 0 n∗i nj i

j

(9) The l.h.s. of Eq. (9) is the sum of probability current densities between the physical state n and the connected states. The stationary condition means that the total current vanishes on each state n, but each single current density may not vanish determining a net probability flow among the communities at the stationary state. However when each current term in the sum (9) vanishes, the stationary solution can be computed by means of a recurrent equation Ei+ P (n)

 (g − a) + a j φj nj /n∗j   P (n) = g (ni + 1)/n∗i

(10)

When this condition holds, the master equation (8) satisfies the DB condition and the stationary distribution corresponds to a thermodynamic equilibrium, since each population is in an equilibrium state with any other population independently. We remark that for a = 0 we recover the master equation associated to the logistic equations (4) P˙ (n, t) = g

 i

(Ei+ − 1)



ni n∗i



(ni − 1)P (n, t) − Ei− ni P (n, t)

 (11)

which is a separable master equation. In such a case theDB condition is always satisfied and the stationary solution has the form P (n) = i Pi (ni ) where Pi (ni ) is a Poisson distribution Pi (ni ) = Pi (0)

(n∗i )ni ni !

(12)

The DB condition (10) plays an important role in the physical interpretation of the master equation, since it can be characterized by a Maximal Entropy Principle [39] with an existence of an internal energy. In the equilibrium state each population minimizes the dynamical effect of the interactions so that the change in a single

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population has a small influence in the ecosystem state (resilience from small perturbations). However the DB condition minimizes the capacity of the ecosystem to change if the environment becomes hostile since it is a stable equilibrium state and a relevant amount of ‘energy’ is necessary to exit from this state. To define a single value distribution from the recurrence (10) the following condition must hold for i = k (g − a) + a

 j

φj (nj + δij )/n∗j (g − a) + a

 j

φj nj /n∗j

g[(nk + 1)/n∗k ] g[(ni + 1)/n∗i ]   (g − a) + a j φj (nj + δkj )/n∗j (g − a) + a j φj nj /n∗j g[(ni + 1)/n∗i ]

=

g[(nk + 1)/n∗k ]

so that ⎛ ⎞ ⎛ ⎞  nj  n φi ⎝ φ j k (g − a) + a φj ∗ ⎠ = ∗ ⎝(g − a) + a φj ∗ ⎠ n∗i nj nk nj j

j

This condition holds only if n∗ φi = i∗ φk nk

(13)

and we recover the same condition for the existence of the fitness potential for the Eq. (6). According to our assumptions we get φi =

n∗i N∗

N∗ =



n∗j

(14)

j

and the mutual fitness (5) reduces to Φ(n) =

1  nj N∗ j

From a biological point of view the DB condition for the master equation (8) means that the different populations contribute in the same manner to the mutual fitness potential. This condition maximizes the independence of communities so that the increment or the decrease of a communities is irrelevant for the other according to the Hubbell’s neutral theory on biodiversity [15], but it changes the global fitness of the ecosystem. From a mathematical point of view the recurrence (10) allows to

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compute explicitly the stationary distribution

P (n) =

 i

   Γ (g/a − 1)N ∗ + j nj  1  an∗ ni i (Ei+ )ni P (0) = P (0) Γ ((g/a − 1)N ∗ ) ni ! gN ∗ i

(15) where P (0) has the role of a normalizing constant  P (0) =

g−a g

(g/a−1)N ∗

The distribution (15) is a multinomial negative distribution with parameters pi =

an∗i gN ∗

i = 1, . . . , N

and p0 =

g−a g

n0 = (g/a − 1)N ∗

The average value of each community is < ni >=

n∗i g n0 pi = (g − a)N ∗ = n∗i p0 g − a gN ∗

in agreement with the equilibrium condition of the average equations (6). But the mutual interaction among the different populations implies the existence of a covariance matrix for the distribution of the populations in the different communities < ni nk > − < ni >< nk >=

an∗i n∗k + n∗i δik (g − a)N ∗

(16)

Therefore we estimate < ni nk > a −1 < ni >< nk > (g − a)N ∗

(17)

that shows as the correlation among the community distributions increases proportionally to the mutual interaction strength (we recall that g − a is fixed) and it decreases as N ∗  1. We remark that the limit a → 0 is singular for the distribution (15) that should reduce to a product of Poisson distributions (see Eq. (12)). A biological interpretation for this fact is that the mutual interactions introduces a source effect in master equation (11) that changes the stationary distribution.

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In the DB condition (13), the recurrence (10) implies that the stationary distribution can be written in the form   n  P (n)  exp −N ∗ U N∗ where N is the total number of particles and it satisfies the condition to apply the Large Deviations Theory [40]. Let xi = ni /N the energy U (x) satisfies the condition (for N ∗  1 with n∗i /N ∗ finite) ⎞ ⎛    ∂U N ∗ xi ⎠ ⎝ = − ln (g − a) + a xj + ln g ∗ ∂xi ni j

so that we have ⎛ ⎞ ⎛ ⎞   g  −1 + xj ⎠ ln ⎝(g − a) + a xj ⎠ + xi ln(xi ) U (x) = − ⎝ a j

(18)

j

The existence of the energy U (x) has also a dynamic interpretation: U (x) is a Ljapounov function for the average dynamics (6) so that ∂U/∂xi = 0 defines the equilibrium solution and it gives the statistical distribution of the fluctuations when N ∗  1. The marginal distributions of (15) have the form of a negative binomial distribution Pk (nk ) =

Γ ((g/N ∗ /a − n∗k ) + nk ) Γ ((gN ∗ /a − n∗ )nk !



an∗k gN ∗

∗ nk   ∗ an∗k g/N /a−nk 1− gN ∗

which represents the expected distribution of fluctuations when one focuses on single community. This result is interesting since the negative binomial distribution is the solution of a 1-dimensional master equation corresponding to a birth and death process for a communities with an external source from the environment. Such a model has been proposed to describe the RSA distribution according to the Hubbell’s neutral theory [19]. Under this point of view when the DB condition is satisfied, the master equation (8) provides a description for the RSA distribution as an overlapping of independent communities with external sources. The overlapping of the marginal distributions weighted by the average number n∗k is given by P (n) =

 n∗ k Pk (n) N∗

(19)

k

Equation (19) evaluates the probability to get a population of n individual extracted randomly from the communities of the ecosystem.

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5 Numerical Simulations We illustrate the previous results with numerical simulations to explain the effect of the control parameter a on the ecosystem evolution. In the simulations we keep constant the difference g − a > 0 so that the birth rate for the linearized system (6) when nk  1 does not depend on the mutual interaction parameter a. Our aim is to study the dynamical properties of the average Lotka-Volterra system (6) and the master equation (8) near the stable equilibrium solution n∗ , since we expect that a possible comparison with the empirical observations could be performed in this dynamic regime. In the simulations we consider the case of two communities with a different equilibrium state n∗1 = 10 and n∗2 = 20 and we initially study the effect of a random perturbation on the average dynamics by integrating the stochastic differential equations

   n1 dn1 = g 1 − ∗ + a (Φ(n1 , n2 ) − 1) n1 dt + dw1 n1

   n2 dn2 = g 1 − ∗ + a (Φ(n1 , n2 ) − 1) n2 dt + dw2 n2

(20)

1

1

0.5

0.5

Fluctuations

Fluctuations

where wi (t) are independent Wiener processes and Φ(n, n2 ) is the fitness potential (5). We remark as the system (20) is physically justified for  1 since the physical space ni ≥ 0 is not invariant for the Gaussian fluctuations and the statistical properties of the solution of the stochastic differential equation (20) approximate the properties of the probability distribution of the master equation (8) only in the limit of the quasilinear approximation [20]. We have numerically solved the Eq. (20) near the equilibrium point in presence and in absence of a mutual interaction. The results are reported in the Fig. 2. The results show clearly as the mutual fitness potential

0

–0.5 –1

0

–0.5

0

50

100

150 Time

200

250

–1

0

50

100

150 Time

200

250

Fig. 2 Left picture: fluctuations of the communities near the stationary state without mutual interactions (a = 0); the values of the other parameters are g = .1, = .13 and n∗1 = 10 and n∗2 = 20. Right picture: fluctuations of the communities near the stationary state with mutual interactions (a = .4); we set the value g = 0.5 to keep constant the reproduction rate at n1,2  1 whereas the values of the other parameters are the same

Probability

Modelling Ecological Systems from a Niche Theory to Lotka-Volterra Equations

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0.006 0.005 0.004 0.003 0.002 0.001 0

0

10

20

30

n1

40

50

60

70 0

10

20

30

40

50

60

70

n2

Fig. 3 Stationary distribution for two interacting communities according to Eq. (15) with parameters g = 0.5, a = 0.4 and n∗1 = 10 n∗2 = 20 0.1

0.15

0.08 Probability

Probability

No interaction 0.1

0.05

0

10

20

30

Population

0.06 0.04 0.02

With interaction 0

No interaction

40

50

0

With interaction 0

10

20

30

40

50

60

70

Population

Fig. 4 Left picture: marginal distributions for the first population with and without mutual interactions; we use the same parameters as in Fig. 2. Right picture: marginal distributions for the second population with and without mutual interactions

introduces a positive correlation between the community fluctuations increasing the overall resilience of the ecosystem. Therefore the correlation effect can be used as a fingerprint for the existence of an interaction among the communities. In the master equation the fluctuations are the consequence of the finite number of individuals in the populations and the stationary distribution describes the population distribution of the ecosystem in presence of two communities. In the Fig. 3 we plot the stationary distribution in the DB condition (i.e. the multinomial negative distribution (15) for two interacting communities). The distribution has a typical bell shape elongated on the positive diagonal according to the existence of a positive correlation between the populations. To point out the effect of mutualism, we have computed the marginal distributions for the two populations with a = 0 and a = .4 keeping fixed the difference g − a. The results are shown in the Fig. 4 We remark as the mutual interaction term in the master equation increases the variance of the distribution enhancing the probability of observing populations with few or many individuals. This effect could be relevant to increase the resilience of the ecosystem and it suggests a role of the mutualism to increase the biodiversity of an ecosystem.

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The understanding of the RSA distribution of an overlapping of binomial negative distributions (see Eq. (19)) is consistent with the theoretical framework discussed in the paper [18].

6 Conclusions Understanding the origin of universal features highlighted by the species abundance distribution in a stationary ecosystem is a key issue to formulate mathematical dynamical models. One of the main problems is that the statistical properties of the equilibrium states may be independent from the details of the interaction structure among the species. This idea is consistent with the Hubbell’s neutral theory, where any difference among species is irrelevant, but it is counterintuitive in the framework of a niche theory as suggested by the fitness landscape paradigm. In this paper we introduce a Lotka-Volterra-like system to model the average dynamics of an ecosystem performed by different communities in mutual interaction by introducing a fitness potential. Each community can be associated to a ecological niche (i.e. a local minimum of the fitness potential) that fixes its average success. We model the population fluctuations inside each community by a master equation under the assumption that the underlying stochastic process is a regular one-step process with probability rates to create or to destroy one representative individual for each population per time unit. This assumption means that the reproduction rate is the same for all the species in each community. Our main result is that the detailed balance condition for the master equation implies a stationary equilibrium state for the population abundance distribution in the form of a multinomial negative distribution. The detailed balance condition implies that all the communities are equivalent for the definition of the mutual fitness potential and it reflects the existence of an equilibrium condition between each couple of communities separately. We recover the negative binomial distribution as the marginal distribution of each community, that provides an interpolation the RSA distribution consistent with the Hubbell’s neutral theory [19]. However the existence of a mutual interaction has a relevant effect on the correlation among the populations of different communities and on the variance of the population distribution. This effect could be used in empirical observations to prove the existence of an interaction among different communities whose RSA distribution can be explained by a neutral theory. We have illustrated this effect by numerical simulations. We finally remark as the detailed balance condition cannot be satisfied when a competitive interaction among the communities is present and in such a case, the master equation model may suggest new statistical properties for the non-equilibrium stationary state of an ecosystem.

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25. Suweis, S., Grilli, J., Banavar, J.R., Allesina, S., Amos Maritan, A.: Effect of localization on the stability of mutualistic ecological networks. Nature Communications 6, 10179 (2015) 26. Landi, P., Minoarivelo, O.H., Brännström, A., Hiul, C., Dieckmann, U.: Complexity and stability of ecological networks: a review of the theory. Population Ecology 60, 319–345 (2018) 27. Narendra, G.S., Samarech, M.C., Elliott, W.M.: On the Lotka-Volterra and other nonlinear models of interacting populations. Rev. Mod. Phys. 43(2), 231–276 (1971) 28. Bazzani, A., Sala, C., Giampieri, E., Castellani, G.: Master equation and relative species abundance distribution for Lotka-Volterra models of interacting ecological communities. Theor. Biol. Forum 109(1-2), 37–47 (2016) 29. Purves, W.D., Pacala, W.S.: Ecological drift in niche-structured communities: neutral pattern does not imply neutral process. In: Biotic Interactions in the Tropics, pp. 107–138. Cambridge University Press (2005) 30. Holt, D.R.: Emergent neutrality. Trends Ecol. Evol. 21(10), 531–533 (2006) 31. Chisholm, R.A., Pacala, S.W.: Niche and neutral models predict asymptotically equivalent species abundance distributions in high-diversity ecological communities. PNAS 107(36), 15821–15825 (2010) 32. Xiao, Y., Angulo, M.T., Liu, Y., Friedman, J., Waldor, M.K., Weiss, S.T.: Mapping the ecological networks of microbial communities. Nature Communications 8, 2042 (2017) 33. Faust, K., Raes, J.: Microbial interactions: from networks to models. Nat. Rev. Microbiol. 10, 538–550 (2012) 34. Suweis, S., Simini, F., Banavar, J.R., Maritan, A.: Emergence of structural and dynamical properties of ecological mutualistic networks. Nature 500, 449–452 (2013) 35. Massen, C.P., Doye, J.P.K.: Power-law distributions for the areas of the basins of attrations on potential energy landscapes. Phys Rev. E 75, 037101 (2007) 36. Bazzani, A, Fani, R., Freguglia, P.: Modeling mutant distribution in a stressed Escherichia coli bacteria population using experimental data. Phys. A Stat. Mech. Appl. 393, 320–326 (2014) 37. Elias, M., Gompert, Z., Jiggins, C., Willmott, K.: Mutualistic interactions drive ecological niche convergence in a diverse butterfly community. PLoS Biol. 6(12), e300 (2008) 38. Dunne, J.A., Williams, R.J., Martinez, N.D.: Network topology and biodiversity loss in food webs: robustness increases with connectance. Ecology Letters 5(4), 558–567 (2002) 39. Chakrabarti, C.G., Ghosh, K.: Maximum-entropy principle: ecological organization and evolution. J. Biol. Phys. 36(2), 175–183 (2010) 40. Smith, E.: Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions. Rep. Prog. Phys. 74, 046601 (2011)

Accurate Recognition of Spatial Patterns Arising in Spatio-Temporal Dynamics of Invasive Species Natalia Petrovskaya and Wenxin Zhang

Abstract Accurate identification of spatial patterns remains a challenging problem in many ecological applications. One example is a problem of biological invasion where distinguishing between patchy spatial density pattern and continuous front spatial density pattern is important for monitoring and control of the invasive species. In this paper we address the problem of pattern recognition in biological invasion in terms of a biologically meaningful mathematical model consisting of two coupled integro-difference equations. The model allows for generating topologically different spatial structures and we employ several topological characteristics of spatial pattern to investigate various spatial density distributions. It is argued that, among the other topological quantities, the number of objects in the visual image of a spatial distribution gives us the most reliable conclusion about spatial pattern when it is required to distinguish between continuous and discontinuous (patchy) spatial structures. Furthermore, sensitivity of the pattern classification above to the definition of a monitoring protocol is discussed in the paper. Two basic properties of the monitoring protocol (i.e. the threshold density value and the number of sampling locations) are investigated and it is demonstrated how their variation affects correct reconstruction of spatial density pattern. Keywords Biological invasion · Spatial pattern · Pattern recognition · Patchy spread

1 Introduction Developing reliable and efficient methods for analyzing spatial pattern is an important stream of research in ecology. Various approaches have been developed in recent decades [9, 22] yet the universal criteria to distinguish between spatial

N. Petrovskaya () · W. Zhang School of Mathematics, University of Birmingham, Birmingham, UK e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_2

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patterns still do not exist. One important problem where the methods of pattern recognition are of great demand is the problem of biological invasion. Understanding the properties of the spatio-temporal patterns of alien species spread during biological invasion is a problem of considerable theoretical and practical importance as it has implications for the invasive species management and control [8, 19]. The seminal results in the theory of biological invasion obtained in [1, 6] and [23] have stated the existence of a continuous population front separating invaded area behind the front from non-invaded area. In recent decades, however, an alternative spatiotemporal pattern of ‘patchy’ invasion has been observed both in practice and theory where no continuous front exists and the alien species spread occurs through the formation and dynamics of patches of high population density [10, 11, 17, 18, 21]. Mathematical theory of patchy invasion is not well developed and it still is unclear how much different patchy spatial structures are from continuous front structures. Primary topological analysis of patchy spatial patterns has been done in [5] for spatial density distributions observed in a reaction–diffusion model of biological invasion. A detailed classification of spatial patterns has been provided in [16] where an integro-difference mathematical model of biological invasion has been considered that allowed for generating a great variety of spatial patterns. Meanwhile a question has been raised in [16] about similarity of continuous front and patchy spatial patterns. Agglomeration of separate patches located close to each other looks often like a continuous front spatial distribution and visual inspection of those patterns may easily result in a wrong conclusion about the type of biological invasion. Thus, introduction of more reliable means of identification of spatial patterns is required and in the present paper we continue with the approach in [16] to investigate several topological characteristics that can potentially help us to distinguish between continuous front and patchy spatial patterns. It will be shown in the paper that, among the other topological quantities such as the fragmentation rate and the density of objects, the number of objects in the binary map of a spatial distribution remains the most reliable topological characteristics when it is required to conclude whether or not spatial pattern is a patchy structure. Employing the number of objects for spatial pattern analysis results in recognition of continuous front density distribution as a single object, while a ‘no front’ patchy structure is considered as several separate objects in a spatial domain. The above classification is reliable and convenient yet it is made under the assumption that information available about the spatial density of invasive species is sufficient for correct reconstruction of the visual image of spatial pattern. Monitoring protocols, however, do not always fulfill the above requirement as financial and labour restrictions may prevent practitioners from collecting data about spatial pattern on the fine resolution scale. One important parameter in the monitoring protocol is the threshold density, i.e. the minimum density imposed when data are collected, as very small values of the population density may be hard to detect due to limitations of monitoring techniques. Clearly, if the threshold density considered in the monitoring routine is inadequately large then patches with the low density may be missed and wrong spatial pattern will be reconstructed from monitoring data. How much information about the population density is detected depends also

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on the number of sampling locations used in the monitoring protocol. The grid of sampling locations must be fine enough to capture sufficient information about the population distribution in order to adequately represent spatial pattern. Patches with the nonzero density will be partially or completely missed if the distance between sampling locations is greater than the patch size, no matter how large the animal density is within the patch (e.g. see [12–14]). The loss of information about spatial pattern because of some unintended flaws in the monitoring protocol may result in misidentification of a spatial structure, i.e. patchy spatial pattern can be confused with a continuous front spatial distribution. Inaccurate recognition of the spatial type of biological invasion may, in turn, lead to a wrong conclusion about the ecological system with possible dangerous consequences for the environment. Thus in the present paper we provide detailed study of parameters that may affect correct recognition of spatial pattern in the model we use for simulation of biological invasion. Namely, we investigate sensitivity of various spatial patterns to the threshold density and the number of sampling locations. The paper is organized as follows. In the next section, we briefly formulate a mathematical model of biological invasion we employ to generate various spatial density distributions. In Sect. 3, we introduce several topological quantities that will be used for classification of spatial patterns. Extensive numerical study will be provided in Sect. 3 in order to investigate reliability of the topological characteristics when they are applied to various spatial patterns appearing in our model. In Sects. 4 and 5, we consider sensitivity of spatial patterns to the conditions of the monitoring routine. It will be demonstrated in Sect. 4 that it is still possible to distinguish between patchy and continuous spatial patterns when an inadequately large value of the threshold density is employed in the problem and some information about the population density is therefore lost. Sensitivity of spatial pattern to the number of sampling locations is checked through numerical study in Sect. 5 where the threshold number of sampling locations required to correctly identify the type of invasion regime will be found in all our numerical test cases. Finally, Sect. 6 provides discussion and concluding remarks.

2 Model and Method In this work we are interested in spatio-temporal density distribution of some species N where the species N is engaged into an agonistic interaction with another species P . That interaction can be thought of as a prey–predator system [21] or a host– parasitoid system [5]. We consider the former case where we assume that the life cycle of both prey and predator species is stage-structured and consists of the demographic stage and dispersal stage. The above assumption results in a timediscrete framework [7] where the population densities of the both species evolve from generation t to generation t + 1. Let Nt (r) and Pt (r) be densities of the prey and the predator in generation t over continuous 2 − D space, i.e. r = (x, y). In our

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model we assume that the species first go through the demographic stage and then disperse. The prey–predator dynamics during the demographic stage is generically described by the following equations:  Nt (r) = f (Nt (r), Pt (r)),

t (r) = g(Nt (r), Pt (r)), P

(1)

where Nt and Pt are the species spatial distributions emerging after the dispersal stage in the previous generation. Following [20, 21] we choose the demographic functions f and g as follows: f (Nt , Pt ) =

A (Nt )2 1 + B 2 (Nt )2

× exp (−κPt ) ,

g (Nt , Pt ) = δNt Pt ,

(2) (3)

where A > 0 is the prey intrinsic growth rate, (1/B) > 0 is the prey density for which its per capita growth rate reaches its maximum, κ > 0 is the predator efficiency and δ > 0 quantifies the predator growth rate. For the purpose of numerical simulation we make functions f and g dimensionless to arrive at f (Nt , Pt ) =

a (Nt )2 1 + b (Nt )2

× exp (−Pt ) ,

g (Nt , Pt ) = Nt Pt ,

(4) (5)

where demographic parameters are a = A/δ and b = (B/δ)2 and functions Nt (r) and Pt (r) now are dimensionless population densities. For a more informed discussion of demographic functions f and g see [20]. After the demographic stage of the given generation is complete, the next generation of species will disperse over domain Ω,  Nt +1 (r) =

 Nt (r ) kN (r, r )dr ,

Ω

 Pt +1 (r) =

t (r ) kP (r, r )dr . P

(6)

Ω

The dispersal kernel ki , i = N, P in Eq. (6) is the probability density function of the event that an individual moves from position r to position r after dispersal. We follow our previous work [16] to consider the Gaussian dispersal kernel given by ki (r, r ) ≡ ki (|r − r |) =

1 |r − r |2 exp(− ), 2παi2 2αi2

i = N, P ,

(7)

where the standard deviation αi is the parameter quantifying the spatial scale of the dispersal. For the rest of this paper we select parameters αN = 0.1 and αP = 0.125 in our computational test cases.

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Finally, substitution of (1) into (6) results in the following model of the two-stage time-discrete spatio-temporal population dynamics given by a system of integrodifference equations:  Nt +1 (r) =  Pt +1 (r) =

       kN |r − r | f Nt r , Pt r dr ,

(8)

       kP |r − r | g Nt r , Pt r dr .

(9)

Ω

Ω

In the model above prey N is thought of as an invasive species and it is assumed that the prey growth can be controlled by predator P . Since the phenomenon of biological invasion implies introduction of the invasive species to a confined spatial domain we consider the following initial conditions N0 (x, y) = N ∗ for l1N ≤ x ≤ l2N and l3N ≤ y ≤ l4N ,

(10)

and N0 (x, y) = 0 otherwise, P0 (x, y) = P ∗ for l1P ≤ x ≤ l2P and l3P ≤ y ≤ l4P ,

(11)

and P0 (x, y) = 0 otherwise, where N ∗ is the prey equilibrium density in the absence of the predator and P ∗ is the predator equilibrium density in the predator–prey system. We refer the interested reader to work [20] where the expressions for N ∗ and P ∗ have been derived. Given the initial conditions (10) and (11), the Eqs. (8) and (9) are considered in the infinite domain Ω = {(x, y) : −∞ < x < ∞, − ∞ < y < ∞} to produce the solution for prey density Nt +1 (r) and predator density Pt +1 (r). Since the solution cannot be obtained in closed form and we have to deal with the numerical solution the original infinite domain is replaced with a finite domain Ω = {(x, y) : −L ≤ x ≤ L, − L ≤ y ≤ L}. For the rest of this paper we always choose the domain size as L = 20. We also use the symmetric initial density distributions where we always take l1N = −1, l2N = 1, l3N = −1, l4N = 1 for prey N and l1P = −0.1, l2P = 0.1, l3P = −1, l4P = 1 for predator P in (10) and (11). The numerical solution to Eqs. (8) and (9) have been carefully studied in our previous work [16, 21] and the interested reader can find the details of numerical simulation there. One important feature of the system (8) and (9) is that the solution depends heavily on parameters a and b in (4) and (5), and several different types of spatial patterns can be generated when a and b are varied. Three basic spatial distributions are shown in Fig. 1 where all of them have been produced at time t = 200. A convex continuous front in Fig. 1a obtained for a = 4.0 and b = 1.8 presents the simplest spatial pattern from a topological viewpoint. An example of a concave continuous front with more sophisticated spatial pattern in the wake is shown in Fig. 1b where the density distribution has been obtained for a = 4.0 and

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Fig. 1 Basic spatial patterns arising in the model. Snapshots of the spatial density distribution at time t = 200. (a) convex continuous front with pattern formation in the wake obtained for parameters a = 4.0 and b = 1.8; (b) concave continuous front obtained for parameters a = 4.0 and b = 0.716; (c) patchy spread (without any continuous front) obtained for parameters a = 4.0 and b = 0.714

b = 0.716. Finally, for a certain range of a and b it is possible to observe spatial pattern appearing as a collection of separate patches of the non-zero density. In the latter case there is no continuous boundary separating the areas with zero and nonzero density and ‘patchy’ invasion occurs as the population spreads over the space [10, 11, 17, 18]. One example of the patchy spatial distribution is shown in Fig. 1c for a = 4.0 and b = 0.714. It is important to notice here that the topological structure of the spatial patterns in the figure does not change as the time progresses, i.e. a patchy spatial distribution will not be transformed into a continuous front distribution at later times, and vice versa. Concave and patchy spatial patterns whose examples are shown in Fig. 1b, c are of particular interest to us. Visual inspection of the pattern in Fig. 1b leads us to the conclusion that concave spatial patterns may not always be easily distinguished from patchy patterns. Indeed, looking at a complex spatial distribution it may be hard to see whether we have a continuous front line or it is broken in some places (cf. also Fig. 6 further in the text). While visual inspection of spatial patterns is not a reliable means of control on its own, the above conclusion is, nevertheless, symptomatic, as it may result in intuitive expectation of spatial patterns being ‘topologically unstable’. That is, we may expect a concave continuous front to break into a collection of separate patches when we slightly change system parameters. Moreover, if continuous front spatial patterns are very similar to patchy patterns, any occasional error in the sampling protocol may result in a wrong reconstruction of the spatial pattern when real-life data are collected. That will then send a wrong practical message to invasion managers about efficient control of invading species. The above issue of ‘topological stability’ has already been investigated in our previous work [16] where we have shown that, despite being topologically close to each other, the spatial patterns resulting from continuous front propagation and from patchy invasion respond very differently to the value of the threshold density in the problem. That result alone, however, is not sufficient to conclude about the robustness of continuous and patchy spatial patterns. In the present work we provide

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a more informed study of this problem where we compare continuous and patchy spatial patterns based on several rigorous topological characteristics.

3 Topological Characteristics of Spatial Pattern We follow our work [16] to develop a comprehensive computational framework that should enable for accurate classification of spatial patterns. First, spatial density distribution N(x, y) is converted into a binary black-white image in the spatial domain (see [16] where a detailed explanation of the conversion procedure is provided). The binary image obtained from the original density distribution is then analysed with the help of the Image Processing Toolbox (IPT) in MATLAB (https:// uk.mathworks.com/help/images/index.html) where all conclusions made about the binary image remain correct for the original spatial pattern. Below we introduce the basic image characteristics used to study spatial patterns in more detail.

3.1 The Number of Objects The number of objects is the most important characteristic of spatial pattern in the problem of biological invasion as it defines the invasion type. We define an object as a region of the non-zero density with a closed external boundary. From the pattern recognition viewpoint, a continuous front density distribution can be classified as a single object, while a ‘no front’ patchy invasion presents a collection of separate objects in a spatial domain. A convex continuous front in Fig. 1a can then be thought of as a single object and a concave continuous front in Fig. 1b is a single object as well. Meanwhile we have five objects in Fig. 1c, where a patchy spatial distribution is shown, as every patch counts as a separate object. It should also be noticed here that in the above classification any density structure behind a continuous front will be ignored as we are only interested in the number of separate patches (cf. Fig. 1b, c). Using the IPT software allows us to count the number of objects n in any binary image obtained from the spatial density distribution. Let us choose the range of parameters a and b (see the discussion of the parametric plane (a, b) in [16, 21]) and compute density distribution N(x, y) at time t = 200 for various pairs (a, b) taken from that range. The IPT software is then used to decide whether spatial density N(x, y) is continuous front (i.e. n = 1) or patchy pattern (i.e. n > 1). The results of pattern analysis are shown as a table in Fig. 2 where the number of objects n has been documented for various pairs (a, b) and both parameters a and b are varied with a small increment. Similar analysis has been done in [16] where it has been shown that, for a certain range of parameters a and b, the evolution of initial conditions (10) and (11) results in a wealth of invasion patterns. Depending on a and b, the population spread can occur either by the propagation of

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Fig. 2 The number of objects in the spatial pattern as a function of parameters a and b in the invasion model (8) and (9) with the initial condition (10)

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26 N. Petrovskaya and W. Zhang

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the continuous population front, convex or concave, or by the dynamics of separate population patches. Let us notice, however, that in the present work we do not need to distinguish between the convex and concave continuous front and the both spatial patterns are considered as one object (n = 1) wherever they appear in the table. We also notice that n = 0 in the table corresponds to the extinction regime (i.e. no objects in the spatial domain). It can be seen from the table that continuous front topology is a prevailing spatial structure whilst patchy spatial patterns are relatively hard to generate. Moreover, patchy spatial patterns appear randomly and there are no threshold values a and b responsible for formation of separate patches in spatial density distributions. For example, choosing a = 6.1 and increasing the value of b from b = 0.708 up to b = 0.718 with the increment 0.002 results in formation of two patchy patterns at b = 0.710 and b = 0.712 following by a continuous front at b = 0.714. Another patchy distribution appears at b = 0.716 and we then see a continuous front again at b = 0.718. Similarly, choosing b = 0.698 and increasing a with the increment 0.1 results in a continuous front at a = 4.6 following by two patchy distributions at a = 4.7 and a = 4.8 and a continuous front again at a = 4.9. The random nature of patchy spatial patterns in parametric plane (a, b) gives the rise to the issue of accurate recognition of a spatial distribution as it remains unclear to what extent we can distinguish between a continuous spatial distribution and a discontinuous (patchy) one. Hence our next step is to introduce other topological characteristics as using various topological properties of the spatial pattern may facilitate our investigation of the above question.

3.2 The Fragmentation Rate and the Density of Objects We could see in our previous discussion that a continuous front invasion scenario corresponds to one object in the spatial pattern while having more than one object in the spatial pattern is related to patchy invasion. The number of objects, however, does not take into account a fine structure of the spatial pattern. Consider, for example, two hypothetical spatial patterns shown in Fig. 3a, b. The number of objects n = 3 is the same in both patterns. Moreover, the total area of the objects is the same and is equal to 11 units where we consider the area of any square cell equal to one in the figure. We therefore introduce the fragmentation rate of the spatial pattern to distinguish between collection of patches in Fig. 3a, b. The image fragmentation is a well-studied topic and various indices can be used in order to measure the degree of fragmentation (e.g. see [4] and references therein). In our definition of the fragmentation rate we follow the paper [2] where in turn the definition was taken from the original work [3]. Let Nb (x, y) be the binary image converted from the original density distribution N(x, y) as explained in the previous section. The fragmentation rate fr (Nb (x, y)) is then defined as fr (Nb ) = 1 −

s(Nb ) , B[p(Nb )]

(12)

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(a)

(b)

Fig. 3 Spatial patterns have the same area of the non-zero density and the same number of objects but their fragmentation rate is different. (a) The fragmentation rate calculated according to (12) is fr = 0.2 (b) the fragmentation rate is fr = 0.4

where p(Nb ) is the population density calculated for the ‘binary’ population, s(Nb ) is a degree of fragmentation for given population Nb and is related to the average number of neighboring objects in the population image, and B[p(Nb )] is the maximum possible value of the index s(Nb ) among all distributions Nb with a given abundance p(Nb ); see [2] for a well-informed explanation of formula (12). The fragmentation rate calculated according to (12) is fr = 0.2 for the image in Fig. 3a and fr = 0.4 for the image in Fig. 3b. The fragmentation rate for all spatial density patterns in Fig. 2 is shown in Fig. 4 where blank spaces are left in the table for pairs (a, b) resulting in the extinction regime. It is seen from the figure that transition from continuous front spatial pattern to patchy spatial pattern is not related to any significant changes in the fragmentation rate of the image. Continuous front patterns and patchy patterns obtained for the same values of a and close values of b may still have very similar fragmentation rate. For example, spatial pattern obtained for a = 6.1 and b = 0.712 is a patchy density distribution (n = 25) with the fragmentation rate fr = 0.0265. Meanwhile, spatial pattern obtained for a = 6.1 and b = 0.714 is a continuous front (n = 1) with the same fragmentation rate fr = 0.0265. Moreover, two continuous front patterns obtained for the same values of a and close values of b may have a distinctly different fragmentation rate. Continuous front pattern at a = 4.7 and b = 0.712 has the fragmentation rate fr = 0.0236, while a small increment in b to have b = 0.714 results in continuous front pattern with fr = 0.0188 which constitutes a 20% drop in the fragmentation rate. It is clear from the data in Fig. 4 that, similarly to the number of objects in spatial pattern, there are no threshold values a and b responsible for formation of separate patches in spatial density distributions. Random variations in the fragmentation rate may stem from the fact that a concave continuous front has a complex density distribution behind the front (cf. Fig. 1) and a continuous front density distribution

2.01 2.17 1.76 1.89 1.87 1.94

1.9

1.9 1.97 2.08 2.34

5

2.1 2.04 2.09

1.66

6.5

6.4

2.22

2.5 2.56

7 7.02 7.04 7.06 7.08

7.1 7.12 7.14 7.16 7.18

2.6

2.5 2.27

2.8 7.3 7.32 7.34 7.36 7.38

2.7 2.59 2.56 7.2 7.22 7.24 7.26 7.28

2.5 2.83 2.39

2.5 2.42 2.44 2.59 2.57

2.7 2.38 2.27 2.57 2.53 2.54 2.67 2.6 2.38 2.49 2.67 2.69

3.91 2.61 2.93 2.61 2.62 2.56 2.27 2.52 2.55 2.64 2.39

2.39 2.66 2.51 2.46 2.73 2.61 2.77

2.4 2.52 2.51 2.76 2.43 2.26 2.65 2.47

2.4 2.42 2.4 2.75

Fig. 4 The fragmentation rate for spatial patterns whose classification is presented in Fig. 2. Blanc spaces in the table correspond to the extinction regime (i.e. no objects in the spatial domain). Fragmentation rate fr is multiplied by the factor of 100 for the sake of convenience

6.9 6.92 6.94 6.96 6.98

2.58

6.3

2.3 2.42 2.58 2.75 2.63 2.65 2.38 2.43

2.3 2.65 2.65 2.69 2.48 2.75 2.49 2.51 2.92 2.65 2.51 2.44 2.29

2.5 2.61 2.82 2.43 2.19 2.66 2.31 2.44 2.49 2.46 2.66 2.69 2.17

2.4 2.47 2.63 2.47 2.68 2.42 2.23 2.44 2.51 2.45 2.33 2.61 2.47 2.37

2.5 2.64 2.59 2.45 2.49

2.5 2.56 2.97 2.59

2.54 2.44 2.56 2.63

6.2

2.85 2.33

2.1 2.38 2.81 2.4 2.39 2.52 2.48

2.81 2.64 2.66 2.23 2.29

2.48 2.14 2.23

2.23 2.01

2.38 2.33 2.11 2.27

6

6.1

2.4 2.32 2.35 2.59 2.47 2.28

2.5 2.37 2.37 2.01 2.57 2.35 2.44 2.66 2.57 2.64 2.46 2.32 2.51 2.42 2.65 2.34 2.63 2.19 2.72 2.31 2.31

2.4 2.32 2.48

2.4 2.66 2.25 2.55 2.34 2.14

2.2 2.29 2.33 2.48 2.47 2.47 2.06 2.25 2.48 2.09

2.1 2.21 2.37 2.35 2.47 2.39 1.95

2.6 2.28 2.41 2.41 2.48 2.56

2.6 2.56 2.44 2.43 2.38 2.44

2.14 2.48 2.39 2.85 2.51 2.43 2.53 2.42 2.66 2.41 2.67 2.18

5.9

2.24

2.5

5.7

5.8

2.4 2.73

5.6

2.4 2.71 2.47 2.52 2.52 2.38 2.32 2.62

2.51 2.13 2.61 2.13 2.21

2.83

5.5 2.47 2.34

2.2 1.98 2.43 2.25 2.48 2.31 2.41 2.34 2.27 2.29 2.29 2.44 2.14

2.3 2.16

2.3 2.27 2.08 2.21 2.3 2.23 2.22

2.2 2.09

2.3 2.03 2.05 2.33 1.91 2.18 2.13

2.3 2.15 2.14 2.31 2.44 2.12

2.43 2.35 2.81 2.36 2.13 2.52 2.25 2.37 2.47 2.24 2.28 2.34 2.28 2.35 2.17 2.23 2.53 2.24 2.45 2.34 2.06

2 2.37 2.14 2.35

5.4 2.87 2.23

2.1 2.28 1.98

2.23

2.51 2.34

5.3 2.41 2.33 2.06

2.2

2.65 2.34 2.16 3.04 1.98 2.22 2.32 2.14 2.55 2.36 2.36 2.27 2.61 2.43 2.38 2.05

2.61

2.56 1.74 2.48 4.15 2.02 2.21 2.03 2.31 2.21 2.24 2.28 2.31 2.38 2.16 2.27 2.03 2.33

5.2 2.25 2.34 2.13

5.1

2.2 2.34 2.09 2.28 2.02 2.33 2.15 1.91 2.07

2.39 2.42 2.13 2.33 2.56 2.24 2.42 1.93 2.72 2.02 2.39 2.37 2.27 2.22 2.17 2.17 2.36 2.16 2.49 2.02 1.92 2.24 2.42 2.16

2.2 1.76 2.23 2.55 2.64 1.71 1.92 2.31 2.26

1.95 2.16

4.9

4.8

2 2.06 2.21 2.19 2.25 2.25 2.08

1.9 2.49 2.54 2.12 2.19 2.03 1.98 2.38 2.11 2.39 2.03

2.3 2.34 2.34 1.94 1.94 1.78 2.12 2.07 2.29

2.5 2.31 1.95 2.51 2.29 1.77 2.06 2.48 1.95 2.08 2.33 2.19 2.08

3 2.92 2.14 2.29

2.1 2.09

2.1 2.18 1.84 1.99 1.86

2.11 2.63 2.11 1.93 2.17 1.89 2.41 2.22 2.03 2.36 1.88 2.09 2.29 2.03 2.23 2.12 2.23 2.22 2.21 2.09 1.89 2.09 2.08

2.35 2.01

1.8 2.15 1.87

4.7

4.6

1.9 1.97 1.89

2.3 1.79 2.05 2.06 2.23 1.94 1.98

1.96 1.91 2.25 2.45 1.98 2.41

1.9 2.01 1.72 2.07 2.16 2.14

2.1 2.01

1.9 1.82 2.03 1.92 2.06 1.93 1.82 1.76

4.5 2.54

2.1

1.7 1.98

2.1 2.19 2.48 1.66 1.99 2.04 1.89 1.96 1.87 2.05

2.76 1.63 2.31 1.92 2.09 1.84 2.17 1.81 2.12 2.34 2.03 2.02 2.09 1.95

2.29 2.69 1.88 1.61 1.88 2.41 2.28 1.87

1.9

1.89 1.76

2.53 2.22 2.33 1.79 1.97 1.89 1.89 2.19 2.41 2.46 2.16 1.81

3.39 2.41

2.6 2.01

2.04 1.97

4.4 2.07 2.04

2.06 2.19

4.2

4.3

2.79

2.06 2.62

4.1

4

3.9

b*10

a

Accurate Recognition of Spatial Patterns 29

30

N. Petrovskaya and W. Zhang

may be very different from patchy pattern with similar values of a and b. Thus, we want to introduce the density of objects to take the density distribution behind the front into account. We define the density of objects as d=

n˜ , A

(13)

where n˜ is now the total number of objects (including the objects behind the continuous front), and A is the total area occupied by those objects. The density of objects for all spatial density patterns in Fig. 2 is shown in Fig. 5. Analysis of data in Fig. 5 leads us to the conclusion that the density of objects is not a defining quantity when it is required to distinguish between continuous and patchy distributions. Similarly to the fragmentation rate, we cannot see any distinct trend in the value of d when continuous front pattern transforms into patchy pattern and vice versa. Consider, for example, patchy spatial pattern obtained for a = 4.4 and b = 0.698. The density of objects for that pattern is d = 0.138. This is a very high value in comparison with d = 0.0303, the density of objects we have for a continuous front spatial pattern obtained for the same a = 4.4 and b = 0.702. However, we cannot conclude that the density of objects for patchy distributions is always distinctly higher than the density of objects for continuous front distributions. Continuous front spatial pattern obtained for a = 6.1 and b = 0.7 has the density of objects d = 0.1426 while patchy pattern with a = 4.1 and b = 0.738 has d = 0.0085. Our study in this section reveals that continuous front and patchy spatial distributions may have a very similar topological structure behind the front and may therefore be easily confused with each other when simple visual inspection of those patterns is made. Moreover, the analysis of the basic topological characteristics such as the number of objects, the fragmentation rate and the density of objects showed that the only reliable index of spatial distributions arising in the model is the number of objects in the pattern. Hence, for the rest of the paper we study what factors can make an impact on the topology of spatial distribution and, given similarity between continuous front and discontinuous patchy distribution, how easy it can be to make a wrong classification of spatial pattern.

4 Sensitivity of Spatial Pattern to the Cut-Off Parameter It has been demonstrated in Sect. 3 that, for any of the topological characteristics we studied there, there are no threshold values that would separate continuous front spatial patterns from patchy distributions. Furthermore, it is clearly seen from Fig. 2 that the solution to (8) and (9) is very sensitive to the choice of parameters and small variation in either parameter a or b may result in transformation of continuous front pattern into patchy pattern and vice versa. Random transition of spatial pattern between continuous front and patchy structures is shown in Fig. 6 where very small

0.57

0.00

0.00

0.00

3.44

4.1

4.2

4.3

4.4

5.82

0.00

0.00

0.00

0.00

0.00

0.00

3.50

0.80

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

0.00

0.00

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5.7 13.77

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

5.6

5.8

5.9

6

6.1

6.2

6.3

6.4

6.5

6.92

0.00

0.00

0.00

0.00

0.00

5.53

6.94

0.00

0.00

2.01

0.00

5.10

0.00

0.00

4.90

0.00

0.00

0.00

0.00

1.00

1.09

0.58

0.52

8.20

7.22

3.14

0.00

0.00

0.00

5.86

0.91

0.00

0.00

0.00

0.00

4.26

1.46

0.48

2.01

0.63

1.56

0.72

2.86

1.52

6.96

0.00

0.00

6.34

0.00

1.17

0.00

0.00

4.75

4.39

6.20

9.78

8.49

1.93

2.17

4.54

3.85

4.34

1.73

4.32

2.77

1.31

6.35

2.85

8.56

6.98

0.00

2.60

0.00

0.00

7

9.04

0.00

2.76

5.34

0.00 14.26

0.00

4.32

0.00 11.53

0.00

0.00 13.23

0.00

0.00

0.00

0.00 12.12

0.00 12.23

7.56

0.70

0.49 10.82

6.97

1.78

0.00

0.56

3.48

0.00

0.00 11.37

0.00

4.74

0.00

0.00

0.00 13.80

0.00

6.35

6.72

0.00

0.00

6.88

0.67

2.80

2.35

8.37

1.01

4.79

4.00

7.35

0.31

0.41

5.58

3.73

4.43

0.07

0.00

0.75

2.72

1.95

7.02

8.59

4.85

5.70

0.88

7.46

6.27

7.66

1.41

3.70

1.69

1.18

2.76

1.11

1.44

8.31

5.06

1.15

1.45

1.34

0.60

0.07

3.22

0.00

0.84

8.53

2.92

0.81

4.28

0.96

3.00

2.51

5.47

4.86

8.78

4.73

0.94

0.87

5.54

0.68

4.27

5.56

4.12

1.17

1.29

0.02

1.49

0.00

0.00

3.28 7.04

7.06

4.65

5.69

3.55

8.43

6.12

3.10

5.35

4.53

4.89

1.32

5.84

1.23

1.88

3.29

3.77

7.42

0.74

1.25

0.40

5.10

6.59

1.54

1.78

0.83

0.68

2.84

0.00

7.08

1.57 7.1

4.36

1.37 10.40

6.34 10.04 13.84

5.45

9.31

2.49

4.54

7.64 13.17

4.71

3.88

5.47

8.95

3.72

3.60

2.66

4.33 11.23

7.83

2.61

8.25

7.02

0.34

2.42

2.75

3.31

4.96

3.66

2.29

2.74

3.03

8.59

0.00

0.00

1.41

0.00

1.34

4.61

4.94

7.44

4.79

2.97

2.95

2.55

7.75

3.81

3.04

4.42

2.36

1.63

4.39

4.60

8.25

2.87

5.12

0.44

8.60

0.55

8.67

7.12

2.82

7.14

3.97

3.91 12.45

1.90 10.42

6.45

6.44 10.97

7.46

4.09

5.50

5.66

3.62

1.18

3.04

4.24

1.32

1.90

3.70

6.72

3.42

3.19

2.37

5.57

0.87

3.57

1.93

2.89

2.88

8.69

5.35

4.66

2.38

2.25

3.71

3.00

1.97

8.81

3.82

3.30

4.39

6.30

3.45

2.35

0.72

1.29

4.45

0.67

0.37

5.31

1.65

7.16

7.72

4.10

7.99

4.40

9.63

3.78

8.06

5.26

2.17

5.46

4.99

1.58

8.91

3.78

3.22

2.47

3.84

1.15

2.47

1.78

0.88

2.70

3.04

3.54

1.30

1.64

3.75

0.28

2.70

3.36

7.18

4.09

7.2

1.71

5.09 12.84

6.93

2.83

3.25

4.37 11.40

4.05

7.86

6.63

7.64

3.10

1.00

3.97

5.91

3.12

6.53

2.83

1.71

1.97

0.69

0.67

4.56

4.55

3.19

1.30

3.48

2.71

7.30

5.97

0.57

4.93

2.11

4.83

4.25

2.55

1.81

3.08

4.03

4.28

2.88

1.28

2.66

2.79

2.81

1.15

2.36

1.90

1.58

9.29

2.70

7.22

2.34

5.83

4.26

7.24

9.85

5.41

5.31

7.09 11.64

5.94

1.29

3.51

6.10

3.15

5.11

1.78

1.43

3.68

5.71

1.15

2.23

0.66

2.73

1.64

4.17

4.58

4.00

3.03

1.99

0.79

3.45

1.38

6.43

2.28

4.05

2.10

3.68

3.95

0.59

3.15

1.57

3.99

1.00

1.93

2.15

5.21

3.19

0.25

1.32

2.03

0.27

1.21

1.95

1.44

7.26

4.87

9.51

1.93

5.45

7.28

6.06

3.26

2.56

5.60

8.32 10.46

2.76

6.95

6.13

4.20

3.76

7.00

1.17

2.55

7.73

2.96

5.20

3.66

2.96

1.94

2.34

4.13

3.80

1.52

2.08

0.40

1.87

3.77

7.3

6.61

3.30

1.50

4.58

2.67

4.08

8.04

5.26

4.46

5.46

5.14

5.42

1.03

0.59

1.90

1.50

1.02

0.63

0.23

1.29

1.97

0.75

1.71

2.20

0.26

3.07

0.40

7.32

9.03

4.70

1.24

3.52

2.76

6.36

4.31

2.84

3.92

2.58

3.07

5.86

3.00

3.39

2.16

3.03

0.58

1.28

2.50

1.44

2.66

1.63

0.79

2.13

1.69

0.92

4.86

7.34

6.11

3.80

5.38

8.52

3.19

5.03

6.95

7.93

6.73

6.15

1.31

3.81

2.63

3.84

1.69

0.94

4.14

2.12

1.62

0.73

2.44

1.33

0.74

2.72

0.93

1.52

2.32

7.36

2.48

7.39

3.01

6.16

4.06

3.37

4.19

3.99

6.96

4.57

4.29

4.04

4.24

2.77

1.16

2.02

3.68

2.97

1.10

4.18

5.46

0.76

2.95

0.74

1.54

0.60

2.51

7.38

6.66

5.98

8.59

6.92

6.74

3.19

7.78

4.70

3.30

3.39

6.99

4.36

2.21

3.04

5.04

2.58

3.06

1.23

2.42

1.59

2.99

1.56

0.84

0.25

0.85

0.32

1.85

Fig. 5 The density of objects for spatial patterns whose classification is presented in Fig. 2. Density of objects d is multiplied by the factor of 100 for the sake of convenience

6.9

6.29

7.78

5.5

0.00

5.17

4.94

5.4 13.91

2.24

6.12

5.97

3.09

0.00

0.00

0.00

0.00

4.5 10.63

5.56

0.00

0.00

0.00

0.00

4

0.00

0.00

3.9

b*10

a

Accurate Recognition of Spatial Patterns 31

32

N. Petrovskaya and W. Zhang

Fig. 6 Sensitivity of spatial patterns. Snapshots of the spatial density distribution are shown at time t = 200 for a = 5.1. A small variation in parameter b results in random transition between continuous front and patchy spatial patterns. (a), (d), (f) continuous front spatial distributions obtained for parameters b = 0.712, b = 0.718 and b = 0.722 respectively; (b), (c), (e) patchy spatial patterns obtained for parameters b = 0.714, b = 0.716 and b = 0.720 respectively

increment δb = 0.002 in parameter b has been selected to generate various density distributions. Consider, for example, spatial patterns in Fig. 6c, d. Those density distributions are very similar to each other, yet one of them is classified as patchy pattern while another is continuous front pattern. In the absence of a well defined boundary between continuous and patchy regimes in the parametric plane (a, b) the question arises as to how strongly spatial patterns can be affected by the occasional loss of information about a spatial density distribution. In other words, how much information about the spatial distribution should we have in order to make a reliable conclusion about its topological characteristics? The above question about sensitivity of spatial patterns is directly related to accuracy of real-life ecological data. In the monitoring routine the information about an invasive species is often collected by sampling where samples taken at certain geographic locations can then be converted into the population density distribution over the whole geographic domain of interest. Small values of the population density may be impossible to detect due to limitations of sampling/monitoring techniques, the minimum detectable density being called the ‘detection threshold’. Hence one way to lose the information about the spatial population is to select a too high value of the detection threshold in the monitoring routine. This scenario can be simulated in our mathematical model by selecting high cut-off density in the problem. The nature of integro-difference equations (8) and (9) is to make the population density positive everywhere in any generation starting from t = 1 due to the dispersal kernel

Accurate Recognition of Spatial Patterns

33

(7) being used in the problem. The visual image of spatial pattern, however, exhibits areas where the population density is equal to zero (cf. Fig. 6). This ‘cleaning’ of the spatial distribution made for the sake of visualisation can be thought of as introducing a modified solution to (8) and (9) as follows: ˆ N(x, y) = N(x, y) for N(x, y) > C,

Nˆ (x, y) = 0

for N(x, y) ≤ C, (14)

where C = const is the cutoff threshold density and modified solution (14) is then used to generate the visual image. In our previous discussion in this paper we have always selected a very low value of the cutoff density (i.e. C = 0.05 which constitute 1% of the maximum density Nmax ). The low cutoff density has been selected in order to preserve as much information about the solution as possible. Meanwhile, it is clear that the topology of spatial pattern will change if we keep increasing the cutoff value and the visual image of the density distribution will finally disappear in the extreme case C > Nmax . Can we then observe transition between the two topological cases of basic interest (i.e. patchy and continuous spatial distributions) at certain values of cutoff C and, if yes, how realistically large are those threshold values of the cutoff? The sensitivity of the spatial pattern to cut-off parameter C has been investigated in our previous work [16]. We have shown in [16] that spatial patterns resulting from continuous front propagation and from patchy invasion respond differently to value C in (14). In both cases, the number of disconnected patches increases along with an increase in the cutoff threshold density, and such an increase starts when cutoff C exceeds a certain critical value. However, the critical threshold is consistently several times larger for continuous front pattern than for patchy pattern, even when visually the patterns may have similar properties (cf. Fig. 6). One example illustrating the results of our study in [16] is given in Fig. 7a, d where the number of objects is shown as a function of cutoff parameter C for the spatial patterns in Fig. 6a, b respectively. It can be seen from the figure that, despite being very similar to patchy spatial pattern in Fig. 6b, a continuous front distribution in Fig. 6a actually breaks into a collection of separate patches when the cutoff density is unrealistically high at C = 0.4. For the sake of our discussion, we briefly investigate how the fragmentation rate and the density of objects depend on the cutoff parameter. The graphs of the fragmentation rate as a function of the cutoff parameter are shown in Fig. 7b, e for continuous front pattern in Fig. 6a and patchy pattern in Fig. 6b respectively. Those graphs have very similar shape when the continuous front and the patchy distribution are investigated as the fragmentation rate increases steadily in both cases with an increase in the cutoff. The density of objects as a function of the cutoff parameter is shown in Fig. 7c, f for the continuous front and the patchy pattern respectively. Again, there is no any significant difference in the shape of the graph when the two spatial patterns are compared. These results confirm our conclusion in Sect. 3 that, despite bringing useful information about a complex structure of the spatial pattern,

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the fragmentation rate and the density of objects cannot be used as reliable indices to distinguish between continuous front and patchy spatial distributions.

5 Reconstruction of Spatial Patterns from Sparse Data We could see in the previous section that continuous and patchy spatial patterns are not sensitive to the loss of information related to increasing the cut-off parameter in the problem. It is still possible to distinguish between spatial patterns of the two invasion regimes when an inadequately large cut-off parameter is employed and some information about the population density is therefore lost. However, variations in the cut-off parameter are not the main source of uncertainty when spatial pattern is reconstructed from data collected in the monitoring routine. Below we discuss sensitivity of spatial patterns to the number N of sampling locations which is crucial when a patchy distribution has to be distinguished from a continuous front distribution. The images of spatial patterns we have analysed in the previous section have been generated from the numerical solution of Eqs. (8) and (9). The solution is defined at nodes of a computational grid where we have used a very fine grid of 1025 nodes at each direction to guarantee that the numerical solution has been computed with sufficient accuracy. Hence, as a result of our computation, we have information about the population density at 1025 × 1025 = 1,050,625 spatial locations (i.e. points (x, y) whose coordinates x and y are prescribed by the grid definition). That information allows us to reconstruct a spatial distribution with very good resolution and clearly distinguish between continuous front and patchy regimes. Let us now assume for the sake of discussion only that spatial distributions we have generated in our model exist in reality and we want to collect information about them. How much information about the population density is collected depends on the number of sampling locations used in a monitoring routine. The grid of sampling locations must be fine enough to capture sufficient information about the population distribution in order to represent spatial pattern adequately. Meanwhile, it is obvious that we cannot use the same grid of 1,050,625 sampling locations and we have to employ a much coarser grid to make our simulation of the monitoring routine realistic. It is also obvious that if we collect data about the population density on a coarse sampling grid then a spatial distribution we reconstruct from that data will be different from the original spatial distribution we have obtained on the grid of 1,050,625 nodes. One example illustrating the above statement is shown in Fig. 8 where we assume that the spatial density distribution in Fig. 6b exists in reality and we want to collect data on the population density by installing a regular ‘sampling grid’ of N × N locations in the domain. We simulate installation of the sampling grid in the domain by considering our original grid and taking every ith node from it. Let us choose N = 65 and take every 16th node from the original grid. As a result we have 4225 spatial locations where information about the population density is available as opposite to 1,050,625 points on the original grid. We then reconstruct a

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Fig. 8 Examples of spatial pattern reconstruction on (a) coarse (N = 65) and (b) very coarse (N = 9) regular Cartesian grids, where N is the number of grid nodes at each direction. The original spatial distribution has been obtained on a fine grid with N = 1025 for parameters a = 5.1 and b = 0.714; see Fig. 6b. The population density values are taken at every 16th node and 128th node of the original grid to simulate the density distribution on the coarse grid and the very coarse grid respectively

spatial distribution from the information we have collected in our ‘sampling routine’ where the result is shown in Fig. 8a. It can be seen from the figure that spatial pattern reconstructed from the information we have at 4225 sampling locations differs from the pattern reconstructed on the original grid yet it remains patchy. We then choose N = 9 to generate a very coarse grid of 81 sampling points and reconstruct spatial pattern on that grid as shown in Fig. 8b. It is clear from the figure that we now have a continuous front density distribution, so the information we collect about the population density on a grid with N = 9 is not sufficient to recognise the type of spatial pattern correctly. We therefore want to understand to what extent any conclusion about spatial pattern can be affected by the number of sampling locations used to collect information and reconstruct a spatial distribution. To investigate this issue we further develop the approach explained in the previous paragraph (see also [15]). We first numerically solve system (8) and (9) on a very fine computational grid G that has N = 2m +1 grid nodes at each direction. We choose m = 10 in our computation and consider the result obtained on the fine grid as the ‘true’ spatial density distribution. We then pretend that the spatial distribution is monitored and information about it can be collected at nodes of a regular ‘sampling’ grid Gs only. Those ‘sampling’ grids Gs are generated from the original grid G by taking the number of nodes at each direction as N = 2s + 1, 1 ≤ s ≤ 9. As in our previous example, if we want to consider a ‘sampling’ grid with N = 26 + 1 = 65 nodes at each direction we choose s = 6 and take every 2m−s th node from the original fine grid. It is clear that information about the population density is readily available at nodes of any coarse ‘sampling’ grid we consider in our procedure because those nodes also belong to the original grid G. Hence we can easily make our ‘sampling’ grid as coarse (fine) as we want by removing (adding) grid nodes, which is difficult in empirical field studies. Correspondingly, we can investigate how the type of spatial pattern we reconstruct

Accurate Recognition of Spatial Patterns

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from the data collected on a given sampling grid depends on the number of grid nodes. Consider continuous front pattern shown in Fig. 6a and let us analyse information we have about the pattern on a sequence of ‘sampling’ grids obtained from the original grid as described above. We first reconstruct it on a grid of 29 + 1 nodes at each direction where the resulting pattern is visually identical to the pattern we have on the original grid. The same can be said when we have spatial pattern reconstructed on a grid of 28 + 1 nodes at each direction. That pattern still appears as a continuous front and the number of objects n in the image is therefore n = 1. We go on with this procedure and coarsen our sampling grid to see whether continuous front pattern breaks into several isolated patches for some threshold number Nt of grid nodes on a sampling grid. The results are shown in Fig. 9a where we generate the number of objects n in the image of spatial pattern as a function of the number of grid points N on a grid where the pattern is reconstructed. Clearly, the number of objects is n = 1 when the number of grid nodes is very large. When we decrease the number N of grid points the pattern transforms as we lose information about it and the image appears as a collection of n = 23 isolated objects when the number of grid points is N = 65 at each direction. Hence the threshold number Nt is identified in this example as Nt = 65. A further decrease in the number of grid points results in an increase in the number of objects and we have n = 35 objects when the pattern is reconstructed on a grid with N = 33 nodes at each direction. However, the number of objects is again n = 1 on a coarser grid of N = 17 nodes. This happens because all information about relatively small patches is now lost on a coarse ‘sampling’ grid and the image appears again as a continuous front spatial structure. We further investigate this problem for continuous fronts in Fig. 6d, f. It can be seen from Fig. 9d, f respectively that the threshold number remains Nt = 65 and a continuous front density distribution is always recognised correctly as a single object on sampling grids with a larger number of grid nodes. Consider now patchy pattern shown in Fig. 6b and let us repeat the analysis above. We are now interested in the threshold number Nt when the patchy distribution is reconstructed as a continuous front based on data taken from a sampling grid of Nt points at each direction. The results are shown in Fig. 9b. It can be concluded from the graph that a patchy spatial pattern is not sensitive to the loss of information as it remains patchy until the grid is coarsened to Nt = 9 nodes at each direction. The other patchy patterns shown in Fig. 6c, e demonstrate similar properties. It follows from the analysis of graphs in Fig. 9c, e that the spatial distributions remain patchy until an extremely coarse grid of N = 3 points is considered.

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38 N. Petrovskaya and W. Zhang

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6 Conclusions Analysis of spatial pattern is of particular importance in the problem of biological invasion as accurate identification of spatial patterns may help practitioners to apply better, more targeted, means of control to prevent an invasive species spreading into the space. Recognition of a spatial distribution as continuous front pattern often implies that the entire area behind the front will be considered for a control action (e.g. application of pesticide), while having a patchy spatial distribution may only require dealing with the areas occupied by patches of the invasive species. In the present paper we have investigated several topological characteristics of spatial pattern that can be employed to distinguish between continuous and discontinuous spatial distributions. The mathematical model we have used to simulate spatio-temporal dynamics of biological invasions allows for generating spatial distributions that are visually very close to each other, yet some of those distributions are continuous fronts while the others are patchy spatial patterns. Hence basic characteristics of spatial pattern such as the number of objects, the fragmentation rate and the density of objects have been analysed for a variety of spatial distributions arising in the model to see if the spatial pattern type can be recognised correctly by using any of the quantities above. It has been shown in the paper that the number of objects in the visual image of a spatial distribution remains the only reliable quantity that allows one to identify patchy patterns of biological invasion. The fragmentation rate and the density of objects do not reflect any change of a continuous front to a patchy distribution occurring as a consequence of changing biologically meaningful parameters in the model and switching back and forth between patchy pattern and continuous one does not result in any predictable change in the fragmentation rate and the density of objects. One important observation about the problem is that the number of objects in the visual image of the spatial distribution presents us with a very generic definition of patchy pattern. Indeed, every separate object in the visual image counts as a patch when the number of objects is analysed, no matter how large the patch size is and how large the density is within the patch. Furthermore, the definition of patch based on the number of objects implies that, if we are allowed to neglect a complex spatial distribution behind the front, a continuous front spatial structure can be formally considered as a single large patch. Under more realistic conditions, however, the definition of patch may incorporate some additional restrictions. Indeed, practitioners may be interested just in identification of large patches with high population density and they may wish to neglect those patches where the population density in the patch and/or the patch size are relatively small. Hence the definition of patch will depend on additional parameters and different definitions of patch will then lead to different conclusions about spatial pattern. One possible approach to decide whether the patch is large or small would be to evaluate the animal density within the patch. An alternative definition, however, could be based on the geometric size of patch where the area occupied by the patch is considered as its main characteristics, no matter how densely the patch

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being populated. In the former case the threshold density is of utmost importance, while in the latter case the distance between sampling locations where information about the population density is collected becomes a very important parameter in the monitoring protocol. Correspondingly, in our model we have studied the sensitivity of spatial patterns to variations in the cutoff density and the number of grid points used to reconstruct the density distribution in the domain. It has been demonstrated in the paper that patchy spatial patterns are very robust and their topology can still be recognised when a lot of information about the population density is missed. In particular, we have demonstrated in all our numerical test cases that patchy spatial patterns are present until the density values are collected on a very coarse sampling grid. While this result is model specific it gives us some useful insight into the accuracy of measurements required to recognise the type of spatial pattern. There are several open questions remaining in our study, the definition of patch size being one of them. Furthermore, the conclusions we have made about spatial patterns arising in a mathematical model cannot be extended straightforwardly to real-life ecological data. The ecologists have to deal with discrete data collected in the monitoring/sampling routine and visualisation of that data is a challenging issue on its own. We anticipate that the definition of spatial pattern will depend on the visualisation algorithm and careful investigation of that question will consist a topic of future work.

References 1. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369 (1937) 2. Garnier, G., Roques, L., Hamel, F.: Success rate of a biological invasion in terms of the spatial distribution of the founding population. Bull. Math. Biol. 74, 453–473 (2012) 3. Harary, F., Harborth, H.: Extremal animals. J. Comb. Inf. Syst. Sci. 1, 1–8 (1976) 4. Hargis, C.D., Bissonette, J.A., David, J.L.: The behavior of landscape metrics commonly used in the study of habitat fragmentation. Landsc. Ecol. 13, 167–186 (1998) 5. Jankovic, M., Petrovskii, S.V.: Gypsy moth invasion in North America: a simulation study of the spatial pattern and the rate of spread. Ecol. Compl. 14, 132–144 (2013) 6. Kolmogorov, A.N., Petrovskiy, I.G., Piskunov, N.S.: A study of the diffusion equation with increase in the quantity of matter, and its application to a biological problem. Moscow Univ. Bull. Math. 1, 1–25 (1937) 7. Kot, M., Schaffer, W.M.: Discrete-time growth-dispersal models. Math. Biosci. 80, 109–136 (1986) 8. Lewis, M.A., Petrovskii, S.V., Potts, J.: The Mathematics Behind Biological Invasions, vol. 44. Springer, Berlin 9. Liebhold, A.M., Gurevitch, J.: Integrating the statistical analysis of spatial data in ecology. Ecography 25, 553–557 (2002) 10. Mistro, D.C., Rodrigues, L.A.D., Petrovskii, S.V.: Spatiotemporal complexity of biological invasion in a space- and time-discrete predator–prey system with the strong Allee effect. Ecol. Compl. 9, 16–32 (2012) 11. Morozov, A.Y., Petrovskii, S.V., Li, B.L.: Spatiotemporal complexity of patchy invasion in a predator–prey system with the Allee effect. J. Theor. Biol. 238, 18–35 (2006) 12. Petrovskaya, N.B., Embleton, N.L.: Evaluation of peak functions on ultra-coarse grids. Proc. R. Soc. A 469, 20120665 (2013). https://doi.org/10.1098/rspa.2012.0665

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13. Petrovskaya, N.B., Embleton, N.L.: Computational methods for accurate evaluation of pest insect population size. In: Godoy, W.A.C., Ferreira, C.P. (eds.) Ecological Modelling Applied to Entomology, pp. 171–218. Springer, Berlin (2014) 14. Petrovskaya, N.B., Petrovskii, S.V.: The coarse-grid problem in ecological monitoring. Proc. R. Soc. A 466, 2933–2953 (2010) 15. Petrovskaya, N.B., Petrovskii, S.V., Murchie, A.K.: Challenges of ecological monitoring: estimating population abundance from sparse trap counts. J. R. Soc. Interface 9, 420–435 (2012) 16. Petrovskaya, N.B., Petrovskii, S.V., Zhang, W.: Patchy, not patchy, or how much patchy? Classification of spatial patterns appearing in a model of biological invasion. Math. Model. Nat. Phenom. 12, 208–225 (2017) 17. Petrovskii, S.V., Morozov, A.Y., Venturino, E.: Allee effect makes possible patchy invasion in a prey–predator system. Ecol. Lett. 5, 345–352 (2002) 18. Petrovskii, S.V., Malchow, H., Hilker, F.M., Venturino, E.: Patterns of patchy spread in deterministic and stochastic models of biological invasion and biological control. Biol. Invasions 7, 771–793 (2005) 19. Petrovskii, S.V., Petrovskaya, N.B., Bearup, D.: Multiscale approach to pest insect monitoring: random walks, pattern formation, synchronization, and networks. Phys. Life Rev. 11, 467–525 (2014) 20. Rodrigues, L.A.D., Mistro, D.C., Petrovskii, S.V.: Pattern formation in a space- and timediscrete predator–prey system with a strong Allee effect. Theor. Ecol. 5, 341–362 (2012) 21. Rodrigues, L.A.D., Mistro, D.C., Cara, E.R., Petrovskaya, N.B., Petrovskii, S.V.: Patchy invasion of stage-structured alien species with short-distance and long-distance dispersal. Bull. Math. Biol. 77, 1583–1619 (2015) 22. Rosenberg, M., Anderson, C.: Spatial pattern analysis. In: Gibson, D. (ed.) Oxford Bibliographies in Ecology. Oxford University Press, New York (2016) 23. Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)

Collocation Techniques for Structured Populations Modeled by Delay Equations Alessia Andò and Dimitri Breda

Abstract Collocation methods can be applied in different ways to delay models, e.g., to detect stability of equilibria, Hopf bifurcations and compute periodic solutions to name a few. On the one hand, piecewise polynomials can be used to approximate a periodic solution for some fixed values of the model parameters, possibly using an adaptive mesh. On the other hand, polynomial collocation can be used to reduce delay systems to systems of ordinary differential equations and established continuation tools are then applied to analyze stability and detect bifurcations. These techniques are particularly useful to treat realistic models describing structured populations, where delay differential equations are coupled with renewal equations and vital rates are given implicitly as solutions of external equations, which in turn change with model parameters. In this work we show how collocation can be used to improve the performance of continuation for such complex models and to compute periodic solutions of coupled problems. Keywords Population dynamics · Numerical continuation · Equilibria · Periodic solutions · Pseudospectral methods

1 Introduction We are interested in different problems regarding equilibria and periodic orbits of complex delay models of structured populations, mainly in view of their stability analysis. The aim is at illustrating how the use of polynomial collocation can help in providing solutions to yet unsolved problems, or improving efficiency of existing techniques.

A. Andò · D. Breda () Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_3

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In particular, we are concerned with systems of the form 

x(t) = F (xt , yt )

(1)

y  (t) = G(xt , yt ),

where F : X×Y → RdX and G : X×Y → RdY are smooth, autonomous, in general nonlinear functionals, with F integral in the x-component. Above, dX and dY are positive integers, τ > 0 and X := L1 ([−τ, 0], RdX ) and Y := C([−τ, 0], RdY ) are Banach spaces endowed with their usual norms. Moreover, the X-component of the state of the dynamical system on the state space X × Y associated to (1) is denoted by xt , defined as xt (θ ) := x(t + θ ),

θ ∈ [−τ, 0],

and the same notation holds for the Y -component. We refer to (1) as a coupled or delay equation or system, where the first is a renewal equation (RE) and the second is a delay differential equation (DDE). The importance of such problems is witnessed by the growing number of recent works, related either to their role in realistic modeling of structured populations [9, 19–22, 28, 29, 35, 37], as well as to numerical approaches for their stability and bifurcation analyses [8, 11–14, 17, 36]. As an illustrative example, let us consider the so-called Daphnia model [21], which describes a structured population b competing for an unstructured resource S through the coupled system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ b(t) = ⎪ ⎪ ⎪ ⎨



amax

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for given β, γ , f and amax . The size ξ of the individuals and their survival rate F are not given in advance as known functions, rather they are defined through the solutions of the initial value problems (IVPs) for ordinary differential equations (ODEs) ⎧ ξ  (α; a, St ) = g(ξ(α; a, St ), St (α − a)), ⎪ ⎪ ⎪ ⎪ ⎨ F  (α; a, S ) = μ(ξ(α; a, S ), S (α − a))F (α; a, S ), t t t t ⎪ ξ(0; a, S ) = ξ ⎪ t 0 ⎪ ⎪ ⎩ F (0; a, St ) = F0

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for given g, μ, ξ0 and F0 . Moreover, the consumers are juveniles until they reach the maturation age a¯ := amat (St ), defined as the age when individuals reach a certain given maturation size ξ¯ , after which they become adults and are able to produce offspring. Namely, ξ(a; ¯ a, ¯ St ) = ξ¯ and, therefore, the problem is state-dependent. Recently, pseudospectral collocation has been proposed to efficiently tackle stability and bifurcation questions of models like (2) or, in general, of coupled problems like (1). In view of the principle of linearized stability, [12] addresses the stability of equilibria, while [8] is the first systematic attempt to attack the stability of periodic solutions of REs and, given the analogous problem for DDEs already solved in [10], it contributed to tackle the same issue for coupled systems in [30]. Breda et al. [13], instead, goes beyond the linearized stability analysis: the pseudospectral approach is used to directly discretize nonlinear delay equations by a finite number of ODEs, and then stability and bifurcations are investigated via available tools such as, e.g., MATCONT [18, 26], based on parameter continuation. Despite the mentioned literature, there are still important open problems and other uses of collocation may help in finding efficient solutions. This is the case, for instance, of the approach of [13] when applied to realistic models as (2), in which the right-hand side is not given through an exact expression, but rather only through solutions of external ODEs. Indeed, when MATCONT continues the system discretizing the original model in the search for, e.g., equilibria, the external ODEs must be solved from scratch every time the continuation parameter is advanced, resulting into a rather computationally demanding procedure (note that the ODEs are solved inside the function defining the right-hand side and not by MATCONT). In Sect. 2 we show on a prototype problem how to use collocation to numerically solve the external ODEs, in order to include these numerical solutions in the continuation framework and thus speed up the overall computation. Another case concerns the computation of periodic solutions of renewal, and hence coupled problems, for which numerical methods still lack. Inspired by Engelborghs et al. [24] for DDEs, in Sect. 3 we illustrate for the first time the application of piecewise collocation to compute periodic orbits of REs as solutions to boundary value problems (BVPs), justifying its implicit use in [14]. Finally, in Sect. 4 we give a summary overview and conclude with a few comments about future work. We close this introduction by illustrating the general idea of collocation methods, which are commonly used for the numerical solution of several kinds of integrodifferential equations (see, e.g., [15, 16, 27, 38]). They are essentially based on choosing a finite-dimensional subspace of the solutions space, such as the polynomial (up to some fixed degree) or the piecewise polynomial space (possibly with adaptive mesh), and looking for an approximation of the solution in such subspace. In practice, one chooses a finite set of collocation points (or nodes) in the domain of an equation and requires that the numerical solution satisfies the

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original equation at such points, taking into account for possible initial or boundary conditions. As a didactic example, consider IVPs for ODEs. Given 

u (t) = g(u(t)),

t ∈ [a, b],

u(a) = u0 and collocation nodes a = t0 < · · · < tn = b, the collocation problem consists in finding a polynomial p of degree n satisfying 

p (ti ) = g(p(ti )),

i = 1, . . . , n,

p(t0 ) = u0 . Such polynomial is unique, as it is the n-degree polynomial interpolating the values ui , i = 0, . . . , n, given as the components of the vector solving a linear system of the form Au = b. Here, b0 = u0 , bi = g(ui ) for all i > 0 and A is obtained from the differentiation matrix of the collocation nodes (see, e.g., [38, 39]) after editing the first row in order to force p(t0 ) = u0 , that is ⎤ 0 ... 0 ⎢d1,0 d1,1 . . . d1,n ⎥ ⎥ ⎢ A=⎢ . .. . . .. ⎥ ⎣ .. . . ⎦ . dn,0 dn,1 . . . dn,n ⎡

1

for di,j := j (ti ) with {0 , . . . , n } the Lagrange basis associated to the collocation nodes.

2 Collocation to Improve Continuation We face the problem of continuing the equilibrium curve of a complex delay model which depends on a parameter λ, having in mind the case of Daphnia and [13]. Let u = f (u, λ)

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be the system of ODEs obtained through the pseudospectral discretization proposed in [13]. An equilibrium curve is, then, a curve u(λ) satisfying f (u(λ), λ) = 0

(3)

for each value of λ. Such a curve exists locally and is unique, given some weak regularity conditions on f . Moreover, it can be approximated by a sequence {(un , λn )}n through numerical continuation as briefly summarized next. Several algorithms for numerical continuation exist (see, e.g., [1, 23, 25, 26]). Mainly, they are predictor-corrector methods. In the following we consider pseudoarclength continuation, based on approximating at each step the arclength of the curve in its tangent space, see [23] for an initial reference. Starting from a known or computed point (u0 , λ0 ) on the curve, the predictor for the following point takes a step of length Δs along the current tangent vector. Then, the corrector looks for the intersection between the curve and the hyperplane which is perpendicular to such vector and contains the predicted point, Fig. 1. This means solving the nonlinear system  u˙ ∗0 (u

f (u, λ) = 0 ˙ − u0 ) + λ0 (λ − λ0 ) = Δs,

which is generally done by resorting to Newton-like methods. This standard approach is at the core of several continuation software packages (e.g., AUTO [3], MATCONT [34], XPPAUT [40]), which basically require an explicitly given right-hand side f . Unfortunately, and as anticipated in the introduction, in realistic models as (2) f can only be expressed through solutions of external IVPs for ODEs. This means that the latter have to be solved from scratch for every value λn of the continuation parameter λ along the continuation sequence. We thus propose to include the solution of the external ODEs into the continuation framework. In this sense we talk about internal continuation, as opposed to the external continuation normally used so far. The aim is to take advantage, when Fig. 1 A predictor-corrector step of pseudo-arclength continuation

0

(λ1 , u1 ) Δs (λ0 , u0 ) 0

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computing a new point of the equilibrium curve, of the solution of the external ODEs already obtained for the previous value of the parameter λ. In order to illustrate this idea and to focus on investigating its performance experimentally, in the forthcoming section we first drop all the technicalities of the original Daphnia model (2) (e.g., dependency of the maturation age on the history of the resource, multi-dimensionality of the external system of ODEs). Then we introduce the state-dependency back again in a subsequent section.

2.1 Prototype Problem It is not difficult to recover that the resource component of any nontrivial equilibrium ¯ S) ¯ of (2) must satisfy (b,  1=

amax ¯ amat (S)

¯ S)F ¯ (a; a, S)da. ¯ β(ξ(a; a, S),

As a prototype including all the relevant features (but for state-dependency, treated in the next section) we concentrate on continuing the curve x(λ) defined implicitly by 

1

φ(a, x, λ)da = 0,

(4)

0

where φ(a, x, λ) = ϕ(a; a, x, λ) with ϕ(α; a, x, λ) solution of 

ϕ  (α; a, x, λ) = g(ϕ(α; a, x, λ), a, x, λ),

α ∈ [0, a],

ϕ(0; a, x, λ) = ϕ0

(5)

for given g and ϕ0 . First, we approximate the integral in (4) by Clenshaw-Curtis quadrature [38], i.e., 

1 0

φ(a, x, λ)da ≈

N 

wj φ(aj , x, λ),

j =0

where N is a fixed positive integer, 0 = a0 < · · · < aN = 1 are the Chebyshev extrema in [0, 1] and w0 , . . . , wN are the corresponding weights. This allows us to express the prototype problem in the form (3) through the use of collocation as follows.

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49

For every quadrature node aj and fixed x and λ, we look for an n-degree polynomial pj (α) := pj (α; aj , x, λ) such that 

pj (αj,i ) = g(pj (αj,i ), aj , x, λ),

i = 1, . . . , n,

pj (αj,0 ) = ϕ0 . for given collocation points 0 = αj,0 < · · · < αj,n = aj . All the collocation variables pj (αj,i ) are then included in the continuation framework (3) by setting u = (p0 (α0,1 ), . . . , p0 (α0,n ), . . . , pN (αN,1 ), . . . , pN (αN,n ), x).

(6)

Now the collocation equations together with the quadrature formula constitute the continuation problem f , given component-wise as ⎧  ⎪ ⎨ pj (αj,i ) − g(pj (αj,i ), aj , x, λ), pj (αj,0 ) − ϕ0 , ⎪ ⎩ N j =0 wj pj (αj,n ).

i = 1, . . . , n,

j = 1, . . . , N,

j = 0, . . . , N,

Let us remark that the drawback of the increased dimension with respect to (5) (which is now O(nN)) is expected to be compensated by the advantage of solving the ODEs internally. Moreover, the continuation framework eliminates the classic problem of searching for a suitable initial guess to start the iterative solution by Newton-like methods. Finally, including the external ODEs inside also eliminates the search of an initial guess for their numerical solution, since one can start from the solution at the previous step. The latter aspect is indeed particularly tedious if one applies collocation to differential equations, and this is why the computation of periodic solutions is in general merged into a continuation framework (see Sect. 3), given the difficulty of finding good estimates to start the iterative solver. The results of some numerical simulations are shown next. We compare the performance of our method with an external continuation where the IVPs are solved by the Python ODE solver from the scipy package, scipy.integrate.odeint. With reference to (4) and (5), the continuation alternatives are compared on both g(ϕ, a, x, λ) = λϕ + xe−λa

(7)

$ g(ϕ, a, x, λ) = −2 λ4 − 4xa(ϕ0 − ϕ),

(8)

and

for which we know the exact solution of the external ODE, as well as the exact expression of the continuation curve. In this way we can evaluate the true error on the continuation curve by varying the number of collocation and quadrature nodes while running a fixed number of continuation steps.

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Fig. 2 Comparison of internal (lines with circles) and external continuation (horizontal lines) for (7): error on the true curve (9) (top) and computational time (bottom) for increasing number of collocation points n and using N = 10 quadrature nodes. See text for more details

10−2

10−9

10−16

4

6

8

10 12 14

4

6

8

10 12 14

100

10−1

n

We remark that for (7) the right-hand side is linear in ϕ, therefore simpler than for (8) in view of solving the relevant ODE. However, the resulting integrand function φ(a, x, λ) has a simpler expression for (8) than for (7), namely a 4-degree polynomial in a. This means that the quadrature is exact for sufficiently large N. Figure 2 (top) shows the error obtained for (7) on the true curve x(λ) =

λ(1 − eλ ) ϕ0 λ + e−λ − 1

(9)

for increasing number of collocation points n and using N = 10 quadrature nodes. The error of internal continuation (line with circles) decays spectrally as n increases: spectral accuracy [38] is the typical expected behavior of collocation when applied to smooth problems, as is the case here. The horizontal lines are the results of external continuation, where the tolerance of the external ODE solver is set respectively to 10−8 , 10−10 and 10−13. For each of these tolerance values, there is (at least) a number of collocation points for which the internal continuation performs better in terms of both time and error, see the computational time in Fig. 2 (bottom), where the diamond markers show that this is true with n = 8, 9 and 12, respectively. A similar test is performed for (8), for which the true curve reads x(λ) = λ2 − ϕ0 ,

(10)

Collocation Techniques for Structured Populations Modeled by Delay Equations

51

10−6

Fig. 3 Comparison of internal (lines with circles) and external continuation (horizontal lines) for (8): error on the true curve (10) (top) and computational time (bottom) for increasing number of collocation points n and using N = 10 quadrature nodes. See text for more details

10−11

10−16 100

10−1

4

6

4

6

8

10

12 14

8

10

12 14

n

Fig. 3. For this case n = 4 collocation points are enough to obtain a better performance of the internal continuation in terms of both time and error. Indeed, as already remarked, the resulting function φ is integrated exactly using N = 10 quadrature nodes being a polynomial of degree 4. The simulations above were replicated using different values for the number N of quadrature nodes, up to 100, keeping fixed the number n of collocation points. Qualitatively speaking, all the tests gave the same results, an illustration of which is shown in Fig. 4. The latter concerns (7) using n = 12 collocation nodes. Lines marked with circles refer to internal continuation, while the ones with squares refer to external continuation with tolerance fixed to 10−13. For N > 10 internal continuation performs better in terms of both time and error. Similar results were obtained for (8) with n = 4.

2.1.1 State-Dependency Taking a further step towards the complete Daphnia model, we test our idea by continuing the curve x(λ) satisfying 

1 a¯

φ(a, x, λ)da = 0,

(11)

where a¯ is defined by a condition of the form φ(a, ¯ x, λ) = ϕ¯

(12)

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Fig. 4 Comparison of internal (solid lines with circles) and external (dashed lines with squares) continuation for (7): error on the true curve (9) (top) and computational time (bottom) for increasing number of quadrature nodes N and using n = 12 collocation points

10−10

10−13

10−16

8

10

12

14

8

10

12

14

100

10−1

N

for a given ϕ¯ and, again, φ(a, x, λ) = ϕ(a; a, x, λ) with ϕ(α; a, x, λ) solving 

ϕ  (α; a, x, λ) = g(ϕ(α; a, x, λ), a, x, λ),

α ∈ [0, a],

ϕ(0; a, x, λ) = ϕ0 . The maturation condition (12) is also included in the continuation framework, adding a¯ to the vector of unknowns (6). This way, and similarly to the numerical solutions of the ODEs, the computation of the maturation age at the current value of the continuation parameter takes advantage of the value already obtained at the previous step. As a further remark, note that in realistic models, the growth rate—corresponding to g in our case—is allowed to change before and after the maturation age. That is, when a > a¯ we have something of the form ⎧ ϕ  (α; a, x, λ) = g2 (ϕ(α; a, x, λ), a, x, λ), ⎪ ⎪ ⎪ ⎪ ⎨ ϕ(a, ¯ a, ¯ x, λ) = ϕ, ¯  ⎪ ⎪ ϕ (α; a, x, λ) = g1 (ϕ(α; a, x, λ), a, x, λ), ⎪ ⎪ ⎩ ϕ(0; a, x, λ) = ϕ0 .

α ∈ [a, ¯ a], α ∈ [0, a], ¯

(13)

In this case, it is natural to approximate the solution of the external ODE with a piecewise polynomial in [0, a], where a¯ is the only break point.

Collocation Techniques for Structured Populations Modeled by Delay Equations Fig. 5 Comparison of internal (lines with circles) and external continuation (horizontal lines) for (14): error on the true curve (9) (top) and computational time (bottom) for increasing number of collocation points n and using N = 10 quadrature nodes. See text for more details

53

10−4

10−10

10−16 101

6

8

6

8

10 12 14 16

100 10 12 14 16 n

As an illustrative example for models including a maturation condition as in (11), we show the results obtained with  g(ϕ, a, x, λ) = −(a + 1)x(λ2 + 1)

ϕ − ϕ0 +1 x(λ2 + 1)(a + 1)

2 (14)

and ϕ¯ = ϕ0 − 1. These choices allows to recover the true curve x(λ) =

2ϕ0 − 1 . λ2 + 1

Figure 5 (top) shows the relevant error obtained for increasing number of collocation points n and using N = 10 quadrature nodes. Again, the error of internal continuation (line with circles) decays spectrally as n increases, while horizontal lines are the results of external continuation with tolerance 10−8 , 10−10 and 10−13 , respectively. As in the cases above, Fig. 5 (bottom) is relevant to the computational time, and shows that respectively for n = 8, 13 and 16 internal continuation performs better in terms of both time and error.

3 Collocation to Compute Periodic Solutions Polynomial collocation has been widely used for computing periodic solutions of ODEs (see, e.g., [2] to start). To this aim, collocation is applied to solve the relevant boundary value problem (BVP) over one period of the solution, together with a

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so-called phase condition to remove translational invariance. The technique has been extended successfully to DDEs in [24]. Here we illustrate a further extension to coupled problems like (1) and then perform some illustrative test on DDEs and REs separately, in view of leaving the investigation of a systematic application to realistic models as (2) to future work. A periodic solution of (1) with (unknown) period T > 0 can be found by solving the BVP ⎧ ⎪ x(t) = F (xt , yt ), t ∈ [0, T ], ⎪ ⎪ ⎪ ⎨ y  (t) = G(x , y ), t ∈ [0, T ], t t (15) ⎪ (x , y ) = (x , ⎪ 0 0 T yT ) ⎪ ⎪ ⎩ p(x, y) = 0 where the last equation is the given phase condition. E.g., a trivial phase condition is one of the form x(0) = x¯

or

y(0) = y, ¯

where x¯ and y¯ are fixed; an integral phase condition is one of the form 

T

(x(t), y(t)), (x˜  (t), y˜  (t))dt = 0,

0

where x˜ and y˜ are given reference solutions [23]. In a continuation framework, such solutions will be the ones found at the previous continuation step. Solving (15) through polynomial collocation means looking for m-degree polynomials u and v in [0, T ] such that ⎧ u(θj ) = F (uθj , vθj ), ⎪ j = 0, . . . , m − 1, ⎪ ⎪ ⎪ ⎨  v (θj ) = G(uθj , vθj ), j = 0, . . . , m − 1, ⎪ (u(0), v(0)) = (u(T ), v(T )) ⎪ ⎪ ⎪ ⎩ p(u, v) = 0 for given collocation points 0 = θ0 < · · · < θm = T . However, due to the presence of delay terms, u and v may need to be evaluated at some point that fall off the interval [0, T ]. This occurs whenever the corresponding right-hand side is evaluated at some t such that θj + t < 0 for some j . In that case, by exploiting periodicity, uθj (t) is to be read as uθj (t + kT ), where k is the least positive integer such that θj + t + kT > 0, and the same for vθj (t). Of course the period T is unknown, but the issue is easily solved by rescaling time according to t → T −1 t, as it is assumed for the sequel. Following [24], the above basic method can be improved by resorting to piecewise polynomial collocation: given a set of mesh points 0 = t0 < · · ·
0. )@ = (P)@ , F )@ ) with The only other possible equilibrium is coexistence, E λ P)@ = P @ = , b

)@ = F

√ Kr b Kr √ = √ √ , a(K b + λ) a(K + P @ )

)@ )22 = 0. J (E

Thus for stability we find the Routh-Hurwitz conditions always satisfied because )@ )) = aλF @ > 0 and det(J (E √ √ P@ Kr 2K + P @ Kr @ @ ) −tr(J (E )) = aF − = > 0. √ √ 2 (K + P @ )2 2 (K + P @ )2 Since the trace is strictly negative, for generic parameter values the equilibrium is stable and no Hopf bifurcation can arise. Within this context, it may seem more consistent to extend the herding effect to the predator-prey interaction. In fact, it is mathematically interesting to analyse the general case 1/2 ≤ α < 1, in which case we reach dP P = Kr − aP α F, dt K + Pα dF = bP α F − λF, dt )0 and coexistence, E ) = (P), F )) with There are again two equilibria: the origin E  1/α λ , P) = b

 1/α b 2 rK λb )= . F aλ(bK + λ)

(6)

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As before, the origin is a saddle: trajectories approach it along P = 0 but all others diverge. For stability of coexistence, we study the Jacobian there, which is ⎞ brK [bK(1 − α) − λ(2α − 1)] aλ − b ⎟ ⎜ (bK + λ)2 ⎟. J) = ⎜ 2 ⎠ ⎝ αb rK 0 a(bK + λ) ⎛

(7)

The determinant is positive for all parameter values, so the coexistence equilibrium cannot be a saddle. The sign of the trace is the same as that of bK(1−α)−λ(2α−1), so that coexistence is stable for parameter values such that λ 1−α > bK 2α − 1

(8)

and unstable otherwise. Note that α = 1/2 is a special case: the trace is then always positive, with the equilibrium unstable for all admissible parameter values. This is a particular case of the fact that, in terms of herding, there is a critical threshold α † below which a stable coexistence equilibrium is impossible, namely α† = 1 −

1 λ > . bK + 2λ 2

(9)

At this threshold, a Hopf bifurcation arises, in view of the sign of the determinant and of the fact that the trace vanishes, Fig. 1.

10

3

8

2.5 2 F

P

6

1.5

4 1

2

0.5

0

0

0

1

2

3

4

0

0.5

1

1.5

2

Fig. 1 One parameter bifurcation analysis of the system (6) varying the parameter α, for the remaining parameter values K = 1, r = 0.2, a = 1, b = 1, λ = 2. The stable equilibria are represented by continuous lines, dashed lines denote the unstable ones while the crosses and dots indicate the minima and maxima of the Hopf bifurcation that arises for α † = 0.6, the value given by (9). The left panel shows the prey population, the right panel contains the predator population, both as functions of α. We denote by Π the region to the left of the bifurcation line, in both frames

Herding Induced by Encounter Rate, with Predator Pressure Influencing Prey Response

69

P

6 4 2 0 0

100

200

300

400

500

600

700

800

900

1000

600

700

800

900

1000

time 3

F

2 1 0 0

100

200

300

400

500

time Fig. 2 The trajectories of the system (6) for a value larger than the Hopf bifurcation point, α = ) for the initial conditions (0.2, 2), a 0.7 > 0.6 = α † converge to the coexistence equilibrium E point that does not belong to Π . Top the prey, bottom the predators. The other parameters are the same as in Fig. 1

Let Π denote the region to the left of the bifurcation line, in both frames of Fig. 1. Outside Π, on the right of the bifurcation line, the coexistence equilibrium ) is stable. If initial conditions are taken in this region, the trajectories converge to E ) E, see Fig. 2. Taking initial conditions and α from within the two regions Π instead, as there are no stable equilibria, the trajectories will eventually reach the coordinate axes and the system will collapse in finite time, as we discuss now. The bifurcation lines will eventually hit axes, which is apparent from Fig. 1, especially in the right frame. This is also evident from the results of the simulations, compare Fig. 3. Irrespective of its stability, the coexistence equilibrium is a focus whenever 4αλa(bK + λ)3 > brK [bk(1 − α)λ(2α − 1)]2 and otherwise it is a node. Since the dP /dt nullcline is strictly above the P axis for all P > 0, a Poincaré-Bendixson box cannot exist and therefore a limit cycle is impossible. This result in a sense is natural, as the basic model for the prey, in the absence of the predators, contains a “modified” Malthusian growth, for which the derivative of P is always nonnegative. Thus on the F = 0 axis, trajectories will grow unbounded.

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P

15 10 5 0 0

10

20

30

40

50

60

70

80

90

100

60

70

80

90

100

time 6

F

4 2 0 0

10

20

30

40

50

time Fig. 3 The trajectories of the system (6) for a value slightly smaller than the Hopf bifurcation point, α = 0.4 < 0.6 = α † and the initial conditions (2, 0.1) ∈ Π and (8, 2) ∈ Π . Trajectories show oscillations of increasing amplitudes and finally attain the value zero in finite time, with the subsequent failure of the numerical integration scheme. In fact for the former IC this occurs at time 90, for the latter at time around 60. Top the prey, bottom the predators. The other parameters are the same as in Fig. 1

)0 and E ) are unstable. Thus for some parameter values, namely α < α † , both E We conjecture that in that case, both populations go extinct, as follows. There is a set Ω containing all the initial values of the trajectories of (6) that end up on the F axis, ,

a Ω = (P0 , F0 ) : P0 ≤ F0 (1 − α) . β

(10)

) solves the Indeed from the second equation in (6) we obtain F  (t) ≥ −λF . If F ) (t) = −λF ), F )(0) = F (0) = F0 , then F (t) ≤ related initial value problem F )(t) = F0 exp(−λt). From the first equation in (6) in turn we get the differential F inequality P  (t) ≤ KrP − aF0 P α exp(−λt). Again, let P) solve the related initial value problem P) (t) = kR P) − aF0 P)α exp(−λt), P)(0) = P (0) = P0 . Letting β = λ + kr(1 − α), we find

   a 1−α Kr(1−α)t ) −βt ) . =e P (t) P0 − F0 (1 − α) 1 − e β

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71

Because the function “1 − exp(−βt)” is increasing, it is enough to require that it “ends up above” P0 to ensure an intersection and therefore a value t ∗ for which P)(t ∗ ) = 0, a P)0 ≤ F0 (1 − α), β

(11)

thereby defining the set Ω. In view of the bounds 0 ≤ P (t ∗ ) ≤ P)(t ∗ ) = 0, it then follows that also P (t ∗ ) = 0, with . / βP01−α 1−α ln 1 − . t =− β aF0 (1 − α) ∗

Thus for (P0 , F0 ) ∈ Ω, P then goes extinct in finite time after which the origin is approached along the F axis. Returning to the behavior of the global dynamics of (6), we observe that because of the Hopf bifurcation at the coexistence equilibrium, based also on the simulations in Figs. 1 and 3, we conjecture that all the trajectories for α < α † eventually will tend to the case of large F and small P , thereby entering Ω, after which as stated above, they will reach the origin. In Table 1, we summarize the situation. If b ≤ λ, i.e. if the predators’ mortality )@ and P @ = P)@ ≤ P). is higher than their gain due to predation, we find F @ ≤ F @ @ @ @ ) and P = P ) > P). We finally compare Conversely, for b > λ, then F > F )@ and F ). Assuming λ > b, let us take α = 1/2 the remaining predators values, F ) so as to make a fair comparison, although in such case equilibrium √ E becomes )@ ≤ F ), because this is equivalent to K b(λ − b) ≥ unstable. It turns out that F √ √ 0 ≥ ( b − λ)λ, which holds unconditionally. Table 1 Summary of the equilibria for the intraspecific interactions: models (2), (4), (6) Model

Equilibrium populations

Feasibility

Stability

Bifurcation

–

Unstable

–

–

Stable

–

–

Unstable

–

–

Stable

–

–

Unstable  α† < α α = α†

–

No prey herding (2)

E0@ : P0@ = 0, F0@ = 0 E@:

(4)

P@

=

λ b,

F@

=

Krb a(Kb+λ)

Prey herding )@ = 0, F )@ = 0 )@ : P E 0 0 0 )@ = P @ , F )@ = )@ : P E

(6)

√

Kr √ b√ a(K b+ λ)

Prey herding and effects on predators )0 = 0, F )0 = 0 )0 : P E ) P )= E:

 λ 1/α b

)= , F

b2 rK

 1/α λ b

aλ(bK+λ)

–

HB

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In sum, we see that an intraspecific competition much weaker than the strict bound imposed by carrying capacity is sufficient to destroy the centre of LotkaVolterra predator-prey dynamics. In the absence of any herding effect on predatorprey interaction, coexistence is unconditionally stable. However, when herding also affects the interspecific interactions, in the case of perfect herding (that is, α = 1/2), coexistence is unconditionally unstable, but if herding is less than perfect the situation is mixed: only some parameter combinations yield stable coexistence, namely when the herding coefficient α exceeds the critical threshold α † .

3 Modification of Well-Mixedness as a Result of Herding Herding introduces a structure in a population: some individuals are near the centre of the herd, others near the edge. Presumably one could model this in a spatially explicit way, with the relative mathematical challenges. However, one can instead think of herding as modifying the encounter rate. Here is a simple way to do so for predator-prey dynamics. The P prey clump into n groups, all of equal size (this can be relaxed, as will be seen below), and these groups exclude predators. The predators have access only to the outside of each group, and since group size is P√ /n, the number of prey per group that each predator encounters is proportional to P /n, assuming that the group has√ a smooth boundary. So over all n groups the encounter rate per predator scales as nP and √ the total encounter rate with predator population F scales as nP F . This generalises to P α F , with 1/2 ≤ α ≤ 1, as follows. Well-mixing is equivalent to α = 1 and also to n = P , that is, no herd effect and all individuals equally exposed to predation. On the other hand, a single large herd with a smooth boundary has n = 1 and predator access limited to the boundary that scales P 1/2 , whence the lower bound on α. Intermediate values of α arise from less than perfect herding. For models with general but constant α, see [16] and [6]. Below, we assume that α is a function of F . When F is very small, the herd is at its loosest, with some herding index β perhaps a bit below 1. As F grows, α gets closer and closer to 1/2. To capture this asymptotic behaviour, we use  α(F ) =

1 −β 2



F + β, h+F

1 ≤ β < 1. 2

(12)

The herding index is trait of the prey population. It somewhat resembles the landscape of fear [5, 10], in that increased herding is a response to increased predator population.

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73

4 Interspecific Interactions We are interested in as simple a model as possible, and so we forego the competitive intraspecific interference from above. The general two-species model can be written as follows P˙ = rf (P ) + γ h(P , F ), P F˙ = mg(F ) + δk(P , F ), F so for the classic Lotka-Volterra model we have that r, δ > 0, γ , m < 0, f = g = 1 for all P and F , and h(P , F ) = F, k(P , F ) = P . In order to account for herding, we need to modify the functions f , h and k from these forms. We assume that the effect of predators on resource acquisition by prey is that resources becomes partially inaccessible. We model this by assuming there is a carrying capacity. In the absence of predators we write this as f (P ) = 1 − bP /r, in their presence this becomes f (P ) = 1 − bP /r − kF P /r. By analogy to the logistic equation, we say the carrying capacity is 1/(b + kF ) in the presence of predators. That is, as predation pressure increases, carrying capacity decreases. This is a stabilising effect similar to but stronger than the intraspecific competition of the previous section. As discussed above, the predators cause the herding index to decline from β to 1/2 as their numbers increase. This determines the interaction functions h(P , F ) and k(P , F ). Below we write the model in classical rather than per capita form, and prefer to specify it there. Finally, the function m specifies predator dynamics in the absence of prey. We use two versions: a specialist predator population that dies out at a constant rate in the absence of prey, and a generalist predator population that grows logistically in the absence of prey, due to possible existing alternative resources.

5 The Specialist Predator Model The model specified above, recalling (12), leads to the following equations: dP = rP − kP 2 F − bP 2 − cP α(F ) F, dt dF = −mF + ecP α(F )F, dt

(13)

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where we assume e < 1.

(14)

Equilibria allowed: the origin E0 , the prey-only point E1 = (P1 , 0), with P1 = rb−1 , the predator-only equilibrium is impossible, as the predator is a specialist, and the coexistence E∗ = (P∗ , F∗ ), with P∗ =

m ce

1 α(F∗ )

,

and where F∗ solves the quadratic e(b + kF )Y 2 − erY + mF = 0,

Y = Φ(F ) =

m ce

1 α(F )

.

(15)

We study this problem as an intersection problem of Φ with F = Ψ (Y ) = eY

r − bY . m + ekY 2

Now 

Φ (F ) = −

m

1 α(F )

ce

m

1 log 2 ce α (F )



1 −β 2



h , (h + F )2

which is positive if and only if m > ce. From Φ(0), this function goes toward the value Φ∞ when F → +∞, with Φ(0) =

 m1

β

ce

,

Φ∞ =

 m 2 ce

.

The function Ψ instead is nonnegative only for 0 ≤ Y ≤ rb−1 , vanishing at the extrema of this interval. Then, a unique positive intersection among the two functions is possible if Φ(0) ≤ rb−1 , which can be restated as β

θ = P1

ce  r β ce = ≥ 1. m b m

(16)

In the opposite case, when (16) does not hold, there could be two or no intersection at all. To have a pair of intersections, arising through a saddle-node bifurcation, the function Φ must be decreasing from its initial value toward its horizontal asymptote, because now Φ(0) > Φ∞ . Thus Φ  (F ) < 0 must be verified. In summary the necessary conditions for the saddle-node bifurcation would then be Φ(0) > Φ∞ , This issue will not be further explored here.

m < ce.

(17)

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  For stability, the Jacobian is J = Jij with J11 = r − 2kP F − cα(F )P α(F )−1 F − 2bP , J12 = −kP 2 − cP α(F ) − cF α  (F )P α(F ) log P , J21 = ecα(F )P α(F )−1 F, J22 = −m + ecP α(F ) + ecF α  (F )P α(F ) log P .

(18) β

At equilibrium E1 we easily find the eigenvalues −r < 0 and ecP1 − m, which provides the stability condition ce < 1. m

β

θ = P1

(19)

Comparing this equation with (16) a transcritical bifurcation is seen to occur when θ crosses from below the value 1, for which the coexistence equilibrium emanates from the prey-only point. Using the equilibrium equations, at equilibrium E∗ we have J11 = cP∗α(F∗ )−1 F∗ (1 − α(F∗ )) − kP∗ F∗ − bP∗ , J22 = ecα  (F∗ )F∗ P∗α(F∗ ) log P∗ .

(20)

Using the Routh-Hurwitz conditions, we find the condition for the trace − ecα  (F∗ )F∗ P∗α(F∗ ) log P∗ > cP∗α(F∗ )−1 F∗ (1 − α(F∗ )) + P∗ (b + kF∗ ), (21) which makes sense, noting that α  (F ) =



1 −β 2



h ce, so that P∗ =

m ce

1 α(F∗ )

> 1,

while conversely (21) is not satisfied. Then, evaluating the determinant, we find % 0 α(F ) α(F )−1 det(−J (E∗ )) = ecF∗ α  (F∗ )P∗ ∗ log P∗ cP∗ ∗ F∗ (1 − α(F∗ )) − bP∗ & α(F )−1 α(F ) α(F ) −kP∗ F∗ ] + α(F∗ )P∗ ∗ [kP∗2 + cP∗ ∗ + cF∗ α  (F∗ )P∗ ∗ log P∗ ] . (24)

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By simplifying and imposing its positivity we obtain the second stability condition   cα  (F∗ )P∗2α(F∗ )−1 F∗ log P∗ + α(F∗ )P∗α(F∗ )−1 kP∗2 + cP∗α(F∗ ) > α  (F∗ )P∗α(F∗ )+1 (kF∗ + b) log P∗ .

(25)

The condition for a Hopf bifurcation is given by the vanishing trace of the Jacobian, giving the critical value for the parameter b:

   ech 1 α(F∗ )−2 α(F∗ )−1 −β bc = F∗ cP∗ (1 − α(F∗ )) + P log P∗ − k . (26) 2 (h + F∗ )2 ∗

5.1 Boundedness Let T = P + F , then T  +μT = (r +μ)P −bP 2 −(m−μ)F −c(1−e)P α F ≤ (r +μ)P −bP 2 −(m−μ)F since 1 > e and for μ < m; hence T  + μT ≤ (r + μ)P − bP 2 ≤ Vm ,

Vm =

(r + μ)2 , 4b

where we have taken the maximum of the parabola on the right hand side. Upon integration, , Vm Vm T (t) ≤ , T (0) . (27) − exp(−μt)) + T (0) exp(−μt) ≤ min (1 μ μ Since the total population is bounded, each individual one is as well.

5.2 The No-Herding Counterpart of (13) It is interesting also to compare the above herding model (13) with a similar model in which hunting is performed on a 1–1 basis, but still accounting for the predators’ effect on the carrying capacity, namely dP = rP − kP 2 F − bP 2 − cP F, dt dF = −mF + ecP F. dt

(28)

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There are only two possible non-trivial equilibria, the predator-free point E1 = (P1 , 0) as for (13), and coexistence, E∗∗ = (P∗ , F∗∗ ), with the prey population level once more given as in (13) and F∗∗ = (cer − bm)(c2e + km)−1 . These equilibria are related via a transcritical bifurcation, as the latter is feasible for cer ≥ bm

(29)

while the stability condition for E1 , in view of the Jacobian Jˇ =



r − 2P (b + kF ) − cF −P (kP + c) ceF ceP − m

 (30)

becomes cer < bm.

(31)

Instead, at E∗∗ , the Routh-Hurwitz conditions are satisfied unconditionally, as they reduce to −tr(Jˇ(E∗∗ )) = (b + kF∗∗ )P∗ > 0,

det(Jˇ(E∗∗ )) = ceF∗∗ (c + kP∗ )P∗ > 0.

Again, the coexistence point is a node if (b + kF∗∗ )2 P∗ > 4ceF∗∗ (c + kP∗ ), conversely it is a focus.

6 The Generalist Predator Model In this case the predator has alternative food sources, a fact that allows it to grow logistically with intrinsic growth rate s and carrying capacity m/s. Its relationship to the prey population is the same as the specialist predator above. Recalling (14) we then have the following model where α(F ) is given in (12): dP = rP − (b + kF )P 2 − cP α(F ) F, dt dF = sF − mF 2 + ecP α(F )F. dt

(32)

The origin is an equilibrium, unstable because trajectories on each axis tend away from it. There are the two points with just one population, E1 = (P1 , 0), P1 = rb−1 , E2 = (0, F2 ), F2 = sm−1 and possibly coexistence: E # = (P # , F # ).

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To find the coexistence equilibrium E # = (P # , F # ), we solve the second equilibrium equation and substitute into the first one, obtaining respectively '



P = Θ(F ) = r − cF

mF − s ce

1−

1 α(F )

(



1 , b + kF

P = ζ (F ) =

mF − s ce



1 α(F )

.

(33) Feasibility requires that both quantities be nonnegative. From ζ we thus obtain F# ≥

s . m

(34)

As for Θ ≥ 0, we need to verify the nonnegativity of the numerator. Observe that this condition can be restated as c mF − s F ≤ r ce



mF − s ce



1 α(F )

and in final form it becomes:

φ1 (F ) =

F (mF − s) re

α(F ) ≤

mF − s = φ2 (F ). ce

(35)

Both functions are positive for F > s/m and φ2 (F ) is a straight line with positive slope m(ce)−1 . We seek to satisfy this condition on the half line [sm−1 , +∞). However, as we shall see below, at the two boundaries of this interval, (35) is not satisfied. Indeed observe that 1 < 2α(F ) < 2β < 2,

(36)

so that for large F , we find that φ1 (F ) behaves like F 2α(F ) > F and therefore φ1 (F ) > φ2 (F ) for large F , so that (35) is not satisfied in this range. On the other hand, for F larger than, but near s/m, we find

   

α(F ) dφ1 F (mF − s) α(F ) dα(F ) F (mF − s) = log + (2mF − s) . dF re dF re F (mF − s)

Taking the limit within the domain, in view of the second term in the bracket and using the upper bound for α in (36), it follows that dφ1  s  m dφ1 = < lims (F ) = +∞. ce dF m F ↓ m dF Thus, near s/m, we also find that φ1 (F ) ≥ φ2 (F ). This means that Θ ≤ 0 at the boundaries of the half line [sm−1 , +∞).

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To find a range where instead Θ ≥ 0, we now seek to satisfy (35) on an interval # , F )# ], for some 0 < F # < F )# . Note that in this situation, by definition, I # = [F # #  ) Θ(F ) = Θ(F ) = 0. For the interval I # to exist we could impose, e.g. φ1 (Fˇ ) ≤ φ2 (Fˇ ), choosing for instance Fˇ = (ce + s)m−1 for which φ2 (Fˇ ) = 1 and then requiring φ1 (Fˇ ) < 1 = φ2 (Fˇ ), namely c(ce + s) − mr < 0.

(37)

Thus (34) is a necessary condition to ensure the nonnegativity ζ(F ) and (37) is a sufficient condition for the positivity of Θ(F ) in some I # . To have the coexistence equilibrium, however we have to require more, namely that an intersection of these curves within this interval I # must occur. Observe that for F > Fˇ , we have dζ = dF



mF − s ce



1 α(F )

 

 1 m dα(F ) 1 mF − s + >0 − log dF α(F )2 ce α(F ) mF − s

using (22). Thus past the point Fˇ the function ζ increases, while Θ must eventually )# . In view of the continuity of these functions, if we require vanish at F r − cFˇ mr − c(ce + s) = Θ(Fˇ ) > 1 = ζ(Fˇ ), = mb + k(ce + s) b + k Fˇ

(38)

)# ]. Together with (34) and (37), (38) a feasible intersection occurs in [Fˇ , F # guarantees the feasibility of E . Again, in view of the shapes of Θ and ζ , the intersections should be two, when they occur, and suitably generated via a saddlenode bifurcation. We do not explore this point any further, though. For stability, observe that in this model there is only one change in the Jacobian with respect to (18), namely J22 = s − 2mF + ecP α(F ) + ecF α  (F )P α(F ) log P . β This entails that at E1 the eigenvalues now are once more −r < 0 and s +ceP1 > 0, showing that it is now unconditionally unstable. Note that in view of the exponent that appears in the last term of the equations in (32), because α(F ) < 1, the Jacobian does not satisfy the Lipschitz condition at P = 0 with consequent loss of well-posedness there, namely uniqueness, which also entails the existence of trajectories that vanish in finite time, as remarked for instance in [16]. However, as mentioned earlier, the dominant terms at the origin show that along the coordinate axes trajectories wander away from it. Further, because J11 is not defined at E2 , this equilibrium is not hyperbolic. But this equilibrium turns out to be stable, because rewriting the right hand side of the first equation in (32) as P [r − (b + kF )P − cP α(F )−1 F ] the dominant term for small P is the last one, implying then that the derivative of P is negative and therefore P → 0 near E2 . In such case the second equation of (32) becomes a logistic equation and F → F2 . Thus trajectories approach this equilibrium point, if they start close enough to it, giving the local asymptotic stability.

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At coexistence the equilibrium equations give once more the previous value (20) for J11 , but now 0 1 # (39) J22 = ecα  (F # )(P # )α(F ) log P # − m F # . Since α  (F ) < 0 by (22), we find J22 < 0. The trace condition for the Routh-Hurwitz criterion gives 1 0 # P∗ (kF∗ + b) + m − ecα  (F # )(P # )α(F ) log P # F # > cP∗α(F∗ )−1 F∗ (1−α(F∗ )). (40)

Imposing the determinant of −J (E # ) to be positive, gives the second condition   log P # + m kF # + b P # F #   # # +ceα(F # )(P # )α(F )−1 k(P # )2 + c(P # )α(F )

c2 eα  (F # )(F # )2 (P # )2α(F

# )−1

  # # > ce kF # + b α  (F # )F # (P # )α(F )+1 log P # + cm(F # )2 (P # )α(F )−1 . (41) The critical value for the parameter b for a Hopf bifurcation is determined by assuming that the trace of the Jacobian vanishes, which gives

   m 1 ech log P # α(F # )−1 # . bc = F # c(1 − α(F # ))P∗α(F )−2 + P − k − −β 2 (h + F # )2 ∗ P#

(42)

6.1 Boundedness Let T = P + F , then T  + μT = (r + μ)P − bP 2 − kF P 2 + (s + μ)F − mF 2 − c(1 − e)P α F

(43)

≤ (r + μ)P − bP 2 − (s + μ)F − mF 2 since 1 > e; then T  + μT ≤ VP + VF ,

VP =

(r + μ)2 , 4b

VF =

(r + μ)2 . 4m

(44)

where we have taken the maximum of the parabolae on the right hand side. Integrating, we finally find , VP + VF VP + VM T (t) ≤ , T (0) . (1 − exp(−μt)) + T (0) exp(−μt) ≤ min μ μ (45) Since the total population is bounded, each individual one is as well.

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6.2 The No-Herding Counterpart of (32) Also in this case we compare the results with the system in which herding is avoided, but which still shows the predator’s influence on the prey carrying capacity, namely dP = rP − (b + kF )P 2 − cP F, dt dF = sF − mF 2 + ecP F. dt

(46)

The equilibria E1 = (P1 , 0) and E2 = (0, F2 ) reappear also in this case, with the same population levels as in (32), and therefore both always feasible. Coexistence ∗∗ = (P ∗∗ , F ∗∗ ) is allowed, with the populations E+ + ∗∗ − s mF+ P ∗∗ = ,

ce

∗∗ = F+



$ 1 2 W + W + 4km(cer + bs) , 2km

W = ks−bm−c2 e,

while the corresponding value F−∗∗ is negative and thus the second possible ∗∗ is always unfeasible. Feasibility for E ∗∗ is given by P ∗∗ ≥ 0, equilibrium E− + i.e. F+∗∗ ≥

s . m

(47)

The value of F+∗∗ is the positive root of the quadratic ψ(F ) = kmF 2 − F W − (bs + ∗∗ , at sm−1 the parabola is negative, cer) = 0. Because of (47), and sm−1 > 0 > E− i.e. we find that feasibility is then equivalent to requiring ψ(sm−1 ) < 0 or cs ≤ mr.

(48)

The Jacobian is  J = †

r − 2P (b + kF ) − cF −P (kP + c) ceF s + ceP − 2mF

 .

(49)

E1 is here unconditionally unstable, in view of the eigenvalues −r and (bs + cer)b −1 > 0. E2 has one negative eigenvalue, −r while the other one gives the stability condition cs > rm,

(50)

which, compared with (48), indicates a transcritical bifurcation for which F+∗∗ arises from E2 as, for instance, r grows past the critical value r † = csm−1 . The

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∗∗ is always ensured, as the Routh-Hurwitz conditions for J † (E ∗∗ ) are stability of E+ + unconditionally satisfied: ∗∗ −tr(J † (E+ )) = (b + kF+∗∗ )P ∗∗ + mF+∗∗ > 0, ∗∗ )) = mF+∗∗ (b + kF+∗∗ ) + ceF+∗∗ P ∗∗ (kP ∗∗ + c) > 0. det(J † (E+

This equilibrium is a focus in case [mF+∗∗ − (b + kF+∗∗ )]2 < 4ceF+∗∗ P ∗∗ (kP ∗∗ + c), otherwise it is a node.

7 Numerical Simulations Table 2 summarizes the theoretical findings on all the models (13), (28), (32) and (46). To further reinforce and confirm the analytical results of these models we now use numerical simulations. Moreover, it is useful to illustrate some of the features of these models and their variants considered here, especially via a few bifurcation diagrams and aspects of the partition of parameter space in terms of types of equilibrium. The softwares Xppaut and Matlab were used. In particular one- and two-parameter bifurcation diagrams are explored.

7.1 The Specialist Predator Model Figure 4 exhibits the phase plane of system (13). The left panel shows the scenario where the coexistence equilibrium E∗ is stable while on the right the prey-only point E1 is stable. In Sect. 3 we determined a critical value for the parameter b, at which a Hopf bifurcation can arise. We now go further with the analysis using one- and twoparameters bifurcation analysis. In Fig. 5 one can see the bifurcation diagram for both the prey and the predator populations, respectively, where b is the varying parameter. The corresponding oscillating system trajectories are shown instead in Fig. 6. Since we work with a biological model for which the prey and predator population can assume only positive real values, the region of our interest is the first quadrant. The other three quadrants are represented in the plots only to give a complete view of the dynamical system behavior (yellow transparent areas). The numerical results are consistent with the analysis: the system has two feasible equilibrium points, the prey-only point E1 and the coexistence equilibrium E∗ respectively. The former undergoes a transcritical bifurcation as b increases beyond

F2 =

E2 = (0, F2 )

, 0 √ 1 1 = 2km W + Δ

F+∗∗ ≥ s m

SC sufficient condition, TB transcritical bifurcation, HB Hopf bifurcation

W = ks − bm − c2 e Δ = W 2 + 4km(cer + bs)

F+∗∗

ce

P ∗∗ =

∗∗ = (P ∗∗ , F ∗∗ ) E+ +

mF+∗∗ −s

Stable

rm < cs

F2 =

E2 = (0, F2 ) –

Unstable Unstable

Generalist predator no prey herding (46) – –

P0 = 0, F0 = 0 P1 = br

E0 E1 = (P1 , 0)

s m

(40), (41)

SC: c(ce + s) < mr

≥

Stable

see the text

s m,

Unstable Unstable

Generalist predator and prey herding (32) – – –

Stable

cer ≥ bm

F#

E#

=

P0 = 0, F0 = 0 P1 = br

E0 E1 = (P1 , 0)

(P # , F # )

P∗ , F∗∗ =

E∗∗ = (P∗ , F∗∗ )

s m

Unstable cer < bm

Specialist predator no prey herding (28) – –

P0 = 0, F0 = 0 P1 = br

E0 E1 = (P1 , 0)

cer−bm c2 e+km

(21), (23) and (25)

E∗ = (P∗ , F∗ )

Unstable β θ = P1 ce m 0.09536 = bc . On the left the initial part of the trajectories, on the right longer simulations, showing that the two trajectories originating outside the region bounded above and below by the lines of the bifurcation in Fig. 9 eventually attain the equilibrium point E2 located on the horizontal axis of the phase plane, i.e. the prey vanish in finite time. These correspond to the continuous and dashed lines, the former vanishing at time 100, the latter at time 300

0.2

P

0.3

0.2

P

0.3

0.1

0.1 0 0

0 5000 5010 5020 5030 5040 5050 5060 5070 5080 5090 5100

100 200 300 400 500 600 700 800 900 1000

time

time

0.2

0.2

F

0.3

F

0.3

0.1

0.1 0 0

100 200 300 400 500 600 700 800 900 1000

time

0 5000 5010 5020 5030 5040 5050 5060 5070 5080 5090 5100

time

Fig. 11 The trajectories of the system (32) for b = 2.5, a value bigger than the Hopf bifurcation point. The other parameter values are r = 1, k = 0.9, c = 2.9, β = 0.75, h = 2, m = 2, e = 1, and s = 0.05. The initial condition values are (0.1, 0.1). Top: prey population in time; Bottom: predator population in time. Left column the initial segment of the solution in time; right column the solution is plotted for much larger times, to show that the coexistence equilibrium is stably attained. This is in line with the findings of Fig. 9, where for b > bc the system settles to coexistence if the initial condition lies in the basin of attraction Σ of the latter

0

1

2

3

4

0

1

2

0

0

1

1

b

2

0 0

0.5

1

1.5

E#

2 b

E#

3

3

0.5

E#

4

4

b

0

1

2

3

4

0

1

2

3

0

0

1

1

1

1.5

b

2

0

0.1

0.2

0.3

0.4

0.5

E#

2 b

E#

0

3

3

0.1

0.2

4

4

0

0

b

0.3

E#

0

1

2

3

4

0

1

2

3

4

0.4

1

1

0.5

b

2

E#

2 b

E#

3

3

4

4

Fig. 12 Two parameter bifurcation analysis of the system (32) varying the parameter b, on the x-axis and r, k, c, β, h, m, e, and s on the y-axis in panels 1–8, respectively. The line is representing the curve of Hopf bifurcation points. With E # we have indicated the part of the plane where the coexistence equilibrium E # is stable

r

3

e

k h

4

s

c m

4

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8 Conclusions So what does this tell us about herding, both as an intraspecific competitive effect and as interspecific predation-reduction effect? Firstly, our model of herding is very simple. A dynamic model, as for example in [9] and its many possible modifications, may well introduce numerous complexities that do not appear here. We use herding only to modify the law of mass action, so obtaining models that are analytically tractable. We suggest that our results apply to prey populations which form fairly long-lasting herds (relative to characteristic times of population dynamics) and interact with individual predators, and where other possible sources of population structure, such as social roles and age-related individual stages are not particularly important to the dynamics. That said, we proved that for intraspecific resource acquisition constrained by herding, the Lotka-Volterra cycles are destroyed. If the herding effect is confined to intraspecific competition and does not extend to predator-prey interaction, predatorprey coexistence is stable for all parameter values. However, when the herd effect is extended to predator-prey interaction, then coexistence may become unstable— and in the extreme of a single, compact herd, with α = 1/2, coexistence is unconditionally unstable. Thus herding has paradoxical effects, depending on whether or not it affects both the acquisition of resources by prey and the capture of prey by predators. For the effect of variable herding on predator-prey interaction, we simplified the intraspecific competition to the logistic law, though with carrying capacity being a declining function of predator pressure. We extended the predation effect to the indirect interaction of increasing the herding parameter α as the predator population grows. We also considered two models of predator dynamics: one in which the predator goes extinct in the absence of prey (the specialist predator model), and one in which it survives in the absence of prey (the generalist predator model). We found that a specialist predator allows a stable predator-free equilibrium, but a generalist predator does not. Likewise, although a Hopf bifurcation occurs in both, the limit cycle in the specialist predator case is stable and in the generalist case it is unstable. Thus in both cases the coexistence equilibrium is only stable for some parameter values. We exhibited two-dimensional slices of parameter space illustrating in more detail the patterns of bifurcation. For comparison, the non-herding analogues were also analysed, where an increase in predators diminishes the carrying capacity but does not influence direct interaction between predators and prey. We found that these cases had simpler dynamics, in that the Hopf bifurcations are absent. Let us now consider possible ecological implications of herding effects as revealed by these models. For the prey, intraspecific costs do not seem to be particularly important, and may even be beneficial, in the sense that the co-existence equilibrium is stabilised and the cycles of pure Lotka-Volterra are suppressed. However, there is the possibility of extinction in finite time if the predator population exceeds prey too far; this may be important if local dynamics in patches are well described by this intraspecific competition model. There is also a curious prediction

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from Eq. (6), in which model the herding causes intraspecific competition but reduces predation pressure. Here, at coexistence, Pˆ = (λ/b)1/α , so that increased herding, in the sense of decreased α, increases prey population size, provided of course that λ > b, and conversely, in the presence of a predators’ gain larger than their mortality, the prey are better off if they scatter around, rather than gathering in herds. On the other hand, when predator pressure causes increased herding, the effects seem to be pronounced. The coexistence equilibrium can be destabilised by a Hopf bifurcation to or from a limit cycle, depending on whether we are looking at a specialist or a generalist predator; in the former case a transcritical bifurcation also appears. For the specialist predator case, these bifurcations are influenced by interactions among the parameters, but in the generalist case the bifurcation seems to depend mostly on the parameter b, which models the predator-independent part of the carrying capacity. Trait-mediated effects such as this have been attracting increasing attention in ecology, both theoretically and observationally [11, 13, 14, 17]. It is intriguing that it can occur in a relatively simple model for two unstructured populations. Acknowledgements This paper was written during a visit of EV at UCT. The invitation of HL and his support are kindly acknowledged. The research of EV has been partially supported by the Dipartimento di Matematica “Giuseppe Peano” research project “Metodi numerici e computazionali per le scienze applicate”. The research of IMB has been partially supported by the INdAM GNCS project “Finanziamento Giovani Ricercatori 2018/2019”. The authors warmly thank the referee for the comments provided, that allowed a substantial improvement of the paper.

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Harvesting Policies with Stepwise Effort and Logistic Growth in a Random Environment Nuno M. Brites and Carlos A. Braumann

Abstract Recently, we have developed optimal harvesting policies based on profit optimization in random varying environments. Namely, we have considered a logistic stochastic differential equation growth model, with the purpose of discussing the use of variable versus constant effort harvesting policies in terms of the expected accumulated discounted profit during a finite time interval. Using realistic parameters, we have concluded that there is only a slight reduction in profit when choosing the applicable constant effort policy instead of the variable effort policy, which presents strong disadvantages. Here, we apply a logistic growth model and a more general profit structure to present alternative policies based on variable effort, named stepwise policies, where the harvesting effort is determined, under the optimal variable effort policy, at the beginning of each year (or of each biennium) but is kept constant during that year (biennium). Replacing the optimal variable effort policy by these stepwise non-optimal policies has the advantage of applicability but, at best, considerably reduces the already small profit advantage the optimal variable effort policy has over the optimal constant effort sustainable policy. Keywords Fisheries management · Stochastic differential equations · Profit optimization · Stepwise effort · Logistic growth

N. M. Brites () Departamento de Matemática, CEMAPRE and REM, ISEG, Universidade de Lisboa, Lisboa, Portugal Centro de Investigação em Matemática e Aplicações, Instituto de Investigação e Formação Avançada, Universidade de Évora, Évora, Portugal e-mail: [email protected] C. A. Braumann Centro de Investigação em Matemática e Aplicações, Instituto de Investigação e Formação Avançada, Universidade de Évora, Évora, Portugal Departamento de Matemática, Escola de Ciências e Tecnologia, Universidade de Évora, Évora, Portugal e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_5

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1 Introduction Stochastic optimal control methods have been applied to derive optimal harvesting policies in a randomly varying environment [1, 9]. Since the population size experiences random fluctuations, it cannot be kept at an equilibrium size. Therefore, the fishing effort, E(t), must be adjusted at every instant, so that the size of the population does not go above some threshold value. So, the optimal harvesting effort will have sudden frequent transitions between maximum or high harvesting efforts and low or null harvesting efforts. These transitions in effort are not compatible with the logistics of fisheries. Besides, the period of low or no harvesting poses social and economical undesirable implications (intermittent unemployment is just one of them). In addition to such shortcomings, these optimal policies require the knowledge of the population size at every instant, to define the appropriate level of effort. The estimation of the population size is a difficult, costly, time consuming and inaccurate task. Therefore, these policies should be considered unacceptable and inapplicable. Braumann [3–5] has considered Stochastic Differential Equation (SDE) harvesting growth models with a constant fishing effort, E(t) ≡ E. For a large class of models it was found that, taking a constant fishing effort, there is, under mild conditions, a stochastic sustainable behaviour. Namely, the probability distribution of the population size at time t will converge, as t → +∞, to an equilibrium probability distribution (the so-called stationary or steady-state distribution) having a probability density function (the so-called stationary density). For the logistic and the Gompertz models, the stationary density was found, and the effort E that optimizes the steady-state yield was determined. The issue of profit optimization, however, was not addressed. In [6] we have determined the constant effort that maximizes the expected profit per unit time at the steady-state for the specific case of the logistic model. One might think that a constant effort policy would result in a substantial profit reduction compared with the optimal variable effort policy, but we have shown that this is not the case. This new policy, rather than switching between large and small or null fishing effort, keeps a constant effort and is therefore compatible with the logistics of fisheries. Furthermore, this alternative policy does not require knowledge of the population size. Since the optimal variable effort policy is not applicable, we present here for the logistic model, intermediate sub-optimal policies, named stepwise policies, where the harvesting effort is determined at the beginning of each year (or of each biennium) under the optimal variable effort policy and kept constant during that year (biennium). These policies are not optimal, but have the advantage of being applicable, since the changes in effort are less frequent and compatible with the fishing activity. Furthermore, although we still need to keep estimating the fish stock size, we do not need to do it so often. Replacing the optimal variable effort policy by these stepwise policies has the advantage of applicability but, at best, considerably reduces the already small profit advantage the optimal variable effort policy has over

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the optimal constant effort policy. In some cases, the optimal sustainable policy even outperforms these stepwise policies in terms of profit (although it might seem so at first glance, there is no contradiction). In [6] a linear price structure was considered. We now generalize that structure, and take a more realistic approach, by considering a quadratic form for the price function. Section 2 presents the variable effort harvesting policy by applying a dynamic programming method. In Sect. 3 we present the sustainable approach based on constant effort. Section 4 shows an application for the Pacific halibut (Hippoglossus hippoglossus) with realistic biological and fishing parameters in which the stepwise effort policies are compared with the others by using numerical and Monte Carlo methods. We end up, in Sect. 5, with the conclusions.

2 Variable Effort Optimal Policy In a random environment the dynamics of a population subject to harvesting and following a logistic growth model can be described, as in [6], by the SDE   X(t) dX(t) = rX(t) 1 − dt −H (t)dt +σ X(t)dW (t), K

X(0) = x > 0,

(1)

where X(t) is the population size at time t, measured as biomass or as number of individuals, r is the population intrinsic growth rate, K is the environment carrying capacity, H (t) is the harvesting rate, σ measures the strength of environmental fluctuations, W (t) is a standard Wiener process and x > 0 is the population size at time 0, which we assume known. We choose the harvesting rate H (t) as H (t) = qE(t)X(t), which is the most traditional form (see, for instance, [7–9]), where q > 0 is a constant representing the fraction of biomass harvested per unit of effort and per unit time and E(t) corresponds to the effort exerted on the population at time instant t. We assume E(t) to be non-anticipating, i.e., it only depends on information available up to time t (included) and to be constrained as 0 ≤ Emin ≤ E(t) ≤ Emax < ∞.

(2)

The profit per unit time can be defined as the difference between sales revenues and fishing costs, i.e., P (t) := R(t) − C(t) = p(H (t))H (t) − c(E(t))E(t),

(3)

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where R(t) are the revenues per unit time from the harvested fish, C(t) is the cost per unit time derived from fishing with effort E(t), p(H (t)) denotes the price per unit yield and c(E(t)) is the cost per unit effort. We assume that the unit prices and costs have, respectively, the form p(H (t)) = p1 − p2 H (t)

and c(E(t)) = c1 + c2 E(t),

where p1 ≥ 0, p2 ≥ 0, c1 ≥ 0 and c2 > 0 are constants, slightly generalizing the price structure appearing in [6]. Thus, (3) becomes   P (t) = p1 qX(t) − c1 − (p2 q 2 X2 (t) + c2 )E(t) E(t). Given the stochastic nature of X(t), we work with the expected profit per unit time 

  E [P (t)] = E p1 qX(t) − c1 − (p2 q 2 X2 (t) + c2 )E(t) E(t) .

(4)

For our purposes, harvesting begins at the time instant t = 0 and the corresponding population size is X(0) = x > 0. Furthermore, harvesting continues up to the time horizon T < +∞ and we work with the profit present value, i.e., future profits are discounted by a rate δ > 0 accounting for interest rate and cost of opportunity losses and for other social rates. For a time t in the time interval [0, T ], we define ⎡ T ⎤ 2  2 −δ(τ −t ) J (y, t) := E ⎣ e P (τ )dτ 22X(t) = y ⎦ ,

(5)

t

which is the expected discounted future profits when the population size at that time is y. The determination of the optimal variable effort harvesting policy is in fact an optimal control problem (OCP), and consists in maximizing the expected accumulated discounted profit per unite time during a finite time interval, i.e., for 0 ≤ τ ≤ T, ⎡ V ∗ := J ∗ (x, 0) = max Ex ⎣

T

E(τ )

⎤ e−δτ P (τ )dτ ⎦ ,

(6)

0

subject to (1), (2) and to the boundary condition J (X(T ), T ) = 0, obtained from (5).

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The above OCP can be solved by the stochastic dynamic programming theory through Bellman’s principle of optimality (as in [2]). The associated HamiltonJacobi-Bellman (HJB) equation (see [2, 10]) is −

,   ∂J ∗ (X(t), t) = max p1 qX(t) − c1 − (p2 q 2 X2 (t) + c2 )E(t) E(t) − δJ ∗ (X(t), t) ∂t E(t)     ∂J ∗ (X(t), t) X(t) + r 1− − qE(t) X(t) ∂X(t) K 1 ∂ 2 J ∗ (X(t), t) 2 2 X (t) . (7) σ + 2 ∂X2 (t)

The optimal variable effort is obtained from the HJB equation (7). Let D be a function that represents the control switching term present in (7), that is,   ∂J ∗ (X(t), t) qE(t)X(t), D(E) = p1 qX(t) − c1 − (p2 q 2 X2 (t) + c2 )E(t) E(t) − ∂X(t) (8)

and denote by Ef∗ ree (t) the unconstrained effort resulting from the maximization carried out in Eq. (8). Thus, Ef∗ ree (t) is obtained by solving the equation dD(E)/dE = 0 with respect to E, which gives Ef∗ ree (t) =

 p1 −

∂J ∗ (X(t ),t ) ∂X(t )



qX(t) − c1   . 2 2 2 p2 q X (t) + c2

(9)

Representing the constrained optimal effort by E ∗ (t) and replacing E(t) by E ∗ (t) in Eq. (7) yields the maximized HJB −

∂J ∗ (X(t), t) = (p1 qX(t) − c1 )E ∗ (t) − (p2 q 2 X2 (t) + c2 )E ∗2 (t) − δJ ∗ (X(t), t) ∂t     ∂J ∗ (X(t), t) X(t) + r 1− − qE ∗ (t) X(t) ∂X(t) K +

1 ∂ 2 J ∗ (X(t), t) 2 2 σ X (t), 2 ∂X2 (t)

where the effort is given by

E ∗ (t) =

⎧ ⎪ ⎪ ⎨Emin ,

Ef∗ ree (t), ⎪ ⎪ ⎩E , max

if

Ef∗ ree (t) < Emin

if

Emin ≤ Ef∗ ree (t) ≤ Emax

if

Ef∗ ree (t) > Emax ,

with Ef∗ ree (t), given by (9), being the unconstrained effort (see [8]).

(10)

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In summary, to determine the optimal variable effort policy, that is, to determine the values J ∗ (x, 0) and E ∗ (t), we need to solve (9) and (10) subject to the growth dynamic given by Eq. (1), and the boundary and initial conditions given above. We have obtained the solutions of (9) and (10) with numerical methods using a CrankNicolson discretization scheme (as in [6, 10]).

3 Constant Effort Optimal Policy To apply a constant effort policy, one considers a particular case of Eq. (1) with E(t) ≡ E, that is,   X(t) dt −qEX(t)dt +σ X(t)dW (t), dX(t) = rX(t) 1 − K

X(0) = x,

(11)

with the assumption r − qE > σ 2 /2 to avoid almost sure extinction (see [4]). From [6], and references therein, we know that the solution of the SDE (11) exists, is unique and is a homogeneous diffusion process. In addition, there exists a stationary distribution for the population size, i.e., an equilibrium probability 1 distribution, with probability density function f (X) = Γ (ρ) α ρ Xρ−1 e−αX (where 2r Γ (·) represents the Gamma function, ρ = 2(r−qE) − 1 and α = Kσ 2 ), towards σ2 which the distribution of the population size converges as t → ∞. We have denoted by X∞ the random variable with density f . It has mean value

  qE σ 2 − E[X∞ ] = K 1 − . r 2r

(12)

and 2 E[X∞ ]

  (ρ + 1)ρ qE = =K 1− E[X∞ ]. α2 r

(13)

The expected sustainable profit per unit time (incorporating a generalization of the price structure presented in [6]) is E [P∞ ] = E [(p1 − p2 H∞ )H∞ − (c1 + c2 E)E] 0 1 = E (p1 − p2 qEX∞ )qEX∞ − (c1 + c2 )E 2     σ2 qE − = p1 qK 1 − − c1 E r 2r      σ2 qE qE 2 2 − − p2 q K 1 − 1− + c2 E 2 , r 2r r

(14)

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and the steady-state optimization problem becomes     qE σ 2 − max E[P∞ ] = p1 qK 1 − − c1 E E r 2r      σ2 qE qE 2 2 − − p2 q K 1 − 1− + c2 E 2 . r 2r r 2

If there is a maximum in the admissible range 0 ≤ E < r−σq /2 , the optimization problem consists in solving the cubic equation dE[P∞ ]/dE = 0, so that the solution satisfies d 2 E[P∞ ]/dE 2 < 0. The resulting optimal sustainable effort, E ∗∗ , is then solution of the equation   qE σ 2 p1 Kq 2 − − c1 − E p1 qK 1 − r 2r r      σ2 qE qE 2 2 − −2E p2 q K 1 − 1− + c2 r r 2r       q σ2 qE  q  qE 2 2 2 2 2 −E p2 q K − − − 1− + p2 q K 1 − = 0. r r 2r r r ∗∗ ], is The correspondent optimal expected sustainable profit per unit time, E[P∞

    qE ∗∗ σ 2 ∗∗ E[P∞ ] = p1 qK 1 − − − c1 E ∗∗ r 2r      ∗∗ σ2 qE ∗∗ qE − 1− + c2 E ∗∗2 . − p2 q 2 K 2 1 − r 2r r

(15)

Finally, at the steady-state, the mean value of the population under the optimal effort E ∗∗ is   qE ∗∗ σ 2 ∗∗ − E[X∞ ] = K 1 − . (16) r 2r Note that the equations presented in [6] correspond to the particular case p2 = 0.

4 Comparison of Policies In [6] we have presented comparisons between the variable effort optimal policy and the constant effort optimal sustainable policy in terms of the effort and profit values and the population size. To perform these comparisons we have used a linear price structure, but here we will apply the quadratic structure using (4) and (15).

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We notice that the profit values given by (6) and (15) can not be directly compared since the optimal policy yields the optimal expected accumulated discounted profit, V ∗ , over a finite time horizon and the optimal sustainable policy yields the optimal ∗∗ ], for a large time horizon T → +∞. However, expected profit per unit time, E[P∞ both profits can be compared by defining the profit per unit time values (∗ refers to the optimal policy and ∗∗ refers to the optimal constant effort sustainable policy) P ∗ (t) := (p1 qX(t) − c1 )E ∗ (t) − (p2 q 2 X2 (t) + c2 )E ∗ (t), 2

P ∗∗ (t) := (p1 qX(t) − c1 )E ∗∗ − (p2 q 2 X2 (t) + c2 )E ∗∗ , 2

and using the following quantities of interest: 1. Expected accumulated discounted profit in the interval [0, T ]: ⎡ V ∗ := Ex ⎣

T

⎤

⎡

e−δτ P ∗ (τ )dτ ⎦ ,

V ∗∗ := Ex ⎣

0

T

⎤ e−δτ P ∗∗ (τ )dτ ⎦ .

(17)

0

2. Expected accumulated undiscounted profit in the interval [0, T ]: Vu∗

⎡ T ⎤  = Ex ⎣ P ∗ (τ )dτ ⎦ ,

⎡ Vu∗∗ = Ex ⎣

0

T

⎤ P ∗∗ (τ )dτ ⎦ .

(18)

0

3. Average expected profit per unit time (average weighted by the discount factors): V∗

P∗ = 

T

,

e−δτ dτ

V ∗∗

P ∗∗ = 

0

T

.

(19)

e−δτ dτ

0

4. Average expected profit per unit time (unweighted average): Pu∗ =

Vu∗ , T

Pu∗∗ =

Vu∗∗ . T

(20)

The above values were computed by performing 1000 Monte Carlo simulations and using a set of parameter values (r, K, q, p1 , c1 and c2 ) from the Pacific halibut (Hippoglossus hippoglossus) stock found in [7, 8]. Other parameters with no information (Emin , Emax , σ, x, p2 and δ) where chosen at reasonable values and the time horizon was set at T = 50 years. The complete set of parameter values is listed in Table 1.

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Table 1 Parameter values used in the simulations. The Standardized Fishing Unit (SFU) measure is defined in [8] Item r K q Emin Emax σ x δ p1 p2 c1 c2 T

Description Intrinsic growth rate Carrying capacity Catchability coefficient Minimum fishing effort Maximum fishing effort Strength of environmental fluctuations Initial population size Discount factor Linear price parameter Quadratic price parameter Linear cost parameter Quadratic cost parameter Time horizon

Values 0.71 80.5 · 106 3.30 · 10−6 0 0.7r/q 0.2 0.5K 0.05 1.59 5 · 10−9 96 · 10−6 0.10 · 10−6 50

Units year−1 kg SFU−1 year−1 SFU SFU year−1/2 kg year−1 $kg−1 $year · kg−2 $SFU−1 year−1 $SFU−2 year−1 year

Table 2 Numerical comparison between policies of the expected profits 1. to 4. (see expressions (17) to (20)). The percent relative difference between the two policies is denoted by Δ. Besides the expected values, we also present the standard deviations (sd). Units are in million dollars for 1. and 2. and in million dollars per year for 3. and 4. 1. 2. 3. 4.

V∗ Vu∗ P∗ Pu∗

Profit value 391.082 1064.048 21.303 21.281

sd 34.396 80.030 1.874 1.601

V ∗∗ Vu∗∗ P ∗∗ Pu∗∗

Profit value 378.514 1025.457 20.618 20.509

sd 31.865 80.777 1.736 1.616

Δ(%) −3.2 −3.6 −3.2 −3.6

For the variable effort policies, the determination of the expected profit values (17) to (20) was based on a Crank-Nicolson discretization scheme (see [6, 10]) using a time and state space grid designed with n = 150 intervals for time (corresponding to a time step Δt = 4 months) and with m = 75 intervals for the state space (corresponding to a space step Δx = 21.47 · 105 kg, with xmax = 2K). The resulting profit values are shown in Table 2, where the left side refers to the optimal variable effort policy, the right side refers to the optimal constant effort policy, and the last column indicates the percent loss in the profit value when using the second policy instead of the first. For each profit value, the respective standard deviation value is also shown. In the first line of Table 2 appears the expected accumulated discounted profits (17), V ∗ and V ∗∗ , over the time horizon T = 50 years. One can see that the second policy implies a reduction in the expected profit of only 3.2% compared to the first policy. Assuming a null value for the depreciation rate, i.e., δ = 0, implies a 3.6% expected profit reduction when comparing the expected accumulated undiscounted profits (18). The percent reductions are the same for the corresponding profits per year (19) and (20), obtained by taking time averages of

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these quantities over the 50 year horizon. The standard deviation values, which measure the variability across the simulated trajectories, are very similar for both policies, with the optimal sustainable policy having a slightly lower variability for the discounted profit and an opposite behaviour for the undiscounted profit. The observed profit reductions that occur when considering a constant effort instead of a variable effort are quite small. Moreover, applying a constant effort policy, drives the fishery manager to maintain across time the same number of vessels, hooks or number of hours worked (just to name a few possibilities). Of course, this is extremely advantageous in terms of implementation, and avoids outof-model costs such as the purchase of new equipment to sustain increased effort periods or payment of unemployment benefits during effort reduction periods. Figure 1 shows what will happen when applying the optimal variable effort harvesting policy (left side) and the optimal constant effort sustainable policy (right side), in terms of the evolution, from time t = 0 to time t = T = 50 years, of the expected population sizes (top), optimal efforts (middle) and profits per unit time (bottom). The thin lines of Fig. 1 show what the harvester would typically observe, i.e., one randomly chosen trajectory corresponding to a possible particular environmental behaviour. The thicker lines represent averages taken over all the simulated trajectories (the one effectively seen and all the others that might have occurred). The dashed lines on the right show the exact values at the steady-state for the population and profit given, respectively, by (16) and (15). From the harvesters point of view (thin lines), the two policies behave quite differently. In fact, while the optimal sustainable effort E ∗∗ is constant across time (regardless of the population size), the optimal variable effort E ∗ (t) changes quite frequently and abruptly, according to the population dynamics, having periods of null effort (meaning that the fishery is closed) and periods with maximum effort (which may involve extra out-of-model costs such as investment in backup equipment or hiring of extra employees not trained in fishing). This sudden and frequent changes in effort are not compatible with the fishing activity, since fishermen cannot accommodate frequent and abrupt changes on the number of vessels, number of gears, number of hours at the sea, among others. In addition, since the population size keeps varying, influenced by the random fluctuations of environmental conditions, a constant evaluation of its size is required. Besides looking at the variation of the effort over time, it is also interesting to look at the time variability experienced by the harvester on the the profit per unit time. If we look at the thin lines at the bottom of Fig. 1 (corresponding to the environmental conditions randomly selected), we see that the optimal policy has frequent periods of zero profit (the periods of zero effort) and a much larger profit variability over time.

Harvesting Policies with Stepwise Effort and Logistic Growth in a Random. . .

Optimal policy (variable effort)

Optimal sustainable policy (constant effort)

Population (mean of 1 000 paths) Population (sample path)

E[X**]= E[X**]= 39118384 39118384 kg kg Population Population (mean (mean of of 11 000 000 paths) paths) Population Population (sample (sample path) path)

Effort (mean of 1 000 paths) Effort (sample path)

E**= 104540 SFU

Profit (mean of 1 000 paths) Profit (sample path)

E[P**]= $ 20545500 Profit (mean of 1 000 paths) Profit (sample path)

6e+07

Population (kg)

105

4e+07

2e+07

0e+00

Effort (SFU)

2e+05

1e+05

0e+00

Profit ($)

6e+07

4e+07

2e+07

0e+00 0

10

20 30 Time (years)

40

50

0

10

20 30 30 Time (years)

40

50

Fig. 1 Mean and randomly chosen sample path for the population, the effort and the profit per unit time. The optimal variable effort policy is on the left side and the optimal constant effort sustainable policy is on the right side

Another disadvantage of the optimal variable effort policy is the exhibition of a possibly dangerous effect near the time horizon, implying a considerable drop of the average population size (see solid line on top left), corresponding to an increase on the average effort (see solid line on middle left). This final effort increase is quite natural. Since “there is no tomorrow”, it is better profitwise to harvest as much as is profitable “now”, without worrying about stock preservation for near future fishing.

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With the optimal sustainable policy, population size is driven to an equilibrium probability distribution with an average population size higher than the one of the variable effort policy. With the constant effort policy, there is no decay of the expected population size near the end of the time horizon. So, the optimal policy leads to a highly variable effort, with occurrence of periods of zero effort and periods of harvesting at maximum effort rates, which imply frequent and abrupt changes on the number of vessels and gears, number of working hours and number of fishermen in activity, among others. Thus, the optimal effort policy is not applicable. We present now a sub-optimal policy, named stepwise policy, based on the variable effort obtained from the optimal policy, but where the effort is kept constant during periods of duration τ , say one or two years. We use τ = vΔt (v is a positive integer) to be a multiple of the time step, Δt, used in the numerical computations and in the Monte Carlo simulations. Therefore, in this stepwise policy, for time t in the period [lτ, (l + 1)τ [ = [tlv , t(l+1)v [, we keep the ∗ (t) = E ∗ (lτ ) constant and equal to the effort of the optimal policy at the effort Est ep beginning of the period. We are aware that this policy is not optimal nor stepwise optimal, but has however the advantage of being applicable, in contrast with the optimal policy. We have focused the study on two scenarios: one with constant effort during periods of one year (annual), denoted by Sa scenario, and the other with constant effort during periods of two years (biennial), denoted by Sb scenario. For the optimal sustainable policy, the effort remains unchanged and it is constant for all time instants, as before. For scenario Sa , we chose Δt = 4 months = 4/12 years and set the effort constant during periods of 1 year, i.e., during 3 consecutive time instants (v = 3). Similarly, for scenario Sb , we kept the effort constant during periods of 2 years, i.e., we set the effort constant during 6 time instants (v = 6). The case v = 1 corresponds to the discretization required to solve the HJB equation concerning the previous comparisons between the optimal effort policy and the optimal sustainable policy. The first and second columns of Table 3 present, for each scenario, the resulting profit values (17)–(20) and their standard deviation values, respectively. The third column shows the percent relative difference between the policy based on stepwise effort and the optimal variable policy presented before (see values in Table 2). Similarly, the percent relative difference between the policy based on stepwise effort and the optimal constant policy (see values in Table 2) is shown in the last column. From Table 3 one can see that, for the scenario Sa , the stepwise and applicable policy gives only slightly lower profit values when compared with the inapplicable variable effort policy (−1.0% and −0.8%, respectively for discounted and undiscounted profits). However, comparing the stepwise policy with the (also applicable) optimal sustainable policy increases the profit values (+2.2% and +2.8%, respectively for discounted and undiscounted profits). When the stepwise

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Table 3 Expected discounted and undiscounted profit values for the stepwise scenarios Sa (annual periods) and Sb (biennial). Besides the expected values, we also present the standard deviations. The percent relative difference between the stepwise policy and the variable effort policy is denoted by Δ∗ and the percent relative difference between the stepwise policy and the constant effort policy ∗ and V ∗ is denoted by Δ∗∗ . Currency values are in million dollars for Vstep step,u , and million dollars ∗ ∗ per year for Pstep and Pstep,u ∗ Vstep

sd

Δ∗ (%)

Δ∗∗ (%)

Sa Sb

387.215 376.844 ∗ Vstep,u

34.687 35.255 sd

−1.0 −3.6 Δ∗ (%)

+2.2 −0.4 Δ∗∗ (%)

Sa Sb

1055.103 1029.606 ∗ Pstep

81.068 82.899 sd

−0.8 −3.2 Δ∗ (%)

+2.8 +0.4 Δ∗∗ (%)

Sa Sb

21.092 20.527 ∗ Pstep,u

1.889 1.920 sd

−1.0 −3.6 Δ∗ (%)

+2.2 −0.4 Δ∗∗ (%)

Sa Sb

21.102 20.592

1.621 1.658

−0.8 −3.6

+2.8 −0.4

effort is applied during a longer biennial periods (scenario Sb ), the profit differences with the optimal effort policy are higher than in the Sa scenario, resulting in profit reductions of −3.6% and −3.2%, respectively for discounted and undiscounted profits. On the contrary, applying the stepwise effort policy instead of the optimal sustainable policy will reduce the profit in −0.4%. In summary, we can conclude that, choosing the applicable policy with stepwise effort causes slight profit losses in comparison with the inapplicable variable effort policy and can be, sometimes, even more profitable than the constant effort policy. The comparison of policies in terms of the profits per year gives differences similar to the accumulated profit differences, since the profits per year are proportional to the accumulated profits. Figure 2 shows the mean and the randomly chosen sample path for the population, the effort and the profit per unit time for both stepwise policies, Sa on the left and Sb on the right. For both scenarios we can see some periods where the population and profit sample paths variability increases in relation to the variable effort policy (see left side of Fig. 1). The increase in variability is more pronounced when we compare the stepwise policy with the constant effort policy (see right side of Fig. 1). Looking at the thick lines of Figs. 1 and 2, corresponding to the mean of the 1000 sample paths, we notice a similar behaviour in terms of variability. At the center part of Fig. 2 one can check the stepwise effort, for both the sample path and the mean of all simulated paths. Their depicted lines in a form of staircase lend the name to the policy: stepwise policy.

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5 Conclusions In this work we have presented numerical comparisons between the optimal policy with variable effort, the suboptimal policy with stepwise effort and the optimal sustainable policy with constant effort. The comparisons were realized in terms of four profit quantities: the expected accumulated discounted profit in a finite time interval, the expected accumulated undiscounted profit in a finite time interval, the average expected profit per unit time weighted by the discount factors and the unweighted average expected profit per unit time.

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To obtain the profit values we have performed 1000 Monte Carlo simulations using a Crank-Nicolson discretization scheme in time and space of the HJB equation and an Euler scheme for the population paths. To compute the simulations we have applied the logistic model to realistic data with parameters from the Pacific halibut (Hippoglossus hippoglossus). The profit differences between the two optimal policies are quite small. Also, we have seen that the optimal policy has frequent strong changes in effort, including periods of null effort, posing serious logistic applicability problems, producing social burdens and out-of-model costs (such as unemployment compensations) and leading to a great instability in the profit earned by the harvester. Furthermore, unlike the optimal variable effort policy, in the optimal constant effort policy there is no need to keep adjusting the effort to the randomly varying population size, and so there is no need to determine the size of the population at all times. This is a great advantage, since the estimation of the population size is a difficult, costly, time consuming and inaccurate task. The optimal policy also can create a possibly dangerous effect near the time horizon implying, on average, a considerable drop on the population size. On the contrary, the optimal sustainable policy does not have these shortcomings, is very easy to implement and drives the population to a stochastic equilibrium. Since the optimal policy in not applicable, we have presented sub-optimal policies, named stepwise policies, based on variable effort but with periods of constant effort. These policies are not optimal, but have the advantage of being applicable, since the changes on effort are not so frequent and can be compatible with the fishing activity. Furthermore, although we still need to keep estimating the fish stock size, we do not need to do it so often. Replacing the optimal variable effort policy by these stepwise policies has the advantage of applicability but, at best, considerably reduces the already small advantages they have over the optimal sustainable policy. In some cases, the much easier to implement optimal constant effort policy even outperforms these stepwise policies in terms of profit. The stepwise policies share with the optimal variable effort policy the disadvantage of having periods of null or low fishing and periods of fishing at the highest rate, with the corresponding social implications and out-of-models costs. Similar work on Gompertz and other population growth models and other population data is under way. Acknowledgements The helpful and valuable comments of an anonymous referee are gratefully acknowledged. Nuno M. Brites was partially supported by the Project CEMAPRE/REM— UIDB/05069/2020—financed by FCT/MCTES through national funds. Both authors are members of the Centro de Investigação em Matemática e Aplicações, Universidade de Évora, funded by National Funds through Fundação para a Ciência e a Tecnologia, under the project UID/04674/2020.

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References 1. Alvarez, L.H.R., Shepp, L.A.: Optimal harvesting of stochastically fluctuating populations. J. Math. Biol. 37, 155–177 (1998) 2. Bellman, R.: Dynamic Programming. Princeton University Press, New Jersey (1957) 3. Braumann, C.A.: Pescar num mundo aleatório: um modelo usando equacões diferenciais estocásticas. In: Proceedings of the XII Congresso Luso-Espanhol de Matemática, Coimbra, pp. 301–308 (1981) 4. Braumann, C.A.: Stochastic differential equation models of fisheries in an uncertain world: extinction probabilities, optimal fishing effort, and parameter estimation. In: Capasso, V., Grosso, E., Paveri-Fontana, S.L. (eds.) Mathematics in Biology and Medicine, pp. 201–206. Springer, Berlin (1985) 5. Braumann, C.A.: Growth and extinction of populations in randomly varying environments. Comput. Math. Appl. 56, 631–644 (2008) 6. Brites, N.M., Braumann, C.A.: Fisheries management in random environments: Comparison of harvesting policies for the logistic model. Fisheries Research (2017). doi: http://dx.doi.org/ 10.1016/j.fishres.2017.07.016 7. Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. Wiley, New York (1990) 8. Hanson, F.B., Ryan, D.: Optimal harvesting with both population and price dynamics. Mathematical Biosciences 148, 129–146 (1998) 9. Suri, R.: Optimal harvesting strategies for fisheries: a differential equations approach. Ph.D. Thesis, Massey University, Albany, New Zealand (2008) 10. Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods, 1st edn. Springer, New York (1995)

Mathematical Modeling of the Population Dynamics of Long-Lived Raptor Species: Application to Eurasian Black Vulture Colonies Casimiro Corbacho, Manuel Molina, and Manuel Mota

Abstract The motivation behind this research is the development of mathematical models to describe the population dynamics of long-lived raptor species. We introduce a class of discrete-time branching models which is indexed by the time instead of the generation. Unlike others classes of stochastic models developed in the scientific literature, we consider the more realistic practical situation where the coexistence in the population of individuals from different generations is assumed. By considering the more general non-parametric statistical setting, we use Approximate Bayesian Computation methods to estimate the most informative reproductive parameters involved in the probability model. Also, using real data, we apply the proposed statistical and computational methodology to describe the population dynamics of two Eurasian black vulture colonies located at the region of Extremadura (Spain), the colonies located at National Park of Monfragüe and at Sierra San Pedro, which appear to be two of the largest and densest breeding colonies worldwide. Comparisons between the results obtained for both colonies are provided and commented. Keywords Biological populations · Long-lived raptor species · Stochastic modeling · Statistical inference · Approximated Bayesian computation methods

C. Corbacho Department of Anatomy, Cell Biology and Zoology, University of Extremadura, Badajoz, Spain e-mail: [email protected] M. Molina () · M. Mota Department of Mathematics and Institute of Advanced Scientific Computation, University of Extremadura, Badajoz, Spain e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_6

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1 Introduction Branching model theory deals with the probabilistic evolution of dynamical systems whose components, after certain life period, reproduce and die in such a way that the transition from one to other state of the system is made according to a certain probability distribution. For background, see e.g. the monographs [1, 2, 10] or [23]. Branching models have been especially developed to describe biological phenomena, playing an important role in studies on population dynamics. Such studies are of great interest in conservation biology, population ecology, or wildlife management, see e.g. the monographs [11] or [12] where several practical applications to cell kinetics, cell biology, chemotherapy, gene amplification, or molecular biology are included. In particular, branching models are appropriate to investigate the population dynamics, over time, of biological species. To this end, significant efforts have been made to develop stochastic models based on the assumption that the demographic dynamics of the species under consideration is well-described through generationindexed branching models, see e.g. [4, 5, 14–16], or [24]. This methodology assumes a non-overlapping situation about the individuals respect to generations. Therefore, the possible coexistence in the population of individuals from different generations is not considered. Clearly, this assumption is not realistic for many biological species. This population dynamics should be described through timeindexed instead of generation-indexed mathematical models. In some epidemic studies, time-indexed models in discrete-time have been introduced, see [19] and [18]. However, discrete-time models based on time-indexed branching processes have not been sufficiently developed yet for demographic dynamics of biological species. Here, we focus on the development of time-indexed branching models in discrete-time to investigate the population dynamics of long-lived raptor species. These types of species are mainly characterized by the lifetime stability of their of breeding pairs and by a marked philopatry. Taking into account these characteristics, in a first mathematical approximation, we will describe the stochastic population dynamics of such raptor species through a time-indexed discrete time multitype branching model. Then, by assuming the more general non-parametric statistical setting, we will investigate several inferential questions of great ecological interest for these species. In particular, in order to forecast possible changes in their population dynamics, it is of great practical importance to determine close estimates for the parameters controlling their reproduction process. The work is organized as follows: In Sect. 2, we provide the mathematical description of the model and its intuitive interpretation. Some probabilistic results are also included. Section 3 is devoted to determining accurate estimates for the most informative reproductive parameters involved in the mathematical model. To this end, an algorithm based on Approximate Bayesian Computation (ABC) methods is proposed. In Sect. 4, using real data, we apply the proposed methodology to describe the population dynamics of two important black vulture colonies located

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at the region of Extremadura (Spain), namely, the colonies located at National Park of Monfragüe and at Sierra San Pedro. A comparative analysis concerning the inferential results obtained in both colonies is then presented. The concluding remarks and some questions for research are included in Sect. 5.

2 The Model Long-lived raptors, including all Bearded, Black, Griffon and Egyptian vultures, as well as large Eagles, form a broad group of species that, in spite of their morphological and ethological differences and also of their wide geographic expansion, share some common features, see for details e.g. [9] or [17]. Among these common features, it is worth noting: 1. The monogamous behaviour and the stability of pairs throughout the long lifespans of individuals. 2. The marked natal philopatry. Such philopatry, where animals tend to return to their birthplace or proximity to reproduce, is present in all the long-lived raptor species, both in territorial (e.g. Bearded or Egyptian vultures and eagles) and colonial (e.g. Black and Griffon vultures) ones. 3. These species of raptors share a similar reproductive strategy. In fact, they are K-selected species, i.e. they possess relatively stable populations and tend to produce relatively low numbers of offspring (maximum 1 in vultures species, and 2–3 in large eagles). 4. The breeding success of the pairs plays an important role in the reproductive dynamics of the population. Success in this case not only relates to the survival of the chicks, but also to their recruitment into the breeding population of their natal area. Usually, the rate of breeding success is assumed to be dependent on the age of the pairs. Taking into account these facts about the population dynamics of these long-lived raptor species, we shall describe their demographic dynamics through the following mathematical model where, for simplicity, using the stability of the pairs, we only consider the female individuals: k be the random variable representing the number of females born in the 1. Let ξn,i year n descending from the i-th female born in the year n − k, i.e., from a female k years old, where k ∈ {af , . . . , al }, n ≥ k + 1, i ≥ 1, with af and al denoting, respectively, the estimated first and last reproduction age for a female. We assume that, k {ξn,i , n ≥ k + 1; i ≥ 1}, k = af , . . . , al

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Let us denote by Ne ≥ 1 and pk ∈ (0, 1), respectively, the clutch size of eggs per year and the probability for a k years old female to have as offspring a female k are assumed to be which stays in the population. For k fixed, the variables ξn,i independent and identically distributed such that, k ξn,i ∼ B(Ne , pk )

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and, from the independence of the distributions B(Ne xn−k , pk ), k = af , . . . , al , V ar [Xn | En ] = Ne

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where m := al − af + 1, i.e., the number of years for the fertile life of a female. Note that, the coordinates of Zn represent, respectively, the numbers of females at the population born in the years n−al , . . . , n−af , i.e, during the reproductive period for a female. Also, from (4), we deduce that the variable: Zn∗ :=

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3 Estimation of p In order to study the population dynamics of a long-lived raptor population, from the necessary and sufficient condition given in (6), it is important to determine accurate estimates for the parameter vector p. In this section, by assuming the more general non-parametric statistical setting, we shall consider the estimation of p from a Bayesian perspective. In practice, the most usual observations in a long-lived raptor population are the annual count of pairs during a certain period of years. Hence, the available sample of data will be: 3 4 CN := Zn∗ , n = al + 1, . . . , N

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for a certain N > al , where Zn∗ are the values corresponding to the variables defined in (5). From the sample CN , the associated likelihood function can not be explicitly determined. In this context, the ABC methodology, see for details [21] and [25], can be applied to solve the estimation of p. Assuming an appropriate prior distribution π(p) for p, the objective of such methodology is to determine a close approximation for the corresponding posterior distribution π(p | CN ). The ABC methodology requires a massive simulation of data from the corresponding mathematical model. To this end, we propose to use an algorithm which accepts those values of p that provide data sets close enough to the observed sample CN . To compare the observed and simulated data, SN = (S1 , . . . , SN ) and SN =   ), we need to define an appropriate metric. The most usual metric, see (S1 , . . . , SN [20] and [27], is defined in the form: 2 2  2 N 2  2 Sn 2S 2 1 Sn 22 2 , − φ(SN , SN ) = 22 + − 122 + 2 S S+ 2 S+ 2 +

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To sum up, in order to estimate the parameter vector p, we propose to apply the following algorithm: 1. To simulate the values X1 , . . . , Xal in such a way that they are consistent with the information provided by CN given in (7). 2. To generate a value for p, from the assumed prior probability distribution π(p). 3. To generate a data set CN , from the mathematical model, taking as parameter vector the generated value for p. 4. To accept the generated value of p if φ(CN , CN ) ≤ ε, where φ is the metric defined in (8) and ε is a suitable tolerance level.

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5. To use all the accepted values for p to approximate the posterior probability distribution π(p | CN ). 6. To determine the corresponding Bayesian estimates for pk , k = af , . . . , al , i.e. the means or/and the medians (depending of the squared error loss function considered) of the approximated posterior probability distribution. The computing programs used to simulate data sets from the mathematical model, to apply the proposed algorithm, and to determine the corresponding Bayesian estimates for the parameters, were developed with the computer language for statistical computing and graphics R, see [22].

4 Application to Eurasian Black Vulture Colonies The Eurasian black vulture (Aegypius monachus) is a long-lived raptor species and one of the largest birds of prey in the world, see for details [7]. The colonies of this species used to occur in places in southern and central Europe. Its habitat changes, poisoning and reduction in food availability due to modern farming practices caused populations to decrease and/or disappear during the twentieth century over most of its breeding range, see e.g. [6]. This species currently shows a highly fragmented distribution and it is listed as near threatened by the International Union for Conservation of the Nature. Although the European population shows an increasing trend in the last decades, the much larger Asian population appears to be in decline, see [3]. The Iberian Peninsula contains one of the largest populations of black vultures, with the highest density (over 2000 breeding pairs); colonies of such species are located in wide areas of the Mediterranean center and southwest regions of Spain (Andalucía, Castilla-La Mancha, Castilla-León and Extremadura). In particular, the two most important black vulture colonies are located in the region of Extremadura, namely, the colonies at National Park of Monfragüe and at Sierra San Pedro, which appear to be two of the largest and densest breeding colonies worldwide, see e.g. [6, 8] or [26]. In the last decades, in both colonies, the number of breeding pairs shows a non-decreasing trend, see Fig. 1. It is known that, in average, in these colonies the first and the last reproduction age for a female vulture takes place around 5 and 30 years, respectively (af = 5 and al = 30). Therefore, the female reproductive period is m =26 years. Also, the clutch size is one egg per year (Ne = 1) and the proportion of females chicks is around 0.44. Table 1 shows the observed data about the number of pairs, for the period of years 1989–2015, in both colonies. In what follows, from the real data sets C (National Park of Monfragüe) and C ∗ (Sierra San Pedro) presented in Table 1, through the mathematical model defined in Sect. 2, we will describe the demographic dynamics in these important colonies. First, it will be necessary to determine accurate estimates for the reproductive parameters p5 , . . . , p30 in both colonies. To this end, we will apply the algorithm described in previous section.

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Fig. 1 Number of black vulture breeding pairs at the colonies located at National Park of Monfragüe and at Sierra San Pedro during the period 1989–2015

Using the fact that the proportion of females is established around 0.44, we have considered as an appropriate prior distribution for the parameter vector p = (p5 , . . . , p30 ), the corresponding product of independent uniform probability distributions in the interval [0, 0.44]: π(p) = (0.44)−26 if pk ∈ [0, 0.44], k = 5, . . . , 30, or 0 otherwise. Then, we ran two massive simulations, obtaining 30,000 accepted values for the parameter p in each colony. In order to perform a realistic comparison of the results in both colonies, we have chosen the following tolerance levels: ε = 1.5 (National Park of Monfragüe) and ε∗ = 0.9 (Sierra San Pedro). For such tolerance levels, we have obtained a similar acceptance rate, approximately 0.035%, in both simulations. By using the information provided from the accepted values of p in each colony, we have approximated the corresponding posterior distributions (posterior densities) for pk , k = 5, . . . , 30. As illustration, in Figs. 2 and 3, we present the estimated posterior densities for π(pi | C ) and π(pi | C ∗ ), i = 5, 15, 20, 30, jointly with the corresponding 95% hight posterior density intervals (dashed lines) for pi , i = 5, 15, 20, 30, in the colonies located at National Park of Monfragüe and at Sierra San Pedro, respectively.

Mathematical Modeling in Long-Lived Raptor Populations Table 1 The observed data about the number of pairs at the colonies located at National Park of Monfragüe and at Sierra San Pedro

Year 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

National Park of Monfragüe 197 212 230 207 216 246 226 220 – 223 241 238 205 216 254 286 287 287 312 299 311 – 285 253 273 307 278

119 Sierra San Pedro 121 154 160 179 167 – 197 195 – 193 237 276 – 315 – 326 290 357 – 326 387 – 352 355 353 339 314

Finally, the corresponding Bayesian estimates for pi , i = 5, . . . , 30, based on the mean and on the median, obtained from the estimated posterior densities are represented in Fig. 4. It is worth noting that: 1. Low values in probabilities are motivated by a very restrictive definition of success, in which we require not only the chick must survive but also return to the colony at the age of breeding. 2. Inexperienced females show a low rate of productivity, which progressively increase in mature females until senescence period where an abrupt drop occurs. 3. Success probabilities are greater in the colony located at National Park of Monfragüe for young females. But, from some age on females in the colony located at Sierra San Pedro present a greater success probabilities than the females in the colony located at National Park of Monfragüe. On the basis of the previous described trend, the small differences between the two areas appear to

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be related to the local features of each area (protection status, human disturbance, resource availability or density of population). 4. According to the estimates obtained for pi , i = 5, . . . , 30, from the condition given in (6), it is deduced that both vulture colonies are not in danger of extinction. 5. Taking into account the population trend showed by the two colonies, we deduce that both areas have reached the carrying capacity to give refuge to more and new breeding pairs.

5 Concluding Remarks The motivation behind this work has been the interest to develop stochastic models for description of the demographic dynamics of long-lived raptor populations. We have introduced a new class of time-indexed discrete time branching processes in

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which the calendar year is taken as unity of time. Using such a model, we have considered Binomial reproduction distributions with success probability depending on the age of the females during their fertile life. By assuming the more general nonparametric statistical setting, we have determined estimates for the most relevant reproductive parameters included in the mathematical model. To this end, we have proposed an algorithm based on ABC methods. Using real data, we have applied the proposed statistical and computational methodology to describe the stochastic population dynamics of two important colonies of black vulture in the region of Extremadura (Spain), the colonies located at National Park of Monfragüe and at Sierra San Pedro. From the corresponding estimates obtained for the parameters governing the reproduction process we deduce that, at the present, these vulture colonies are not in danger of extinction. Population dynamics of long-lived raptor species are dependent on a general and natural biological trend related to the age of female, local density of population mediated by carrying capacity in each colony, and environmental features of areas.

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These factors are related to protection status, human disturbance and resource availability, and they, acting together, entirely affect to the rate of productivity of the species. Next, we indicate some questions for research: 1. To determine, under a parametric statistical setting (assuming some suitable mathematical functions for pk ) estimates for the reproductive parameters involved in the mathematical model. 2. To investigate the probability distribution corresponding to the time to the extinction when condition given in (6) for the almost sure extinction of the population is verified. 3. To apply the mathematical model introduced in Sect. 2 to the ecological problem concerning the conservation of raptor populations possibly in danger of extinction, or re-establishment of populations that have previously gone extinct. Acknowledgements We are grateful to the reviewer for his/her helpful comments and corrections which have improved the paper. This research has been supported by the Gobierno de Extremadura (grants IB16013, IB16099, GR18103), the Ministerio de Economía y Competitividad of Spain (grant MTM2015-70522-P) and the FEDER.

References 1. Asmussen, S., Hering, H.: Branching Processes. Birkhauser, Boston (1983) 2. Athreya, K.B., Ney, P.E.: Branching Processes. Dover, Mineola (2004) 3. BirdLife International: Species factsheet: Aegypius monachus. IUCN Red List for birds (2018). http://www.birdlife.org

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4. Bruss, F.T., Slatvchova–Bojkova, M.: On waiting times to populate and environment and a question of statistical inference. J. Appl. Probab. 36, 261–267 (1999) 5. Corbacho, C., Molina, M., Mota, M., Ramos, A.: Birth-death branching models. Application to African elephant populations. J. Theor. Biol. 332, 108–116 (2013) 6. Cramp, S., Simmons, K.E.L.: Hanbook of the Birds of the Western Palaearctc, vol. 2. Oxford University Press, Oxford (1980) 7. Del Hoyo, J., Elliott, A., Sargatal, J. (eds): Handbook of the Birds of the World, vol. 2. New World Vultures to Guineafowl. Linx Edicions, Barcelona (1994) 8. Del Moral, J.C., De la Puente, J.: Buitre negro-Aegypius monachus. In: Salvador, A., Morales, M.B. (eds.) Enciclopedia virtual de los vertebrados. Museo Nacional de Ciencias Naturales, Madrid (2017). http://www.vertebradosibericos.org/ 9. Donázar, J.A.: Los buitres ibéricos: Biología y Conservación. Reyero, Madrid (1993) 10. Guttorp, P.: Statistical Inference for Branching Processes. Wiley, New York (1991) 11. Haccou, P., Jagers, P., Vatutin, V.: Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press, Cambridge (2005) 12. Jagers, P.: Branching Processes with Biological Applications. Wiley, London (1975) 13. Mode, C.J.: Multitype Branching Processes. Elsevier, New York (1971) 14. Molina, M., Mota, M., Ramos, A.: Mathematical modeling in biological populations through branching processes. Application to salmonid populations. J. Math. Biol. 70, 197–212 (2015) 15. Molina, M., Mota, M., Ramos, A.: Statistical inference in two-sex biological populations with reproduction in a random environment. Ecol. Complex 30, 63–69 (2017) 16. Mota, M., del Puerto, I., Ramos, A.: The bisexual branching process with population-size dependent mating as a mathematical model to describe phenomena concerning to inhabit or re-inhabit environments with animal species. Math. Biosci. 206, 120–127 (2007) 17. Newton, I.: Population Ecology of Raptors. T and AD Poyser, London (1979) 18. Penisson, S.: Estimation of the infection parameter of an epidemic modeled by a branching process. Electron. J. Stat. 8, 2158–2187 (2014) 19. Penisson, S., Jacob, C.: Stochastic methodology for the study of an epidemic decay phase, based on a branching model. Special Issue on Applications of Stochastic Processes in Biology and Medicine. Int. J. Stoch. Anal. 2012 598701 (2012) 20. Plagnol, V., Tavare, S.: Approximate Bayesian computation and MCMC. In: Niederreiter, H. (ed.) Proceedings of Monte Carlo and Quasi-Monte Carlo Methods, vol. 2002, pp. 99–114. Springer, Berlin (2004) 21. Pritchard, J.K., Seielstad, M.T., Perez-Lezaun, A., Feldman, M.W.: Population growth of human Y chromosomas: a study of Y chromosome microsatellites. Mol. Biol. Evol. 16, 1791– 1798 (1999) 22. R Development Core Team: A language and environment for statistical computing. R Foundation for Statistical Computing (2009). http://www.r-project.org 23. Sankaranarayanan, G.: Branching Processes and its Estimation Theory. Wiley, New Delhi (1989) 24. Slatvchova–Bojkova, M.: Computation of waiting time to successful experiment using agedependent branching model. Ecol. Model. 133, 125–131 (2000) 25. Tavare, S., Balding, D.J., Griffiths, R.C., Donnelly, P.: Inferring coalescence times for molecular sequence data. Genetics 145, 505–518 (1997) 26. Villegas, A., Sánchez-Guzmán, J.M., Costillo, E., Corbacho, C., Morán, R.: Productivity and fledging sex ratio in a Cinereous Vulture (Aegypius monachus) population in Spain. J. Raptor Res. 38, 361–366 (2004) 27. Wilkison, R.D., Tavare, S.: Estimating primate divergence times by using conditioned birthand-death processes. Theor. Popul. Biol. 75, 278–285 (2009) 28. Yakovlev, A.Y., Yanev, N.M.: Relative frequencies in multitype branching processes. Ann. Appl. Probab. 19, 1–14 (2009)

On the Role of Inhibition Processes in Modeling Control Strategies for Composting Plants Giorgio Martalò, Cesidio Bianchi, Bruno Buonomo, Massimo Chiappini, and Vincenzo Vespri

Abstract We introduce a mathematical model for the composting process in biocells where several chemical phenomena, like the aerobic biodegradation, the hydrolysis of insoluble substrate and the biomass decay, occur. We investigate the best aeration strategies in presence of inhibition processes due to high concentrations of oxygen. Optimal stategries are obtained as result of a suitable optimal control problem. The dynamics exhibits an enhanced level of the oxygen concentration that guarantees the aerobic feature of the biodegradation process. Then, a nonlinear bioeconomic term is included in the objective functional to take into account of the external operational cost. The role of the economic cost in the control policy is analyzed and discussed. Keywords Waste · Composting · Bioreactor · Inhibition · Bioeconomic cost · Optimal control

1 Introduction Waste management is an important challenge, especially for local authorities [32], since the traditional stockage is not an efficient technique at long term [17]. In fact, in a containment vessel the risk of soil and aquifers contamination is high due to the

G. Martalò · B. Buonomo () Department of Mathematics and Applications, University of Naples Federico II, Naples, Italy e-mail: [email protected]; [email protected] C. Bianchi · M. Chiappini Istituto Nazionale di Geofisica e Vulcanologia, Rome, Italy e-mail: [email protected]; [email protected] V. Vespri Department of Mathematics and Computer Sciences, University of Florence, Florence, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_7

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formation of leachate. In addition, new stocakage sites are often required in order to face the increasing amount of waste [5]. In order to overcome such critical issues a different approach has been proposed by conceiving the vessel as a bioreactor. From this perspective, the containment site can be seen as a biologically active environment where the stocked matter can be degraded by suitable natural processes [27], possibly improved by means of external operation like mechanical aeration [11, 13]. A significant fraction of solid waste is given by the organic matter, i.e. a mixture of green, food, agricultural waste, biodegradable matter. Such portion can be used to produce both an agricultural fertilizer and biogas (similar to methane but with low calorific value). As a consequence, a consistent reduction of the total amount and an increasing capacity of landfills follow [30]. The degradation process is a natural and spontaneous phenomenon but it can be improved by external operations, like aeration or leachate recicrculation [4, 11, 13, 19, 33]. The degradation takes place by means of a bacterial population, that uses the waste as a nutrient, and it can occur both in absence or in presence of oxygen, respectively in anaerobic and aerobic processes [9, 10]. Anaerobic digestion is usually involved in the biogas production and in the leachate treatment [14, 18] while the aerobic one is more useful in composting techniques, like aerated static pile, in-vessel or windrow composting [17]. The benefits of both approaches can be combined in the so-called integrated systems where the organic matter is conceived as a source of energy for biogas production, and as raw material for the high quality composting [22]. The waste is pressed and mashed and the liquid part is separated by the solid one. The first is treated by an anaerobic process in a biodigester by producing burnable biogas. The second is used mainly to produce a high quality compost. The product of this process is not still usable as fertilizer and it has to undergo an additional maturation phase out of the bioreactor [17]. In this paper, we focus our attention to the aerobic phase involved in biocell composting; this means that aerobic biodegradation occurs in a closed system where additional oxygen can be injected by external operation. Inspired by the investigation in [20], a new mathematical model has been proposed in [24] to describe the digestion process in biocell. Such description tries to ensure the aerobic feature of the digestion phenomenon. As well described in [1], an aerobic bacterial population requires a sufficient level of oxygen for its survival; if the oxygen concentration in the system atmosphere goes under a given threshold part of the process can become anaerobic. The optimal control proposed in [24] shows an aeration strategy to maintain the oxygen concentration level close to an optimal operational value, identified as the value corresponding to the fastest degradation of the organic matter. In this paper we are interested in including other distinctive features. In particular, first we would like to model and discuss possible inhibition effects due to high oxygen concentration values. As indicated in [1], a high oxygen fraction in the cell atmosphere can overdry the organic matter, so that a necessary level of moisture cannot be maintained in the entire evolution and the aerobic degradation

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is not guaranteed. Such effects can be introduced in the model by a suitable modification of the biodegradation term in the evolution equations governing the biological system. More precisely, the dependence of the degradation rate on the oxygen variable has to be modified in order to have small contributions for high oxygen concentration. A second characteristic feature to be taken into account is the economic cost related to the artifical aeration. Such operation corresponds to an economic contribution in term of technology, electricity or working hours and its cost has a crucial role in the decision policy, since any intervention requires to be cost effective [2]. The paper is organized as follows: after a presentation of the mathematical description of the aerobic process in Sect. 2 and of the optimal control problem in Sect. 3, we will introduce the inhibition effect due to excessive aeration in Sect. 4 and present a comparison with the results obtained in [24] in Sect. 5. In Sect. 6 we will introduce the bioeconomic cost due to artificial aeration and discuss the control strategy for varying weigth of the cost in decision policy in Sect. 7.

2 The Aerobic Biodegradation Model We indicate by S the soluble substrate, i.e. the waste fraction ready to be degraded, by I the insoluble one, that is not yet available for the digestion process and has to be decomposed, by X the biomass, by L the liquid part and by M the inert mass, i.e the pre-compost. The time evolution of these variables is driven by three different chemical phenomena; more precisely, (1) the aerobic biodegradation of the soluble substrate in presence of the oxygen Ω, where the biomass concentration increases and water and inert mass are produced, (2) the hydrolysis, where the insoluble substrate is decomposed giving the soluble one, (3) the biomass decay, that converts part of the biomass in pre-compost and insoluble substrate. Such phenomena are mathematically described by the following system of nonlinear ordinary differential equations 1 S˙ = − g˜ (S, Ω) X + Kh I YS 1 bX I˙ = −Kh I + YI X˙ = g˜ (S, Ω) X − bX 1 L˙ = g˜ (S, Ω) X YL     1 1 1 ˙ + g˜ (S, Ω) X + 1 − bX M = − 1− YS YL YI

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where the upper dot denotes the derivative with respect to the time variable t˜, g˜ is the biomass growth function describing the soluble substrate degradation, Kh and b are respectively the hydrolitic and biomass decay constant and YS , YI and YL are the yield coefficients. All the parameters are given positive constants. The time variation of the oxygen is given by 1 Ω˙ = − g˜ (S, Ω) X . YΩ We can notice that the inert mass variation is obtained by imposing the conservation of the total (solid+liquid) mass   M˙ = − S˙ + I˙ + X˙ + L˙ . Let us denote by M˜ the total mass, M˜ = S + I + X + L + M. It follows that M˜ (t) = M˜ (0) = S (0) + I (0) + X (0) + L (0) + M (0) =: M˜0 for all t ≥ 0. The system can be rewritten by introducing the non-dimensional variables t = μt˜ , s =

I X L M Ω S ,i = ,x = , = ,m = ,ω = , Ω0 M˜0 M˜0 M˜0 M˜0 M˜0

where Ω0 is the optimal operational value of the oxygen concentration for the biodigestion process. As indicated in [38] the aerobic process occurs in presence of a suitable oxygen level. The aerobic biomass can survive in presence of as little as 5% oxygen concentration in the system atmosphere but if the oxygen level goes under 10% part of the biodegradation can become anaerobic. Moreover an oxygen level around 10% guarantees a fast degradation of the organic matter. It follows that such level can be considered an optimal operational value for the oxygen concentration. The non-dimensional differential equations have the following form s˙ = −

1 g (s, ω) x + ch i YS

1 i˙ = − ch i + βx YI x˙ = g (s, ω) x − βx

(1)

1 ˙ = g (s, ω) x YL     1 1 1 g (s, ω) x + 1 − βx m ˙ = − 1− + YS YL YI and ω˙ = −γ g (s, ω) x.

(2)

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where now the upper dots denote the derivative with respect to the non-dimensional variable t, g is the non-dimensional version of the bacterial growth function g, ˜ ch and β are respectively the non-dimensional hydrolitic and decay coefficients, γ = T0 / (YΩ Ω0 ), 0 ≤ s , i , x , , m ≤ 1, ω > 0 and its optimal operational value is equal to 1. From Eq. (2) immediately follows that ω decreases and the oxygen will be, at least partially, consumed. In order to ensure the survival of the composting process for a long time, it is necessary to inject additional oxygen in the system by an external operation, like mechanical aeration, since a sufficient level of oxygen can not be guaranteed.

3 Optimal Aeration Control Problem The aeration operation can be modeled by introducing the control function u = u (t) that describes the addition of oxygen in the cell atmosphere. The model (1) and (2) is modified by substituting Eq. (2) by ω˙ = −γ g (s, ω) x + u (t) ,

(3)

where 0 ≤ u(t) ≤ umax ∀t ≥ 0. The positive value umax is an upper bound for the control variable u corresponding to the maximal value of oxygen that can be introduced in the biological system. We assume that the optimal time profile is in the admissible control set 3 4 U = u Lebesgue measurable on (0, tf ) | 0 ≤ u(t) ≤ umax , ∀t ≥ 0 and the objective functional is given by 2    J (u) = −m(tf ) + ω tf − 1 +



tf

(ω (t) − 1)2 dt

(4)

0

5 6 where 0, tf is the time range, m(tf ) is the inert mass final concentration and the other terms express the deviation of the oxygen level from  the operational optimal one at the final time tf (first term) and in the time range 0, tf (integral part). The aim is to determine the state (s  , i  , x  ,  , m , ω ) associated to an admissible control u ∈ U satysfying (1)–(3) and minimizing the objective functional (4), i.e.   J u = min J (u) . u∈U

(5)

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To determine a solution of the optimal control problem we consider the Pontryagin’s principle [12, 29] that converts the problem (5) into the problem of minimizing the following Hamiltonian function H (X, u, , t) = (ω (t) − 1)2  

 λs λ 1 1 λm + γ λω − g (s, ω) x − λx − + 1− + YS YL YS YL   

1 λi + ch i (λs − λi ) + βx − λx + 1 − λm + u (t) λω YI YI where X = (s, i, x, , m, ω) and  = (λs , λi , λx , λ , λm , λω ) are, respectively, the sets of state and adjoint variables. The adjoint variables solve a system of six ordinary differential equations which can be described in vectorial form by ˙ = A + b  with final condition        tf = (0, 0, 0, 0, 0, 2 ω tf − 1 )T where b = (0, 0, 0, 0, 0, 2 (1 − ω (t)))T ,   ⎤ 1 αs 1 αs 1 − α 0 −α − + γ α s s s ⎢ YS ⎥ YL YS YL ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −ch ch 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥     ⎢ αx ⎥ 1 β αx 1 1 ⎢ 1− αx − 1 − β γ αx ⎥ − −αx + β − + ⎢ ⎥ YI YL YS YL YI ⎢ Y ⎥ A = ⎢ s ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥   ⎣ αω ⎦ 1 αω 1 1− αω 0 −αω − + γ αω YS YL YS YL ⎡

and αs =

∂g ∂g (s, ω) x , αx = g (s, ω) , αω = (s, ω) x . ∂s ∂ω

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We can give a characterization of the control when the Hamiltonian is linear in the control variable u. For such reason we recall some definitions. 5 6 The optimal control u is called a singular control on t, t¯ if  ∂H    X , u , , t = 0 , ∂u 5 6 for every t ∈ t, t¯ and the corresponding solution (X , u ) is called singular arc. If u∗ is a singular control, the problem order is the smallest number q  such that the 2q-th derivative  d 2q ∂H    X , u , , t , 2q ∂u dt explicitly contains the control variable u (if no derivative satisfies this condition then q = ∞). In our case, the switching function is given as σ (t) =

∂H = λω ∂u

and we can give the following characterization to the control function ⎛

⎞ 0 u (t) = ⎝ singular ⎠ umax

⎛

if

⎞ > λω ⎝ = ⎠ 0 .
0 a fixed constant, is the set of positive bounded Lebesgue integrable functions. The right-hand side term in (1) results to be continuously differentiable w.r.t both n and E. The goal is to minimize the environmental damage at the final time at the minimum cost in terms of the effort allocation for the species removal. A penalty term for the budget constraint E ≤ B is also imposed for the implementation of the eradication program. That translates into searching for a control E ∗ ∈ U which realizes . ' 3 ( /   T E(t) dt (2) min ν e−δT n(T ) + e−δt E 2 (t) + c E∈U B 0 subject to the state equation (1). Here ν is a weight for the final population density and δ ∈ (0, 1) is the discount factor. The constant value c ≥ 0 represents a weight for the penalty term associated with the budget constraint E ≤ B. In the case when c is sufficiently large, the penalty term induces the budget bound on the optimal solution E(x, t) ≤ B. The adopted cubic function for the penalty term in (2) corresponds to the case of quadratic control (q = 2) in the general model introduced in [4].

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2.2 First Order Necessary Conditions for Optimality The basic framework of an optimal control problem is to prove the existence and characterize the optimal solution. Existence, necessary conditions, and uniqueness of the solution have been stated in [4] for the more general spatially explicit dependent model. The following theorem, based on the Pontryagin’s Maximum Principle [16], gives necessary conditions to characterize the optimal solution. Theorem 1 Let E ∗ ∈ U be a solution of the optimal control problem (2), with density n∗ satisfying the state equation  n μn E ∗ n˙ = r n 1 − − , k 1 + hμn

n(0) = n0 .

(3)

Then, for all t > 0, there exists a piecewise differentiable current costate variable λ(t) ≥ 0 such that H (n∗ (t), E ∗ (t), λ(t)) ≤ H (n∗ (t), E, λ(t)) for all the admissible controls E, where the current Hamiltonian H is  H (u, E, λ) = E + c 2

E B

3

   n μn E . + λ rn 1− − k 1 + hμn

(4)

and λ˙ = δλ +

  μ E∗ 2r n + −r + λ k (1 + hμn)2

λ(T ) = ν

(5)

Furthermore, ⎧ ⎪ E ∗ (t) = 0 if ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 ≤ E ∗ (t) ≤ B if ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ E ∗ (t) = B if

 ∂H  ∗ n (t), 0, λ(t) > 0 ∂E  ∂H  ∗ n (t), E ∗ (t), λ(t) = 0 ∂E  ∂H  ∗ n (t), B, λ(t) < 0, ∂E

(6)

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Denote α =

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3c and define the function ϕα which maps R2+ into R+ so that B3 ⎧ 8  1 αμs z ⎪ ⎪ 1 + − 1 , if α > 0, ⎪ ⎪ ⎨α 1 + hμs ϕα (s, z) = (7) ⎪ ⎪ μ s z ⎪ ⎪ ⎩ , if α = 0, 2 (1 + h μ s)

for each s, z ≥ 0. Let the triplet (n(t), λ(t), E(t)), with E ∈ U , solve the equation ∂H μn λ = 0, (n, E, λ) = 2 E + α E 2 − ∂E 1 + hμn

(8)

i.e. E(t) = ϕα (n(t), λ(t)). Whenever the existence of the optimal solution is guaranteed, the necessary conditions can be applied to solve the optimal control (2) subject to (1): search for those n(t) and λ(t) which satisfy the following statecurrent costate optimality system: ∗  n  μ n En,λ n˙ = r n 1 − − , k 1 + hμn   ∗ μ En,λ ˙λ = δλ + −r + 2 r n + λ, k (1 + hμn)2

(9) (10)

with n(0) = n0 and λ(T ) = ν, equipped with ∗ (t) = min {max {ϕα (n(t), λ(t)), 0} , B} , En,λ

for t ∈ [0, T ].

3 The State-Control Optimality System By properly rearranging condition (8) and Eqs. (3)–(5), one can obtain a system of differential equations, alternative to the state-current costate optimality system, whose variables are the population density n(t) and the control E(t), as in [11]. In this way, a more direct interpretation of the qualitative behavior of the optimal solution E(t) will be given. By totally differentiating Equation (8) with respect to time we get (1 + αE)E˙ =

μn μλ n. ˙ λ˙ + 2(1 + h μ n) 2(1 + h μ n)2

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1 + hμn E (2 + αE) and, by plugging in the above μn expression the derivatives of n and λ given in (3) and (5) respectively, we have

From (8) we also find λ =

E˙ =

E (2 + α E) Q(n) , 2(1 + α E) (1 + h μ n)

where Q(n) is the quadratic polynomial defined by    2r n n . Q(n) = δ − r + (1 + h μ n) + r 1 − k k

(11)

Therefore the necessary conditions given by (3), (5) and (8) can be rearranged to give the following system:  n μ n E∗ n˙ = r n 1 − − , k 1 + hμn E˙ =

E (2 + αE) Q(n) 2(1 + αE) (1 + h μ n)

with boundary conditions n(0) = n0 and E(T ) = ϕα (n(T ), ν). As a consequence of (6), it can be shown that the optimal solution E ∗ (t) corresponds to ⎧ ⎪ 0, if E(t) ≤ 0 ⎪ ⎪ ⎪ ⎪ ⎨ E ∗ (t) = min {max {E(t), 0} , B} = E(t), if 0 < E(t) < B ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ B, if E(t) ≥ B We will refer to the following system as to the state-control optimality system:  n  μ n min {max {E(t), 0} , B} n˙ = r n 1 − − , k 1 + hμn E˙ =

E (2 + αE) Q(n) 2(1 + αE) (1 + h μ n)

(12) (13)

with boundary conditions n(0) = n0 , E(T ) = ϕα (n(T ), ν ).

3.1 Phase-Space Analysis in the State-Control Plane We aim to study the dynamics of the above state-control optimality system for 0 ≤ n ≤ k by using traditional phase-plane methods. We limit our study to the

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trajectories that verify the constraint 0 ≤ E(t) ≤ B. Hence, we focus on the following system:  μnE n − , n˙ = r n 1 − k 1 + hμn E˙ =

E (2 + αE) Q(n) 2(1 + αE) (1 + h μ n)

(14) (15)

with boundary conditions n(0) = n0 and E(T ) = ϕα (n(T ), ν). By looking at the phase-plane for system (14)–(15), we first observe that the E axis is an invariant set for the dynamics, since the trajectories move along it with δ E (2 + αE) . Also n(t) evolves on its axis, following the logistic equation E˙ = 2(1 + αE) law and tending to its maximum k. As a consequence, the whole first quadrant of the (n, E) phase-plane is invariant for the dynamics, since the system trajectories never cross the axis. The zero-growth isoclines for the variable n are given by the axis n = 0 and the curve r  n E(n) = 1− (16) (1 + h μ n) μ k The isoclines for the variable E are given by the axis E = 0 and the straight lines n = ni , where ni are the roots of Q(n). In order to guarantee the existence and positivity of two distinct and positive roots, we adopt the following assumptions on the model parameters: √ δ 3 − 2 2 + 2khμ 1.

(17) (18)

Among all the possible scenarios (that might deserve to be analyzed in forthcoming papers), here we focus on the special one determined by constraints (17)–(18). The interest in this specific case is justified by the fact that the realistic values related to species invasions, object of our research activity within the ECOPOTENTIAL project, obey the above conditions. As examples, we cite the case study 1 in [4] and the eradication of the exotic plant Ailanthus Altissima in Alta Murgia National Park. Under conditions (17)–(18) , Q has two distinct positive roots 9 −k h μ (δ − r) − r − k ΔQ n1 = , 4r h μ

9 −k h μ (δ − r) − r + k ΔQ n2 = , 4r h μ (19)

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2  where ΔQ = h μ (δ − r) + kr − 8kr h μ δ. Indeed, condition (18) assures the discriminant of Q, ΔQ , to be positive, while (17) assures 0 < n1 < n2 < k. The point P0 = (0, 0) is an equilibrium for the system. As Q(0) = δ, the linearized system defined by the Jacobian ⎛

μE 2r n ⎜ r − k − (1 + h μ n)2 , ⎜ J (n, E) = ⎜   ⎜ ⎝ r E (2 + α E) 2 (1 + h μ n)2 − 1 − k h μ − k 2 k (1 + α E)(1 + h μ n)2

−

μn 1 +hμn

⎞

⎟ ⎟ ⎟   ⎟ (1 + α E)2 + 1 Q(n) ⎠ 2 (1 + α E)2 (1 + h μ n)

at P0 has two positive eigenvalues r, δ > 0, this revealing the instability nature of the equilibrium at the origin. Further equilibria of the system are Pk = (k, 0), P1 = (n1 , E1 ) and P2 = n1/2 r (n2 , E2 ), where n1 , n2 are defined in (19), and E1/2 = (1 − ) (1 + h μ n1/2) μ k are obtained by the isocline n˙ = 0 given by (16). The equilibrium Pk is a saddle since the Jacobian J (k, 0) has −r < 0 and δ + r > 0 as eigenvalues. The Jacobian evaluated at (n1 , E1 ) ⎛

δ

⎜ ⎜   J (n1 , E1 ) = ⎜ ⎜ ⎝ rE1 (2 + αE1 ) 2(1 + hμn1 )2 − 1 − hμk 2 k (1 + αE1 )(1 + hμn1 )2

−

μn1 ⎞ 1 + hμn1 ⎟ ⎟ ⎟ ⎟ ⎠ 0

has characteristic equation λ2 − δλ + γ1 = 0 where γ1 = γ (n1 , E1 ) and γ (n, E) =

  r μ nE (2 + αE) 2(1 + hμn)2 − 1 − hμk 2 k (1 + αE)(1 + hμn)3

.

Similarly, at (n2 , E2 ) the characteristic equation is given by λ2 − δλ + γ2 = 0 where γ2 = γ (n2 , E2 ). This means that the equilibria P1 and P2 cannot be stable since at least one eigenvalue is positive or it has a positive real part. The trend of the flow in the (n, E) phase-plane, depicted in Fig. 1, suggests that P1 is a saddle equilibrium point and P2 is an unstable focus. In this figure, the loci n˙ = 0, given by Eq. (16), and the lines n = n1/2 , where E˙ = 0, partition the first quadrant into six regions labeled from I to VI. In Region I the solution trajectories are increasing in the E direction and decreasing in the n direction. After crossing the line n = n2 , they enter Region II and become decreasing also in the E direction. Some of these trajectories can intersect the line n = n1 ; after that, they reach Region III and start to growth again in the E direction, approaching the axis n = 0. The other ones cross the isocline n˙ = 0 and enter Region V, changing the n direction. Subsequently, these trajectories reach Region VI, where both n and E directions are increasing, and they enter again Region I. In that way they run around the unstable equilibrium P2 , until

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they are able to reach Region III and continue to growth in the E direction. Finally, the trajectories starting from Region IV may cross the n isocline or the line n = n1 , and reach Region III or VI, respectively. Then they evolve in the corresponding Regions as described above.

3.2 Analysis of the Optimal Solution Through the Phase-Space Analysis We can use the phase-plane diagram to qualitatively characterize the solution for the state-control optimality system. We stress that the budget constraint E ≤ B limits our analysis on trajectories entirely contained in [0, k] × [0, B]. Consider the curve of the final control values ϕα (n, ν), drawn with a red-dashed line in Fig. 1. This curve nullifies at the origin and monotonically grows with respect to n, tending to its upper limit

Emax,α =

III

⎧ $ 1 ⎪ ⎪ 1+ ⎪ ⎨α

αν h

⎪ ⎪ ⎪ ⎩ ν 2h

−1

 if α > 0, if α = 0.

I

II

IV

V

VI

0

Fig. 1 Trajectories of the state-control optimality system (14)–(15) for a variety of initial conditions in the first quadrant of the (n, E) phase-plane. The curve of the final control values (dashed red line), the n isocline (dotted cyan line), the E isoclines (green lines) and the unstable manifold of the saddle Pk = (k, 0) (black line) are also shown. Regions I–VI are defined by the n isocline (cyan-dotted line) and the E isoclines (green lines). The curve of the final control values (dashed-red line) and the unstable manifold of the saddle Pk = (k, 0) (black line) are also shown. Parameters: r = 1.92, k = 1, μ = 9, h = 1, δ = 0.1, ν = 5 e0.1 , c = 1, B = 2.25. Logarithmic scale

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ν . As we better show in the 2h followings, the optimal solution will lay under the curve ϕα (n, ν). Therefore it ν is sufficient to assume that B ≥ to have that E(t) ≤ ϕα (n, ν) < B and, 2h ∗ consequently, E (t) = min {max {E(t), 0} , B} = E(t), for all t ≥ 0. Under the above constraint, the optimality system (12)–(13) is equivalent to the system (14)– (15) so far analyzed, this providing an a posteriori justification of our analysis. ν Notice that, if B < , it may happen that E(t) > B for some t. In these points, 2h the optimal control function assumes values E ∗ (t) = B < E(t); furthermore, E ∗ turns out to be not differentiable in [0, T ]. In order to use our approach based on the analysis of the dynamics (14)–(15), we have to increase the weight c of the penalty term (which in turn corresponds to increase the parameter α) so that Emax,α ≤ B is verified, with this assuring that E(t) < B. This aspect will be object of further future investigations. Given a fixed project length T ≥ 0, the optimal trajectory will start from a value (n0 , ), with 0 < ≤ B, and will stop, at time t = T , on the curve of the final control values ϕα (n, ν) at the ending point (n(T ), ϕα (n(T ), ν)). Observe that the optimal solutions related to shorter project length will correspond to the ones that are more distant from the zero-growth isocline n˙ = 0. This is because the more the curves approach the isocline, the slowly the variable n decreases. Since our aim is to reduce the invasive population density, we limit our search of the optimal solution among all the admissible trajectories entirely contained in Regions I, II and III, where n is decreasing in time. Hence we neglect project lengths large enough to provide trajectories starting in Regions IV, V or VI, where n is increasing. The trend of the flow in the phase-plane shown in Fig. 1 suggests that the intersection between the curve ϕα (n, ν) and the n isocline (16) detects the lowest value n¯ reachable by an optimal control action. Notice that this intersection always exists since ϕα (n, ν) nullifies at n = 0 and reaches the value ϕα (k, ν) > 0, while the r curve (16) starts at n = 0 with the positive value and then it vanishes at n = k. μ However, according to the position of that intersection with respect the equilibria P1 and P2 and to the initial density n0 , we can establish when that minimum value can be achieved. For example, suppose that the intersection holds at n¯ with n1 ≤ n¯ ≤ n2 as in Fig. 1. In this case, the admissible solutions, which are required to be decreasing in the n direction, are wholly included in Regions I and II, under the curve ϕα (n, ν) (see Fig. 3). As a consequence, there are no admissible solutions for n0 < n. ¯ If n¯ < n0 < k it could exist a project length T > 0 that allows to achieve the minimum value n¯ at t = T . A different behaviour occurs if n0 = k. In this case, as the final time grows, the solution will be more and more close to the unstable manifold of the saddle equilibrium point Pk (black line in Fig. 1), which lays above the n isocline (16). Consequently, the limit value for the population density is given by the value nk > n, ¯ detected by the intersection between the curve of final control values and the unstable manifold of Pk . Even incrementing the project length the final density value It results that ϕα (n, ν) < Emax,α ≤ Emax,0 =

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Fig. 2 Optimal trajectories for different project lengths in the state-control phase-plane (n,E), starting from n0 = k. The curve of the final control values (dashed-red line), the n isocline (dottedcyan line), the E isoclines (green lines) and the unstable manifold of the saddle Pk = (k, 0) (black line) are also shown. Parameters: r = 1.92, k = 1, μ = 9, h = 1, δ = 0.1, ν = 5 e0.1 , c = 1, B = 2.25

cannot be lower than nk . It results that all admissible solutions are wholly included in Regions I and II, under the curve ϕα (n, ν) and above the unstable manifold of Pk (see Fig. 2). Similar reasoning can be done for different positions of n¯ with respects n1 and n2 , but we do not enter in the details as, in this paper, our main purpose is to analyze the temporal dynamics of the model described in [4]. However, we stress that the phaseplane analysis of the state-control optimality system allows to localize the optimal trajectories and this represents a useful information for numerically detecting the solutions of the optimal control problem (1)–(2).

4 Numerical Approximation of the Temporal Dynamics We use the forward-backward sweep approach described in [10] joint with an exponential Lawson symplectic scheme [6]. This choice is motivated by the fact that the ODE system (9)–(10) is what is referred in literature as a nearly Hamiltonian system: n˙ =

∂H , ∂λ

λ˙ = δλ −

∂H . ∂n

It has been proven that a partitioned method which consists of a Runge–Kutta scheme for the state variable n and the exponential Lawson [9] symplectic counterpart for the current costate λ, is an effective alternative to the standard ODE solvers in the framework of optimal control [6].

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In more details, given a Runge–Kutta scheme defined by coefficients ai,j , bi , with bi = 0 and its symplectic counterpart defined by coefficients aˆ i,j , bi , with bk bj = bk aˆ kj + bj aj k for each k, j and ck = cˆk = sj =1 aˆ k,j with cs = 1, the scheme applied to (4) proceeds according to the following steps: • make an initial guess for λlk on the mesh time tlk = tl + ck Δt for l = 0, . . . L − 1 and k = 1, . . . s, with Δt = T /L; • using initial conditions n0 and the guess values λlk , solve n forward in time according to the Runge–Kutta scheme: nlk = nl + Δt

s 

akj F (nlj , λlj ),

j =1 s 

k = 1, . . . , s, (20)

nl+1 = nl + Δt

bk F (nlk , λlk )

k=1

for l = 0, 1, . . . L − 1, where  n  μn ϕα (n, λ) F (n, λ) = r n 1 − − k 1 + hμn and ϕα (n, λ) is defined in (7); • using the transversality condition, set λL = λ(T ) = ν and solve λ backward in time by means of the exponential Lawson symplectic counterpart λlk = eδΔt (ck −1) λl+1 + Δt

s  (bj − aˆ kj )eδΔt (ck −cj ) G(nlj , λlj , tlj ) j =1

λl = e−δΔt λl+1 + Δt

s 

bk e−δΔt ck G(nlk , λlk , tlk )

k=1

for l = L − 1, L − 2, . . . , 0 and k = s, . . . , 1, where G(n, λ, t) = rλ −

μλ ϕα (n, λ) 2r nλ − ; k (1 + hμn)2

• check for convergence; • evaluate E(tl ) ≈ El = ϕα (nl , λl ), for l = 0, . . . , L that approximate the values of the control function E(t) at the temporal grid tl = l Δt. bj aj k . This implies that, if the Runge–Kutta scheme used bk on the state variable n is explicit (i.e. aj k = 0 for j ≤ k), the corresponding Lawson backward scheme for the costate variable λ is explicit too. Indeed, for k = s, since Notice that (bj − aˆ kj ) =

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cs = 1, it results that λls = λl+1 . Moreover, λlk = eδΔt (ck −1) λl+1 + Δt

s  bj aj k eδΔt (ck −cj ) G(nlj , λlj , tlj ) bk

j =k+1

for l = L − 1, L − 2, . . . , 0 and k = s − 1, . . . , 1. The exponential Lawson symplectic approximation of the nearly Hamiltonian system (4) generates symplectic discrete flows approximating the Hamiltonian dynamics n˙ =

 ∂H , ∂λ

Ψ˙ = −

 ∂H , ∂n

(21)

 given by: in terms of the Hamiltonian function H     n μn E α 3 −δt 2  H (n, Ψ, t) = e E + E + Ψ rn 1− − , 3 k 1 + hμn where the costate variable Ψ evolves according to the following dynamics μΨ E 2r Ψ˙ = − r Ψ + n Ψ + k (1 + hμn)2

(22)

with Ψ (T ) = ν and E = E(t) defined by    ∂H μn Ψ = e−δt 2 E + α E 3 − = 0. ∂E 1 + hμn The system (21) is a time dependent Hamiltonian system that preserves the  ˙ = ∂ H = 0, the Hamiltonian function  symplecticity of the flow. However, as H ∂t is not an invariant of the system. In this case, a conserved quantity can be identified by considering the time as an additional variable τ evolving according the dynamic dτ = 1 and constructing an augmented Hamiltonian dt     n μn E α 3 −δτ 2 ) E + E + Ψ rn 1− − − ξ (23) H (n, τ, Ψ, ξ ) = e 3 k 1 + hμn in terms of the new variable ξ , which evolves according to the dynamics ξ˙ =

 ∂ Hˆ α  = −δ e−δτ E 2 + E 3 . ∂τ 3

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The augmented system n˙ =

) ∂H , ∂Ψ

ξ˙ =

) ∂H , ∂τ

Ψ˙ = −

) ∂H , ∂n

τ˙ = −

) ∂H ∂ξ

(24)

with n(0) = n0 , ξ(0) = 0, Ψ (T ) = ν, τ (0) = 0, τ (T ) = T , is still a Hamiltonian )=H )(t) (see, for example system which preserves the augmented Hamiltonian H [5]). We will exploit this conservation property in order to check the qualitative behavior of the numerical approximations. When the symplectic RK pairs underlying the exponential Lawson symplectic schemes are applied to the augmented Hamiltonian system (24), we can check the correctness of the solution by monitoring the preservation of the invariant (23). Indeed, it is well known in literature (see, for example, [8]) that symplectic schemes, while preserving the symplecticity of the flow, also retain the invariant bounded in the long run approximation. For this reason, the inner stages nlk and λlk can be used for approximating the dynamics of ξ(t), for t > 0, by considering ξl+1 = Δt

l  s  j =0 k=1

 α  −δ bk e−δtlk El2k + El3k 3

for l = 0, . . . , L − 1, where we set Elk = ϕα (nlk , λlk ). Provided that E0 = ϕα (n0 , λ0 ), and    n0  μn0 E0 α 3 2 ) − H (n0 , 0, λ0 , 0) = E0 + E0 + λ0 r n0 1 − , 3 k 1 + hμn then )(nl , tl , e−δtl λl , ξl ) − H )(n0 , 0, λ0 , 0)| )(tl ) = | H ΔH for all l = 0, . . . L, represents the error in the preservation of the augmented Hamiltonian. The first order symplectic pair is given by the Euler method defined by coefficients a1,1 = 0, b1 = 1 and the Implicit Euler scheme defined by aˆ 1,1 = 1, b1 = 1 and set c1 = cˆ1 = 1. Take an initial guess for λl , for l = 1, . . . L − 1 (λL = λ(T ) = ν) and solve nl+1 = nl + Δt F (nl , λl+1 )

(25)

for l = 0, . . . L − 1. The backward step for λ, starting from λL , proceeds for l = L − 1, . . . , 0, evaluating λl = e−δΔt (λl+1 + Δt G(nl , λl+1 , tl+1 )) . Then, check for the convergence and evaluate the solution El = ϕα (nl , λl ).

(26)

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As second example, let us consider the second order Runge–Kutta pair 0 0 0 1 1 0 1/2 1/2

0 1/2 −1/2 1 1/2 1/2 1/2 1/2

make an initial guess for λl1 and λl2 = λl+1 and evaluate: nl2 = nl + Δt F (nl , λl1 ),

nl+1 = nl +

 Δt  F (nl , λl1 ) + F (nl+1 , λl+1 ) 2 (27)

for l = 0, . . . L − 1. Then solve, for l = L − 1, . . . 0: λl1 = e−δΔt λl+1 + Δt G(nl2 , λl+1 , tl+1 ) λl =

e−δΔt λl+1

 Δt  + G(nl , λl1 , tl ) + e−δΔt G(nl2 , λl+1 , tl+1 ) 2

(28)

Check for the convergence and evaluate the control values on the temporal grid El = ϕα (nl , λl ), for l = 0, . . . , L. In a similar way, we built schemes up to order four, by coupling explicit Runge– Kutta methods for forward integrating the state variable n and the corresponding symplectic Lawson scheme for the costate variable λ (see [6]). For dealing with the spatial explicit version of the reaction-diffusion optimal control model given [4], we built the COINS code written in R language and available at GitHub repository [15]. It treats the diffusive flow by means of classical finite difference approximations and uses the symplectic partitioned Runge Kutta pair described above, for approximating the reaction dynamics.

4.1 Numerical Results In order to verify both the effectiveness of symplectic Lawson schemes and the qualitative analysis on the state-control optimality system, we detect the optimal trajectories for cases analyzed in Sect. 3.2. For the parameters involved in the model we use the same numerical values considered in the case study 1 in [4], where the control of an invasive plant species population in a spatial domain used as a parking area is described. The parameters are r = 1.92, k = 1, μ = 9, h = 1, δ = 0.1, ν = 5 e0.1, c = 1, B = 2.25, and they satisfy conditions (17)–(18). In Fig. 2 we plot the approximated optimal trajectories for different project lengths in the state-control phase-plane (n, E), starting from n0 = k, obtained with the second order scheme described in (27)–(28). As the final time grows, the solution

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Fig. 3 Optimal trajectories for different project lengths T in the state-control phase-plane (n,E), ¯ k). The curve of the final control values (dashed-red line), the n starting from n0 = 0.7 ∈ (n, isocline (dotted-cyan line), the E isoclines (green lines) and the unstable manifold of the saddle Pk = (k, 0) (black line) are also shown. Parameters: r = 1.92, k = 1, μ = 9, h = 1, δ = 0.1, ν = 5 e0.1 , c = 1, B = 2.25

approaches the unstable manifold of the saddle equilibrium point Pk . The limit value for the population density is given by nk ≈ 0.0595. In Fig. 3 the same results are reported starting from n0 = 0.7. For all T > 1.56, the final values of the population density n(T ) are lower the threshold value nk . Observe that the lowest density value, starting from the initial n0 = 0.7, is at T = 4.37 where n(T ) = 0.0181. Accordingly with the analysis developed in Sect. 3.2, this value results to be greater than the density n¯ ≈ 0.01045 at the intersection between the curve of the final control values and the unstable manifold of Pk . In Fig. 4 we also report the evolution of the population density n(t) and the current  versus the time in order costate λ(t) against time. We plot the conserved quantity H to check the correctness of the approximations. The numerical results obtained in both the cases n0 = k and n0 < k, confirm the qualitative analysis developed in Sect. 3.2.

5 Conclusions In this paper we face the problem of determining the best strategy for reducing the density of the invasive species when the effort available to the control actions is limited. We focus on the temporal dynamics of the spatially explicit model developed in [4]. We perform a qualitative analysis of the state-control optimality system useful to detect the regions of the phase-plane where optimal trajectories are entirely contained. Then we determine a threshold value for the budget B, that assures that the optimal control action is always within a given budget constraint. This value is directly proportional to the weight ν of the final population density

Optimal Control of Invasive Species with Budget Constraint n0

n

0.6

5

0.4

λ(t)

n(t)

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0.2 nk n 0.1 0.5

4 3 2

1

1.56

2.5

1

4.37

0.1 0.5

1

1.56

t 2

2.5

4.37

0 ∼ H(t)

E(t)

4.37

0.5

1.5 1 0.5 0

2.5 t

–0.5 –1 –1.5 –2

0.1 0.5

1

1.56

2.5 t

4.37

0.1 0.5

1

1.56 t

Fig. 4 Time-dependent behavior of the density n(t), the current costate λ(t), the control E(t) (t), corresponding to optimal solutions at project times T = and the conserved quantity H 0.1, 0.5, 1, 1.56, 2.5, 4.37 and inital density n0 = 0.7. Parameters: r = 1.92, k = 1, μ = 9, h = 1, δ = 0.1, ν = 5 e0.1 , c = 1, B = 2.25

and inversely proportional to the average time h for removing an item of population. Moreover, we find that when starting from an initial value n0 equal to its carrying capacity k, the optimal density concentration, for increasing value of T , tends to a limit value greater than the minimum value that can be reached in a finite time T starting from any n0 < k. For confirming the qualitative results and approximating the optimal solutions, we implement a partitioned method based on a Runge–Kutta scheme and its exponential Lawson symplectic counterpart [6]. The description of the optimality state-costate system in terms of an augmented Hamiltonian allows us to derive a conserved quantity useful to check the correct qualitative behaviour of the approximated solution. Acknowledgements This work has been carried out within the H2020 project ECOPOTENTIAL: Improving Future Ecosystem Benefits Through Earth Observations’. The project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 641762.

References 1. Baker, C.M.: Target the source: optimal spatiotemporal resource allocation for invasive species control. Conserv. Lett. 10, 41–48 (2016) 2. Baker, C.M., Bode, M.: Placing invasive species management in a spatiotemporal context. Ecol. Appl. 26(3), 712–725 (2016)

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3. Baker, C. M., Diele, F., Lacitignola, D., Marangi, C., Martiradonna, A.: Optimal control of invasive species through a dynamical systems approach. Nonlinear Anal. Real World Appl. 49, 45–70 (2018) 4. Baker, C.M., Diele, F., Marangi, C., Martiradonna, A., Ragni, S.: Optimal spatiotemporal effort allocation for invasive species removal incorporating a removal handling time and budget. Nat. Resour. Model. 31(4), e12190 (2018) 5. Diele, F., Marangi, C., Ragni, S.: SB3 A splitting for approximation of invariants in timedependent Hamiltonian systems. Appl. Math. Comput. 217, 2798–2807 (2010) 6. Diele, F., Marangi, C., Ragni, S.: Exponential Lawson integration for nearly Hamiltonian systems arising in optimal control. J. Math. Comput. Simul. 81(5), 1057–1067 (2011) 7. European Commission Staff, WORKING DOCUMENT IMPACT ASSESSMENT Accompanying the document Proposal for a Council and European Parliament Regulation on the prevention and management of the introduction and spread of invasive alien species, 2013, http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:52013SC0321 8. Hairer, E., Lubich, C., Wanner G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006) 9. Lawson, J.D.: Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372–380 (1967) 10. Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models, 1st edn. Chapman & Hall/CRC, Boca Raton (2007) 11. Leonard, D., Van Long, N.: Optimal Control Theory and Static Optimization in Economics. Cambridge University Press, Cambridge (1992) 12. Magnea, U., Sciascia, R., Paparella, F., Tiberti, R., Provenzale, A.: A model for high-altitude alpine lake ecosystems and the effect of introduced fish. Ecol. Model. 251, 211–220 (2013) 13. Marangi, C., Casella, F., Diele, F., Lacitignola, D., Martiradonna, A., Provenzale, A., Ragni, S.: Mathematical tools for controlling invasive species in Protected Areas. In Mathematical Approach to Climate Change and its Impacts. Springer INdAM Series. Springer Cham. 38, 211–237 (2020) 14. Martiradonna, A., Diele, F., Marangi, C.: Analysis of state-control optimality system for invasive species management. In Anal. Prob. Appl. and Comput. Trends in Mathematics. Birkhäuser, Cham. 3–13 (2019) 15. Martiradonna, A., Diele, F., Marangi, C.: COINS (COntrol of INvasive Species); R routine for the optimal control of invasive species, ECOPOTENTIAL Project (2018). https://github.com/ CnrIacBaGit/COINSvlabrepo 16. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelize, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)

A Shape Optimization Problem Concerning the Regional Control of a Class of Spatially Structured Epidemics: Sufficiency Conditions Sebastian Ani¸ta, Vincenzo Capasso, Marta Lipnicka, and Andrzej Nowakowski

Abstract In a series of papers by the first two authors a stabilization problem had been considered for an epidemic model, described by a reaction-diffusion system, including a feedback operator. This paper deals with a related optimal control problem based on functional (shape functional) and sanitation controls; a dual dynamic programming method is constructed for deriving sufficient conditions for an optimal solution as well as ε-optimality conditions in terms of dual dynamic inequalities. Approximate optimality and a related conceptual algorithm are presented as well. Keywords Reaction-diffusion systems · Epidemic systems · Optimal control · Regional control · Shape optimization

1 Introduction Given a spatial region Ω hosting an epidemic phenomenon due, as in our model below, to environmental pollution produced by a human population, the public health concern consists of providing methods for the eradication of the disease in the relevant population, as fast as possible. This has led the first two authors of this paper to suggest that implementation of such programs might be done only in a

S. Ani¸ta Faculty of Mathematics, University Alexandru Ioan Cuza, Ia¸si, Romania e-mail: [email protected] V. Capasso () Department of Mathematics, University of Milano, Milano, Italy e-mail: [email protected] M. Lipnicka · A. Nowakowski Faculty of Mathematics and Computer Science, University of Łód´z, Łód´z, Poland e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_9

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given subregion ω ⊂ Ω, conveniently chosen so to lead to an effective eradication of the epidemic in the whole habitat Ω (“Think globally, act locally”). This practice may have an enormous importance in real cases with respect to both financial and practical affordability. The epidemic problem with local control applied to a subregion of the whole habitat we are interested in has been described and analyzed in [1–3]. However in those papers mainly stabilization problem had been considered. Our aim here is to formulate an optimal control problem with respect to some functional depending on sanitation parameters—controls applied to a suitable subregion (also controlled), of the whole habitat, i.e. our controls are parameters and a subregion. Thus we are looking not only for an optimal control of parameters but also for a shape of subregion which may consist of a finite number of mutually disjoint subdomains. The paper is organized as follows. In Sect. 2 a typical epidemic system with a spatial structure is presented, in the frame of models studied by the authors in previous papers. In Sect. 2.2 a paradigmatic regional control problem is presented. For such a problem a motivation for further analysis on the related optimal control problem is given. In Sect. 3 the dual programming method is presented; accordingly sufficient optimality conditions are then provided in Sect. 4. Section 5 is devoted to the approximate optimality (ε-optimality), which eventually leads to a conceptual algorithm for possible numerical computations. Unfortunately the concrete construction of a numerical code for a significant example of the above mentioned optimal control problem based on the proposed methods has required unexpected efforts, so that we have left the relevant numerical analysis and computational implementations to future investigations.

2 Preliminaries 2.1 A Spatially Structured Man-Environment-Man Epidemic Model A widely accepted model for the spatial spread of epidemics in an habitat Ω, via the environmental pollution produced by the infective population, e.g. via the excretion of pathogens in the environment, is the following one, as proposed in [7, 8] (see also [9], and references therein). The model below is a more realistic generalization of a previous model proposed by one of the authors (V. C.) and his co-workers [10– 12] to describe fecal-orally transmitted diseases (cholera, typhoid fever, infectious hepatitis, etc.) which are typical of the European Mediterranean regions; it can anyhow be applied to other infections, and other regions, which are propagated by similar mechanisms (see e.g. [13]); schistosomiasis in Africa is a typical additional

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example [21]. The model is described by the following system:  ⎧ ∂u ⎪ ⎨ 1 (t, x) = d1 Δu1 (t, x) − a11 u1 (t, x) + k(x, x  )u2 (t, x  )dx  ∂t Ω ⎪ ⎩ ∂u2 (t, x) = −a22 u2 (t, x) + g(u1 (t, x)) ∂t

(1)

in Ω ⊂ RN (N ≥ 1), a nonempty bounded and open set with a sufficiently smooth boundary ∂Ω; for t ∈ (0, +∞), where a11 ≥ 0, a22 ≥ 0, d1 > 0 are constants and • u1 (t, x) denotes the concentration of the pollutant (pathogen material) at a spatial location x ∈ Ω, and a time t ≥ 0, • u2 (t, x) denotes the spatial distribution of the infective population at a spatial location x ∈ Ω, and a time t ≥ 0. There is no diffusion for the population, • the terms −a11u1 (t, x) and −a22u2 (t, x) model natural decays, • the total susceptible population is assumed to be sufficiently large with respect to the infective population, so that it can be taken as constant. For this kind of epidemics the infectious agent is multiplied by the infective human population and then sent to the sea through the sewage; because of the peculiar eating habits of the population of these regions the agent may return via some diffusion-transport mechanism to any point of the habitat Ω, where the infection process is further activated; thus the integral term 

k(x, x  )u2 (t, x  )dx  Ω

expresses the fact that the pollution produced at any point x  ∈ Ω of the habitat is made available at any other point x ∈ Ω; when dealing with human pollution, this may be due to either malfunctioning of the sewage system, or improper dispersal of sewage in the habitat. Linearity of the above integral operator is just a simplifying option. The Laplace operator takes into account a simplified random dispersal of the infectious agent in the habitat Ω, due to uncontrolled additional causes of dispersion (with a constant diffusion coefficient to avoid purely technical complications); we assume that the human infective population does not diffuse, thus modeling the fact that it is unlikely that human infectives may move around (anyhow the case with diffusion would be here a technical simplification). As such, system (1) can be adopted as a good model for the spatial propagation of an infection in agriculture and forests, too. Finally, the local “incidence rate” at point x ∈ Ω, and time t ≥ 0, is given by (i.r.)(t, x) = g(u1 (t, x)), depending upon the local concentration of the pollutant. The parameters a11 and a22 are intrinsic decay parameters of the two populations.

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2.2 A Regional Optimal Control Problem By controls in our optimal control problem we understand functions γ ∈ Γ Γ = {γ = (γ1 , γ2 ) : γ1 , γ2 ∈ L∞ (Ω), γ1 (x), γ2 (x) ∈ [δ, Θ] a.e. x ∈ Ω}

(2)

and sets ω ∈ Λ Λ = {ω  Ω : int ω = ∅, ω measurable }, where 0 < δ < Θ are given numbers. We deal with the following Problem (P): Minimize 

T

J (u1 , u2 , γ , ω) =

β(γ1 (x)u1 (t, x))χω (x)dx dt 0



T

+

 Ω





0

λ(γ2 (x)u2 (t, x))χω (x)dx dt + Ω

l(u2 (T , x))dx

(3)

Ω

over all controls (γ , ω), γ ∈ Γ , ω ∈ Λ and states u = (u1 , u2 ) subject to the system ∂u1 (t, x) − d1 Δu1 (t, x) + a11 u1 (t, x) ∂t 

k(x, x  )u2 (t, x  )dx  − γ1 (x)χω (x)u1 (t, x),

=

(4)

Ω

∂u2 (t, x) + a22u2 (t, x) + g(u1 (t, x)) = −γ2 (x)χω (x)u2 (t, x) ∂t

(5)

for (t, x) ∈ Q =]0, T [×Ω with u1 (0, x) = u01 (x), u2 (0, x) = u02 (x) for x ∈ Ω, u01 , u02 ∈ L∞ (Ω),

(6)

u01 (x), u02 (x) ≥ 0 a.e. x ∈ Ω,

∂u1 (t, x) + αu1 (t, x) = 0 ∂ν

for (t, x) ∈ Σ =]0, T [×∂Ω.

(7)

Here d, a11 , a22 and α are positive constants; χω is the characteristic function of ω. We assume g : R+ → R+ to be Lipschitz continuous and increasing with respect to its variable in R+ . Finally we assume that k ∈ L∞ (Ω × Ω) is such that k(x, x  ) ≥ 0 a.e. in Ω × Ω and  k(x, x  )dx > 0, a.e. x  ∈ Ω. (8) Ω

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The idea underlying the choice of the above cost functional is the following: 

T

 β (γ1 (x)u1 (t, x)) χω (x)dx dt

0

Ω

represents the cost of the programme of sanitation of the environment in the subregion ω; 

T

 λ (γ2 (x)u2 (t, x)) χω (x)dx dt

0

Ω

represents the cost of treatment of the infected population in the subregion ω;  l (u2 (T , x)) dx Ω

is included to represent the overall loss, in the entire habitat Ω, due to the residual level of the infective human population evaluated at the end of the observation/control period [0, T ]. β, λ, l are suitably chosen continuous functions, bounded from below. We remind that, by Babak [5, 6], under the assumption u01 , u02 ∈ L∞ (Ω), u01 (x), 0 u2 (x) ≥ 0 a.e. in Ω, we have the existence, uniqueness and nonnegativity of solutions to (4)–(7), i.e. u1 (t, x), u2 (t, x) ≥ 0, (t, x) ∈ Q. In [4] for a different epidemic problem—no control being a set—existence and necessary optimality conditions, as well as two gradient type algorithm are derived. The essential point in the convergence of the gradient algorithm (using the necessary optimality conditions—Pontryagin’s maximum principle) is that it starts from an arbitrary control function and stops when the difference between two computed controls in next two steps is smaller than a given ε > 0. However we do not know whether the calculated sequence of controls converges to the optimal control or the values of the cost functional for those controls and corresponding states converge to optimal values. To the knowledge of the authors, in the literature there is no optimal control theory providing sufficient conditions which can be applied to the control problem proposed here above. The main reason being that we deal with state equations with controls being sets. Optimal control problems in which the characteristic function of an unknown subset ω ⊂ Ω appear in many papers. In [19] the optimum design for two-dimensional wave equation is studied, in [20] an optimal location of the support of the control for one-dimensional wave equation is determined, in [16, 17] the optimal geometry for the support of the controls in stabilization problem is considered. The standard techniques in classical optimal control theory are based on the lower semicontinuity of some physical quantity (functional) with respect to control and on the compactness of the set of admissible controls. In the optimum design problem the location of the source is optimized. Thus, the lower semicontinuity of the shape functional is required with respect to some family of sets. So, except for very particular cases, there is no

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optimal location in optimum design problem. That is why among the techniques and tools used to analyze this type of optimum design problems, the methods of relaxation, homogenization and appropriate variational formulations have played a very important role. A problem similar to (P), but with all integrals taken over ω and without control γ , has been discussed from the point of view of the existence of an optimal shape in the book [14] and from a geometrical point of view in [18]. The existence of optimum design is essential if we have not at hand any sufficient optimality conditions. From the beginning of the last century, under strong influence of Hilbert, the existence issue became one of the fundamental questions in many branches of mathematics, especially in the calculus of variations as well as in its branch, the optimal control theory. Of course, following the existence proof, the next step is derivation of necessary optimality conditions and evaluation of the minimum argument. However, it should be pointed out that for many variational problems the existence of a solution accompanied by some necessary optimality conditions are not sufficient to find the argument of minimum in practice. Actually, existence of an optimum design is essential if we do not have at hand any sufficient optimality conditions; the stronger result, i.e. sufficient optimality conditions for a minimum in a specific problem, may replace the existence requirement and significantly help to calculate it. We may like to point out that for many optimal control problems the existence of a solution accompanied by some necessary optimality conditions are not sufficient to find the argument of minimum in practice. On the other hand, in the calculus of variations it had already been pointed out by Weierstrass that, from a practical point of view, the most important issues for the solution procedure are the so-called sufficient optimality conditions for a relative minimum, i.e. optimality conditions relative to some possibly smaller set of arguments of functional which is determined by additional (practical) conditions. Some steps in this direction were done in [25], where the functional is quadratic and the state equation is linear and elliptic with Dirichlet boundary condition. In the next section we develop a new dual dynamic programming theoretical approach to derive a verification theorem, i.e. sufficient optimality conditions for problem (P). However the main advantage of this paper is that we also develop sufficient conditions for ε-optimality i.e. we formulate conditions which allow us to assert that for the calculated control (e.g. numerically) we know how far we are from the optimal value. Just this approximate theory is fundamental for our numerical algorithm. We call the controls γ ∈ Γ and ω ∈ Λ admissible shape controls and the solution (u1 , u2 ) corresponding to it admissible state. We denote by Ad the set of admissible states and controls.

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3 Dual Dynamic Programming Method for Problem (P) In classical dynamic programming (i.e. in one dimensional case) we have a value function S(t, s) depending on time t and the state variable s. Having possibility to perturb a given point (t, s) we are able to calculate the full derivative of S(t, s): St (t, s) + Ss (t, s)˙s and using some properties of the value function we can derive Hamilton–Jacobi equation. Essential point in that approach is that we can perturb S(t, s) at each point of the open domain of the definition of S. In the case of problem (P) we have not possibility to perturb the optimal value of (P). That is why in [23] a proposition of a dual dynamic programming method was described and then developed for different optimal control problems, including shape control in [15, 22, 24, 25]. In [23], instead of considering objects of dynamic programming such as the value function S(t, s) or Hamilton–Jacobi equation in the space (t, s), new space—dual space is proposed and new objects of dual dynamic programming are defined: an auxiliary function, a dual optimal value and a dual Hamilton–Jacobi equation which the auxiliary function should satisfy. The dual space in [23] is, in fact, defined by conjugate (dual) functions (variables) which appear in Pontryagin maximum principle. An essential feature in dual method is that we do not deal directly with the value function but with some auxiliary function, defined in a dual domain, satisfying some dual relations which imply sufficient optimality conditions for the primal functional. Such an approach has some advantages. We do not need any properties of the function’s value such as smoothness or convexity. It has some disadvantages too: the auxiliary function must satisfy a kind of generalised transversality conditions which is somehow restrictive. The approach we present here was inspired by the paper [25], where the simple model of shape optimisation (elliptic equation) was investigated. The new challenge in the control problem (P) derives from the fact that the problem under consideration is nonlinear and with a Neumann type boundary condition. Therefore we need really to construct a new dual dynamic programming type approach for problem (P). Thus let us start first withthe definition Let P ⊂ R1+N+3 be a set  of a dual set. 0 2 of the variables (t, x, p) = t, x, y , y with y ∈ R , y 0 ≤ 0 and all (t, x) ∈ Q (Q =]0, T [×Ω (see the formulation of Problem (P)). Denote by Y = {(y 0 , y) = p; there exists (t, x, p) ∈ P }, by clP the closure of P (with all (t, x) ∈ clQ) and by Pb its subset: Pb = {(t, x, p) ∈ clP ; (t, x) ∈ Σ}, and & % clY = (y 0 , y) = p; there exists (t, x, p) ∈ clP ,

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where Σ is defined in (7). Let u be the vector of pairs coordinates (u1 , u2 ), Δx u = (d1 Δx u1 , 0), f (t, x, u, γ , ω) = (f1 (t, x, u, γ1 , ω), f2 (t, x, u, γ2 , ω)), where 

k(x, x  )u2 (x  , t)dx  − γ1 (x)χω (x)u1 (t, x) for (t, x) ∈ Q,

f1 = −a11u1 (t, x) + Ω

(9) f2 = −g(u1 (t, x)) − a22 u2 (t, x) − γ2 (x)χω (x)u2 (t, x) for (t, x) ∈ Q,

(10)

h = h(t, x, u), where h = −αu1 (t, x) for (t, x) ∈ Σ.

(11)

Now let us introduce an auxiliary function V : clP → R belonging to H 2 (P ) (Sobolev space having function, first and second weak derivatives in L2 (P )) and satisfying the “transversality condition”: V (t, x, p) = y 0 Vy 0 (t, x, p) + yVy (t, x, p) for (t, x, p) ∈ clP ,

(12)

where Vy 0 , Vy are partial derivative of V , y = (y1 , y2 ) and p = (y 0 , y). By u : clP → R2 we denote a function of 1 + N + 3 variables (t, x, y 0 , y). In the sequel we shall assume that u(t, x, p) = −Vy (t, x, p) for (t, x, p) ∈ clP .

(13)

At the end of Sect. 2 we have denoted by Ad the set of all admissible states and controls. However, we need also the dual states p(t, x) which should relate somehow to the primal states u(t, x). For that relation we choose the formula (13). This implies that actually, we shall consider not all admissible controls and corresponding to them admissible states but only those primal states which satisfy the relatetion (13) for some dual states. To this effect we introduce for given fixed ξ(·) ∈ (H 2 (Ω))2 the following set of primal states (together with controls) for which there exist dual states satisfying the initial condition y(0, x) = ξ(x) and both satisfy (13): %   Adu = (u(·), γ (·), ω) ∈ Ad; there exist p(t, x) = y 0 , y(t, x) , (t, x) ∈ clQ, 2  y(·) ∈ H 2 (Q) , y 0 ≤ 0, (t, x, p(t, x)) ∈ clP , y(0, x) = ξ(x), x ∈ Ω, u(t, x) = u(t, x, p(t, x)), (t, x) ∈ clQ}. In fact, we shall study our optimal control problem just on the set Adu . We consider condition (12) and function (13) on the set clP . The function p : Q → R3 we call dual trajectory while u : Q → R2 we call primal trajectory. Next define a dual

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u optimal value SD u SD =

inf

(u,γ ,ω)∈Adu

−y 0 J (u, γ , ω).

(14)

u the dual optimal value in contrast to the optimal value We named SD

S=

inf

(u,γ ,ω)∈Ad

J (u, γ , ω)

u depends strongly upon dual trajectories p(t, x) which in fact determine as SD the set Adu . Moreover, essential point is that the set Adu is, in general, smaller u is greater than the than Ad i.e. Adu ⊂ Ad and thus the dual optimal value SD u 0 0 optimal value S i.e. SD ≥ (−y¯ )S (y¯ corresponds to an optimal dual trajectory). However, the construction of the set Adu is in the spirit of Weierstrass strong relative minimum, i.e. relative to some smaller set covered by the family of extremals (one parameter family of the solutions to Euler–Lagrange equation) which, in practice, is greater than the global minimum but significancy smaller than a local minimum. Of course, to find that one parameter family means we have to solve the Euler– Lagrange equation which, in the nonlinear case, is not easy. But we should have in mind that local minimum means that there exists a neighborhood of suspected extremal in which we want to assert some minimization theorem. However we do not know that neighborhood as well, in general, the type (in which space) of this neighborhood. In our problem to find the set Adu , first we must find the function V i.e. to solve Eqs. (15)–(18) with (12) (see below) and then to define the set of admissible dual trajectories. It is not easy task, but then we will have a possibility to assert that the suspected trajectory is really optimal with respect to all trajectories lying in Adu . This fact for our problem is presented in literature for the first time. Of course, one can wonder are we able to find V or is the set Adu nonempty? The answer is not simple. In some cases we can solve that problem, in many cases we cannot do it, similarly as in classical calculus of variation with Weierstrass approach. This is why in Sect. 5 we develop approximate theory for Problem (P) which works in practice. Notice that in spite of the fact that our problem depends on time we cannot perturb it with respect to initial data and time (they are fixed) as it is usually done in classical optimal control theory to derive the Hamilton Jacobi equation. Thus let us introduce, for our problem, a dual Hamilton–Jacobi equation in P which an auxiliary function V must satisfy:

sup

{ Vt (t, x, p) − Δx V (t, x, p) + yf (t, x, −Vy (t, x, p), γ , ω)

γ1 ,γ2 ∈[δ,Θ],ω∈Λ

+ y 0 (β (γ1 (−Vy1 (t, x, p)))χω (x) + λ(γ2 (−Vy2 (t, x, p)))χω (x))} = 0

(15)

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with terminal condition Vy 0 (T , x, p) = −l(−Vy2 (T , x, p)) for (T , x, p) ∈ P

(16)

and a dual Hamilton–Jacobi type equation on Pb −

∂Vy1 (t, x, p) + yh(t, x, −Vy1 (t, r, p)) = 0 for (t, x, p) ∈ Pb ∂ν

(17)

with boundary condition x Vy 0 (t, x, p) · ν = 0 for (t, x, p) ∈ Pb .

(18)

Remark 1 Notice that for the auxiliaray function V we have two dual Hamilton– Jacobi equations: one in P and second in Pb . The reason is not that we are working with dual approach but because the Problem (P) is very difficult to study by any dynamical programming approach. It seems to us that the line presented here is the only one applicable to Problem (P).

4 Sufficient Optimality Conditions The dual approach to the dynamic programming described in the former section allow us to formulate and to prove a kind of verification theorem ensuring sufficient optimality conditions for our problem (14). We would like to stress that we are working now in the dual space clP and with the auxiliary function V defining, by (13) the set Adu . Let us fix y¯ 0 < 0 and ξ(x), x ∈ Ω, ξ(·) ∈ (H 2(Ω))2 . Define the set P of dual trajectories, which is in fact determined by V ,   P = {p(t, x) = y¯ 0 , y(t, x) ; (t, x) ∈ clQ, (t, x, p(t, x)) ∈ clP , y(·) ∈ (H 2 (Q))2 , y(0, x) = ξ(x), x ∈ Ω, there exists (u(·), γ (·), ω) ∈ Adu , u(t, x) = −Vy (t, x, p(t, x)), (t, x) ∈ Q ∪ Σ}. We see that there is a strict relation between the set of primal trajectories u (determinated by controls (γ , ω)) from Adu and the set of dual trajectories p ∈ P. Theorem 1 Assume that there exists V ∈ H 2 (P ) satisfying (15)–(18), (12) on ¯ x, p) = −Vy (t, x, p) for (t, x, p) ∈ P and take (u(·), P . Put u(t, ¯ γ¯ (·), ω) ¯ ∈ ¯ (y¯ 0 , y(·)) Adu¯ , (Adu¯ determined by u), ¯ = p(·) ¯ ∈ P, y(0, ¯ x) = ξ(x), x ∈ Ω, such that ¯ x)), for (t, x) ∈ Q ∪ Σ u(t, ¯ x) = −Vy (t, x, p(t,

(19)

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and for (t, x) ∈ Q, Vt (t, x, p(t, ¯ x)) − Δx V (t, x, p(t, ¯ x)) + y(t, ¯ x)f (t, x, −Vy (t, x, p(t, ¯ x)), γ¯ (x), ω) ¯ ¯ x))))χω¯ (x) + λ(γ¯2 (x)(−Vy2 (t, x)))χω¯ (x)) = 0, + y¯ 0 (β(γ¯1 (x)(−Vy1 (t, x, p(t, (20) ∂V (t, x, p(t, ¯ x)) +y(t, ¯ x)h(t, x, −Vy1 (t, x, p(t, ¯ x))) = 0 for (t, x, p(t, ¯ x)) ∈ Pb ∂ν (21) hold. Then (u(·), ¯ γ¯ (·), ω) ¯ is an optimal triple relative to all (u(·), γ (·), ω) ∈ Adu¯ i.e. −y¯ 0 J (u, ¯ γ¯ , ω) ¯ ≤ −y¯ 0 J (u, γ , ω) for all (u(·), γ (·), ω) ∈ Adu¯ . Proof The transversality conditions (12) are the basis in the proof and we start it just applying operator ∂t∂ − Δ to both sides of (12) along any dual trajectory p(·) ∈ P generating the triplet (u(·), γ (·), ω) ∈ Adu¯ . We calculate for (t, x, p(t, x)) ∈ clP d (having in mind that dt means total derivative in t and Δ means the Laplace operator of x → Vy 0 (t, x, p(t, x)) or x → Vy (t, x, p(t, x)) and Δx means the Laplace operator in x—second variable) Vt (t, x, p(t, x)) + Vy (t, x, p(t, x))yt (t, x) − Δx V (t, x, p(t, x)) −Vy (t, x, p(t, x))Δx y(t, x) = y¯ 0 (

d V 0 (t, x, p(t, x)) dt y

−ΔVy 0 (t, x, p(t, x))) + (yt (t, x) − Δx y(t, x))Vy (t, x, p(t, x)) +y(t, x)(

d Vy (t, x, p(t, x)) − ΔVy (t, x, p(t, x))). dt

Reducing similar terms on both sides we get for (t, x, p) ∈ clP Vt (t, x, p(t, x)) − Δx V (t, x, p(t, x)) = y¯ 0 (

d V 0 (t, x, p(t, x)) − ΔVy 0 (t, x, p(t, x))) dt y

(22)

+y(t, x)(

d Vy (t, x, p(t, x)) − ΔVy (t, x, p(t, x))). dt

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We know that u(t, x) = −Vy (t, x, p(t, x)), (t, x) ∈ Q ∪ Σ, thus Vy (t, x, p(t, x)) satisfies (4), (5) (remembering (9), (10)) i.e. for (t, x) ∈ Q ∪ Σ −

d Vy (t, x, p(t, x)) + ΔVy (t, x, p(t, x)) = f (t, x, −Vy (t, x, p(t, x)), γ (x), ω). dt (23)

Applying (23) to (22) and then using (15) we come to y¯ 0 [β (γ1 (x)(−Vy1 (t, x, p(t, x))))χω (x) + λ(γ2 (x)(−Vy2 (t, x, p(t, x))))χω (x)] (24) +y¯ 0 (

d V 0 (t, x, p(t, x)) − ΔVy 0 (t, x, p(t, x))) ≤ 0 for (t, x) ∈ Q. dt y

Integrating (24) over Q we get 



T 0

y¯ 0 {β (γ1 (x)(−Vy1 (t, x, p(t, x))))χω (x)

Ω

+λ(γ2 (x)(−Vy2 (t, x, p(t, x))))χω (x)}dx dt 

T

≤− 0

,

 y¯

0

Ω

d V 0 (t, x, p(t, x)) − ΔVy 0 (t, x, p(t, x)) dx dt. dt y

Applying terminal condition (16), boundary condition (18) and the properties of y at (0, x) i.e. y(0, x) = ξ(x), x ∈ Ω, we obtain that  − y¯

T

0 0

 Ω

3 β (γ1 (x)(−Vy1 (t, x, p(t, x))))χω (x)

(25)

4 +λ(γ2 (x)(−Vy2 (t, x, p(t, x))))χω (x) dx dt  % & ≥ −y¯ −l(−Vy2 (T , x, p(t, x))) − Vy 0 (0, x, (y¯ 0 , ξ(x))) dx. 0

Ω

Now we follow along p(t, ¯ x) in the same manner as above for p(t, x) but now applying (19)–(21), then we have  − y¯

T

0 0

 Ω

{β (γ¯1 (x)(−Vy1 (t, x, p(t, ¯ x))))χω¯ (x)

+λ(γ¯2 (x)(−Vy2 (t, x, p(t, ¯ x))))χω¯ (x)}dx dt = −y¯ 0

 % & ¯ x))) − Vy 0 (0, x, (y¯ 0 , ξ(x))) dx. −l(−Vy2 (T , x, p(t, Ω

(26)

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Combining (25) with (26) gives −y¯ 0 J (u, ¯ γ¯ , ω) ¯ ≤ −y¯ 0 J (u, γ , ω) which completes the proof.



5 Sufficient Conditions for ε-Optimality The theory presented in the last sections being in terms of dual dynamic programming gives us a possibility to find at least theoretically the optimal value. However, in practice, it is difficult to solve equations stated there in exact form—the equations are nonlinear. In fact, we solve such a system using different approximate (numerical) methods. Therefore what we can get then is eventually approximate optimality. This is why in this section we present a dual dynamic approach to sufficient conditions for an approximate (ε-optimality) optimality. Just the dual ε optimality conditions are the base for the construction of a computational algorithm of approximate optimality.

5.1 ε-Optimality In this section we use the same notations and definitions concerning the dual notions, i.e. the sets P , Y, Pb as in Sect. 3. Let us fix any yε0 < 0, ξ(·) ∈ (H 2 (Ω))2 and ε > 0. As for the definition of an ε-optimal value we will use an inequality (see the definition below in (32)) it suggests that expressions allowing to derive assertions like Theorem 1 should also satisfy suitable inequalities. Thus we require that an auxiliary function V˜ , V˜ : clP → R belonging to H 2 (P ), satisfying (12), in the case of ε-optimality must fulfill the following system of inequalities: the dual Hamilton– Jacobi inequality in P εyε0 ≤

sup

(t, x, p) + yf (t, x, −V˜y (t, x, p), γ , ω) { V˜t (t, x, p) − Δx V

γ1 ,γ2 ∈[δ,Θ],ω∈Λ

+ y 0 (β (γ1 (−V˜y1 (t, x, p)))χω (x) + λ(γ2 (−V˜y2 (t, x, p)))χω (x))} ≤ 0

(27)

with terminal condition, for (T , x, p) ∈ P , −yε0l(−Vy2 (T , x, p))+εyε0 ≤ yε0 Vy 0 (T , x, p) ≤ yε0 l(−Vy2 (T , x, p))−εyε0

(28)

and the dual Hamilton–Jacobi type inequality on Pb : εyε0 ≤ −

∂Vy1 (t, x, p) + yh(t, x, −Vy1 (t, r, p)) ≤ 0, (t, x, p) ∈ Pb ∂ν

(29)

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with boundary condition εyε0 ≤ x Vy 0 (t, x, p) · ν ≤ −εyε0 ,

(t, x, p) ∈ Pb .

(30)

Thus let us assume that there exists a solution V˜ ∈ H 2 (P ) to (27)–(30) satisfying also (12). Then we can define similarly as in Sect. 3 uε (t, x, p) = −V˜y (t, x, p), (t, x, p) ∈ clP

(31)

and 3   Aduε = (u(·), γ (·), ω) ∈ Ad; there exists p(t, x) = yε0 , y(t, x) , (t, x) ∈ clQ, y(·) ∈ (H 2 (Q))2 , (t, x, p(t, x)) ∈ clP , y(0, x) = ξ(x), x ∈ Ω, u(t, x) = uε (t, x, p(t, x)), (t, x) ∈ clQ}. We need also (see the beginning of Sect. 4) the set   Pε = {p(t, x) = yε0 , y(t, x) ; (t, x) ∈ clQ, (t, x, p(t, x)) ∈ clP , y(·) ∈ (H 2 (Q))2 , y(0, x) = ξ(x), x ∈ Ω, there exists (u(·), γ (·), ω) ∈ Aduε , u(t, x) = −V˜y (t, x, p(t, x)), (t, x) ∈ Q ∪ Σ}. For given V˜ ∈ H 2 (P ) satisfying (27)–(30), (12) let uε be defined by (31). Let pε ∈ Pε and the state uε (t, x) = uε (t, x, pε (t, x)), (t, x) ∈ Q be corresponding to some γε , ωε accordingly to the definition of Pε . We call the triplet (uε , γε , ωε ) ε-optimal with respect to all triplet (u(·), γ (·), ω) ∈ Aduε if the inequality − yε0 J (uε , γε , ωε ) ≤ −yε0 J (u, γ , ω) − 4εyε0

(32)

is satisfied for all (u(·), γ (·), ω) ∈ Aduε . Notice that as −yε0 > 0 the ε-optimality is in fact with ε˜ =4ε-optimal. This relates to the fact that in (27) we have ε and in the proof of the next theorem then we obtain the inequality (32) just with 4ε. Now we are ready to formulate the verification theorem for the ε -optimality, i.e. we give conditions under which the trio (uε , γε , ωε ) satisfies (32). Theorem 2 Let the trio (uε , γε , ωε ), the dual trajectory pε and the auxiliary function V˜ be as described above and satisfy the dual Hamilton Jacobi inequalities (27), (29) along the dual trajectory pε : εyε0 ≤ V˜t (t, x, pε (t, x)) − Δx V˜ (t, x, pε (t, x))

(33)

+yε (t, x)f (t, x, −V˜y (t, x, pε (t, x)), γε (x), ωε ) +yε0 (β(γ1ε (x)(−V˜y1 (t, x, pε (t, x)))χωε (x) + λ(γ2ε (x)(−V˜y2 (t, x)))χωε (x)) ≤ 0

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for (t, x) ∈ Q and ∂ V˜ (t, x, pε (t, x)) +yε (t, x)h(t, x, −V˜y1 (t, x, pε (t, x))) = 0 for (t, x, pε (t, x)) ∈ Pb . ∂ν

Then (uε , γε , ωε ) is an ε-optimal triplet with respect to all (u(·), γ (·), ω) ∈ Aduε , i.e. we have inequality (32) satisfied. Proof The proof is similar to the proof of Theorem 1, this is why we only highlight the main steps. Thus take any triplet (u(·), γ (·), ω) ∈ Aduε and p(·) ∈ Pε such that u(t, x) = −V˜y (t, x, p(t, x)), (t, x) ∈ Q ∪ Σ. Next, since V˜ satisfies the transversality condition so applying to that relation the operator ∂t∂ − Δ we get the analogous equality for V˜ as in the former proof for V : d V˜t (t, x, p(t, x))−Δx V˜ (t, x, p(t, x)) = yε0 ( V˜y 0 (t, x, p(t, x))−ΔV˜y 0 (t, x, p(t, x))) dt

+ y(t, x)(

d ˜ Vy (t, x, p(t, x)) − ΔV˜y (t, x, p(t, x))). dt

(34)

Again using the fact that −V˜y (t, x, p(t, x)) satisfies our state equation from (34) and (27) we infer that (compare (24)) yε0 [β (γ1 (x)(−V˜y1 (t, x, p(t, x))))χω (x) + λ(γ2 (x)(−V˜y2 (t, x, p(t, x))))χω (x)] + yε0 [

d ˜ V 0 (t, x, p(t, x)) − Δx V˜y 0 (t, x, p(t, x))] ≤ 0 for (t, x) ∈ Q. dt y

(35)

Similarly, for triplet (uε , γε , ωε ) and pε , but now applying (33), we get the inequality εyε0 ≤ yε0 [β(γ1ε (x)(−V˜y1 (t, x, pε (t, x))))χωε (x) + λ(γ2ε (x)(−V˜y2 (t, x, pε (t, x))))χωε (x)] +yε0 [

d ˜ V 0 (t, x, pε (t, x)) − ΔV˜y 0 (t, x, pε (t, x))] for (t, x) ∈ Q. dt y

(36)

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Integrating (35) and (36) over Q we come to 



T 0

yε0 [β(γ1 (x)(−V˜y1 (t, x, p(t, x))))χω (x)

Ω

+λ(γ2 (x)(−V˜y2 (t, x, p(t, x))))χω (x)]dx dt 



T

≤− 0

yε0 [

Ω

d ˜ V 0 (t, x, p(t, x)) − ΔV˜y 0 (t, x, p(t, x))]dx dt dt y

and  εyε0



T

− 0

Ω

yε0 [β(γ1ε (x)(−V˜y1 (t, x, pε (t, x))))χωε (x)

+λ(γ2ε (x)(−V˜y2 (t, x, pε (t, x))))χωε (x)]dx dt  ≤ 0

T

 Ω

yε0 [

d ˜ V 0 (t, x, pε (t, x)) − ΔV˜y 0 (t, x, pε (t, x))]dx dt. dt y

Applying the terminal and the boundary conditions for V˜ and the initial condition for y we infer from the above two inequalities the following relation 



T

− 0

Ω

yε0 [β(γ1ε (x)(−V˜y1 (t, x, pε (t, x))))χωε (x)

+λ(γ2ε (x)(−V˜y2 (t, x, pε (t, x))))χωε (x)]dx dt − yε0  ≤− 0

T

 Ω

 Ω

l(−V˜y2 (T , x, pε (t, x)))dx

yε0 [β(γ1 (x)(−V˜y1 (t, x, p(t, x))))χω (x)

+λ(γ2 (x)(−V˜y2 (t, x, p(t, x))))χω (x)]dx dt−yε0

which ends the proof.

 Ω

l(−V˜y2 (T , x, p(t, x)))dx−4εyε0 ,



5.2 Computational Algorithm The sufficient conditions formulated for the ε-value state that if we are, in any way, to find a function V˜ , the trio (uε , γε , ωε ) and the dual trajectory pε such that V˜ , γε , ωε and pε satisfy (33) then the trio (uε , γε , ωε ) is ε-optimal i.e. satisfies

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(32). Of course to guess these functions is not easy. Usually one applies some numerical methods to solve inequalities (27)–(30) and Eqs. (4)–(7) and then from (31) try to determine pε . Below we describe some procedure which allow us to calculate numerically a suboptimal pair (uε , γε , ωε ). The algorithm, we present below, ensures that we find, in finite number of steps, suboptimal pair, however may be with larger ε. We explain it after presentation of the algorithm. Algorithm 1. Fix ε > 0 and calculate an auxiliary function V˜ from (27)–(30). ˜ uε consisting from a finite family of M triplet (u(·), γ (·), ω): 2. Form Ad a. define sets ωn and controls γn defined in these sets n = 1, . . . , M, b. for each given γn , ωn calculate un , n = 1, . . . , M solving state equations. 3. Find minimal value of J (un , γn , ωn ), n = 1, . . . , M and corresponding to it triplet denote by (u(·), ˆ γˆ (·), ω). ˆ 4. Assume yε0 = −1 and determine y(·) ˆ from the relation uˆ (t, x) = −Vˆy (t, x, −1, y(t, ˆ x)). 5. For V˜ , (u(·), ˆ γˆ (·), ω) ˆ and y(·) ˆ check the inequality (33), a. if V˜ , (u(·), ˆ γˆ (·), ω) ˆ and y(·) ˆ satisfy (33) then (u(·), ˆ γˆ (·), ω) ˆ is an ε-optimal triplet and J (u, ˆ γˆ , ω) ˆ is an ε-optimal value, i.e. J (u, ˆ γˆ , ω) ˆ satisfies (32), ˆ do not satisfy (33) then go to 2. b. if V˜ and (u(·), ˆ γˆ (·), ω) ˆ and y(·) ˜ uε in point 2. of the Algorithm is not the same set Remark 2 Note that the set Ad ˜ uε is to help to find the trio (u(·), as in (32). The aim of the forming Ad ˆ γˆ (·), ω) ˆ and y(·). ˆ The verification theorem only asserts that if V˜ , (u(·), ˆ γˆ (·), ω) ˆ and y(·) ˆ satisfy the inequality (33) then the value J (u, ˆ γˆ , ω) ˆ satisfies (32). Let us observe that fast calculation of the triplet (u(·), ˆ γˆ (·), ω) ˆ being an ε˜ uε in step 2. optimal (see 5. a.) depends strongly upon construction of the set Ad If we are not lucky i.e. after calculation we are in case 5. b. then we have to repeat ˜ uε . In the worst case it may happen that we should do it even forming the set Ad infinitely many times. However the advantage of the procedure described in the above Algorithm is that we can stop it after a finite number of steps, by assuming larger εˆ > 0 at point 1., i.e. if V˜ and γˆ (·), ω, ˆ y(·) ˆ satisfy (33) with larger εˆ and we are content with the inequality (32) for this εˆ . The main advantage of the approach described above is that we do not know the value inf(u,γ ,ω)∈Aduε J (u, γ , ω) (here Aduε is full set not finite) and we are able to estimate J (u, ˆ γˆ , ω) ˆ how far we are from that value. In most numerical approaches to optimal control theory we only know that there exists an approximate convergent sequence to the value inf(u,γ ,ω)∈Ad J (u, γ , ω). But in nonlinear problems as it is the problem considered here we know nothing about the calculated convergent sequence even whether it is convergent to inf(u,γ ,ω)∈Ad J (u, γ , ω) or when we should stop our calculations.

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6 Conclusions In the paper we develop the sufficient optimality conditions for the epidemic problem with controls being parameters and subregions. We develop also the ε-optimal theory of approximate solutions and build on this base a numerical algorithm. That algorithm allows to find numerically concrete value of parameters γ and subregion ω such that the functional considered is ε-optimal with given ε. As anticipated in the Introduction, the concrete construction of a numerical code for a significant example of the above mentioned optimal control problem based on the proposed methods has required unexpected efforts, so that we have left the relevant numerical analysis and computational implementations to future investigations. Acknowledgements The paper was inspired by the lecture given by V. Capasso during the conference Nonlocal Aspects in Mathematical Biology in Bedlewo 2016. The authors are indebted to the referees for the valuable comments and suggestions to improve the paper.

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13. Codeço, C.T.: Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infect. Dis. 1, 1–14 (2001) 14. Delfour, M.C., Zolesio, J.P.: Shapes and Geometries: Analysis, Differential Calculus and Optimization, Advances in Design and Control. SIAM, Philadelphia (2001) 15. Galewska, E., Nowakowski, A.: A dual dynamic programming for multidimensional elliptic optimal control problems. Numer. Funct. Anal. Optim. 27, 279–289 (2006) 16. Hebrard, P., Henrot, A.: Optimal shape and position of the actuators for the stabilization of a string. Syst. Control Lett. 48, 199–209 (2003) 17. Hebrard, P., Henrot, A.: Spillover phenomenon in the optimal locations of actuators. SIAM J. Control Optim. 44, 349–366 (2005). 18. Henrot, A., Pierre, M.: Variation et optimisation de formes: une analyse geométrique (French). Mathématiques et Applications, vol. 48. Springer, Berlin (2005) 19. Münch, A.: Optimal design of the support of the control for the 2-D wave equation: a numerical method. Int. J. Numer. Anal. Model. 5, 331–351 (2008) 20. Münch, A.: Optimal location of the support of the control for the 1-D wave equation: numerical investigations. Comput. Optim. Appl. 42, 443–470 (2009) 21. Näsell, I., Hirsch, W.M.: The transmission dynamics of schistosomiasis. Comm. Pure Appl. Math. 26, 395–453 (1973) 22. Nowakowska, I., Nowakowski, A.: A dual dynamic programming for minimax optimal control problems governed by parabolic equation. Optimization 60, 347–363 (2011) 23. Nowakowski, A.: The dual dynamic programming. Proc. Amer. Math. Soc. 116, 1089–1096 (1992) 24. Nowakowski, A.: Sufficient optimality conditions for Dirichlet boundary control of wave equations. SIAM J. Control Optim. 47, 92–110 (2008) 25. Nowakowski, A., Sokolowski, J.: On dual dynamic programming in shape control. Commun. Pure Appl. Anal. 11, 2473–2485 (2012)

The Interplay Between Voluntary Vaccination and Reduction of Risky Behavior: A General Behavior-Implicit SIR Model for Vaccine Preventable Infections Alberto d’Onofrio and Piero Manfredi

Abstract The onset in the last 15 years of behavioral epidemiology has opened many new avenues for epidemiological modelers. In this manuscript we first review two classes of behavioral epidemiology models for vaccine preventable diseases, namely behaviour-implicit SIR models with prevalence-dependent vaccination (at birth and among older individuals), and prevalence-dependent contact rate. Subsequently, we briefly propose a general framework of behavior–dependent nonlinear and linear Forces of Infection (FoI) valid for a vast family of infectious diseases, and including delays and ‘epidemic memory’ effects. Finally and mainly, we develop a new general behavioral SIR model. This model combines the two aforementioned types of behavioral phenomena, previously considered only separately, into a single unified model for behavioral responses. The resulting model allows to develop a general phenomenological theory of the effects of behavioral responses within SIR models for endemic infections. In particular, the model allows to complete the picture about the complicate interplay between different behavioral responses acting on different epidemiological parameters in triggering sustained oscillations of vaccine coverage, risky behavior, and infection prevalence. Keywords Behavior · Epidemics · Memory · Delay · Vaccine · Force of infection · Contact rate · Transmission rate

A. d’Onofrio () International Prevention Research Institute, Lyon, France e-mail: [email protected] P. Manfredi Department of Economics, Universitá di Pisa, Pisa, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_10

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1 Introduction The birth of mathematical and computational epidemiology dates back to a century ago about, when a few mathematical pioneers developed the main cornerstone ideas and models of the new discipline [21, 29]. Their ground-breaking idea lied in the description of the key process namely, infection transmission from an infected to a susceptible individual, by the law of mass action imported from Statistical Mechanics. Accordingly, contagion is abstracted as a chemical reaction that can or cannot occur (with a certain probability) upon the random encounter of two individuals. Social contacts between individuals are in their turn abstracted: individuals contact each other at random, as the particles of a perfect gas colliding in a box [7]. In particular, still owing to the Statistical Mechanics paradigm, the two key parameters, namely the contact rate per individual, and the transmission rate per contact, are taken as natural constants of human behavior, possibly mirroring the social characteristics of a given community or setting, at a certain time moment. Building on extensions of this simple idea, more recent pioneering contributions aiming to better integrate models with data [2, 19, 20], have allowed mathematical models of infectious diseases to leave their traditional, abstract, bio-mathematical environment, to become central supporting tools for public health decisions. Main instances are for example the determination of the duration of school closure during a pandemic outbreak, or the fraction of new-born children to be immunized for a vaccine-preventable infection, as is the case of measles and pertussis. This critical role has conferred to mathematical epidemiology a prominent role in policy making, by allowing a substantial advance of public health as a scientific discipline. Some of contemporary models are highly sophisticated in both their mathematical/computational structure, and in their data requirements [26]. In these sophisticated models, the patterns of social contacts with which individuals contact each other, classified according to a range of characteristics (e.g. age, level of social/sexual activity, etc.) are the key determinants of the transmission of both close-contact infections, as influenza or measles, and of sexually transmitted infections (STIs), such as HIV/AIDS. Nonetheless, as it was pointed out [17], even in such highly sophisticated models there remained a key missing layer: the humans’ behavior. Indeed, even these realistic models continue to treat contact and transmission rates as natural constants, exactly in the same way as the simple SIR model. This means that individuals’ social behavior is totally unaffected by the state of the disease. Briefly, this means for example that during an epidemic outbreak individuals will continue to contact each other at the same rate regardless of how low or high is the perceived risk of acquiring infection or even of dying from it. As contact patterns are usually measured from normal situations [27], the resulting models are therefore unlikely to apply under the complicate and stressed social conditions that might result during a dangerous epidemic or a during a period of panic raised by a pandemic threat [17].

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Similarly, the models used in the current public health practice to evaluate the impact of childhood immunization programs most often treat vaccine uptake as a constant [2]. This implies to postulate that the vaccination coverage prevailing in the community is totally unaffected by individuals’ risk perceptions about the disease and the vaccine. Such an hypothesis is at odd with the fact it is the degree of adhesion of the public that will ultimately determine the success of the program, especially when the program is voluntary or when laws for mandatory vaccines are not carefully enforced. Clearly, this static human behavior, which is the ultimate legacy of the Statistical Mechanics paradigm, is an unrealistic abstraction, which at best can apply in some particular situations (e.g. an epidemics of a non-threatening and non-costly infection). Indeed, by their very nature, humans are neither static nor passive. For example, they can decide to spontaneously change their social behavior in response to a pandemic threat, can redirect their sexual activity towards partners perceived as less-at-risk in response to news about a dangerous STI known to circulate in the population, or can decide not to vaccinate their children after having compared perceived costs and benefits of a vaccination program, thereby threatening its success. A central role in these decisions is played by the way communication technologies affect the shape and the speed of spread of the relevant information. The need to seriously account for human behavior has led in the last 10 years to a deep rethinking of the mathematical modeling of infectious diseases. This has in turn led to the birth of a new branch of mathematical epidemiology, which we termed the behavioral epidemiology (BE) of infectious diseases [24]. As argued in [4, 24], BE has an intrinsically multi-disciplinary core, aiming to combine classical epidemiological modeling [2, 3, 7] and behavioral sciences, namely sociology, psychology, economics, anthropology etc, to improve our understanding of the complex interplay between infection dynamics and the related underlying human behavior. Behavioural epidemiology has now grown rapidly and a summary of its many different facets can be found in a number of reviews appeared on the subject [18, 24, 32] to which the interested reader can refer. This work has a twofold goal. First, we aim to introduce some of the basic ideas of behavioural epidemiology and their mathematical and public health implications for the dynamics and control of vaccine preventable infectious diseases such as e.g., measles and pertussis. In particular we do this by presenting (1) some basic SIR models with voluntary vaccination where individuals change their propensity to vaccinate their children depending on the perceived changes in the risks of infection and disease, and (2) some basic SIR models where instead individuals change their social contact patterns—modulating their behaviour at risk—still based on their perceived risks of infection and disease. In relation to this we rely on work from [9, 11, 24] where individuals’ responses are modulated from awareness of the current and past trend of infection summarised by suitable phenomenological information indexes. Second, we aim at combining the previous two issues—which have typically been considered separately in the available behavioural epidemiology literature—by proposing a general model where awareness of risks can affect both

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the propensity to vaccinate as well as the contact rate. This double feedback, though not well documented for current vaccine preventable endemic infections, is likely to occur in many circumstances. Surely a well documented example has been represented by the alert arising from the doubling in the number of deaths from invasive meningococcal disease observed in the Tuscany region of Italy between 2015 and 2016 [8, 28, 31]. As meningitis is perceived as a very serious disease, the scaring public news appearing at the time jointly with the offer of free vaccination from the local public authorities, were able to dramatically increase the vaccination coverage in all population age groups i.e., the newborn (those typically targeted by the Italian public health system for free vaccination against meningococci), the young, the adolescents as well as the adults [28]. On the other hand it is known that the concurring—worried—public health communications recommending avoidance of possible risky behavior especially among adolescents (including e.g., avoidance of exchanges of cigarettes, glasses, etc.) were also taken very seriously, resulting also in a reduction in the contact and transmission rates relevant for meninogocci transmission. However, we expect such effects to arise in many other circumstances. For example an ongoing measles epidemics might—at least at the local level—stimulate an upward pressure on vaccine coverage as well as protective behaviours such as e.g., not to send to schools non–vaccinated children. Motivated by these considerations, we propose, as a first step, a general SIR model for vaccine preventable endemic infections where individuals can respond to changes in their perceptions of risks by modulating both their propensity to vaccinate among newborn but also, and mainly, among older individuals, as well as their contact rate. This article is organised as follows. In Sect. 2 we review the behavior of SIR models of endemic infections for mandatory vs voluntary—behaviour dependent— vaccination. In Sect. 3 we discuss SIR models with behavior—dependent contact rates. In Sect. 4 we present and investigate our general SIR model with behavior responses in both vaccination propensity and contact patterns. Concluding remarks follow.

2 The SIR Model: Mandatory vs. Voluntary Vaccination 2.1 The Case of Constant Vaccination Coverage The basic SIR model for the control of endemic infections assumes vaccination at birth at constant coverage p, which is reminiscent of a situation where a mandatory immunization program exists. The resulting model is as follows: S  = μ (1 − p) − μS − β(t)SI

(1)

I  = I (β (t) S − (μ + ν))

(2)



R = μp + νI − μR

(3)

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where we denoted by S, I, R (S + I + R = 1, allowing to omit the third equation) the fractions of individuals who are, respectively, susceptible to acquiring infection, infective, i.e. able to retransmit infection to others, and removed because of e.g. immunity acquired after recovery. The infective fraction I is also called the infection prevalence. The function β(t) denotes the transmission rate which is typically timedependent. The other demo-epidemiological parameters are: μ > 0 which denotes both the birth and death rates, assumed identical to ensure that the population is stationary over time, and ν > 0 which is the rate of recovery from infection. In the most well-known case of constant transmission rate β(t) = β, the previous SIR model, which always admits a disease-free equilibrium point DF E = (1 − p, 0, p), has a simple threshold behavior depending on the interplay between the basic reproduction number "0 = β/(μ + ν) and the vaccine uptake p. Assuming that the basic reproduction number "0 exceeds one (ensuring a globally stable endemic equilibrium in absence of immunization) then, if the vaccine-reduced reproduction number "0 (1 − p) > 1 then the DFE is unstable and the infection continues to persist endemically about its endemic equilibrium, while if the vaccine coverage is large enough to ensure "0 (1 − p) ≤ 1 then the infection will be eliminated i.e., the DFE is globally attractive. The condition "0 (1 − p) < 1 can be rewritten as: p > pc , where pc = 1 −

1 "0

is the so called critical immunization coverage, which we also term the MayAnderson threshold [2]. Finally, if one also takes into account the presence of vaccination at ages older than birth S  = μ (1 − p) − μS − μφS − β(t)SI, where μφS is the vaccination rate of adults. the parameter φ is non–dimensionalised and, given the average lifespan L = 1/μ the average age at adult vaccination is L/φ. One can easily find the following disease elimination condition: "0

1−p < 1, 1+φ

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which also reads p+φ > pc . 1+φ

2.2 Voluntary Vaccination: A Phenomenological Model The hypothesis of constant p is clearly an approximation which roughly mirrored mandatory immunization systems enacted by many countries in the past, but it is no more valid under many recent scenarios. Consider for simplicity a voluntary vaccination system where parents take their decisions on whether to immunize or not their children based primarily on perceived costs and benefits of that immunization. These perceived costs and benefits in turn depend on the available information about the state of the infection—and related serious disease—and about the risks that are perceived to be connected with vaccination, i.e., suffering serious side effects from immunization. In such circumstances available information might feedback on the current vaccine uptake thereby affecting infection dynamics. For example, during epoch of low infection incidence individuals might perceive a quite high relative cost from vaccine side effects, therefore reducing their propensity to vaccinate, and the opposite during epidemic phases. Our behavior-implicit framework for information related immunization [10–13] considers the following SIR model for a non-fatal childhood infectious disease in a stationary homogeneously mixing population (we omit the R equation since R = 1 − S − I ): S  = μ (1 − p(M)) − μS − φ(M) − β(t)SI

(4)

I  = I (β (t) S − (μ + ν))

(5)

where the transmission rate β(t) > 0 is taken either constant or periodically varying with period θ equal to 1 year [2], while functions p(M) ≥ 0 and φ(M) ≥ 0 denote the vaccination coverage at birth and the coverage at subsequent immunizations respectively, both which are taken here to be increasing functions of a suitable information index M [11]. The index M summarizes the information on benefits and costs of immunization used by parents to take their vaccination decisions. Thus, M might be any function of the current, or past, infection prevalence (or incidence), taken as measures of the perceived cost of suffering infection or its serious sequelae, or of the prevalence (or incidence) of vaccine adverse events (VAE), taken as measures of the perceived cost of suffering VAEs. Here we focus on the perceived risk of infection (and related disease) as the driving force of immunization decisions, on the simplifying assumption that the perceived risk of vaccine adverse events is coarsely constant over time.

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The forms actually adopted for the vaccine uptake functions p(M) and φ(M) are such that, first of all p0 = p(0) > 0 and φ0 = φ(0) > 0 where the fixed components p0 and φ0 mirror the presence of a sub-population vaccinating independently of the state of information on infection and disease. Moreover, we assume that p1 (M) = p(M)−p0 and φ1 = φ(M)−φ0 are increasing functions mirroring, respectively, parents’ and adults’ reaction to increasing perceived risk from the disease. For example, taking M to be the current infection prevalence I , previous formulation amounts to state that when infection prevalence increases, people in the group influenced by information react by increasing their children and/or their own vaccine uptake, and vice-versa. Of course, for very large levels of M we assume p1 to saturate to some level p1sat ≤ 1 − p0 . A saturating level is not required for φ1 (M), although it is reasonable. The functions p1 and φ1 are continuous and differentiable, except at a finite number of points.

2.3 Modeling the Information Index M(t) The index M can be taken to represent a measure of the perceived risk following infection (including serious sequelae), which summarizes the way information on infection and its serious sequelae, and ensuing perceptions on benefits and costs (of measures to be adopted to reduce risks), affect perceptions about risks. We can assume that M is given by a continuous function ω(S, I ) with ∂I ω(S, I ) > 0. In particular, M might be any function of the current, or past, infection prevalence or incidence, e.g. of the form M = h(β(t)S(t)I (t)) (e.g. M = k∗ β(t)S(t)I (t) with k∗ > 0), or M = g(I ) (e.g. M = kI with k > 0). For example, if M = kI might define the perceived risk of serious disease as the product of the perceived risk of infection, given by some linear function of prevalence I , times the perceived risk of serious disease given infection. For the sake of simplicity since now on we shall deal with functions M = g(I ). Following the review in [18], this hypothesis amounts to build a prevalence-based model, as opposite to belief-based, where the use of information is global, i.e. homogeneously available to everyone, as opposite to local, as is the case for spatially structured models. More realistically M also depends on past values of state variables, as for many infections information typically becomes available only with a delay, due to a number of procedures (such as laboratory confirmations, reporting to public health

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authorities, and diffusion by the available channels), and moreover awareness in the population requires time. In this case M will take the more general form: t M(t) =

g(I (τ ))K(t − τ )dτ

(6)

−∞

where K is a probability density function called the delaying kernel [23]. As for function g(I ), we assume that g(0) = 0 and g  (I ) > 0. Here, besides the trivial kernel K(t) = δ(t), where δ is the Dirac function, yielding the unlagged case M(t) = g(I (t)), we consider two main types of delaying kernels, i.e. the well-known exponentially fading memory kernel K(t) = a exp(−at), with expectation < t > given by the fading time scale T = 1/a [23], and the kernel:   1 K(t) = e−t /T1 − e−t /T2 . (7) T1 − T2 The latter kernel, introduced in [16], represents a parsimonious way to model the effects of information handling by individuals, as it accounts for two sub-processes possibly occurring independently and at different time-scales: (1) formation and acquisition of information, with time-scale T1 , and (2) memory fading of acquired information, with time-scale T2 . Often the first process is much faster than the second. Note that if T1 ≈ 0 then K(t) ≈ (1/T2)e−t /T2 , i.e. K(t) collapses into an exponentially fading memory with time scale T2 . This kernel has expectation < t >= T1 + T2 and V ar(t) = T12 + T22 . Compared to the exponentially fading memory, which assigns maximum weight to current information—usually unavailable—this kernel satisfies K(0) = 0, mirroring negligible use of current information, as in the commonly used Erlang kernels of higher order [23]. However, unlike the latter kernels, which consider sub-processes having the same time scale, the kernel (7) is much more flexible. Note indeed that the first order Erlang kernel K(t) = a 2 t exp(−at), corresponds to (7) in the case where T1 = T2 . We term (7) the acquisition-fading kernel. As for Erlang kernels, also (7) is reducible to ordinary differential equations (ODEs). Under the exponentially fading memory the Eq. (6) reduces to the single ODE: M  = a(g(I ) − M)

(8)

Finally, under the acquisition-fading kernel (7) Eq. (6) reduces to the following pair of ODEs: M1 = a1 (g(I ) − M1 )

(9)

M2 = a2 (M1 − M2 ) ,

(10)

where a1 = 1/T1 , a2 = 1/T2 , M(t) = M2 (t).

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The Erlang first order kernel i.e., the exponentially fading memory, corresponds to the particular case a1 = a2 .

3 Behavior–Modulated Contact Rate In epidemic model, a key concept is the infection incidence, which represents the absolute number of new infection cases per unit of time. Although intuitive from the epidemiological point of view, the infection incidence constitutes a modelling challenge [5, 25]. For a generalized family of SIR model with one class of susceptible and one of infectious, denoting as X the total number of susceptible, Y the total number of infectious and N the total population size, and with J the incidence, generalizing Begon and coworkers [5] one has: J = X × C × π1 × π2

(11)

where: C is the average number of general contacts per time unit, π1 the probability that a contact is with an infectious subject and π2 is the probability that a contact with an infectious subject induces the infection of the susceptible subject. Clearly, π1 is a function of the state variables [5]: π1 = π1 (X, Y, N) whereas, in the classical epidemiology view C and π2 were considered constant. The product F oI = Cπ2 π1 (X, Y, N)

(12)

is the force of infection (FoI), and represents the per capita rate at which susceptible individuals acquire infection per unit of time. As far as the function π1 (X, Y, N) is concerned, for the SIR model the two most popular choices are: (1) the classical mass action law π1 = qY leading to the following force of infection; F oI = βY (2) the frequency dependent mass action law π1 = hY/N. leading to the following force of infection F oI = β

Y = βI N

In both cases the term β, called the transmission rate, is taken as a natural constant of human behavior.

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In the mathematical epidemiology literature, the first step beyond the Statistical mechanics paradigm yielding the first epidemiological model with behavioral change was the behavior-implicit, prevalence-dependent SIR epidemic model proposed by Capasso and Serio in the seventies [6]. In [6] the contact rate β, until then taken as constant, is allowed to be a decreasing function of infection prevalence I . This implies that the Force of Infection (FoI), in the case of ‘frequency dependent mass action law’ takes the following non-linear form [6]: F oI (I ) = β(I )I

(13)

with: β  (I ) < 0. The authors pointed out that, unlike standard mass action formulations, this could make the FoI to become a non-monotone function of the prevalence (e.g. if β(I ) = β0 (1 + hI 2 )-1 ). The authors motivated their formulation with the possibility of behavioral changes in response to the changing epidemiological conditions that appear as the epidemic out in the population. For example, during epochs in which disease prevalence is perceived to be high, also the risk of infection might be perceived as high, thereby inducing changes in individuals’ contact behavior to reduce risks, thereby ultimately affecting also the actual risk of getting infected. Today Capasso and Serio’s formulation would be classified as a behavior-implicit [4] formulation, to mirror the fact that behavior is embodied into the mathematical model in an implicit manner, i.e. via a nonlinear specification of the FoI possibly mirroring individuals responses to changing epidemiological conditions, rather than incorporating rules explicitly describing the agents’ behavior. Since [6], several other works have investigated epidemic models with a non-linear FoI [1, 22, 30]. With reference to the expression of the incidence rate (11) and of the generalized force of infection (12), we propose here the following form of FoI: )(M)π1 (X, Y, N) F oI = C(M)π2 (M)π1 (X, Y, N) = β

(14)

)(M) = C(M)π2 (M) and, as a consequence, where C  (M) < 0, π2 (M) < 0, β  ) β (M) < 0. In defining the generalized behaviour-dependent FoI of (14) we have taken into the account that: (1) behaviour can modify (with different patterns and intensity, of course) both the average number of contacts per time unit and the probability of getting the infection when in contact with an infectious; (2) behaviour is not only based on the knowledge of current stage of spread of the disease but also on the memory of past epidemic history. Finally, note that (11) and (14) can be easily extended to more complex models where multiple epidemic state variables and more complex patterns of transmission are considered.

3.1 Extending the Capasso–Serio Behavioral Model By noting that behavior changes in turn require changes in the individuals’ information endowment, in [9] we also attempted to generalise previous behavior-

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implicit models of endemic infections (such as e.g., measles), by representing the contact rate β along the same notion of information-dependent behavior we developed in [11] for the vaccination coverage. This led us to consider simple SIR dynamic models of recurrent endemic infections where the contact rate is a phenomenological function of an information index M sharing the above described characteristics, yielding the following FoI: F oI (M) = β(M)I,

(15)

where β  (M) < 0. This assumption yields the following SIR model with behaviourdependent contact rate [9]: S  = μ(1 − S) − β(M)I S 

I = β(M)I S − (μ + ν)I

(16) (17)

completed by Eq. (6), governing the dynamics of M, and by the balance equation of the removed fraction R(t): R(t) = 1 − S(t) − I (t).

4 A General SIR Model Embedding Behavioral Feedbacks on Both Vaccination Propensity and the Contact Rate As discussed in the Introduction, in the available behavioural epidemiology literature dealing with endemic, vaccine preventable, infectious diseases, the feedback that the awareness of changes in trends of infection prevalence—as modulated by information—might yield on behavior towards the disease, has been investigated separately i.e., either for its effects on vaccination coverage or for those on the contact rate. However, it is reasonable to expect that in many circumstances behavioral changes might involve both the propensity to vaccinate as well as the contact rate. Consistently, in this section we propose a new general SIR model to investigate the synergy between these two feedbacks, on the assumption that the information background on which decisions to switch to a different behaviour are taken is summarised by the same information index M. In particular we include in the model both prevalence-dependent vaccination of newborn as well of older individuals to allow the possibility, for older individuals who avoided vaccination at birth, to consider later vaccination during epochs of increasing perceived risks. The dependence of the behaviours on the prevalence is mediated by the information index M. This yields to the following model S  = μ(1 − p(M) − S) − μφ(M)S − β(M; t)I S 

I = β(M)I S − (μ + ν)I

(18) (19)

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to be complemented by a model linking the information index to the spread of the diseases and by R = 1 −S −I . We assume there that β is both a decreasing function of M and a constant or periodic function of the time. In the general case, thus, the integro-differential system (18)–(19)–(6) forms a family of models. By specific choices of the delay kernel K(τ ), a range of models can be derived from the general family of models (18)–(19)–(6). The results we previously obtained in [11] and [9] will thus become particular subcases of the more general results we now derive for model (18)–(19)–(6). In relation to the proposed new model there are two main substantive questions, i.e. (1) how perceptions of risks related to the disease might affect behaviordependent vaccination as well as contact behaviour, and how this in turn affects infection control, and (2) how behavior might affect the dynamical pattern of infection, e.g. by triggering oscillations.

4.1 Modelling Human Behavior and Its Implications for Infection Control We recall here the first question: ‘How perceptions of risks related to the disease might affect behavior-dependent vaccination as well as contact behaviour? And how this in turn affects infection control?’ We start noticing that the family of models (18)–(19)–(6) always admits the disease-free equilibrium (DFE):  DF E =

1 − p0 , 0, 0 1 + φ(0)

 (20)

The stability properties of the DFE are provided by the following theorem which holds regardless the actual form of the information index M Theorem 1 Under θ -periodic β(0, t), the DFE (20) of (18)–(19)–(6) is globally asymptotically stable (GAS) if: 1 − p0 1 1 Q= 1 + φ(0) μ + ν θ

θ β(0, u)du < 1. 0

If instead Q > 1, then the DFE is unstable. The proof of Theorem 1 is based on the fact that the differential inequality S  ≤ μ(1 − p(0) − (1 + φ(0))S) implies that asymptotically S(t) ≤ S∞ =

1 − p0 . 1 + φ(0)

(21)

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Thus, asymptotically I  ≤ I (β(0, t)S∞ − (μ + ν)). As a consequence if Q < 1 then I (t) → 0+ . Moreover, the linearization equation for I (t) by setting I = 0 + i + O(i 2 ) is i  = i(β(0, t)S∞ − (μν )) As a consequence: (1) Q < 1 guarantees both the local and global stability of the DFE; (2) Q > 1 implies the unstability of the DFE. Note that if β(0, t) is constant then condition (21) becomes the well-known one reported in Sect. 2 i.e., "0

1 − p0 ≤1 1 + φ(0)

where "0 = β(0)/(μ + ν) is the basic reproduction number of the SIR model for endemic infections [2, 7]. The interpretation of condition (21) follows from the proper understanding of Q. Quantity Q represents indeed the appropriate vaccine reproduction number computed in the correspondence of the baseline vaccine coverage for newborn (p0 ) and older individuals (φ(0)) respectively, and in presence of the normal social contact rate (β(0, t)) which are associated to situations of minimal perceived risk (M = 0). In particular, the previous result recall us that elimination turns out to be feasible only if the baseline risks conditions that are perceived under circumstances of minimal infection circulation (M = 0) are capable to stimulate an overall vaccination coverage (of both newborn and older individuals) already in excess of the critical threshold pc . Otherwise, elimination can never be achieved even if the overall uptake p(t) = p(M(t)) could temporarily reach values as high as 100% during epochs of high prevalence and therefore high perceived risks. Moreover, if ∂t β(M, t) = 0 and it holds: 1 − p0 "0 > 1 1 + φ(0)

(22)

the system has a unique endemic equilibrium EE = (Se , Ie , Me ), where Me = g(Ie ), Se =

1 K(Me ) "0

and Ie is the unique solution of the equation μ μ+ν

  1 (1 + φ(g(I )))K(g(I )) = I 1 − p(g(I )) − "0

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4.2 Behavioural Responses and Infection Dynamics We recall here the second main question namely, How behavior might affect the dynamical pattern of infection (e.g. by triggering oscillations)? As we showed in the previous section, the existence and stability of the DFE, as well as the existence and location of the endemic equilibrium hold for the general family of models, independently of the form of the delaying kernel. On the contrary, the stability properties of the endemic state critically depend on #(τ ). Thus, to answer the second substantive question it is necessary to consider specific models of the memory kernel. For the unlagged case M = g(I (t)), and under the assumption that ∂t β(M, t) = 0 is constant (meaning that, besides behavioural effects, we rule out other time effects, such as periodicities, on the contact rate) and p(M) and φ(M) are differentiable, it holds that: Theorem 2 Let condition (22)holds. Then the unique endemic state EE of system (18)–(19) in absence of delays is GAS in the positively invariant set: Ω

∗∗

, 1 − p0 = (S, I ) | S ≥ 0, I > 0, S + I ≤ 1, S ≤ . 1 + φ(0)

(23)

The proof follows by applying the Poincare-Bendixon theorem with weight function 1/I . The previous result indicates that inclusion of current information only is not sufficient to trigger oscillations. It is therefore interesting to look at whether the EE can be destabilised when agents base their decisions on past information as well. In the case of the exponentially fading kernel, we obtain the following threedimensional family of models: S  = μ(1 − p(M) − S − φ(M)) − β(M)I S

(24)

I  = β(M)I S − (μ + ν)I

(25)



M = a (g(I ) − M)

(26)

Computing the Jacobian matrix at the unique endemic equilibrium, and defining the following quantities: −b11 = ∂S S  = −μ(1 + φ(M)) − β(M)I < 0 −b12 = ∂I S  = −β(M)S < 0 σ = ∂M S  = −μp (M) − μφ  (M)S − β  (M)I S

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b21 = ∂S I  = β(M)I > 0 ∂I I  = 0 −b23 = ∂M I  = β  (M)I S < 0 ∂S M  = 0 ∂I M  = ag  (I ) > 0 ∂M M  = −a < 0 one gets a characteristic equation of the form c0 +

3

i 1 ci λ

= 0 where c3 = 1 and:

c2 = a + b11 > 0 c1 = a(b11 + g  (I )b23 ) + b12 b21 > 0   c0 = a b11 g  (I )b23 + (b12 g  (I ) − σ )b21 All the bhk are positive, and σ has no pre-defined sign in the general case. Thus if σ > σ ∗ = b12 g  (I ) +

b11 g  (I )b23 >0 b21

it can be c0 < 0 and the equilibrium point is unstable. If instead it is σ ∗ < 0 then from the Ruth-Hurwitz condition c1 c2 − c0 > 0 we have the following second-order inequality in a q2 a 2 + q1 a + q0 > 0 where: q2 = b11 + g  (I )b23 > 0 2 q1 = b21 (σ + b12 (1 − g  (I ))) + b11

q0 = b11 b12 b21 > 0 Thus, if Δ = q12 − 4q2 q0 ≤ 0

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then the endemic equilibrium is locally asymptotically stable, whereas if q1 < 0 and Δ > 0, i.e. if √ −q1 > 2 q2 q0 i.e. 9 2 b21(−σ − b12 (1 − g  (I ))) − b11 > 2 (b11 + g  (I )b23 ) b11 b12 b21

(27)

then there exist an interval (a1 , a2 ) such that if a ∈ (a1 , a2 ) EE is unstable and Yakubovitch oscillations arise through two Hopf bifurcations at a = a1 and at a = a2 ; if a < a1 and a > a2 then EE is locally stable. Specific subcases where only of the three rates p(M), φ(M) and β(M) was nonnull have been investigated in [9–14]. When only vaccination is present, the EE can be destabilized in the presence of an exponentially fading information memory. On the contrary, as shown in [9] for the scenario where human behavior only affects the contact rate β, the EE is locally stable for all a > 0 i.e., independently of the magnitude of the average information delay. Thus it is of interest to investigate the role of β  (M) in possibly stabilizing the EE. Although inequality (27) is quite complicate, taking into the account that the derivative of the contact rate appears in b23 = −β  (M)SI > 0 and in σ and by rewriting the latter as follows σ = −ω0 − β  (M)SI it yields 9 2 b21 (ω0 + β  (M)SI − b12 (1 − g  (I ))) − b11 > 2 (b11 − β  (M)SIg  (I )) b11 b12 b21 (28)

In other words, the presence of the negative term β  (M)SI decreases the l.h.s. of inequality (27) and the presence of the positive term −β  (M)SIg  (I ) increases its r.h.s. Overall, roughly speaking, this makes less likely the fulfillment of the inequality (27), i.e. less likely the onset of oscillations. In other words the reduction of risky behaviour as a response to perceptions of increasing risk from the disease has beneficial effects, at least as far as the onset of recurrent epidemics is concerned. Note that, focusing instead on the role of p(M) and of φ(M), one can read in the reversed direction the inequality (27) and conclude that the initiation of voluntary vaccination in a scenario of behavior-dependent contact rate will, on the one hand, contribute to reduce the average infection prevalence Ie (as it is easy to verify) but, on the other hand, may induce recurrent epidemics. As indeed noted in [11, 15] the recurrent oscillations induced by prevalence-dependent vaccination behaviour can be very complicate—even when purely periodical—as they have the potential to generate huge amplitude oscillations with extremely long periodicities, possibly very difficult to predict and handle in a real public health context.

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Finally, under the acquisition-decay kernel, one yields the following system: S  = μ(1 − p(M) − S − φ(M)S) − β(M)I S 

I = β(M)I S − (μ + ν)I M1 = a1 (g(I ) − M1 ) 

M = a2 (M1 − M) .

(29) (30) (31) (32)

Though in principle it is possible to analytically characterise the local stability of the endemic state for the above four-dimensional system, the problem becomes analytically cumbersome also for simple choices of p(M), φ(M), g(I ) and β(M). This is also true for the case a1 = a2 .

5 Concluding Remarks In the first part of this manuscript we have reviewed two main classes of behavioral epidemiology models. The first one is that of SIR models for vaccine preventable endemic infectious diseases with prevalence-dependent, behaviour-implicit, vaccine coverage at birth and among older individuals in a regime of voluntary vaccination. The second one is that of SIR models for endemic infections with prevalence-dependent contact rate. These classes of models [9, 11] replace critical epidemiological parameters typically taken as constant in basic models namely, the vaccine coverage and the contact rate, by general functions of available current and past information on the infection and related serious disease. The underlying idea is that human agents actively respond to changes observed in the (current or past) infection prevalence by adapting their immunization decisions and contact patterns. These models bring a number of relevant novelties compared to classic models [9, 11]. In the second part of the manuscript we developed a new general SIR model combining the two aforemention types of behavioral phenomena, previously considered only separately in the literature, into a single unified model. In this new model individuals respond to changes in the information indexes by continuously adapting (1) the propensity at which they immunize their children at birth, (2) the propensity at which they choose immunization at later ages, and (3) their contact rate, which summarises their behaviour at risk. This combination of behavioral responses, though rarely documented for vaccine preventable infectious diseases of childhood, has certainly occurred for example during the recent alert occurred in Tuscany during 2015–2016, when rates of invasive meningococcal disease and deaths dramatically increased [8, 28, 31]. The alert pushed the Regional public health system towards offering supplementary immunization in a wide range of age groups which was largely accepted by the population, while at the same a number of

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possibly risky behaviour surely declined, especially among adolescents and young adults [8, 28]. The resulting model was not designed to specifically describe the meningococcal disease alert in Tuscany, which would have required a more complicate model, but rather to develop a general phenomenological theory of the effects of behavioral responses within SIR models for endemic infections. The model allows to complete the picture offered in previous separate works, in particular suggesting the complicate interplay between different behavioral responses acting on different epidemiological parameters in triggering sustained oscillations of vaccine coverage, risky behavior, and infection prevalence. Acknowledgement We thank the Anonymous Referee that helped us to greatly improve this manuscript, and the Editors for their support and patience.

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PC-Based Sensitivity Analysis of the Basic Reproduction Number of Population and Epidemic Models Francesco Florian and Rossana Vermiglio

Abstract The basic reproduction number, simply denoted by R0 , plays a fundamental role in the analysis of population and epidemic models. However in mathematical modelling the specification of the input parameters can be crucial since, due to some limitations in experimental data available, they can be uncertain and often represented as random quantities in a suitable probabilistic framework. In this context the Polynomial Chaos Expansions (PCEs), coupled with suitable numerical methods, furnish important tools for the sensitivity analysis and the uncertainty quantification of the random model response. The aim of this paper is to describe how the variability of R0 is affected by the variability of the input parameters, through the evaluation of Sobol’ indices by PC-based methods. The use of a suitable and new computational model of R0 allows also to consider more complex epidemic models, where R0 is defined as the spectral radius of the infinite-diminensional next generation operator. The efficiency and versatility of the numerical approach are confirmed by the experimental analysis of two examples of increasing complexity.

The author “Rossana Vermiglio” was a member of the INdAM Research group GNCS. F. Florian University of Zürich, Institute of Mathematics, Zürich, Switzerland Member of CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy e-mail: [email protected] R. Vermiglio () CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_11

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Keywords Population dynamics · Epidemic models · Basic reproduction number · Global sensitivity analysis · Sobol’ indices · Analysis of variance · Polynomial chaos

1 Introduction In population dynamics the Malthusian parameter, which gives the exponential growth rate of the population, was the first mathematical tool used to investigate the asymptotic behaviour of populations, even if Matlthus himself has declared that “it can describe quite well the evolution only on short time” [23]. As of Ross’s monography on the malaria diffusion [26], an alternative approach has been proposed, which it is based on the “basic reproduction number” or simply R0 , which gives the expected number of newborns generated by a single typical individual during its entire life. In epidemiology, the focus is on the disease and so the “newborns” are the new infected individuals, “lifetime” is the infectious period and R0 is the number of secondary cases produced by a “typical” infected individual during its infectious period, assuming to consider a completely susceptible population. Nowadays R0 plays a significative role in epidemiology, since it directly relates to the amount of control effort needed to eliminate an infection from a population and the literature on this subject is quite large (see e.g. [1, 10, 11, 15, 16, 18, 19]). In the formulation of more sophisticated structured population models R0 is the spectral radius of the linear operator, called “next generation operator”, which is infinite-dimensional. So in many situations it cannot be explicitly calculated and it is necessary to elaborate suitable numerical methods to evaluate it. Despite the importance of the basic reproduction number to study the stability of populations, the numerical approximation of R0 has been only recently faced in [5, 13, 21], where, by means of suitable discretization techniques, the “next generation operator” is turned into the “next generation matrix”, whose special radius furnishes an approximation of R0 . In particular in [21] the author considers one-dimensional finite-element method with piecewise constant polynomials, whereas finite-element with higher degree polynomials and pseudospectral methods have been introduced and experimentally investigated in [5, 13]. Another crucial aspect in mathematical modelling concerns the specification of the input data, i.e. model constants and parameters, which, due to some limitations in experimental data available or inherent variability of the system under scrutiny, are often uncertain. The sensitivity analysis (SA) aims to study how the uncertainty in the output of the model (either mathematical or numerical) can be distributed to the different sources of the uncertainty of model input. In other words it determines which parameter or group of them is more important in influencing the output. Moreover SA can also allow the modellers to improve the quality of their model identifying the critical regions in the space of inputs and the unimportant parameters to reduce the dimension of the problem. A probabilistic framework is well suited to this purpose so that the input data and as a consequence the model output are

PC-Based Sensitivity Analysis of the Basic Reproduction Number of. . .

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often considered as random quantities on a suitable probability space. Nowadays numerous approaches are available to perform the SA and a good state-of-the-art of the techniques can be found in [20, 27]. Here we focus on the global SA (variance decomposition technique), which decomposes the variance of the model output in terms of variance contributions of each single input parameter or group of them by computing the so called Sobol’ indices [30–32]. A related problem is the uncertainty analysis, which focuses on the quantification of the effect of the propagation of the uncertainty of input data through the determination of the mean and the variance of the random model response, the quantiles of its distribution and confidence bounds. In this probabilistic context the Polynomial Chaos Expansions (PCE) are powerful tools to represent random variables and, in many situations, they are a valid alternative to Monte Carlo (MC) methods to perform uncertainty quantification (UQ) and SA of the random model solution. In fact, the implementation of MC methods is usually straight forward, but it provides estimates with limited accuracy and moreover it can be difficult to use when computationally expensive models are considered. On the contrary the use of PCEs allows one to exploit the possible regularity of the model solution with respect to the random parameters and to get more accurate results with less computation cost. Moreover, when the PCE of the model response is available, the mean, the variance and the Sobol’ indices can be analytically obtained from its expansion coefficients with limited costs. So the total computation cost is mainly determined by the evaluation of the PC coefficients. Nowadays the literature on the applications of PCEs for UQ and SA together with the numerical methods to efficiently compute the PC coefficients is quite wide. We select the monographs [22, 38] and the report [25], where the interested readers can find the basic results, the fundamental numerical approaches and a rich list of references. In particular for the PC-based approach for SA, we remind the papers [9, 22, 33]. Finally for the relevance of delay models in populations dynamics, we also recall that the uncertain delay differential equations have been recently introduced and both UQ and SA of the random stability indicator has been performed to study the effect of uncertainty on the stability of equilibria respectively in [35, 36]. In particular the PCE of the stability indicator is obtained by the stochastic collocation methods in [35], whereas in [36] the authors propose the non-intrusive spectral approach based on the Padua quadrature points as a new alternative to tensored Gaussian grids and Smolyak sparse grids [22, 38] when the uncertain parameters are limited. The aim of this paper is to show the efficiency and versatility of PC-based methods for the global SA of reproduction number R0 . It is no worthless to emphasise again that the determination of the model inputs as well as of the group of them which contribute most to the variance of R0 allows a better understanding of the model under scrutiny and to refine it. Moreover in the combination with an efficient computational model for R0 , SA can be extended to more complex population and epidemic models, for which the next generator operator is infinitedimensional. The paper is organised as follows. Section 2 focuses on the general definition of R0 and on the pseudospectral discretization of the next generation operator

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to approximate it. In Sect. 3 we summarise the fundamental results on PCEs as well as their application to the computation of Sobol’ indices. Section 4 concludes with two specific examples of increasing complexity, namely a system of ordinary differential equations (ODEs) describing the dynamics of influenza with vaccination and diffusion [29] and a partial differential equation (PDE) modelling the infective population with age-structure [21]. The numerical results are obtained by using the Matlab software available at UQLab [24] and the Matlab code for the approximation of the next generation operator [5, 13].

2 The Basic Reproduction Number In this section after giving a quick sketch of the definition of the basic reproduction number R0 , we briefly present the efficient and useful computational method for its evaluation recently proposed in [5, 13]. The basic idea is to turn the infinitedimensional next generator operator into a matrix by using the pseudospectral discretization technique and then compute its spectral radius. More details on the definition and computation of R0 can be found in [5, 11, 21].

2.1 Definition For many structured mathematical models in epidemiology, the linearization around an equilibrium can be written as the following linear abstract differential equation on a Banach space X I  (t) = BI (t) − M I (t), t ≥ 0,

(1)

where I (t) ∈ X represents the infective population at t ≥ 0, B : X → X is the linear, bounded and positive birth operator and M : D(M ) ⊂ X → X is the linear mortality operator. We assume that M is injective and that −M generates a C0 semigroup {e−M t }t ≥0 with σ (−M ) ⊂ C− . Given an initial condition I (0) = I0 ∈ D(M ) it can be easily seen that the solution satisfies the equation I (t) = e−M t I0 +



t

e−M (t −s)BI (s)ds, t ≥ 0.

(2)

0

The meaning of the basic reproduction number is the number of newborns generated by a single individual during the entire lifetime and so, following [11], it can be defined as the spectral radius of the operator K = BM −1 which is known as the next generation operator. In many situations K is compact. In this abstract setting R0 cannot usually be explicitly calculated and some numerical methods are needed.

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2.2 Approximation Given a discretization index N and a finite-dimensional space XN , we introduce the restriction operator RN : X → XN and the prolongation operator PN : XN → X, such that RN PN is the identity on XN , whereas PN RN φ furnishes a suitable approximation of φ ∈ X. Now we can define the finite-dimensional operator BN : XN → XN BN = RN BPN ,

(3)

and the finite-dimensional operator MN : DM ⊂ XN → XN MN = RN M PN ,

(4)

where DN = {ΦN ∈ XN : PN ΦN ∈ D(M )}. By solving the finite-dimensional eigenvalue problem BN ΦN = λMN ΦN ,

(5)

we construct an approximation R0,N of R0 . In the paper [21] the piecewise constant approximation has been proposed for φ ∈ X := L1 (0, a† ) and the convergence of R0,N to R0 has been analysed. In [13] finite-element with higher degree polynomials and pseudospectral methods have been introduced and experimentally investigated, whereas the convergence analysis is ongoing and out of the purpose of this paper. In what follows we conclude with a brief introduction of the pseudospectral 3 4 discretization for X := L1 (0, a† ) and D(M ) = φ ∈ X : φ  ∈ X, Cφ = 0 , with C : X → R a linear functional. By choosing in the interval [0, a†] the mesh ΔM = {a0 = 0 < a1 < · · · < aN = a† } of the Chebyshev extrema points, namely a† ai = 2

   iπ 1 − cos , N

i = 0, . . . N,

(6)

and defining XN := RN+1 , the restriction operator RN : X → XN results to be RN φ = (φ(a0 ), . . . , φ(aN )) ∈ XN , while the prolongation operator PN : XN → X N is PN ΦN = for (6). Note i=0 i ΦN,i , where i are the Lagrange polynomials N that PN RN is the interpolation operator and CPN ΦN = i=0 Ci ΦN,i = 0, ΦN = (ΦN,0 , . . . , ΦN,N ) ∈ XN . In this case we expect to obtain the spectral accuracy in the approximation of the eigenvalues as it happens for the infinitesimal generator of linear delay equations [2, 4] and age-structured population models [3].

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3 PC-Based Global Sensitivity Analysis In this section we set the probabilistic framework and we briefly introduce the PCEs along with their application to the computation of Sobol’ sensitivity indices. For easy notation we consider random variables only, but the same representations and approximations can be extended to complex-valued random variables, random vectors and stochastic processes.

3.1 PCEs Let (Ω, Σ, P) be a complete probability space, where Ω is the event space, Σ is the σ -algebra, and P is the probability measure. We denote by L2 (Ω, Σ, P) the Hilbert space of square-integrable random variables on it. The theory of Polynomial Chaos goes back to the pioneeristic works of N. Wiener [37] and R.H. Cameron W.T. Martin [7], according to which any a random variable Y ∈ L2 (Ω, Σ, P) can be expressed into a convergent series of polynomials in a sequence of Gaussian random variables {ξi }∞ i=1 of the form Y (ω) = y0 H0 +

∞  α1 =1

yα1 H1 (ξα1 (ω)) +

α1 ∞   α1 =1 α2 =1

yα1 ,α2 H2 (ξα1 (ω), ξα2 (ω)) + . . . , (7)

where Hj ’s are the multivariate Hermite polynomials of degree j, which are mutually orthogonal with regards to the Gaussian measure associated with {ξi }∞ i=1 [7, 37]. The series (7) is the PCE of Y and its deterministic coefficients, simply named PC coefficients, capture the probabilistic description of Y. In the applications only PCEs of finite dimension M can be considered. Here M is given by the sources of uncertainty in the model. Moreover for computational purposes it is also necessary to truncate the PCE to order P , so that it results also finite. Specifically all the infinite sums in (7) are replaced by sums over M dimensions with polynomials of degree less or equal to P . The resulting truncated expansion of order P and dimension M converges in the mean square sense as both M and P go to infinity [7]. Nowadays polynomial families that are orthogonal with respect to non Gaussian probability measures have been used to construct the socalled generalised PCEs. In particular Legendre polynomials are used for uniform distribution, Laguerre polynomials for gamma distribution, Jacobi polynomials for beta distribution, and so on. For convergence results of generalised PCEs we refer to [12] and to the books [14, 22, 38] for further details and references. Hereafter we consider finite-dimensional generalized PCEs, simply called PCEs. In what follows we briefly explain the tensored construction of the multidimensional polynomials as well as the numerical approximation of the expectation and variance of Y.

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Let ξ = (ξ1 , . . . , ξM ) be a vector of M independent standard random variables defined on (Ω, Σ, P) with joint probabilistic density function pξ , i.e. pξ (ξ ) =

M 

(8)

pi (ξi ),

i=1

where pi (ξi ) is the probability density function of ξi . We denote by Ξ the support of ξ and by L2 := L2 (Ξ, BΞ , pξ ) the Hilbert space of square-integrable random variables defined on the probability space (Ξ, BΞ , pξ ), where BΞ is the Borel set of Ξ. The inner product on L2 is given by  Y, Z :=

Y (ξ )Z(ξ )pξ (ξ )dξ, Ξ

√ and #Y # := Y, Y  is the induced norm. The random variable Y ∈ L2 (Ω, Σ, P) can be approximated by the truncated expansion YP given by  YP (ω) = yα Ψα (ξ(ω)), (9) α∈AP

where {Ψα (ξ )} are the multivariate orthogonal polynomials induced by the density function pξ of the random vector ξ , α = (α1 , . . . , αM ) ∈ NM 0 is a M-dimensional multi-index, # · # is a suitable norm on NM , and A = {α ∈ NM P 0 0 : ||α|| ≤ P }. M+P Let nP be the cardinality of AP . By using the || · ||1 we have that nP = ( ), P whereas np = (P + 1)M for the || · ||∞ . In both cases np rapidly increases with respect to the stochastic dimension M and the expansion degree P . Other truncation sets AP have been proposed in literature [22, 24, 38]. The polynomial basis {Ψα } is built via the tensor product of one-dimensional polynomials, i.e. Ψα (ξ ) = Ψα1 (ξ1 ) · · · ΨαM (ξM ), α ∈ NM 0 ,

(10)

where Ψαi is the univariate orthogonal polynomial w.r.t. pi (ξi ) of degree αi . Note that the tensored construction of the PC basis (10) allows to consider random variables ξi associated with different probability laws pi (ξi ). Now we can operate in the stochastic space L2 and approximate, for instance, the expectation and the variance of Y as follows 7 7  yα Ψα (ξ )pξ (ξ )dξ = y0 , E[Y ] = Y (ω)dP(ω) ≈ E[YP ] = α∈AP

Ω

V ar[Y ] = E[(Y

− E[Y ])2 ]

≈

V ar[YP ] =

Ξ



α∈AP ,α=0

|yα |2 #Ψα #2 . (11)

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We remark that the quantities #Ψα # are known for the classical families of orthogonal polynomials. PCEs coupled with suitable numerical methods to compute the PC coefficients provide a convenient framework to represent random variable and to analyse and quantify the propagation of data uncertainties in the model solution. Once the uncertain input data are considered as a random vector X with components in L2 (Ω, Σ, P), one can tackle the propagation of uncertainty induced by X into the random model response Y by determining its PC coefficients {yα }. Several numerical approaches have been proposed in the literature, which are usually classified into two categories: intrusive methods, which require ad hoc and usually heavy implementation, and non-intrusive methods, which make use of a set of model evaluations obtained from a suitable sampling of the input random variables. In the latter case, the computational cost is determined by the size of sample of the random variable Y . Among the class of non-intrusive spectral methods, we recall that the Padua points have been recently proposed as sample set and applied for SA of the random stability indicator of uncertain delay differential equations with small M[36].

3.2 Sobol’ Decomposition and PC-Based Sobol’ Indices In this subsection, after a brief introduction of the Sobol’ decompositions, we define the Sobol’ indices [30–32] and we show how the orthogonality property of the decompositions and the tensored construction of the PC basis (10) allow one to compute them analytically with almost no additional cost [9, 22, 33]. Without loss of generality, we assume that all the random variables ξi , i = 1, . . . , M, are uniformly distributed in [0, 1] and, therefore, that the support of ξ is Ξ = [0, 1]M . Generalized Sobol’ decompositions can be derived for more general independent variables [34] as well as for dependent ones [8]. Any square integrable function g in [0, 1]M admits a unique orthogonal decomposition, known as Sobol’ decomposition, of the form g(ξ ) =

 σ ⊆{1,...,M}

gσ (ξσ ),

(12)

where σ = {i1 , . . . , im }, ξσ = {ξi1 , . . . , ξim } and g∅ = E[g(ξ )]. The integral of each gσ over any of its independent variables is zero, i.e. 1 gσ (ξσ )dξj = 0, j ∈ σ. 0

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Moreover all the summands of the expansion (12) can be computed recursively as follows: g∅ = gσ (xσ ) =

71

71

g(ξ )dξ,

0

71  . . . g(ξ )dξ∼σ − gσ  (ξσ  ),

0

σ  ⊂σ

0

where the notation ∼ σ indicates that the elements of σ are excluded, i.e. ∼ σ = {1, . . . , M} \ σ. Due to the orthogonality nature of (12), the total variance of Y = g(ξ ) can be decomposed as 

V ar[Y ] =

V arσ [Y ]

σ ⊆{1,...,M},σ =∅

where V arσ [Y ] = gσ , gσ  is the contribution to V ar[Y ] of the interaction of the set of random parameters {ξi : i ∈ σ }. To compare the impact of the different sources of uncertainty the Sobol’ sensitivity indices can be defined as follows Sσ =

V arσ [Y ] , V ar[Y ]

σ ⊆ {1, . . . , M}.

Each Sσ measures the amount of the total variance V ar[Y ] due to the uncertainties in ξσ . It is often convenient to consider also the i-th total sensitivity index, which is the  sum of all the sensitivity indexes to which the parameter ξi contributes, i.e. Ti = Sσ . i∈σ

We now link the Sobol’ decompositions and the PCEs, to show how we can approximate the Sobol’ indices by using the truncated PCE YP of Y (9). From (10), we get that the decomposition of the chaos polynomial Ψα involves a single term, namely Ψα (ξ ) = Ψσα (ξσα ), σα = {i : αi = 0}. For instance we have that gσ (ξσ ) =



yα Ψα (ξ )

α∈Aσ

/ σ }. Therefore (9) can be where Aσ := {α ∈ NM 0 : αi > 0, i ∈ σ, αi = 0, i ∈ recast as follows   YP = y0 + y α Ψα , σ ⊆{1,...,M},σ =∅ α∈Aσ

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and V ar[Y ] ≈ V ar[YP ] = V arσ [Y ] ≈ V arσ [YP ] =

 α∈AP ,α=0



α∈Aσ ,α=0

yα2 #Ψα #2 , yα2 #Ψα #2

.

(13)

From (13), we can conclude that the computation cost of the PC-based approach for SA is due to the evaluation of the PC coefficients of a suitable approximation YP of Y (see the end of Sect. 3.1).

4 Examples Aim of this section is to show the utility, the efficiency and the versatility of the PC-based approach in Sect. 3 to perform the global SA of the basic reproduction number, through the analysis of two test problems with increasing complexity. First we consider a system of ordinary differential equations, describing the influenza epidemic model with vaccination and diffusion [29], for which the explicit formula of R0 is available. Second we consider the age-structured epidemic model in [21], which is a linear partial differential equation. In the latter case we need to use the computational model for R0 , which is based on the psuedospectral discretization. The numerical simulations make use of the Matlab software in UQLab [24] and of the code for the approximation of R0 in [13]. We remind that in UQLab only nonintrusive methods are used, so that the total computation cost is mainly due to the model evaluations on the sample of the input random variables.

4.1 Influenza Epidemic Model with Vaccination [28, 29] As first example, we consider a system of nonlinear ordinary differential equations, modelling the dynamics of influenza [28, 29]. The model reads as follows ⎧ S  (t ) = −ββE S(t )E(t ) − ββI S(t )I (t ) + αS(t )I (t ) − rS(t ) + δR(t ) + θ V (t )− φS(t ) + r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V  (t ) = −ββE βV V (t )E(t ) − ββI βV V (t )I (t ) + αV (t )I (t ) − rV (t ) − θ V (t ) + φS(t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ E  (t ) = −(r + k + σ )E(t ) + ββE S(t )E(t ) + ββE βV V (t )E(t ) + αE(t )I (t ) ⎪ +ββI S(t )I (t ) + ββI βV V (t )I (t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I  (t ) = αI (t )2 − (r + α + γ )I (t ) + σ E(t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R  (t ) = −(r + δ)R(t ) + kE(t ) + γ I (t ) + αR(t )I (t ) ⎩

(14)

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where S, V , E, I and R denote the dimensionless proportions w.r.t. the total population of susceptible, vaccinated, exposed, infectious and recovered individuals respectively. Note that S + V + E + I + R = 1. The biological meaning of the parameters are given in Tables 1 and 2. By linearizing (14) around the disease-free φ r+θ equilibrium ( r+θ+φ , r+θ+φ , 0, 0, 0), some easy computations give the following explicit formula for R0 R0 =

β(r + θ + βV Φ)(rβE + αβE + γβE + σβI ) (r + α + γ )(r + σ + γ )(r + θ + φ)

Here R0 represents the average number of secondary infection cases produced by an infectious individual during its entire infectious period and it is given by the spectral radius of the next generation matrix. First, under the assumptions on the uncertain parameters given in Table 2 and by choosing the remaining ones as in Table 1, we compare the performance of PC-based methods in Sect. 3 and MC methods in computing the first order Sobol’ indices. The results in Table 3 (see also Fig. 1 for a visual overview) show that both methods identify that, among the sources of uncertainties, the contact rate β has the larger impact on the total variance of R0 of the model (14). Then the Table 1 Description of the parameters for the model (14) and their values [29] Parameter βI σ −1 γ −1 δ −1 μ r k α θ −1

Description Ability to cause infection by infectious individuals Mean duration of latency Mean recovery time for clinically ill Duration of immunity loss Natural mortality rate Birth rate Recovery rate of latents Flu induced mortality rate Duration of vaccine-induced immunity loss

Value (unit day where applicable) 1 2 5 365 5.500e+08 7.140e–05 1.857e–04 9.300e–06 365

Table 2 Description of uncertain parameters for the model (14) and their distributions [29] Parameter β

Description Contact rate

βE 1 − βV

Ability to cause infection by exposed individuals Vaccine efficacy

φ

Rate of vaccination

Distribution Gaussian distribution with mean 0.514 and standard deviation 0.1 Triangular distribution with minimum 0.00, maximum 0.33 and mode 0.25 Triangular distribution with minimum 70%, maximum 70% and mode 80% Triangular distribution with minimum 0.001, maximum 0.003 and mode 0.002

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Table 3 Sobol’ indices of the uncertain parameters β, βE , βV , φ for the model (14) computed with different methods Method MC MC PCE 5 PCE 5 PCE 10 PCE 10

Sample size 104 106 200 300 200 300

Sβ

SβE

SβV

Sφ

8.799735e–01 8.798158e–01 8.798823e–01 8.798816e–01 8.798999e–01 8.798824e–01

1.524643e–02 1.577289e–02 1.583885e–02 1.583860e–02 1.584269e–02 1.583854e–02

1.631913e–02 1.475846e–02 1.457150e–02 1.457163e–02 1.456267e–02 1.457187e–02

8.151178e–02 8.543602e–02 8.503733e–02 8.503806e–02 8.503266e–02 8.503723e–02

Fig. 1 First order Sobol’ indices of the uncertain parameters β, βE , βV , φ for the model (14) computed with different methods

rate of vaccination φ follows, while the uncertainty in the parameters βE , βV have weaker influence. But, from the point of view of the computation cost, the PC-based methods are superior since they furnish accurate estimates by using smaller sample sizes, confirming to be a valid and efficient alternative to MC methods. As second step we complete the SA of R0 by computing the Sobol’ indices of all orders and the total indices, but, accordingly to the above considerations, in Table 4 we show only the values furnished by the PCE of order 5 with 200 realizations. The results in Table 4 indicate that the Sobol’ indices whose subsets contain both β, φ have the larger values. Moreover the first order Sobol’ indices in Table 3 computed with PCE of order 5 and 200 realizations and the total indices in Table 4 have similar values. The great importance of the effect of β is confirmed and the analysis of the total indices gives Tβ ≈ 88%, whereas Tφ ≈ 9%, and the influence of the parameters βV and βE are equivalent and negligible.

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Table 4 Sobol’ indices of different orders and the total indices of the uncertain parameters β, βE , βV , φ for the model (14) computed with PCE 5 with 200 realizations Order 2 Value Sβ,βE 5.9956e–04 Sβ,βV 5.5159e–04 Sβ,φ 3.2184e–03 SβE ,βV 9.9279e–06 SβE ,φ 5.7927e–05 SβV ,φ 2.2143e–04

Order 3 Sβ,βE ,βV Sβ,βE ,φ Sβ,βV ,φ SβE ,βV ,φ

Value Order 4 3.7918e–07 Sβ,βE ,βV ,φ 2.1786e–06 8.4108e–06 1.5168e–07

Value Total Value 5.6047e–09 Tβ 8.8426e–01 TβE 1.6509e–02 TβV 1.5363e–02 Tφ 8.8546e–02

4.2 Age-Structured Epidemic Model [21] We consider the following linear equation for the infective population resulting from the linearization around the disease-free steady state  a ⎧ ⎨ (∂ + ∂ )i(t, a) = S 0 (a) †β(a, σ )i(t, σ )dσ − (μ(a)+γ (a))i(t, a), t > 0, a ∈ (0, a ) t a † 0 ⎩ i(t, 0) = 0, t > 0,

(15) where i(t, a) is the infective population of age a at time t, S 0 is the total population, i.e. the susceptible population in the disease free state, a† is the maximum age, β is the transmission coefficient, μ is the human mortality and γ is the recovery rate. The vertical transmission is excluded since i(t, 0) = 0, t > 0. Let X := L1 (0, a† ). By defining I (t) := i(t, ·) ∈ X, t ≥ 0, Eq. (15) can be recast in the abstract form (1), where M : D(M ) ⊂ X → X is M ϕ(a) = ϕ  (a) + (μ + γ )(a)ϕ(a)

(16)

3 4 D(M ) = φ ∈ X : φ  ∈ X, φ(0) = 0 ,

(17)

with

and B : X → X is 

a†

0

Bφ(a) = S (a)

β(a, σ )φ(σ )dσ.

(18)

0

So that the next generation operator K reads as K φ(a) = BM −1 φ(a) = S 0 (a)





a†

β(a, σ ) 0

σ

e−

7σ ρ

(μ+γ )(η)dη

φ(ρ)dρdσ,

0

(19)

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which is compact under the additional assumption [17, Assumption 4.4]  lim

h→0 0

a†

|S 0 (a + h)β(a + h, σ ) − S 0 (a)β(a, σ )|da = 0, uniformly for σ ∈ R,

where the domain of S 0 (a)β(a, ·) is extended by S 0 (a)β(a, ·) = 0 for all a ∈ R \ [0, a† ], by applying the Kolmogorov-Riesz-Fréchet theorem in [6]. In particular the next generation operator (19) is compact, when S0 , β, μ and γ continuous, strictly positive and uniformly bounded functions on [0, a† ] as considered in [21]. To approximate R0 , which is the spectral radius of K , we now construct the pseudospectral discretization of M and B. Given N ∈ N,, we consider the extrema Chebyshev nodes (6), the associated Lagrange polynomials j , j = 0, . . . , N and the weights bj , j = 0, . . . , N of the Clenshaw-Curtis quadrature formula. Taking into account the domain condition in (17), we obtain the following N-square matrices ⎞ ⎛  N (a1 ) 1 (a1 ) − (μ + γ )(a1 ) . . . ⎟ ⎜ .. .. .. (20) MN = ⎝ ⎠ . . . 1 (aN )

. . . N (aN ) − (μ + γ )(aN )

and ⎞ S 0 (a1 )b1 β(a1 , a1 ) . . . S 0 (a1 )bN β(a1 , aN ) ⎟ ⎜ .. .. .. BN = ⎝ ⎠. . . . 0 0 S (aN )b1 β(aN , a1 ) . . . S (an )bN β(aN , aN ) ⎛

(21)

By solving the matrix eigenvalue problem (5) with MN and BN given in (20) and (21) respectively, we compute the approximation R0,N of R0 . When the parameters in (16) and (18) are uncertain and represented by random variables, also R0,N is a random variable. By proceeding along the same line in [35], we can prove that R0,N is a square-integrable function and so it admits a PCE. By using its PC coefficients, we can evaluate the Sobol’ indices and perform the SA. In what follows we choose N = 20 and, in accordance with [13, 21], we assume S 0 (a) = s0 + s1 a + s2 a 2 , β(a, σ ) = 3 ∗ 10−5 (β1 ∗ 10−7 (102 − (a − σ )2 ) + β2 ∗ 10−3 ), γ (a) = γ 1 μ(a) = μμ 2 −a

(22)

where the parameters are given in Table 5. Moreover, in order to reduce the execution time we consider a† = 1 and we omit the MC simulations. The results in Table 6 confirm again the efficiency of PC-based methods (see also Fig. 2). The Sobol’ indices of different orders and the total indices shown in Table 7 are

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Table 5 Parameters’ values and distributions in (22) for the model (15) Parameter μ1 (dimensionless) μ2 (year) s0 (dimensionless)

Value 0.3675 110 5187

s1 (year−1 ) s2 (year−2 )

226.438 2.77

Parameter β1 (dimensionless) β2 (dimensionless) γ (year−1 )

Distribution Uniform distribution in [5, 7] Uniform distribution in [0.5, 1.5] Gaussian distribution with mean 52 and variance 10

Table 6 First order Sobol’ indices of the uncertain parameters β1 , β2 , γ for the model (15) computed with different methods Method PCE 5 PCE 5 PCE 10 PCE 10

Sample size 200 300 200 300

Sβ1

Sβ2

Sγ

1.237710e–01 1.237220e–01 1.258303e–01 1.235116e–01

3.075348e–02 3.103270e–02 3.095852e–02 3.086210e–02

8.386314e–01 8.380924e–01 8.323927e–01 8.386063e–01

Fig. 2 First order Sobol’indices of the uncertain parameters β1 , β2 , γ for the model (15) computed with different methods

Table 7 Sobol’ indices of different orders and the total indices of the uncertain parameters β1 , β2 , γ for the model (15) computed with PCE of order 5 and 200 realizations Order 2 Sβ1 ,β2 Sβ1 ,γ Sβ2 ,γ

Value 0.0000e+00 5.5737e–03 1.2702e–03

Order 3 Sβ1 ,β2 ,γ ,

Value 1.2816e–07

Total Tβ1 Tβ2 Tγ

Value 1.2934e–01 3.2023e–02 8.4547e–01

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computed with PCE 5 with 200 realizations. The results in both Tables 6 and 7 clearly indicate that the recovery rate γ is the more important parameter, followed by β1 , whereas β2 is almost irrelevant. Note that the pairs (β1 , β2 ) doesn’t play any role and that the effect of the combinations of the parameters is not significant. Indeed also in this case the first order and the total indices have similar values.

5 Conclusions In this paper, by combining the PC-based methods and the pseudospectral discretization of the next generator operator, when needed, we have proposed an efficient numerical tool for performing the global SA of the basic reproduction number also for more complex population and epidemic models. The identification of the more significative parameters allows a better understanding of the model under scrutiny and offers the possibility to refine it or to implement a suitable control strategy. The numerical approach can be extend to other relevant quantities in mathematical modelling.

References 1. Barril, C., Calsina, A., Ripoll, J.: A practical approach to R0 in continuous-time ecological models. Math. Methods Appl. Sci. 41(18), 8432–8445 (2017) 2. Breda, D., Maset, S., Vermiglio, R.: Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM J. Sci. Comput. 27(2), 482–495 (2005) 3. Breda, D., Maset, S., Iannelli, M., Vermiglio, R.: Stability analysis of the Gurtin-MacCamy model. SIAM J. Numer. Anal. 46, 980–995 (2008) 4. Breda, D., Maset, S., Vermiglio, R.: Stability of Linear Delay Differential Equations. A Numerical Approach with MATLAB. SpringerBriefs in Electrical and Computer Engineering. Springer, New York (2015) 5. Breda, D., Florian, F., Ripoll, J., Vermiglio, R.: Efficient numerical computation of the basic reproduction number for structured populations. Int. J. Non. Sci. Num. Sim. (2019) 6. Breziz, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011) 7. Cameron, R.H., Martin, W.T.: The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals. Ann. Math. Sec. Ser. 48(2), 385–392 (1947) 8. Chastaing, G., Gamboa, F., Prieur, C.: Generalized Hoeffding-Sobol decomposition for dependent variables - application to sensitivity analysis. Electron. J. Stat. 6, 2420–2448 (2012) 9. Crestaux, T., Le Maître, O., Martinez, J.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94, 1161–1172 (2009) 10. Cushing, J.M., Diekmann, O.: The many guises of R0 . J. Theor. Biol. 404 295–302 (2016) 11. Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990) 12. Ernst, O.G., Muglera, A., Starkloff, H.-J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM: Math. Model. Numer. Anal.. 46(2), 317–339 (2012)

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Linear Dynamics of mRNA Expression and Hormone Concentration Levels in Primary Cultures of Bovine Granulosa Cells Malgorzata J. Wieteska, John A. Hession, Katarzyna K. Piotrowska-Tomala, Agnieszka Jonczyk, Pawel Kordowitzki, Karolina Lukasik, Dariusz J. Skarzynski, and Leo Creedon

Abstract Gene Regulatory Matrices (GRMs) are a well known technique for modelling the interactions between genes. This technique is used here, but with genes and hormones, to create Gene and Hormone Regulatory Matrices (GHRMs). In addition, a network (a directed weighted graph) is constructed from the underlying interactions of several mRNA encoding enzymes and receptors and two hormones: estradiol (E2) and progesterone (P4). This also permits comparison of the impact of each given environmental condition on E2 and P4 production, as well as mRNA expression levels. Apart from differential equations techniques (which require knowledge of rates of decay of a given hormone and mRNA) there is no existing technique to accurately predict the concentration of hormones based on the concentration of mRNA. This novel approach using GHRMs permits the use of nodes to accurately model the concentrations of the remaining ones. Experiments were performed to collect data on the gene expression and hormone concentration levels for primary bovine granulosa cells under different treatments. This data was used to build the GHRM models. Keywords Gene expression · Bovine · Regulatory matrices · Hormone

M. J. Wieteska · J. A. Hession · L. Creedon () Mathematical Modelling Research Group, Institute of Technology Sligo, Sligo, Ireland e-mail: [email protected] K. K. Piotrowska-Tomala · A. Jonczyk · P. Kordowitzki · K. Lukasik · D. J. Skarzynski Institute of Animal Reproduction and Food Research, Polish Academy of Sciences, Olsztyn, Poland © Springer Nature Switzerland AG 2020 M. Aguiar et al. (eds.), Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Springer Series 21, https://doi.org/10.1007/978-3-030-41120-6_12

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1 Introduction Bovine ovarian follicles contain a layer of granulosa cells (usually several cells thick) which produce two hormones: estradiol (E2) and progesterone (P4). E2 and P4 are among the main hormones responsible for regulation of the growth and decay (atresia) of the follicles and for ovulation. The information required for the production of these two hormones is communicated through mRNA encoding enzymes (genes) involved in the steroidogenesis pathway [11, 14]. To illustrate the interactions between expression values of assessed genes and the concentration of E2 and P4 the networks based on Gene and Hormone Regulatory Matrices were created. A genetic network can be defined as a weighted directed graph consisting of nodes representing genes/hormones, and of edges representing the impact of one node on another node. The interactions can be described using trigonometric, polynomial or linear functions [15]. We have used the linear approach to model the dynamics of mRNA and hormone concentrations in the granulosa cells. Gene Regulatory Matrices (GRMs) are a well known technique for modelling the interactions (promotion and inhibition) between genes. This assumes that the process is Markovian. This technique is used here, but with genes and hormones, to create Gene and Hormone Regulatory Matrices (GHRMs). This is the first time that we are aware that the concept of gene regulatory matrices (GRMs) has been applied to genes and hormones. This novel approach using GHRMs permits the use of three nodes/variables to accurately model the concentration of the remaining ones over 24 hours. This permits comparison of the strength of the impact of each of the three genes (or hormones) on the concentration of the genes responsible for the atresia of the follicles (BAX, BCL2), the gene directly responsible for progesterone production (HSD 3B2) and genes directly responsible for estradiol production (HSD 17B1, CYP 19A1). This also permits comparison of the impact of each given environmental condition on E2 and P4 production. This will also help to understand the role of IGF-1 in twinning since IGF-1 peripheral concentration is about 47% higher in twinning cows compared to control [4]. Apart from differential equations techniques (which require knowledge of rates of decay of a given hormone and mRNA) there is no existing technique to accurately predict the concentrations of hormones based on the concentration of mRNA. These differential equations are difficult to construct due to the different half-lives of mRNA and protein [1, 10]. The new Gene and Hormone Regulatory Matrix (GHRM) approach overcomes this problem as it can predict the concentration of genes and hormones at the next time point and model the interactions between genes and hormones. We have used only 3 nodes at a time to predict the concentrations of two hormones and 11 genes. This research investigates the possibility of predicting hormonal concentration and mRNA expression of chosen genes based on their initial values. Moreover, experiments were performed to change the environment of the cells (different treatments) by addition of hormones. These experiments show the metabolic shift of the cells (as the chosen supplements are utilized by the cells).

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This paper also aims to assess the impact of insulin-like growth factor 1 (IGF1) and follicle stimulating hormone (FSH) on the production of estradiol (E2) and progesterone (P4) and their effect on atresia (follicle death).

2 Experimental Materials and Methods Several cows of different breeds and ages were slaughtered in an abattoir and the ovaries were harvested and transported to the laboratory on dry ice. Then the follicles were isolated from the ovaries and the granulosa cells were isolated from the follicles. Follicles were divided into two classes according to their diameter: secondary (5 − 8mm) and dominant (> 8mm). At around 8mm a shift occurs from FSH dependent growth (for follicles < 8mm) to LH dependent phase of follicle growth (for follicles > 8mm) [9]. Granulosa cells from dominant follicles were pooled together and granulosa cells from secondary follicles were pooled together. This led to two cell cultures, one for dominant follicles and one for secondary follicles. Bovine granulosa cell cultures were performed according to the modified procedure developed by Gutierrez et al. [6]. Cells were cultured in culture medium for approximately 24h (dominant) and 48h (secondary follicles) in a collagen coated 1 × 48 well (in both classes of cultures for assessing hormone concentrations) and a 3 × 6 well (for dominant follicles) or a 4 × 6 well (for secondary follicles) for assessing mRNA expressions. The cells were washed and resuspended in culture media with respective stimulators/treatments labelled as described in Table 1. The media were collected every 4h and genes (mRNA) were isolated with Fenozol every 8h for 24h and all samples were stored at −80◦ C until use. Reverse Transcription was carried out using Reverse Transcriptase Kit in thermocycler (Biorad PCR). Real-time Polymerase Chain Reaction (PCR) analysis used Power SYBER Green Master Mix and specific designed primers (Table 2). Relative mRNA

Table 1 Treatments labels of cell cultures of dominant and secondary follicles Dominant follicles D1 D2 D3 D4

D5 D6

Secondary follicles S1 S2 S3 S4 S5 S6 S7 S8

Treatment Culture medium only 25ng/ml IGF-1 50ng/ml IGF-1 100ng/ml IGF-1 50ng/ml IGF-1 + 25ng/ml FSH 100ng/ml IGF-1 + 25ng/ml FSH 25ng/ml FSH 50ng/ml FSH 50ng/ml IGF-1 + 2ng/ml LH 100ng/ml IGF-1 + 2ng/ml LH

Gene CYP11A1 CYP10A1 BCL2 BAX IGF1R FSHR STAR HSD3B2 LHR HSD17B1 RIPK3 GAPDH ACTB

Primer sequence 5’GGGGACATCAAGGAAGACCT’3 5’ATGAAGGTCGTCCTGGTCAC’3 5’GAGTTCGGAGGGGTCATGTG’3 5’GTGCCCGAGTTGATCAGGAC’3 5’GATCCCGTGTTCTTCTACGTTC’3 5’GCCAAGTCAACTTACCGCTT’3 5’GGTGGTGGCACGTTTTCAAT’3 5’CTAATGGGTGGGCTCTGAAA’3 5’AATGGGACAACGCTGATTTC’3 5’ACACACCTGACCTTTGAACCAA’3 5’CCAGAGAGAGCAGGTTCCAC’3 5’CACCCTCAAGATTGTCAGCA’3 5’GAGGATCTTCATGAGGTAGTCTGTCAGG’3

Table 2 Gene-specific primers used for amplification Primer sequence 5’TGAGCAGAGGGACACTGGT’3 5’CTCCTGCCTAGCTGACAACC’3 5’GCCTTCAGAGACAGCCAGGA’3 5’CCATGTGGGTGTCCCAAAGT’3 5’AAGCCTCCCACTATCAACAGAA’3 5’TGACCCCTAGCCTGAGTCAT’3 5’CCTTGTCCGCATTCTCTTGG’3 5’CACGCTGTTGGAAAGAGTCA’3 5’GGCCTGCAGTTTAGTGGAAG’3 5’CTCTGTCCCAGCAAGCCTCT’3 5’AATCAGGCGGTTGTTGTTTC’3 5’GGTCATAAGTSCCTCCACGA’3 5’CAACTGGGACGACATGGAGAAGATCTGGCA’3

Size of PCR product 160 bp 326 bp 203 bp 126 bp 101 bp 193 bp 79 bp 473 bp 126 bp 110 bp 219 bp 103 bp 349 bp

GenBank BC133389.1 NM174305 U92434.1 U9292569.1 X54980 NM174061 Y17259.1 NM174343 NM174381 NM001102365.1 NM001101884.2 BC102589 AY141970

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quantification data was then analysed with the Real-time PCR Miner algorithm. P4 and E2 concentrations were analysed using radioimmunoassay (RIA). The levels of the following eleven genes and two hormones were assessed at each of the 4 times: Genes responsible for the atresia of the follicles: BAX, BCL2, RIPK3; gene directly responsible for progesterone production: HSD 3B2; and genes directly responsible for estradiol production: HSD 17B1, CYP 19A1, STAR, CYP11A1. Furthermore, receptors: IGF1r, LHr, FSHR; and hormones: E2, P4 were also assessed. There were 13 genes and hormones whose expression and concentration was measured at 4 different times, giving a 13 × 4 array for each of the 14 treatments.

2.1 Datasets The cell cultures of dominant and secondary follicles originated from the same cows (the same ovaries) and were collected at the same time. This allowed the comparison of the characteristics of GHRMs of those 2 classes of follicles. The experimental procedure gives us three datasets. 1) The first dataset (labeled D), contains data from the cell culture of dominant follicles, from 6 different treatments (subdatasets labeled D1 to D6). 2) The second dataset (labeled S), contains data from the cell culture of secondary follicles, from 8 different treatments (subdatasets labeled S1 to S8). 3) The third dataset (labeled Dtest), contains data from the cell culture of dominant follicles, from 6 different treatments. This is similar to dataset D, but data was collected from cultures of different granulosa cells (from different ovaries). These subdatasets are labeled Dtest1 to Dtest6. The created GHRMs were built based on the subdatasets (D1 to D6) of group D and their performance was assessed on the results observed for group D and also on corresponding treatments of the subdatasets from group Dtest. In addition GHRMs were built based on the raw data of Group S and their performance was assessed on the results observed for group S. A logarithmic normalisation was performed and GHRMs were built for raw data, log2 normalised data of Group D and for log2 normalised Group S data. The comparison between raw and log2 transformed data for 5 genes at the times recorded in group S3 is shown in Fig. 1.

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unnormalised data 0.4

BCL2 CYP19A1 HSD3B2 IGF1R BAX

0.35 0.3

0

BCL2 CYP19A1 HSD3B2 IGF1R BAX

–2 –4

0.25

–6

0.2 0.15

–8

0.1 –10

0.05 0

0

5

10

15

20

25

–12

0

5

10

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20

25

Fig. 1 Unnormalised and normalised mRNA expression values for treatment S3

3 Regulatory Function Approach: Gene and Hormone Regulatory Matrices (GHRMs)—Effect of Treatment on the Strength of the Interactions We have used a system of linear equations to describe the level of gene expression / hormone concentration at given time points. These equations are determined by the constant matrix GH RM describing interactions between respective genes / hormones and the vector Et which is a vector of the (log2 normalised) expression values at time t. In the literature there are many examples using Et +1 = MEt + Et to describe dynamical systems of gene expression [2, 3, 5], where M is an (n × n) constant matrix and E is the (n × 1) vector of the expression levels of a given gene at a given time. This assumes that the process is Markovian. A similar approach was taken in this paper and the matrix GH RM is defined to be an approximate solution of Et +8 − Et = GH RM Et at all times t (in hours), where GH RM is a constant matrix of size 13 × 13 and the Et are (13 × 1) vectors of expressions/concentrations. GH RM entries eij can be regarded as weights of the interactions. An approximate matrix for GH RM was found using QR decomposition with column pivoting. Note that using this approach due to the nature of the algorithm there are only 3 non-zero rows in the 13 × 13 matrix GH RM. Define E to be the rectangular matrix of expression/concentration data where: E = [ei (tj )] where ei (tj ) is the expression/concentration of gene/hormone i at time j. So E consists of the 4 columns E0 , E8 , E16 and E24 .

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The concentration of mRNA and hormones at the next time Et +8 is calculated based on the previous state Et (the concentration of mRNA and hormones at time t). The minimisation can be solved separately for each gene/hormone. For each gene/hormone minimisation of the error was based on the least squares method. Obtained results were compared to the measured/observed data.

3.1 GHRM: Numerical Example We will demonstrate the performance of this technique on the subdataset S3, log2 transformed. The expression and concentration values are shown in Table 3. We have expression levels of 11 genes and concentrations of 2 hormones at 4 times, as described in the Sect. 2. Following the notation of Gebert et al. [5], define E˙ 8(n) = E8(n) − E8(n−1) , for n = 0, 1, 2, 3. Thus, the difference quotients become: E˙ 8 = [0.05, 1.14, 1.44, −1.16, 0.66, 0.92, 0.00, 0.79, −0.39, −0.21, 2.22, 101.35, 102.20]T E˙ 16 = [−2.31, −1.81, −2.84, −3.50, −1.24, −1.39, −1.46, −1.91, −2.76, −1.21, −3.26, 1.04, 2.34]T E˙ 24 = [0.41, 1.05, 1.11, −0.45, 0.16, −0.34, −2.04, 1.38, 0.29, −0.23, 1.43, −0.45, −0.98]T To find the GH RM, we have to solve the following minimization problem. Minimise 3 

||GH RM E8(n) − E˙ 8(n) ||2

n=1

by varying the entries of GH RM. Due to the method used, the GH RM which was produced has only 3 non-zero rows, namely for IGF1R, RIPK3 and P4, respectively. These non-zero rows are shown below: ⎡

−1.18 −1.18 −1.66 −1.40 −0.61 −0.51 0.07 −1.34 −1.33 −0.46 −1.98 0.00

⎢ ⎣ 0.48

0.43

0.63

0.64

0.25

0.24

0.12

0.48

0.55

0.21

0.76

⎤

⎥ 0.74 −0.29 −0.59 ⎦

−0.03 −0.04 −0.06 −0.04 −0.02 −0.03 −0.01 −0.04 −0.04 −0.01 −0.07 −0.98 −0.97

230 Table 3 Observed expression levels for treatment S3 (log2 transformed). This is the matrix E

Table 4 GH RM predicted expression levels for treatment S3

M. J. Wieteska et al. Gene/Hormone BCL2 CYP19A1 HSD3B2 IGF1R BAX CYP11A1 FSHR HSD3B2 RIPK3 LHR StAR E2 P4

0h −7.85 −3.21 −3.04 −1.37 −11.35 −5.60 −10.92 −9.88 −10.53 −9.54 −6.65 −100 −100

8h −7.80 −2.07 −1.60 −2.53 −10.69 −4.68 −10.91 −9.08 −10.92 −9.75 −4.44 1.35 2.20

16h −10.11 −3.87 −4.43 −6.03 −11.93 −6.07 −12.37 −10.99 −13.68 −10.96 −7.70 2.38 4.54

24h −9.70 −2.83 −3.32 −6.49 −11.73 −6.41 −14.41 −9.61 −13.39 −11.19 −6.27 1.93 3.56

Gene BCL2 CYP19A1 HSD3B2 IGF1R BAX CYP11A1 FSHR HSD3B2 RIPK3 LHR StAR E2 P4

8h −7.80 −2.07 −1.60 −2.53 −10.69 −4.68 −10.92 −9.09 −10.92 −9.75 −4.44 1.35 2.20

16h −10.11 −3.88 −4.44 −6.03 −11.93 −6.07 −12.38 −11.01 −13.69 −10.96 −7.71 2.38 4.54

24h −9.71 −2.83 −3.34 −6.49 −11.76 −6.42 −14.42 −9.62 −13.40 −11.19 −6.28 1.93 3.56

By evaluating Et +8 = (GH RM + I )Et , the predictions for the gene expression and hormone concentration levels were calculated and are shown in Table 4. The errors between Tables 3 and 4 were < 1%.

4 Correlation Matrices To investigate the dependence of each gene/hormone on the others we have used correlation matrices containing correlation coefficients between nodes. Correlation matrices are widely used to study the gene environment networks [13]. Correlation matrices were built to test the differences in correlations between gene expression and hormone concentration levels in different experimental conditions and for different sizes of follicles.

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BCL2 (G1) CYP19A1 (G2) HSD3B2 (G3)

Genes/hormones

IGF1R (G4) BAX (G5) CYP11A1 (G6) FSHR (G7) HSD17B1 (G8) RIPK3 (G9) LHR (G10) StAR (G11) E2(H1) P4 (H2) G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 H1 H2 Genes/hormones

Fig. 2 Correlation matrix for treatment D1

Visual representation of the correlation matrix for Treatment D1 is shown in Fig. 2. Correlation coefficient values are shown using colors. Light yellow denotes value of 1, which means that for every increase in one gene/hormone, there is an increase in the other gene/hormone. Correlation coefficient of dark blue color denotes the value −1, which means that an increase in a given gene/hormone results in a decrease in the other gene/hormone. Zero means that the two genes/hormones are not correlated and is show as green. The absolute value of the correlation describes the strength of the relationship between given genes/hormones. The genes/hormones used to construct the GHRM for treatment D1 are underlined in Fig. 2. In the correlation matrices for each treatment the correlation coefficients near zero are rare, so there is high linear dependency between the expression levels of most genes/hormones, so the linear model can give a good approximation of the dynamics of the system, Fig. 2. The GHRM constructed for subdataset D1 was based on CYP11A1 (G6), FSHR (G7) and E2. Note that the numerical example was based on the results for group S3 (for which GHRM was based on IGF1R, RIPK3 and P4). The correlation matrix for treatment D1 shown in Fig. 2 reveals that there was a strong correlation between CYP11A1 (G6) and a majority of genes (G2 and G4 to G10). A similar pattern was observed for correlations between FSHR (G7) and genes G2 and G4 to G10. However, strong positive correlations occur between CYP11A1 (G6) and genes G2, G4 and G6, but for these genes there were strong negative correlations with FSHR (G7). Conversely, when the strong negative correlations occur between CYP11A1 (G6) and genes G5, G7, G8, G9 and G10, there were strong positive correlations

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with FSHR (G7). The correlations between E2 (the third non-zero row used in that GH RM) and genes G2, G4 to G10 were weak. However, the correlation between G1 and E2 is strongly negative and the correlation between E2 and (G11, E2 and P4) are strongly positive. Looking at Fig. 2, it appears that genes G2 and G4 to G10 are strongly correlated with many other genes and hormones. The algorithm used to create the GHRM chooses the 3 genes/hormones G6, G7 and E2. So the two techniques (GHRMs and correlation matrices) reveal different interaction patterns.

5 Comparison Between GHRMs for Different Treatments Each GHRM is a 13 × 13 matrix, but with many zero entries. For each of the treatments, there were 10 rows of zeros in the GHRM (usually different rows for each treatment). Removing the zero rows gives a 3 × 13 GHRM. Table 5 shows the genes and hormones used in the nonzero rows of the GHRMs for the different treatments. The GHRM usually contains 2 genes and 1 hormone. The addition of 25ng/ml of FSH and/or 50ng/ml IGF-1 has a maximum stimulatory effect on the E2 production [12]. The hormone used in the GHRM is E2 for dominant follicles and P4 for secondary follicles. The correlation coefficients between E2 and other genes/hormones are close to the correlation coefficients between P4 and other genes/hormones (for example the bottom 2 rows of the matrix in Fig. 2 look alike). 11 genes were investigated: genes used as a marker of atresia of the follicles (BAX, BCL2, RIPK3); the gene directly responsible for progesterone production (HSD 3B2); genes directly responsible for estradiol production (HSD 17B1, CYP 19A1); other genes involved in E2 and P4 synthesis (CYP11A1, STAR); and genes encoding receptors (IGF1R, LHR and FSHR). Most of the genes used for the 14 GHRMs are associated with either growth (FSHR, LHR) or decay (RIPK3, BAX, BCL2) of the follicle. Due to the obtained accuracy (error < 1%) the comparison of GHRMs calculated for different treatments of the secondary and dominant follicles was possible. The GHRMs for the

Table 5 Genes and hormones used in the GH RM for the 14 treatments Treatment S1 S2 S3 S4 S5 S6 S7 S8

3 nodes in GHRM P4, HSD17B1, RIPK3 P4, FSHR, BCL2 P4, IGF1R, RIPK3 P4, HSD17B1, RIPK3 P4, FSHR, LHR P4, E2, RIPK3 P4, HSD17B1, RIPK3 P4, HSD17B1, RIPK3

Treatment D1 D2 D3 D4 D5 D6

3 nodes in GHRM E2, FSHR, CYP11A1 E2, LHR, RIPK3 E2, FSHR, P4 E2. FSHR, BAX E2, FSHR, LHR E2, FSHR, IGF1R

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secondary follicles in most cases were based on one of the atretic genes (except S5). This trend was also noticed for GHRMs of dominant follicles, but with more exceptions. Exceptions were D3 (where P4 replaced the atretic gene (although it should be noted that P4 increases in atretic cells)), D5 and D6 (where atretic genes were replaced by genes associated with growth (LHR in D5 and IGF1R in D6). In D1 the nodes used were genes responsible for growth (FSHR) and estradiol production (E2 and CYP11A1). Each GHRM is visualised using star-shaped graphs in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. GHRM entries were considered as weights / strengths of interaction and were used to depict the star-shaped diagrams. The color bars were used to show the weight of the given interactions, where yellow was used to describe positive and dark blue was used to describe negative interactions. The thicknesses of the lines in the star graphs are proportional to the strengths of interactions represented by the absolute values of the entries in the GHRMs. Each figure has 3 nodes in the centre (corresponding to the 3 nonzero rows of the GHRM), with arrows pointing out to the other 10 nodes on the outside. Cells adapt to different experimental conditions resulting in the metabolic shift (change in the required enzymes) of the cells (as the chosen supplements are utilized by the cells). The built GHRM for each treatment reflects that metabolic shift. This change was investigated by comparison between nodes needed to build the GHRMs and by investigating the impact of a given gene/hormone on the expression/concentration levels of the remaining nodes.

Control: 5-8mm follicles 1.5

1 BAX CYP19A1

IGF1R

0.5 E2

LHR

0

P4 HSD17B1

-0.5

RIPK3 FSHR StAR

-1 HSD3B2

BCL2 CYP11A1

-1.5

-2

Fig. 3 GHRM network for treatment S1

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25ng/ml IGF-1: 5-8mm follicles 3

2

IGF1R CYP11A1 E2

1 RIPK3

0 HSD3B2

FSHR P4

-1

BCL2

StAR

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-2 BAX

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Fig. 4 GHRM network for treatment S2

50ng/ml IGF-1: 5-8mm follicles E2

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0.5 BAX

0

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CYP11A1

-0.5

IGF1R LHR

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-1 HSD17B1 CYP19A1

-1.5 StAR

Fig. 5 GHRM network for treatment S3

HSD3B2

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100ng/ml IGF-1: 5-8mm follicles 1.5 BAX

1

CYP19A1

IGF1R

0.5 E2 LHR P4

0 HSD17B1

RIPK3 FSHR

StAR

-0.5

-1 HSD3B2

BCL2 CYP11A1

-1.5

-2

Fig. 6 GHRM network for treatment S4

50ng/ml IGF-1 + 25ng/ml FSH: 5-8mm follicles 2

RIPK3

BAX

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1 FSHR

0.5

HSD3B2 P4

IGF1R LHR

0

-0.5

HSD17B1 BCL2

-1 CYP19A1

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Fig. 7 GHRM network for treatment S5

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100ng/ml IGF-1 + 25ng/ml FSH: >8mm follicles FSHR

0.8

StAR

BCL2

0.6 LHR

0.4 RIPK3

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0.2

P4 CYP19A1

E2

0 -0.2

IGF1R CYP11A1 HSD3B2

-0.4

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Fig. 8 GHRM network for treatment S6

25ng/ml FSH: 5-8mm follicles 0.8 0.6 BAX CYP19A1

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0.4 0.2 E2

LHR P4

0 HSD17B1

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StAR

-0.2 -0.4

HSD3B2

BCL2

-0.6

CYP11A1

-0.8 -1

Fig. 9 GHRM network for treatment S7

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50ng/ml FSH: 5-8mm follicles 1.5

1

CYP11A1 StAR

0.5

CYP19A1 LHR

0 RIPK3 HSD17B1 HSD3B2

-0.5

P4 BAX

-1 BCL2

-1.5

FSHR IGF1R

E2

-2

Fig. 10 GHRM network for treatment S8. The E2 levels were not available so the GHRM was constructed based on 12 instead of 13 nodes

Control: >8mm follicles

3

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StAR

0.5

2 CYP19A1

BAX

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0 E2 IGF1R

0

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FSHR

-1 -1 LHR

HSD17B1

-2 RIPK3

-3 -4

-3

-2

-1

Fig. 11 GHRM network for treatment D1

0

-1.5

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1

2

3

4

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25ng/ml IGF-1: >8mm follicles 0.8 0.6 BAX P4

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HSD3B2 RIPK3

0

LHR

E2 CYP19A1

HSD17B1

-0.2 -0.4

StAR

FSHR

-0.6

IGF1R

-0.8

Fig. 12 GHRM network for treatment D2

50ng/ml IGF-1: >8mm follicles 1 CYP11A1

HSD3B2

BAX

0.5 LHR

0 P4

HSD17B1

FSHR BCL2

E2

-0.5

CYP19A1 IGF1R StAR

RIPK3

-1

-1.5

-2

Fig. 13 GHRM network for treatment D3

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100ng/ml IGF-1: >8mm follicles 1.5

P4

1

HSD3B2

RIPK3 BCL2

0.5

BAX

CYP19A1

FSHR HSD17B1

E2

0

CYP11A1

-0.5

LHR IGF1R

StAR

-1

Fig. 14 GHRM network for treatment D4

50ng/ml IGF-1 + 2ng/ml LH: >8mm follicles 3 IGF1R

RIPK3

2 HSD17B1

CYP19A1

1

0 FSHR E2

StAR

HSD3B2

-1

LHR

-2

P4

BAX

-3

-4 CYP11A1

BCL2

-5

Fig. 15 GHRM network for treatment D5

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100ng/ml IGF-1 + 2ng/ml LH: >8mm follicles 0.5 HSD3B2 HSD17B1

0

StAR BCL2

-0.5 RIPK3

IGF1R FSHR E2

BAX

-1

CYP11A1 LHR

-1.5

P4 CYP19A1

-2

Fig. 16 GHRM network for treatment D6

5.1 Comparison Between GHRMs for Different Treatments for the Secondary Follicles The GHRMs of the secondary follicles were based on the interactions with three genes/hormones, two of which were P4 and RIPK3 (except in treatment S5 and treatment S2 where RIPK3 was replaced by BCL2). RIPK3 plays a role in the necroptosis of the corpus luteum (CL) [8] and it is hypothesised that it also plays a role in the atresia of the follicle, while the role of BCL2 in follicle atresia is well established. The third gene/hormone used in 4 cases to build the GHRM was HSD17B1. In 2 cases (S2 and S5) the third gene/hormone was FSHR. In treatment S3 the third gene/hormone was IGF1R and in treatment S6 the third gene/hormone used was E2. Unfortunately, for treatment S8 the E2 concentration data was incomplete, so the GHRM used only the data from the 12 other nodes. The strength of the impact of described genes/hormones on the expression level/hormone concentration (as measured by the maximum and minimum entries in the GHRM) varied between treatments. The weakest impact was recorded for treatments S6 and S7. The strongest impact was recorded for treatment S2. In the secondary follicles, interactions in all treatments between P4 and remaining genes/hormone were weak. In the treatments S4 and S8, the interactions between RIPK3 and the three genes BCL2, IGF1R and BAX were strong (slightly stronger in treatment S4).

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In the secondary follicles in treatment S7, nearly all interactions between RIPK3 and remaining genes/hormones (except with HSD17B1) were negative. Similarly, in treatment S3 (in which IGF1R was one of the nonzero rows in the GHRM) almost all interactions with IGF1R (except FSHR and P4) were negative. Lastly, in treatment S2, almost all interactions between BCL2 and other genes/hormones (except FSHR and P4) were negative.

5.2 Comparison Between GHRMs for Different Treatments for the Dominant Follicles In the dominant follicles’ GHRMs for all treatments were built using E2. FSHR was also used (except in treatment D2). The remaining genes/hormones used to build GHRMs varied between treatments: two treatments (D2 and D5) used LHR and treatments D1, D2, D3, D4 and D6 used CYP11A1, RIPK3, P4, BAX and IGF1R, respectively. The strength of the impact of the interaction was weakest in treatment D2 and strongest in treatment D5. The comparison between different treatments of the dominant follicles shows that in treatment D3 there was a strong positive interaction between E2 and IGF1R (GHRM entry = 0.88). This was not observed in other treatments where this interaction was weak (GHRM entry between −0.0996 and 0.0231). Furthermore, the GHRM entries from the cell cultures of the dominant follicles show that interaction between atretic genes (BCL2 and BAX) and E2 was in almost all cases weak (except treatments D4 and D6) and the impact of IGF-1 was not observed. In treatments D4 and D6 the weights of the interactions (GHRM entries) between E2 and BAX were 1.1379 and 0.6881, respectively.

5.3 Comparison Between Secondary and Dominant Follicles for Different Treatments Using GHRMs The genes/hormones used to build the GHRMs in the cultures of secondary and dominant follicles differ. The hormone used in GHRMs for the secondary follicles was P4, whereas the hormone used in GHRMs for the dominant follicles was E2. The GHRMs for the secondary follicles (in most treatments) used RIPK3, which we hypothesize may play a role in follicle atresia. GHRMs of dominant follicles used FSHR in most cases. This is surprising as FSH is important for growth of small (< 5mm) and secondary (5 − 8mm) follicles. In the cultures of the secondary follicles the remaining node used to build GHRMs for treatments S1, S4, S7 and S8 was HSD17B1, whereas in the dominant follicles there was more variation in the remaining node used.

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Furthermore, it was observed that for treatments S5 and D5, both gene nodes in the GHRMs corresponded with each other (both were based on FSHR and LHR). As usual, the hormone node was either P4 or E2, respectively. Interesting observations were also made in cell cultures of both classes of follicles for treatments S2 and D2. In these treatments (S2 and D2) the GHRMs were based on either FSHR (in secondary follicles) or LHR (in dominant follicles); this difference is probably the result of the metabolic shift between secondary and dominant follicles (FSH dependent and LH dependent phases of growth). The gene with an established role in atresia (BCL2) was used in S2 and the gene which we hypothesise to play a role in atresia (RIPK3) was used in the GHRM for D2. As usual, the third node was either P4 (for S2) or E2 (for D2).

6 Conclusions The GHRM matrices yielded an error < 1% on the training datasets based on the log2 normalised data (the GHRMs built based on raw data were less accurate and for this reason this approach is not shown here). The numerical example given in subsection GHRM: Numerical Example gives a very accurate prediction. This was typical for datasets D and S. Unfortunately, using the GHRMs obtained from the training dataset D for corresponding treatments in the test dataset Dtest, shows that it is not possible to obtain reliable results using the GHRM matrices built on different cell cultures. This is not surprising as there is a difference between transcription profiles of different cells, which seems to occur in stochastic pulses in mammalian cells [7]. This error could be minimized by using a larger training dataset which would allow creation of a matrix with more robust values. Harper et al. [7] concluded that the active phase length was similar whereas the inactive transcriptional phase differs between “different reporter genes controlled by identical promoters in the same living cell”. Likewise, although Gene Regulatory Matrices (GRMs) are widely used in the literature, they often do not give reliable predictions. The Gene and Hormone Regulatory Matrices (GHRMs) were used to estimate the concentration of genes/hormones at specific times. The results yielded 13 × 13 constant matrices (GHRMs) with three nonzero rows. We found that the GHRMs varied between different treatments, especially in the weight of the interaction between different genes/hormones. However the following trends were noted: (1) the GHRMs can play a role in explaining patterns of interactions between genes and hormones (2) the GHRMs for dominant follicles always used E2, whereas GHRMs of secondary follicles always used P4 as one of the nonzero rows (3) most of the genes used for the 14 GHRMs are associated with either growth (FSHR, LHR) or decay (RIPK3, BAX, BCL2) of the follicle.

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In addition, this method can be of value when estimating missing values in a dataset as it would give a better prediction than the presently applied method (averaging previous and next data values). This would result in a better understanding of the dynamics of the given protein translation. Acknowledgements Study was supported by the Grant OPUS of National Sciences Center No UMO-2018/29/B/NZ9/00391. The authors wish to acknowledge funding from IT Sligo’s President’s Bursary Award and from IT Sligo’s Research Capacity Building Fund.

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