Advanced Mathematical Methods in Biosciences and Applications [1st ed. 2019] 978-3-030-15714-2, 978-3-030-15715-9

Table of contents :
Front Matter ....Pages i-xii
Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design (Faina Berezovskaya, Georgiy P. Karev)....Pages 1-25
Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments (Alexander S. Bratus, Artem S. Novozhilov, Yuri S. Semenov)....Pages 27-51
A Survey on Quasiperiodic Topology (Roberto De Leo)....Pages 53-88
Combining Bifurcation Analysis and Population Heterogeneity to Ask Meaningful Questions (Irina Kareva)....Pages 89-110
Polyvariant Ontogeny in Plants: When the Second Eigenvalue Plays a Primary Role (Dmitrii O. Logofet)....Pages 111-130
Recurrence as a Basis for the Assessment of Predictability of the Irregular Population Dynamics (Alexander B. Medvinsky)....Pages 131-145
Total Analysis of Population Time Series: Estimation of Model Parameters and Identification of Population Dynamics Types (Lev V. Nedorezov)....Pages 147-157
Modelling Cancer Dynamics Using Cellular Automata (Álvaro G. López, Jesús M. Seoane, Miguel A. F. Sanjuán)....Pages 159-205
Analytical Solutions for Traveling Pulses and Wave Trains in Neural Models: Excitable and Oscillatory Regimes (Evgeny P. Zemskov, Mikhail A. Tsyganov)....Pages 207-219
Numerical Study of Bifurcations Occurring at Fast Timescale in a Predator–Prey Model with Inertial Prey-Taxis (Yuri V. Tyutyunov, Anna D. Zagrebneva, Vasiliy N. Govorukhin, Lyudmila I. Titova)....Pages 221-239
Within Host Dynamical Immune Response to Co-infection with Malaria and Tuberculosis (Edme Soho, Stephen Wirkus)....Pages 241-261
Back Matter ....Pages 263-264

Citation preview

STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health

Faina Berezovskaya Bourama Toni Editors

Advanced Mathematical Methods in Biosciences and Applications

STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health

STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health Series Editor Bourama Toni Department of Mathematics Howard University Washington, DC, USA

This interdisciplinary series highlights the wealth of recent advances in the pure and applied sciences made by researchers collaborating between fields where mathematics is a core focus. As we continue to make fundamental advances in various scientific disciplines, the most powerful applications will increasingly be revealed by an interdisciplinary approach. This series serves as a catalyst for these researchers to develop novel applications of, and approaches to, the mathematical sciences. As such, we expect this series to become a national and international reference in STEAM-H education and research. Interdisciplinary by design, the series focuses largely on scientists and mathematicians developing novel methodologies and research techniques that have benefits beyond a single community. This approach seeks to connect researchers from across the globe, united in the common language of the mathematical sciences. Thus, volumes in this series are suitable for both students and researchers in a variety of interdisciplinary fields, such as: mathematics as it applies to engineering; physical chemistry and material sciences; environmental, health, behavioral and life sciences; nanotechnology and robotics; computational and data sciences; signal/image processing and machine learning; finance, economics, operations research, and game theory. The series originated from the weekly yearlong STEAM-H Lecture series at Virginia State University featuring world-class experts in a dynamic forum. Contributions reflected the most recent advances in scientific knowledge and were delivered in a standardized, self-contained and pedagogically-oriented manner to a multidisciplinary audience of faculty and students with the objective of fostering student interest and participation in the STEAM-H disciplines as well as fostering interdisciplinary collaborative research. The series strongly advocates multidisciplinary collaboration with the goal to generate new interdisciplinary holistic approaches, instruments and models, including new knowledge, and to transcend scientific boundaries.

More information about this series at http://www.springer.com/series/15560

Faina Berezovskaya • Bourama Toni Editors

Advanced Mathematical Methods in Biosciences and Applications

123

Editors Faina Berezovskaya Department of Mathematics Howard University Washington, DC, USA

Bourama Toni Department of Mathematics Howard University Washington, DC, USA

ISSN 2520-193X ISSN 2520-1948 (electronic) STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health ISBN 978-3-030-15714-2 ISBN 978-3-030-15715-9 (eBook) https://doi.org/10.1007/978-3-030-15715-9 Mathematics Subject Classification: 92Dxx, 92B05, 92C80, 37B20, 35B34, 70K43, 70K30, 65P30, 37N25, 34C60 © Springer Nature Switzerland AG 2019 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, express 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

Mathematical Biology has gained indeed tremendous interest over the last decades and is rapidly expanding in an attempt to duplicate the success of mathematical methods of inquiry to the physical science. This volume in the STEAM-H series features recent advances in mathematical biosciences selectively exposed in the following chapters: Chapter “Arnold’s Weak Resonance Equation as a Model of Greek Ornamental Design” by F. Berezovskaya and G. Karev presents a mathematical model of Greek ornamental designs based on parameterized complex differential equations describing weak resonance. In chapter “Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments,” A. Bratus, A. Novozhilov, and Y. Semenov review the classical Eigen and Crow-Kimura models, in particular, with emphasis on outstanding biological- and mathematical-related questions. Chapter “A Survey on Quasiperiodic Topology” by R. De Leo surveys the Novikov problem on the structure of leaves of the foliations induced by a collection of closed 1-forms in a compact manifold. Chapter “Combining Bifurcation Analysis and Population Heterogeneity to Ask Meaningful Questions” by I. Kareva discusses a combination of methods to include Hidden Keystone Variable to improve understanding of evolving ecological systems. Chapter “Polyvariant Ontogeny in Plants: When the Second Eigenvalue Plays a Primary Role” by D. Logofet reports on a recently developed theory of rankone corrections of nonnegative matrices and its impact on matrix model of stagestructured population dynamics. In chapter “Modelling Cancer Dynamics Using Cellular Automata,” A. Lopez, J. Seoane, and M. Sanjuan use a hybrid cellular automaton model to explore the dynamics of tumor growth in the presence of an immunological response. Chapter “Recurrence as a Basis for the Assessment of Predictability of the Irregular Population Dynamics” by A. Medvinsky is an overview of methods to assess predictability based on the recurrence nature of population fluctuations using field observations and mathematical modeling.

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Preface

In chapter “Total Analysis of Population Time Series: Estimation of Model Parameters and Identification of Population Dynamics Types,” L. Nedorezov revisits statistical and mathematical models of population dynamics. Chapter “Analytical Solutions for Traveling Pulses and Wave Trains in Neural Models: Excitable and Oscillatory Regimes” by E. Zemskov and M. Tsyganov considers the Tonnelier-Gerstner model describing mathematically the structure of a solitary pulse and the form of a periodic sequence of pulses. In chapter “Numerical Study of Bifurcations Occurring at Fast Time Scale in a Predator-Prey Model with Inertial Prey-Taxis,” Y. Tyutyunov, A. Zagrebneva, V. Govorukhin, and L. Titova uncover complex bifurcation transition in the simple model of prey-taxis in predator-prey system leading to periodic, quasiperiodic, and chaotic spatiotemporal dynamics. Chapter “Within Host Dynamical Immune Response to Coinfection with Malaria and Tuberculosis” by E. Soho and S. Wirkus analyzes the malaria-tuberculosis coinfection nine-variable model and the impact of the order of introduction of the two diseases. The book as a whole certainly enhances the overall objective of the series, that is, to foster the readership interest and enthusiasm in the STEAM-H disciplines (Science, Technology, Engineering, Agriculture, Mathematics, and Health) to include statistical and data sciences, stimulating graduate and undergraduate research through effective interdisciplinary collaboration. The STEAM-H series is now hosted at Howard University, Washington, DC, USA, an area that is socially, economically, intellectually very dynamic and home to some of the most important research centers in the USA. This series, by now well established and published by Springer, a world-renown publisher, is expected to become a national and international reference in interdisciplinary education and research. Washington, DC, USA Washington, DC, USA

Faina Berezovskaya Bourama Toni

Acknowledgments

We are grateful to the many reviewers for their professionalism; they raised many questions and challenged the materials featured in this book with insightful comments and suggestions from concepts to exposition. We would like to express our most sincere appreciation to all the contributors for the excellent research work. They all made this volume a reality for the greater benefit of the community of Mathematical Sciences.

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Contents

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Faina Berezovskaya and Georgiy P. Karev

1

Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander S. Bratus, Artem S. Novozhilov, and Yuri S. Semenov

27

A Survey on Quasiperiodic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roberto De Leo Combining Bifurcation Analysis and Population Heterogeneity to Ask Meaningful Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irina Kareva

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Polyvariant Ontogeny in Plants: When the Second Eigenvalue Plays a Primary Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Dmitrii O. Logofet Recurrence as a Basis for the Assessment of Predictability of the Irregular Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Alexander B. Medvinsky Total Analysis of Population Time Series: Estimation of Model Parameters and Identification of Population Dynamics Types. . . . . . . . . . . . . . 147 Lev V. Nedorezov Modelling Cancer Dynamics Using Cellular Automata. . . . . . . . . . . . . . . . . . . . . . 159 Álvaro G. López, Jesús M. Seoane, and Miguel A. F. Sanjuán Analytical Solutions for Traveling Pulses and Wave Trains in Neural Models: Excitable and Oscillatory Regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Evgeny P. Zemskov and Mikhail A. Tsyganov

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Numerical Study of Bifurcations Occurring at Fast Timescale in a Predator–Prey Model with Inertial Prey-Taxis . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Yuri V. Tyutyunov, Anna D. Zagrebneva, Vasiliy N. Govorukhin, and Lyudmila I. Titova Within Host Dynamical Immune Response to Co-infection with Malaria and Tuberculosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Edme Soho and Stephen Wirkus Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Contributors

Faina Berezovskaya Department of Mathematics, Howard University, Washington, DC, USA Alexander S. Bratus Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow, Russia Vasiliy N. Govorukhin Vorovich Institute of Mathematics, Mechanics and Computer Sciences, Southern Federal University, Rostov-on-Don, Russia Georgiy P. Karev National Centre for Biotechnology Information, National Institutes of Health, Bethesda, MD, USA Irina Kareva Mathematical and Computational Sciences Center, School of Human Evolution and Social Change, Arizona State University, Tempe, AZ, USA Roberto De Leo Department of Mathematics, Howard University, Washington, DC, USA Dmitrii O. Logofet A.M. Obukhov Institute of Atmospherics Physics, Moscow, Russia Álvaro G. López Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain Alexander B. Medvinsky Institute of Theoretical and Experimental Biophysics, Pushchino, Moscow Region, Russia Lev V. Nedorezov Research Center for Interdisciplinary Environmental Cooperation RAS, Saint-Petersburg, Russia Artem S. Novozhilov Department of Mathematics, North Dakota State University, Fargo, ND, USA

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Contributors

Miguel A. F. Sanjuán Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain Department of Applied Informatics, Kaunas University of Technology, Kaunas, Lithuania Institute for Physical Science and Technology, University of Maryland, College Park, MD, USA Yuri S. Semenov Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow, Russia Jesús M. Seoane Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain Edme Soho Department of Mathematics, Hostos Community College, Bronx, NY, USA Lyudmila I. Titova Vorovich Institute of Mathematics, Mechanics and Computer Sciences, Southern Federal University, Rostov-on-Don, Russia Mikhail A. Tsyganov Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Russia Yuri V. Tyutyunov Federal Research Center The Southern Scientific Centre of the Russian Academy of Sciences (SSC RAS), Rostov-on-Don, Russia Southern Federal University, Rostov-on-Don, Russia Stephen Wirkus School of Mathematical & Natural Sciences, Arizona State University, Glendale, AZ, USA Anna D. Zagrebneva Department of Computer and Computer-Based System Software, Faculty of IT Systems and Technologies, Don State Technical University, Rostov-on-Don, Russia Evgeny P. Zemskov Federal Research Center for Computer Science and Control, Russian Academy of Sciences, Moscow, Russia

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design Faina Berezovskaya and Georgiy P. Karev

Abstract We propose and study a mathematical model that reproduces qualitatively several ancient ornamental designs that one can see in archeological and historical museums of Crete and Athens. The spiraling wave pattern can be seen throughout the full spectrum Classical tradition. The designs contain several rings that circumscribe a fixed number of “flowers” (centers or spirals), specific to each design. The model proposed is based on a complex differential equation of “weak resonance” (Arnold 1977). We analyze the role of the model parameters in giving rise to different peculiarities of the repeated designs, in particular, the “dynamical indeterminacy.” The model allows tracing design changes under parameter variation, as well as to construct some new ornamental designs. We discuss how observed ornamental design may reflect some philosophical ideas of ancient inhabitants of Greece.

1 Introduction 1.1 The Model as an Equation with Complex Variable Patterns exist in ancient ornamental designs, such as the ones that one can see in historical museums of Crete and Athens. The designs contain bands of different but fixed numbers of “flowers” (mathematically, centroids or spirals), “spider-

F. Berezovskaya () Department of Mathematics, Howard University, Washington, DC, USA e-mail: [email protected] G. P. Karev National Centre for Biotechnology Information, National Institutes of Health, Bethesda, MD, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_1

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F. Berezovskaya and G. P. Karev

nets,” and various types of cycles up to “stars” (see below the definitions). The boundaries of the bands may be smooth or star-shaped curves; the bands are “connected” by these lines. The designs reminded us of some phase portraits of quasi-Hamiltonian dynamical systems, invariant under rotations by the specific angles 2π/n for integer n. The aim of this paper is to find the simplest mathematical model that can describe key features of these ornaments. To this end, we consider the equation proposed by V. Arnold to analyze the problem “Loss of stability of self-sustained oscillations” (see [1, 2] and references therein). The Arnold’ equation is a complex differential equation that describes an equivariant vector field, i.e., symmetric vector field invariant with respect to rotation to the angle 2π /n z = z + zA (|z|) + B zn−1 ,

(1.1)

where z = x + iy is a point in the complex plane, integer n ≥ 4,  =  1 + i  2 ; the function A(|z|) is given by the formula A (|z|) =

s k=1

Ak |z|2 k , Ak = A1k + i A2k , B = B1 + iB 2 ,

(1.2)

n−1 n−3 where integer s = n2 − 1 = n−2 2 if n is even, s = 2 − 1 = 2 if n is odd. Evidently, Eq. (1.1) has equilibrium z = 0 for any Ak , B, and  1 ,  2 are its ∗ eigenvalues; the equation  can have also “peripheral” equilibria E(z ) if for some  n−1 2 ∗ ∗ ∗ ∗ ∗ = 0. z = 0, z + z A |z | + Bz Originally, Eq. (1.1) was constructed for describing the loss of stability of selfoscillations in maps; Eq. (1.1) describes “strong resonance” if n = 1,2,3,4 and “weak resonance” if n≥5. Cases for n ≤ 4 were studied in [1–8] (and discussed in many other works (see, for example, [9])) using both analytical and numerical methods of bifurcation theory. To the best of our knowledge, cases of n ≥ 5 are still not investigated completely. The main problem considered in [1, 2] was description of phase-parameter portraits of Eq. (1.1) in a neighborhood of co-dimension 2 bifurcation for  = 0. Specifically, the goal was to reveal sequences of all co-dimension 1 bifurcations that arise in the vicinity of (0,  2 ), for any fixed coefficients A = 0, B = 1. Notably, all cases n ≤ 4 demonstrate essentially different phase behaviors, and corresponding equations have different “organizing centers.” In what follows we consider cases for n≥4 and show that Eq. (1.1) demonstrates different kinds of phase behaviors depending on whether n is even or odd. We analyze the role of parameters B and As for even or odd s in the genesis of patterns and repeated designs for different n. Phase portraits of the equation with n≥4 have patterns that mimic the qualitative features of some of Greek ornamental designs. Figures of Appendix 2 show several phase portraits of Eq. (1.1) for different n; all of these portraits contain some of the four main types of annular patterns observed in Greek ornamental designs.

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

3

Fig. 1 Example of model (1.1) with n = 5. The phase portrait contains centroids around O and five peripheral equilibria; the peripheral equilibria compose one “5-flower ring” bounded by 2 separatrix cycles; any two neighboring saddles contain “spider-nets”

Definitions (See Fig. 1) • “n- cycle” is a separatrix limit cycle composed of n saddles “connected” by their separatrices; we call it an “n-star” if it is not convex and “convex n-cycle” otherwise; • “centroid” is a pattern composed of a center or spiral equilibrium together with a set of orbits around it; • “n-flower ring” is a pattern consisting of n centroids and n saddles together with their separatrices; each centroid is placed inside the “leaf” composed by a separatrix cycle or separatrix loop; • “spider-net” is a pattern consisting of one or two neighboring saddles together with their separatrices (outgoing to infinity) and hyperbolic-shape orbits between the separatrices. Next, we show below that for a wide range of parameters, Eq. (1.1) is Hamiltonian, and its phase portraits represent certain collections (unions) of patterns described above. Detailed descriptions of phase portraits of Hamiltonian system are given in s.3. We also present some portraits, which cannot be reproduced by Hamiltonian equations but can be considered as small perturbations of portraits of Hamiltonian equation under variations of coefficients of (1.1).

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F. Berezovskaya and G. P. Karev

Statement 1 A For any integer n ≥ 4 and small values of parameters 2 ≥ 0, Ak , k = 1, . . . , s , the phase-parameter portrait of Eq. (1.1) can contain the four patterns defined above. In Discussion, we try to explain and comment on the “dynamical indeterminacy” that seems to us to be the interesting peculiarity of the Greek designs. In terms of dynamical systems, it implies the presence of domains in the phase plane such that the system shows essentially different limiting behaviors for close initial values. For example, we observe such domains in Figs. 12 and 13, where trajectories from the neighborhood of the origin or the limit cycle can reach any of the peripheral points.

2 Some Properties of the Model 2.1 The Equation in Different Coordinate Systems In order to analyze Eq. (1.1), it is convenient to write it in both polar and Cartesian coordinates. In polar coordinates (r, ϕ) : z = reiϕ (x = r cos (ϕ), y = r sin (ϕ)), Eq. (1.1) becomes the System:  s r  = r 1 +

k=1

ϕ  = 2 +

s k=1

A1k r 2k + r n−2 (B1 cos (nϕ) + B2 sin (nϕ)) ,

(2.1)

 A2k r 2k + r n−2 (−B1 sin (nϕ)) + B2 cos (nϕ) .

In (x, y) Cartesian coordinates (x, y) Eq. (1.1) becomes the System: x  = 1 x − 2 y +

s k=1

 k  A1k x − A2k y x 2 + y 2 + (B1 p (x, y) + B2 q (x, y)) , (2.2)

y  = 2 x + 1 y +

s k=1

 k  A2k x + A1k y x 2 + y 2 + (−B1 q (x, y) + B2 p (x, y)) ,

where p(x, y), q(x, y) are polynomials of degree n − 1, such that px (x, y) = qy (x, y), py (x, y) = −qx (x, y). Rotating the phase plane by angle β such that cos β =  B2 1 2 , sin β =  B2 2 2 , B1 +B2

we can rewrite Systems (2.1) and (2.2), correspondingly, as follows:

B1 +B2

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

 s r  = r 1 +

k=1

ϕ  = 2 + where B =



s k=1

 A1k r 2k + B r n−2 cos (nϕ) ≡ P (r, ϕ) ,

5

(2.3)

A2k r 2k − Br n−2 sin (nϕ) ≡ Q (r, ϕ) ,

B12 + B22 , and

x  = 1 x − 2 y +

s k=1

 k  A1k x − A2k y x 2 + y 2 + B p (x, y) ≡ F1 (x, y) , (2.4)

y  = 2 x + 1 y +

s k=1

 k  A2k x + A1k y x 2 + y 2 − Bq (x, y) ≡ F2 (x, y) .

Notice that P (r, ϕ) = F1 (x, y) cos (ϕ) + F2 (x, y) sin (ϕ) ,

(2.5)

Q (r, ϕ) = (F2 (x, y) cos (ϕ) − F1 (x, y) sin (ϕ)) /r,

x = r cos (ϕ) , y = r sin (ϕ) .

(where)

Equation (1.1) and Systems (2.3) and (2.4) possess many specific properties, some of which we discuss below.

2.2 Case B = 0 If B = 0, then Eq. (1.1) reads z = z + zA (|z|) ,

(2.6)

where the function A is defined by (1.2). This differential equation serves as an approximation of the Poincaré map, which was applied to analysis of loss of stability of a closed orbit (limit cycle) [9, 10]. Let us recall some important properties of Eq. (2.6).

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F. Berezovskaya and G. P. Karev

In polar coordinates, Eq. (2.6) becomes  s r  = r 1 +

k=1

ϕ  = 2 +

s k=1

 A1k r 2k ≡ rP1 (r),

(2.7)

A2k r 2k ≡ Q1 (r).

In particular, for small  1 and 2 = 0, A11 = 0 truncation of the system:   r  = r 1 + A11 r 2 , ϕ  = 2 serves as a model system for the Andronov–Hopf bifurcation of changing stability of equilibrium O(0,0), where A11 is the first Lyapunov value. The bifurcation is supercritical, accompanied by appearance/disappearance     of a stable limit cycle if A11 < 0, 1 > 0 ,and subcritical if A11 > 0, 1 < 0 ,accompanied by appearance/disappearance of an unstable limit cycle. If A11 = 0, then the number and stability of limit cycles is defined by the second Lyapunov valueA12 ; if A12 = 0, 2 = 0, then the next truncation system, containing term A12 r 4 ,describes the generalized Andronov–Hopf bifurcation. Thus, the first Eq. (2.7) defines the number and stability of limit cycles of System (2.7). For  1 ∼ = 0, equilibrium O is a “weak spiral” (a center in linear approximation); the direction of orbit rotation of (2.7) close to O is defined by the sign of  2 , and in a general case, by the sign of Q1 . The direction of rotation can change for r = r∗∗ ,  where r∗∗ is a root of the function Q1 , where ϕ = 0. The point (r∗ , ϕ) such that P1 (r∗ ) = 0, Q1 (r∗ ) = 0 is called a “quasi-equilibrium” of (2.7). Quasi-equilibria are always composed of a circle with radius r∗ such that  ϕ = Q1 (r∗ ) = 0; we call to this circle a quasi-equilibrium cycle. In a vicinity of every quasi-equilibrium, the orbits of the system change direction of rotation (see Fig. 2). Taking into consideration that limit cycles of (2.7) correspond to the roots of polynomial P1 , and quasi-equilibrium cycles correspond to the roots of polynomial Q1 and applying the Descartes’ rule of signs (“The number of positive roots of the polynomial is either equal to the number of sign of differences between consecutive nonzero coefficients, or is less than it by an even number”), we prove the following statement: Proposition 1 The number nc of limit cycles and the number nq of quasiequilibrium cycles of System (2.7) does not exceed s, where s is defined in (1.2). Let us study the role of the term B zn−1 in Eq. (1.1). In Cartesian coordinates, it becomes Bp(x, y) in the first equation of System (2.4), and −Bq(x, y) in the second equation of (2.4). Here, p(x, y), q(x, y) are polynomials of (n−1)-th power and

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

7

Fig. 2 Quasi-equilibria of System (2.7), with n = 5, ε1 = .0001, ε2 = .1, A21 = −.1. Equilibrium O is “weakly unstable” in all three panels. (a) Quasi-equilibrium point asA11 = 0, B = 0; (b) appearance of peripheral equilibria for A11 = 0, B = .01; (c) appearance of a limit cycle and peripheral equilibria for A11 = −.01, B = .01

px (x, y) = qy (x, y), py (x, y) = − qx (x, y). In polar coordinates, this term corresponds to Brn − 1 cos (nϕ) in the first equation of System (2.3) and to −Brn − 2 sin (nϕ) in the second equation of (2.3). We show below that these terms are responsible π for appearance of “peripheral” equilibria E(rk , ϕk ), where ϕk = ± 2n + 2π k/n, k = 0, . . . n−1.

3 Hamiltonian Model 3.1 Hamiltonian It is known [10] that a system in Cartesian coordinates is Hamiltonian if its divergence vanishes. The divergence of Eq. (1.1) taken in form (2.4) Div =

∂F2 ∂F1 + = ∂x ∂y

           2 1 + x 2 + y 2 2A11 + x 2 + y 2 3A12 + x 2 + y 2 4A13 + x 2 + y 2 . . . =

      2 1 + r 2 2A11 + r 2 3A12 + r 2 4A13 + r 2 5A14 . . . vanishes as 1 = 0, A1k = 0, k = 1, . . . , s. Then the following statement is true:

(3.1)

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F. Berezovskaya and G. P. Karev

Proposition 2 If conditions (3.1) hold, then Eq. (1.1) is Hamiltonian and can be written in polar coordinates in the form

r  = B r n−1 cos (nϕ) = −r ϕ  = 2 +

s k=1

∂H ≡ PH (r, ϕ) ∂ϕ

A2k r 2k − Br n−2 sin (nϕ) =

(3H)

1 ∂H ≡ QH (r, ϕ) r ∂r

with Hamiltonian H (r, ϕ) = 2

rn r 2 s r 2k+2 + − B sin (nϕ) + h, A2k k=1 2 2k + 2 n

(3.2)

where h is an arbitrary constant. In (x, y)-coordinates, Eq. (1.1) as given in the form of System (2.4) becomes  k ∂H ≡ F1H (x, y) , A2k x 2 + y 2 + B p (x, y) = − k=1 ∂y (4H)

 s x  = − 2 y + y

y  = 1 y +

k  ∂H ≡ F2H (x, y) A2k y x 2 + y 2 − Bq (x, y) = k=1 ∂x

s

with Hamiltonian  2  s A2 x 2 + y 2 k x + y2 k + + BR (x, y) + h, (3.3) H (x, y) = 2 k=1 2 2k





∂ where R (x, y) = q (x, y) + ∂x p (x, y) dy dx − p (x, y) dy, and h is an arbitrary constant. System (3H) has equilibrium O(0, 0) and can have “peripheral” equilibria E∗ (r∗ , ϕ∗ ), r∗ > 0, whose coordinates (r∗ , ϕ∗ ) satisfy the system cos (nϕ) = 0, 2 +

s k=1

A2k r 2k − Br n−2 sin (nϕ) = 0.

(3.4)

Equation (3.4) defines 2n rays ϕ = ϕ ∗ = ϕj± , where ϕj± = ±

2πj π + , j = 0, . . . , n − 1. 2n n

(3.5±)

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

9

  Then sin nϕj± = ±1, and coordinates r∗ of equilibria E∗ are the roots of one of the polynomials Q + ≡ 2 +

s k=1

A2k r 2k − Br n−2 , Q− ≡ 2 +

s k=1

A2k r 2k + Br n−2 , B > 0. (3.6±)

Let us consider polynomials Q+ , Q− along the rays ϕj+ , ϕj− for some fixed j = 0,   . . . n − 1. The corresponding equilibria E ± r ± , ϕj± can be only saddles or centers (see Figs. 1, 6, 7, 8, 9, 10, and 11)becausethe system is Hamiltonian. Clearly, if the equilibrium E + r + , ϕj+0 is a saddle (center) for some j0 , then all   equilibria E + r + , ϕj+ are saddles (centers), j = 0, . . . n − 1. Similar assertion   is valid for equilibria E − r − , ϕj−0 . These properties allow us to omit index j in

the notation of peripheral equilibrium and consider only points E+ (r+ , ϕ+ ) and E− (r− , ϕ− ) because topological type of equilibrium E does not depend on j.For example, if E+ (r+ , ϕ+ ) (E− (r− , ϕ− )) is a center/a saddle, then the phase plane contains n centers/saddles with the same coordinates r+ (r− ) and ϕ = ϕ + + 2πj n ,j = 2πj − 0, . . . , n − 1 /ϕ = ϕ + n , j = 0, . . . , n − 1. The number and characteristics of equilibria essentially depend on whether n is odd or even. Proposition 3 Let n≥5 be odd. Then 1. polynomials P+ , P− have no more than s = n−1 2 real positive roots; 2. if polynomial P+ (P− ) has M real roots and J of them are positive, then the polynomial P− (P+ ) also has M real roots and M-J of them are positive; 3. let r∗ , r∗ be maximal and minimal roots among all positive roots of polynomials P+ and P− . Then corresponding equilibria E∗ and E∗ are saddles; 4. every two equilibria Ej+ ,Ej++1 corresponding to consequent roots of polynomial P+ are saddle/center or center/saddle; same statement is valid for polynomial P− . In order to study model behavior as x, y → ∞, it is useful to consider the system on the Poincare sphere (see for example [10]). The Poincaré sphereis defined by two transformations of Cartesian coordinates (x, y), given by formulas u = x1 , w = yx and {u = y1 , v = xy }. Analyzing equilibria in the equators of the Poincare sphere as u = 0, we get the following.

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F. Berezovskaya and G. P. Karev

Proposition 4 In the equator of the Poincaré sphere, System (3H) with odd n has n equilibria, which are alternating stable and unstable nodes (see Fig. 3) Using Propositions 3 and 4, we can prove the following. Theorem 2 Let conditions (3.1) hold and assume Eq. (1.1) is Hamiltonian. Then for odd n ≥ 5, the phase portrait of the Equation contains only centroids, n-cycles and spider-nets and can contain no more than n−3 2 flower rings. Example 1 System (3H) with n = 5 (see Fig. 4). In this case, only one ring can exist due to Proposition 3. The right-hand sides of System (3H) are PH (r, ϕ) = Br 4 cos (5ϕ) , QH (r, ϕ) = 2 + A21 r 2 − Br 3 sin (5ϕ) , B > 0. Polynomial Q0 = 2 + A21 r 2 has one positive root if 2 A21 < 0and has no real roots if 2 A21 > 0.In the first case for B > 0, one of polynomials Q± = 2 + A21 r 2 ∓ Br 3 has one real positive root, and second polynomial has one negative and two positive roots. Common number of positive roots of both Q± is n−1 2 = 3, and two  3 of them are saddles. LineC : 272 B 2 + 4 A21 = 0 serves as the boundary between two domains of different phase behaviors. Let, for example, P+ has two positive roots r1 + = 1, r2 + = 2.Then coefficients 7B 2 + of P+ are 2 = − 4B 3 , A1 = 3 .For any B > 0 polynomial P also has a negative root. In Fig. 4 we present the bifurcation diagram of the model for 2 = −4, A21 =     7, B = 3. Here Q+ /Q− has roots r1 + , r2 + , r3 + = 1, 2, − 23 , r1 − , r2 − , r3 − = {−1, −2, −2/3}. Additional example is given in Fig. 7 of Appendix 2. Now let us consider the case, when in Model (3H) n≥4 is even. Then s = n2 − 1 (see (1.2)) and System (3H) can be rewritten in the form r  = B r n−1 cos (nϕ) ,

Fig. 3 Equators of Poincaré sphere, (a) n is odd, (b) n is even,B < A2s

(3.7)

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

11

Fig. 4 Bifurcation diagram of Example 1 for n = 5,  2 = − 4, B = 3; Domain 1:A21 = 7,Domain 2: A21 = −1; boundary  3/2 C : B = A21 /27

ϕ  = 2 +

s−1 k=1

  A2k r 2k + A2(n−2)/2 − B sin (nϕ) r n−2 .

Let E∗ (r∗ , ϕ∗ ) be a “peripheral” equilibrium of (3.7). Then its ϕ∗ -coordinate is given by (3.5±), and r∗ −coordinate is a positive root of even polynomials Q± , Q± ≡  2 +

s−1 k=1

  A2k r 2k + A2n−2 ∓ B r n−2 .

(3.8±)

2

Each polynomial P± can have from 0 up to n−2 2 positive roots. s−1 Consider even polynomial Q0 ≡ 2 + k=1 A2k r 2k + A2n−2 r n−2 that differs 2

from polynomials P± only by the last coefficient. Polynomial P0 has no more than s = (n − 2)/2 positive roots. Proposition 5 Let n≥4 be even, and B > 0. Then 1. polynomials Q+ , Q− have no more than s = n−2 2 real positive roots; 2. polynomials Q± have the same number of positive roots as polynomial Q0 if A2n−2 >−B > 0 and if 0 < B < −A2n−2 ; 2 2

2

3. if B > A n−2 , then at least one of polynomials P− , P+ has a real positive root 2 r∗ and this r∗ is r-coordinate of a saddle equilibrium of (3.7); 4. every two equilibria Ej+ , Ej++1 corresponding to subsequent roots of polynomial Q+ are saddle/center or center/saddle; similar statement is valid for the polynomial Q− . Analyzing “infinite” equilibria at the Poincaré sphere (similar to Proposition 4), we get the following. Proposition 6 In the equator of the Poincaré sphere, System (3.7) has at

least 2

2

equilibria, which are alternating stable and unstable nodes if B > A n−2

and has 2 no equilibria otherwise (see Fig. 3b).

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F. Berezovskaya and G. P. Karev

Fig. 5 Bifurcation diagram for Example 2: n = 4,  1 = 0,  2 = 1. Domain 1: A21 = 1, B = .7; Domain 2: A21 = 1, B = 1.2;Domain 3 : A21 = −1, B = .7. Boundaries between Domains: B = ±A21

Propositions 4 and 6 are special cases of Proposition 8 given below. Theorem 3 Let conditions (3.1) hold and assume Eq. (1.1) is Hamiltonian. Then for even n≥4, the phase portrait of Eq. (1.1) contains

centroid, n-cycles, and may

2 n−2 contain no more than 2 flower rings; for B>

A n−2

the phase portrait additionally 2 contains spider nets. Proof of the Theorem is given in Appendix 1. 3 Example  22 System (3H) with n = 4. Here PH (r, ϕ) = Br cos (4ϕ) , QH (r, ϕ) = 2 1 + r A1 − B sin (4ϕ) , B > 0.

In this case only one ring is possible. Polynomial P0 = 1 + A21 r 2 has no roots when A21 > 0, and has one positive root whenA21 < 0. PolynomialsQ± = 1 +  r 2 A21 ∓ B have no roots if 0 < A21 + B and A21 − B > 0, correspondingly. They can have two positive roots if A21 < 0, B < −A21 and only one root if |B| > A21 (see Figs. 5, 6, and 8).

3.2 Rearrangements of Phase Portraits of Hamiltonian Model Now we consider how the “repeated structures” appear in a phase plane of Hamiltonian Eq. (1.1) and (3.1), assuming for simplicity that  2 > 0. Firstly, let us assume that all coefficients A2k > 0. For any n, phase curves starting in the vicinity of the equilibrium O result in a “centroid,” i.e., a family of closed cycles. Next, we need to differentiate between the cases of odd and even n. Let n be odd. Then there exists a “separatrix cycle” for any fixed B = 0 and some A2k that serves as a boundary of a family of closed curves due to Proposition 3 (see Figs. 1, 4, 9, 10, and 11). The equilibria in the equator of Poincaré sphere are stable (attractive) and unstable (repulsive) nodes (Proposition 4). So, the phase plane of such systems contains “spider-nets” (see Fig. 1). Notice now that the described picture is complete if one of polynomials P± has only one positive root; it happens, for example, if all coefficients A2k are positive. If for the some B several A2k are

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

13

negative, then the “flower ring” (a ring of centroids inside the separatrix cycles) can appear. The number of “flower rings” cannot be more than n−3 2 . The appearance of any such ring is accompanied by change of the sign of coefficient A2k for some k; it   implies change of a number of alternating sign in sequence A2k , B , k = 1, . . . s. Remark that the structure of phase portraits for odd n depends also on whether s = n−3 2 is odd or even. For odd s (see left panels of Fig. 9 (n = 7), A2.6 (n = 11)), “specific” flower rings can arise, which are similar to those that meet in some Greek designs (see, for example, Fig. A3.5) in [11]). Let n be even. Then the number of flower rings as well as the number of centers

2 ±

is no more than n−2 2 .Let  2 > 0. If 0 ≤ B < A n−2 , then polynomials (3.8±), Q ≡ 2   s−1 2 2k s−1 2 2k 2 + k=1 Ak r + A2n−2 ∓ B r n−2 , and R ≡ 2 + k=1 Ak r + A2n−2 r n−2 have 2

2

the same number of sign changes in their sequences of coefficients. If polynomial R has no real roots, then polynomials Q± also have no real roots; if R has 0 ≤ K ≤ n−2 2 real positive roots, then polynomials Q± also have K real positive roots. So, the number of positive roots of polynomials Q± is even (or zero). Notice that in this case the equator of Poincaré sphere has no equilibria, and so the phase plane contains a “center” for large r. If > A2n−2 > 0, then one of the polynomials P± , say P+ , gets one more positive 2

root, r+ . We can verify that equilibrium E+ (r+ , ϕ+ ) is a saddle. The number of real positive roots of the second polynomial, Q− , does not change. So, both polynomials Q± have an odd number of equilibria. The equator of the Poincaré sphere in this case contains nodes of alternating stability, and so the phase plane contains “spider-net” structures for large r. Lastly, if A2n−2 < 0 and 0 < B < −A2n−2 , then Q− gets one more positive root, 2

2

corresponding to a center, and the number of real positive roots of the polynomial Q+ does not change. Polynomials Q± both have an even number of positive roots. Examples of phase portraits of Hamiltonian systems (1.1) with even n are given in Appendix 2 (Figs. 6 and 8).

3.3 The Hamiltonian as a Generalized Lyapunov Function A derivative of Hamiltonian function (3.2) on the solutions of system (2.3) is dH dt =  ∂H  s s ∂H  ∂H  r n−1 1 1 2 2k 2k 1 + k=1 Ak r 1 + k=1 Ak r sin (nϕ) . ∂r r + ∂ϕ ϕ = r ∂ϕ =B n Then

n−1

  s

dH

<  B r

if

1 +

A1k r 2k < ,

dt k=1 n where  is a small positive number and k = 1, . . . , s.

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F. Berezovskaya and G. P. Karev

Thus, for small enough coefficients 1 , A1k , the Hamiltonian (3.3) serves as Lyapunov function of model (1.1), Therefore, the system keeps its main asymptotic properties described above.

4 On General Model 4.1 Finite Equilibria In the general (non-Hamiltonian) case, the (x,y)-coordinates of equilibria are defined by the system F1 (x, y) = 0, F2 (x, y) = 0,

(4.1)

where F1 (x, y), F2 (x, y) are given in (2.4) and the polar coordinates (r, ϕ) of equilibria are defined by the system P (r, ϕ) = 0, Q (r, ϕ) = 0,

(4.2)

where P(r, ϕ), Q(r, ϕ) are given in (2.3):  s P (r, ϕ) = r ε1 +

k=1

Q (r, ϕ) = ε2 +

s k=1

 A1k r 2k + B r n−2 cos (nϕ) , A2k r 2k − Br n−2 sin (nϕ) .

System (2.3) has equilibrium O. It can have equilibrium Ek (rk, ϕk ), where coordinate r = rk is a root of the equation  s F (r) ≡ ε1 +

k=1

A1k r 2k

2

 s + ε2 +

k=1

A2k r 2k

2

= B 2 r 2(n−2) .

(4.3)

By solving Eq. (4.3) together with equation Fr (r) = 0,

(4.4)

one can find r-coordinates of multiple equilibrium points of the model and then, using one of Eq. (4.2), find the corresponding equilibrium value of ϕ. Notice now that generally in non-Hamiltonian case, equilibria Ek (rk , ϕk ) can be saddles, stable and unstable nodes/spirals. In practical computations, it is more convenient to search peripheral equilibrium points in a neighborhood of quasi-equilibria described in s.2.3. 1 Model (1.1) can have limit cycles (see also 2.3). It was shown in [3, 4, 7, 8] that equilibrium points of general (non-Hamiltonian) system (1.1) as n = 4 can appeared

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

15

outside, inside, and on a limit cycle. In [3, 4, 7, 8], the sequences of bifurcations of co-dimension 1 that are realized in the system under variation of parameters A1,2 k ,B have been found. Notice that we observed similar behaviors for general non-Hamiltonian model (1.1) as n = 5, n = 6 (see Figs. 12 and 13).

4.2 Stability of Equilibria In a neighborhood of peripheral equilibrium point, the Jacobian matrix J (r, ϕ) =   Pr Pϕ of system (2.3) consists of the elements Qr Qϕ Pr =

Qr =

s k=1

s k=1

2kA1k r 2k + (n − 2) B r n−2 cos (nϕ) , Pϕ = −n Br n−2 sin (nϕ) ,

A2k 2kr 2k−1 − (n − 2) Br n−3 sin (nϕ) , Qϕ = −n B r n−2 cos (nϕ) .

In the Hamiltonian case, peripheral points are centers or saddles for which T race (J (r, ϕ)) = 0,

Det (J (r, ϕ)) > 0, Det (J (r, ϕ)) < 0, correspondingly. In the general case (if coefficients ε1, A1k are small enough), saddles remain saddles but centers become spirals. It is possible to verify that these spirals are unstable if O(0,0) is unstable. Domains of repelling of all spirals are divided by separatrices (see Fig. 12 and 13).

4.3 Equilibria at Infinity (See Fig. 3a, b) Proposition 8 In the equator of Poincare sphere, equilibrium points of System (2.3) are alternating stable and unstable nodes for odd n; the same is true for even n if  2  2  2  2 B 2 > A1s + A2s . When B 2 < A1s + A2s , the system has no equilibria in the equator of the Poincare sphere. Proofs of Propositions and Theorems are given in Appendix 1.

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F. Berezovskaya and G. P. Karev

5 Discussion In this work, we studied complex Eq. (1.1) with ≥4, which describes behaviors of equivariant vector field, i.e., the vector field that is invariant under rotations by the specific angle 2π/n. More precisely, we worked with the Systems (2.3) (2.4) that are equivalent to (1.1); System (2.3) writes Eq. (1.1) in polar coordinates, and System (2.4) writes Eq. (1.1) in Cartesian coordinates. Notice that Systems (2.3) and (2.4) are Hamiltonian for specific (explicitly formulated) values of coefficients. We constructed and described the phase-parameter portraits of Eq. (1.1) as “perturbations” of phase-parameter portraits of its Hamiltonian version. The value of n (odd or even), as well as coefficients B and As , serves as the bifurcation parameters. The structures of portraits essentially depend on these parameters. We denoted the principle characteristics of considered portraits, which we referred to as “flower rings”; by definition, “n-flower ring” is a pattern consisting of n centroids and n saddles together with their separatrices; each center is placed inside the “leaf” composed by a separatrix cycle or separatrix loop (see Fig. 1). We showed that the number s of flower rings in the model (1.1) is s ≤ n2 − 1 if n is even and s ≤ n−1 2 − 1 if n is odd. We described also the possible sequences of appearance and disappearance of flower rings in the model under variations of various parameters. We compared the qualitative structures in the obtained portraits with key features of the ancient Greek ornamental designs that one can see in historical museums of Crete and Athens. In our opinion, some of the phase portraits of the model and the ornamental designs have many common characteristics. For this reason, one can consider Eq. (1.1) as a kind of mathematical “blueprint” for these ornaments. The ornaments are characterized by rings containing a certain number of points connected by spiral-like lines. This ornamental motif usually dating to the Bronze Age (1900–1700 B.C.) can be found in many artifacts presented in the Herakleion and Athens archeological museums. A remarkable property of these ornaments is the “dynamical indeterminacy,” known in the theory of dynamical systems: under a small change in the initial conditions of the “starting point,” trajectories lead to different regions, and so one can visit all regions of the ornament with a very small shift of an initial point. This peculiarity remind us the phase-parametric portraits of complex Eq. (1.1); notice that this equation was suggested by V. Arnold for investigation of essentially different problem, namely, resonances in mechanical systems [1, 2]. We found the main patterns of the phase portraits of this equation; some of them are similar to the ornaments, but the portraits may also contain other patterns that appear with parameter variation. Together with “spiral-like” lines and patterns, the ornaments may have patterns composed of closed cycles around centers. Such kinds of patterns usually arise in a Hamiltonian system. The “spiral-like” ornamental design may reflect the ancient philosophical idea of harmonic unity of the world, where various aspects of world phenomena are

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

17

interconnected. According to some authors (see, e.g., [12], ch. 2), ancient Greeks did not favor the idea of “progress.” In contrast, they believed that initially there existed a perfect “golden age,” and mankind, in their development down from the Golden Age, is destined to degenerate (Hesiod). This degradation may imply destroyed connections between different parts of the world, causing it to disintegrate into separate parts. The process may be reflected in ornaments by transition from “spirallike” to “center-like” patterns; we looked for a corresponding exhibit that shows such a transition and have found it in National Archaeological Museum, Athens (see corresponding photos of artifacts in [11]). Mathematically, this process corresponds to the transition from general Eq. (1.1) to Hamiltonian (1.1), (3.1), which happens when particular model coefficients vanish. Variation of model parameters results in the change of the phase-parametric portraits, whereby one can “animate the evolution” of ornaments and perhaps on some level reflect the idea of destruction of the “golden age” in the language of mathematics and art.

Appendix 1 Proof of Propositions 3 Coordinates (r± , ϕ± ) of peripheral equilibria E± of Hamiltonian system (3H) satisfy the system (3.4). So ϕj± = ±

2πj π + , j = 0, . . . , n − 1 2n n

(A1.1)

and r± are the roots of the polynomials P± where Q+ (r) = Q0 (r) − Br n−2 , Q− (r) = Q0 (r) + Br n−2 , B > 0, Q0 (r) = 2 +

s k=1

(A1.2)

A2k r 2k .

An even polynomial Q0 (r) of degree 2s, where s = n−3 2 ,has 0 ≤ m ≤ s changes of signs of its coefficients, so it has at most m positive roots (or less than m by an even number) due to Descartes theorem. Then one of the polynomials Q+ , Q− has at most m+1 changes of signs and the second has at most m. So, for m = s one of the polynomials Q+ , Q− has at most n−1 2 positive roots and the second has at most n−3 n−1 2 . Now we show that the number of saddles in the system is at most 2 ,and the n−3 number of centers is at most 2 . The second assertion of proposition is obvious (evidently, if r∗ is a root of P+ then −r∗ is a root of Q− ). The total number of positive roots of both polynomials is an odd number k ≤ n − 2. Note also that P+ (0) = P− (0) =  2 = 0, and the

18

F. Berezovskaya and G. P. Karev

equilibrium O(r = 0, ϕ = 0) is a center of the system. The system is Hamiltonian, so its equilibria can be only saddles and centers that alternates in each ray ϕ = ϕ ± + 2πj + n , j = 0, . . . , n − 1. Let r1 < r2 < . . . < rk be positive roots of polynomials Q , 1 − Q . Thus the “closest” to O peripheral equilibria Ej (r1, ϕj ) j = 0, . . . , n − 1 can be only saddles that together with their separatrices, the saddles compose separatrix cycle in (r, ϕ)-plane; the cycle contains inside the point O. Next, the largest root rk of polynomials P± also corresponds to a saddle because the number k is odd; together with their separatrices the saddles Ej k (rk, ϕj ) j = 0, . . . , n − 1 also compose separatrix cycle in (r, ϕ)-plane. We have proven that the system has at most n−1 2 separatrix cycles containing n n−3 saddles and at most 2 flower rings containing n peripheral centers.

Proof of Proposition 5 In this case Q (r, ϕ) = 2 +

s−1

2 2k k=1 Ak r

  + A2(n−2)/2 − B sin (nϕ) r n−2 . After

substitution sin(nϕ) = ± 1 we get even polynomials Q± (r, ϕ) of the form ±

Q (r, ϕ) = 2 + Q0 (r) = 2 +

s−1

A2 r 2k k=1 k

s−1

2 2k k=1 Ak r

Q±

  2 + A n−2 ∓ B r n−2 = Q0 (r) ∓ Br n−2 , (A1.3) 2

+ A2n−2 r n−2 ,B > 0, s = 2

n 2

− 1.

n−2 2

has at most real positive roots. So, the common Each even polynomial number of positive roots is at most n − 2. The equilibrium O(0, 0) is the center in the system, so the peripheral equilibria Ej 1 (r1, ϕj ), j = 0, . . . , n − 1 “closest” to O (if any) are saddles; together with their separartices they compose n-separatrix cycle in (r, ϕ)-plane. It has been discussed in the proof of Proposition 4 that saddles and centers are alternating in the (ϕj ) − rays, j = 0, . . . , n − 1. Let 0 ≤ m ≤ n−2 2 be the number of the changes of signs of coefficients of polynomial Q0 (r). It is evident that for A2n−2 >−B > 0 and for 0 < B < −A2n−2 2

2

both polynomials Q± have the same signs of coefficients of the highest exponent as Q0 (r). Then the coefficients of each polynomial Q± have m changes of signs, and so a number of positive roots equal to m or less by an even number; it means that the total number of positive roots in both polynomials is even. Then the number of saddles is equal to the number of peripheral centers. Note that the points Ej ∗ (r∗ , ϕj ), j = 0, . . . , n − 1 with the largest root r∗ are centers. The union of centroids corresponding to these centers composes n-flower ring. For |B| > A2n−2 one of polynomial Q− , Q+ “looses” one change of sign in 2

the sequence of coefficients. Then the number of positive roots of this polynomial becomes odd, so total number of positive roots in both polynomials is odd. Then the

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

19

number of saddles is one more than the number of peripheral centers. Note that in this case the points Ej ∗ (r∗ ϕj ), j = 0 . . . .n-1 with the largest root r∗ are saddles (see Fig. 4).

Proof of Proposition 8 Lemma 1 System (2.4) in equators of Poincare sphere (i.e., in coordinates (u, w) : x = 1/u, y =  w/u and (v, u) : x = v/u, y = 1/u) is equivalent to system (2.3) in coordinates ρ = 1r , ϕ . Proof of these statement follows from formulas: ϕ = arctan (w) = arccot (v),   w  √ u 2 and ϕ  = − v 2 , ρ  = , thus ϕ  = 1+w ρ =  1 2) , ρ = ( (1+v ) (1+w ) x 2 +y 2

√



u . (1+v 2 ) According to this lemma, we consider system (2.3) in (ρ, ϕ)-coordinates. Changing independent variable dt → ρ n − 2 dτ we get the system:

 s ρ  = −ρ 2 1 ρ n−2 +

k=1

ϕ  = 2 ρ n−2 +

 A1k ρ n−2−2k + B cos (nϕ) ≡ P i (ρ, ϕ) ,

s k=1

(A1.4)

A2k ρ n−2−2k − B sin (nϕ) ≡ Qi (ρ, ϕ) .

The equator of Poincare sphere is defined by ρ = 0. For n odd system (A1.4) can be written as ρ  = −ρ (B cos (nϕ) + o1 (ρ)) ≡ P ii (ρ, ϕ) ,

(A1.5)

ϕ  = −B sin (nϕ) + o2 (ρ) (ρ, ϕ) ≡ Qii (ρ, ϕ) where o1 (ρ), o2 (ρ) are polynomials such that o1 (0) = o2 (0) = 0. Thus, equilibria in the equator are Ij (0, ϕj ), where ϕj satisfy to the equation sin (nϕ) = 0, i.e., ϕj =

2πj , j = 0, . . . , n − 1. n

Notice that cos(nϕj ) = ± 1 for two neighboring j. Jacobian of the considering system     Pρii Pϕii −Bcos (nϕ) Bρsin (nϕ) J(ρ, ϕ)= . = ii Qii 0 −Bncos (nϕ) ρ Qϕ

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F. Berezovskaya and G. P. Karev

     B 0 −B 0 Then J(0, ϕj )= . , J 0, ϕj +1 = 0 Bn 0 −Bn Thus, one of these points is a stable node, and another is an unstable node. The first statement is proven. If n is even, then system (A1.4) in the Poincare’ sphere equators can be written in the form   ρ  = −ρ A1n−2 + B cos (nϕ) ρ + o1 (ρ) ≡ P ii (ρ, ϕ) , (A1.6) 2







ϕ =

A n−2 − B sin (nϕ) + o2 (ρ) ≡ Qii (ρ, ϕ) . 2

2

Equilibrium values of ϕ in system (A1.7) (if they exist) satisfy to the equation A2n−2 − B sin (nϕ) = 0. 2

  2 arcsin A n−2 /B + πj n , j =0, . . . , n − 1 if 2



2

A n−2 /B 1.



j This equation has real roots ϕj = (−1) n

2

2

It is easily to verify that in the second case the equator does not contain equilibria, and in the first case equilibria are alternated stable and unstable nodes. Indeed, J (0, ϕj ) =

   − A1n−2 + Bcos nϕj 2

⎛

⎛

0

⎜ − ⎝ A1n−2 ⎜ 2 =⎜ ⎜ ⎝

0



 −Bncos nϕj ⎞  2  ⎠ ± B 2 − A2n−2 2

0

⎞ 0

 ∓n A2n−2

⎟ ⎟ ⎟ ⎟ ⎠

.

2

Statements are proven. Propositions 4 and 6 can be easily proved by the same methods as Propositions 3 and 5 were.

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

21

Appendix 2: Phase Portraits of Several Hamiltonian Models

1 Fig 6 Phase-parameter portraits of Hamiltonian model (3H)   for ε1 = 0, A1 = 0, ε2 =1. 1 2 2 “center” A1 = 1, B = 0.7, 2a, 2b “center + spider-net”, 2a A1 = 1, B = 1.2); 2b (A21 = −1, B = 1.2). Portraits 2a and 2b are topological orbital equivalent, 3 “flower ring+spider-net” (ε2 = 1, A21 = −1, B2 = 0.7). The lines B = A21 , B = −A21 are parameter boundaries between domains

Fig 7 Phase portraits of Hamiltonian model (3H) for ε1 = 0, A11 = 0, ε2 = 1.1 “center+spidernet” (A21 = 1, B = 2), 2a, 2b “flower ring + star + spider-net”, 2a (A21 = −1, B = 0.3), 2b (A21 = −1, B = 0.2). Portraits 2a and 2b are topological orbital equivalent. The parameter boundary  3 between domains 1 and 2 has equation: C : 272 B 2 + 4 A21 = 0

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F. Berezovskaya and G. P. Karev

Fig 8 Phase portraits of Hamiltonian model (3H) for ε1 = 0, A11 = 0, A12 = 0.1, 2 “center   + spider-net”, 1 ε2 = 0, A21 = 1, A22 = 0, B = −0.5 ; 2 ε2 = 0, A21 = 1, A22 = 0, B = 0.5 ;  2 2 3 “centers + one flower ring”, ε2 = 1, A1 = −1, A2 = 0.1, B = 0.102 ;4  center+one flower ring” “center”, ε2 = 1, A21 = −1, A22 = −0.25, B = 0.05 ;5  ε2 = −1, A21 = −1, A22 = 0.04, B = 0.04 ; 6 “center + star + two flower rings”, (ε2 = − 1,  A21 = −1, A22 = 0.06, B = 0.1

Fig 9 Phase portraits of Hamiltonian model (3H) for 1 = A11 = A12 = A13 = 0,2 = −0.56;1 “center + two flower rings + spider-nets”, (A21 = 3, A22 = −3.5, B = −1.6 ; 2 “center +  one flower ring + spider − net" ; A11 = 0, A21 = 3, A22 = −3.5, B = −1

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

23

Fig 10 Phase portraits of Hamiltonian model (3H) for 1 = A11 = A12 = A13 = 0, 2 = 8, A21 = −33, A22 = 23.765, A23 = −3.5 and different B. (a) B = 0, (b) B = 0.1, (c) B = 0.045. To better distinguish placing and shapes of flower rings in the phase portrait, figure c is consisted of figures with different window scales: (x, y)[−7, 7] in upper window c, (x, y)[−3.2, 3.2] in window c1, (x, y)[−2., 2.] in window c2, (x, y)[−0.7, 0.7] in window c3

Fig 11 Phase portraits of Hamiltonian model (3H) for 1 = A11 = A12 = A13 = A14 = 0, 2 = 14.4, A21 = −55.6, A22 = 54.6, A23 = 14.4, A24 = 1; both portraits contain “center+two flower-rings”, (a) B = 0.05, (b) B = − 0.01

Fig 12 Phase portraits of non-Hamiltonian model (2.4) for n = 5,  1 = 0.005. “Flower ring” outside (a) and inside (b) of the limit cycle, (c) without a limit cycle, a: 2 = 1, A11 = −0.01, A21 = −1, B = 0.1; b: 2 = −0.1, A11 = 0.045, A21 = 1, B = 1; c: 2 = −0.1, A11 = −0.045, A21 = 1, B = 1

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

25

Fig 13 Phase portraits of non-Hamiltonian model (2.4) for n = 6,  1 = − 0.001. “Flower ring” of peripheral equilibria are placed outside (a) and inside (b) of unstable limit cycle, (a) 2 = −0.1, A11 = 1.3, A21 = 0.1, A22 = −0.1, B = 0.05. (b) 2 = 0.1, A11 = 1, A21 = −0.1, A22 = −0.1, B = 0.05

References 1. V.I. Arnold, Loss of stability of self-oscillation close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11, 85–97 (1977) 2. V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York, Heidelberg, Berlin, 1983) 3. F.S. Beresovskaya, A.I. Khibnik, On separatrices bifurcations in the problem of autooscillations stability loss at resonance 1: 4. Prikl. Math. Mech. 44, 663–667 (1980) 4. F.S. Beresovskaya, A.I. Khibnik, On the problem of bifurcations of self-oscillations close to a 1:4 resonance. Selecta Mathematica formerly Sovietica 13(2), 197–215 (1994) 5. R.I. Bogdanov, A versal deformation of a singular point of a vector field in the plane in the case of zero eigenvalues. Selecta Math. Sov. 1(4), 389–421 (1981). (Transl. from Russian, “Proceeding of Petrovskii Seminar” v.2 Moscow University, 37-65, 1976) 6. E.I. Horozov, Versal deformations of equivariant vector fields for the cases of symmetries of order 2 and 3. Trudy Sem. I. G. Petrovskogo 5, 163–192 (1979) 7. A.I. Neistadt, Bifurcations of phase portraits of a system of differential equations, arising in the problem of auto-oscillations stability loss at 1: 4 resonance. Prikl. Math. Mech. 42, 830–840 (1978) 8. B. Krauskopf, The bifurcation set for the 1:4 resonance problem. Exp. Math. 3(2), 107–128 (1994) 9. Y. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 1995) 10. A. Andronov, E. Leontovich, I. Gordon, A. Maer, Theory of Bifurcations of Dynamical Systems on a Plane (NASA TT F-556, Israel Program for Scientific Translations, Jerusalem, 1973) 11. F. Berezovskaya, G. Karev, Arnold Weak Resonance Equation as the Model of Greek Ornamental Design. arXiv preprint arXiv:1811.09880 (2018) 12. K.R. Popper, The Open Society and its Enemies, The Spell of Plato, vol I (Routledge and Kegan Paul, London, 1966)

Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments Alexander S. Bratus, Artem S. Novozhilov, and Yuri S. Semenov

Abstract We review the major progress in the rigorous analysis of the classical quasispecies model that usually comes in two related but different forms: the Eigen model and the Crow–Kimura model. The model itself was formulated almost 50 years ago, and in its stationary form represents an easy to formulate eigenvalue problem. Notwithstanding the simplicity of the problem statement, we still lack full understanding of the behavior of the mean population fitness and the quasispecies distribution for an arbitrary fitness landscape. Our main goal in this review is twofold: first, to highlight a number of impressive mathematical results, including some of the recent ones, which pertain to the mathematical development of the quasispecies theory. Second, to emphasize that, despite these 50 years of vigorous research, there are still very natural both biological and mathematical questions that remain to be addressed within the quasispecies framework. Our hope is that at least some of the approaches we review in this text can be of help for anyone embarking on further analysis of the quasispecies model. Keywords The quasispecies · Crow–Kimura model · Error threshold · Mean population fitness AMS Subject Classification 15A18, 92D15, 92D25

A. S. Bratus Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow, Russia A. S. Novozhilov () Department of Mathematics, North Dakota State University, Fargo, ND, USA e-mail: [email protected] Y. S. Semenov Applied Mathematics–1, Moscow State University of Railway Engineering, Moscow, Russia © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_2

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1 Introduction In 1971 Manfred Eigen published a groundbreaking paper “Selforganization of matter and the evolution of biological macromolecules” [15], in which he considered various aspects of the problem of the origin of life. As a part of his comprehensive approach he introduced a system of ordinary differential equations, nowadays commonly called the quasispecies model, which since then became one of the classical models in the field of mathematical biology. Despite its intrinsic simplicity (the model is “almost” linear) and despite almost 50 years of vigorous research, there are still open and deep mathematical questions about this model. Our goal in this review is to discuss both classical and relatively recent progress in the mathematical aspects of the analysis of the quasispecies model and also highlight some open problems. We exclusively concentrate on the mathematical side of the story and refer the interested reader to the earlier reviews [1, 16, 20, 31] and to the whole recent volume [13] devoted to the various other sides of the quasispecies theory. While the original quasispecies model was formulated in continuous time we start with a more natural discrete time settings. We consider a population of individuals such that there are l different types. Let ni (t) denote the number of individuals of type i at the time moment t. The individual reproduction success is described by the constant fitness coefficients wi ≥ 0, which we put together into the diagonal matrix W = diag(w1 , . . . , wl ) or vector w = (w1 , . . . , wl ) ∈ Rl , to both of which we refer as fitness landscape. Moreover, the reproduction is error prone such that the probability that an individual of type j begets an individual of type i is given by qij ∈ [0, 1], and hence we have the stochastic mutation matrix  Q = [qij ], where qii = 1 − lj =1,j =i qij is the probability of faithful reproduction. Now simple bookkeeping yields the following recurrence equation: ni (t + 1) =

l 

wj qij nj (t),

i = 1, . . . , l,

(1.1)

j =1

or, in the matrix form n(t + 1) = QW n(t),

 n(t) = n1 (t), . . . , nl (t) ∈ Rl .

(1.2)

Since the model (1.1), (1.2) is linear it is possible to have three different outcomes: either the total population size will explode to infinity, or tend to zero, or, in some exceptional cases, will stay constant. From the evolutionary point of view we are mostly interested in the population composition and therefore it is natural to consider the system for the corresponding frequencies p(t) = l

n(t)

i=1 ni (t)

,

Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred. . .

29

which takes the form p(t + 1) =

QWp(t) , w(t)

(1.3)

where w(t) =

l 

wi pi (t) = w · p(t)

i=1

is the mean population fitness that guarantees that p(t) is the probability distribution for any time moment t, that is, it belongs to the simplex Sl for any time moment: p(t) ∈ Sl = {x ∈ Rl :

l 

xi = 1, xi ≥ 0}.

i=1

The dot denotes the usual dot product in Rl . Before moving forward we note that if we assume that no mutations occur the wi system becomes pi (t + 1) = w(t) pi , and hence wi pi (t) pi (t + 1) = , pj (t + 1) wj pj (t) which immediately implies that, assuming without loss of generality that w1 is the strict maximum of the fitness landscape, p1 (t) → 1 and for all the rest pj (t) → 0. Moreover, the change in the mean population fitness is given by l

w(t) = w(t + 1) − w(t) =

i=1



2 wi − w(t) pi (t) 1 = Vart (w) , w(t) w(t)

which is arguably the simplest form of Fisher’s theorem of natural selection (see, e.g., [8] for the discussion and mathematical underpinnings of this “theorem”). That is, the behavior in case of no mutations is very simple: all the types of individuals except for the most fit one are being washed out from the population; moreover, the mean population fitness is increasing at each step, and the magnitude of the increase is proportional to the variance of the fitness landscape at each time moment. To be able to write an analogous system in continuous time one must separate the processes of selection (as described by the fitness coefficients) and mutation because, strictly speaking, only one elementary event can occur during a sufficiently small period of time. Hence, assuming that μij is the mutation rate of an individual of type j into an individual of type i, and mi ∈ R is the (Malthusian) fitness of individuals of type i (which is the difference of the birth and death rates and can be negative in this case), then the change of the population numbers is described by ˙ = (M + M)n(t), n(t)

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where we introduce the notations for the fitness landscape M = diag(m1 , . . . , ml ) and mutation matrix M = [μij ]. Similarly to the above, it is more natural (see, e.g., [21] for the discussion) to consider the equations for the frequencies, which in this case take the form  ˙ = M − m(t)I p(t) + Mp(t), p(t)

(1.4)

where m(t) =

l 

mi pi (t) = m · p(t)

(1.5)

i=1

is the mean (Malthusian) fitness of the population and I is the identity matrix. Model (1.4), (1.5) was dubbed by Baake and co-authors as a paramuse model, due to the parallel mutation selection scheme, see [1]. This model also got some treatment in an influential population genetic textbook by Crow and Kimura [11], and therefore is often called the Crow–Kimura model. Models (1.3) and (1.4) are intrinsically related since it can be shown [19] that model (1.4) is a limit for small generation time of model (1.3). The same limiting procedure helps relate the Wrightian and Malthusian fitnesses wi = emi t ≈ 1 + mi t,

t → 0,

and the mutation probabilities and corresponding mutation rates qij = δij + μij t,

t → 0,

where δij is the Kronecker delta. For model (1.4) also a version of the Fisher’s theorem of natural selection holds since, by elementary manipulations, ˙ m(t) = Vart (m) ≥ 0, and similarly only one type of individuals survives in the long run (assuming as before that there is a strict maximum of the fitness landscape). Both models (1.3) and (1.4) can be called the quasispecies models, but the fact is that Eigen in his 1971 paper considered yet another mathematical model, which takes the form (we use the notations introduced before, but now a care should be exercised since clearly in the continuous setting one needs to talk about rates and not probabilities) ˙ = QWp(t) − w(t)p(t). p(t)

(1.6)

Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred. . .

31

We are not aware of any elementary mechanical derivation of (1.6) but note that the equilibrium point of this model coincides with the fixed point of (1.3). Mathematical models (1.3), (1.4), and (1.6) share several common features. In particular, 1. Selection does not lead to a homogeneous population, as it happens in the systems with no mutations. More precisely, under some mild technical conditions necessary to apply the Perron–Frobenius theory for nonnegative matrices, models (1.3), (1.5), and (1.6) possess the only globally asymptotically stable equilibrium lim p(t) = p,

t→∞

which is the positive eigenvector of the eigenvalue problem QWp = λp,

(1.7)

for (1.3) and (1.6), and of the eigenvalues problem (M + M)p = λp,

(1.8)

for (1.4). Moreover, this eigenvector corresponds to the dominant real eigenvalue of the matrices QW and M + M, respectively, which is equal to the mean population fitness at the equilibrium p: λ = w = w · p,

or

λ = m = m · p.

This vector p was called by Eigen the quasispecies, which is the target of selection in these models. 2. The mean population fitness w(t) or m(t) are not necessarily increasing functions of time; that is, the evolution in these quasispecies models does not imply the steady climbing of the fitness landscape. (This was first noted, to the best of our knowledge, in [32], where it was shown that it is possible to have (a) w(t) is nondecreasing along trajectories, (b) w(t) is non-increasing along trajectories, and (c) w(t) may increase or decrease or even pass through extremum.) A detailed discussion and additional references to this quite frequent phenomenon can be found in [7]. 3. For some specific fitness landscapes there exists a sharp transition of the equilibrium distribution p as a function of mutational landscape which separates the phase where the quasispecies vector generally concentrates around the fittest type (the so-called selection phase) and the uniform distribution of the quasispecies (no selection, or random, phase). This transition was called the error threshold and have clear connections to the phenomenon of phase transition in statistical physics. We discuss this phenomenon in more details below.

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We emphasize that mathematically the asymptotic behavior of the quasispecies models boils down to the analysis of high-dimensional eigenvalue problems (1.7) and (1.8). Due to the Perron–Frobenius theory we always know that the quasispecies distribution exists, it is a different, and much more complicated, question how to calculate p and the mean population fitness for some specific cases of fitness landscape and mutation matrix. In the rest of this review we survey various specific types of matrices QW and M + M, for which at least partial information about the leading eigenvalues and the corresponding quasispecies eigenvectors can be obtained. Our presentation is necessarily biased since, especially in the second part of the review, we concentrate on our own results, which are discussed, together with detailed proofs, at length in [6, 33–36].

2 Sequence Spaces and Exact Solutions To make progress in mathematical analysis of eigenvalue problems (1.7) and (1.8) one needs to specify the structure of the matrices involved. We deliberately did not specify what we understand by “individual type” in the previous section. In the language of population genetics the models considered above describe evolution in one-locus haploid population with l alleles. Since Eigen was interested primarily in the problem of the origin of life, his interpretation of “individual type” was quite different. By analogy with RNA and DNA molecules he considered the population of sequences of fixed length N , where each different sequence is composed from the letters of some finite alphabet. For example, one can take the four letter alphabet of nucleotides {A, T , G, C}, and therefore there will be l = 4N different sequences in the population or 20-letter alphabet of amino acids, and hence there will be l = 20N different polypeptides. The simplest choice, however, is to consider initially two letter alphabet {0, 1} and hence deal with the population of binary sequences of fixed length N having total l = 2N different sequence types. Such underlying space of binary sequences possesses a nice geometric interpretation: Different types of binary sequences of length N are in one-to-one correspondence with the vertices of an N -dimensional hypercube (see Fig. 1). Moreover, now we can, under some additional assumptions, describe in much more detail the corresponding mutation matrices. Namely, assume that mutations occur independently at each site of the sequences with the same probability q and introduce the Hamming distance Hij between sequences of types i and j . For the following it is convenient to use the lexicographical order of different sequences such that sequence of type i, where i = 0, . . . , 2N − 1, has exactly the binary representation of integer i, supplemented, if necessary, by additional zeros. Now, the elements of the mutation matrix Q are given by qij = (1 − q)N −Hij q Hij ,

i, j = 0, . . . , 2N − 1,

that is, matrix Q is defined in terms of only one scalar parameter q.

(2.1)

Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred. . .

33

Fig. 1 The binary sequence spaces for sequences of length N = 2, 3, 4, 5. The vertices of the hypercube correspond to various sequences, which are listed along the x-axis, and the Hamming distance between any two sequences is equal to the least number of edges connecting the corresponding vertices

In continuous time (model (1.4)) the mutations during short time interval are only possible to the neighboring sequences, and therefore the mutation rates are given by

μij =

⎧ ⎪ ⎪ ⎨μ,

−Nμ, ⎪ ⎪ ⎩0,

Hij = 1, Hij = 0,

(2.2)

Hij > 1.

Using the introduced sequence spaces, the eigenvalue problems (1.7) and (1.8) now depend only on the fitness landscapes W and M and mutation parameters q and μ, respectively, and hence it is customary to emphasize this dependence by writing p(q) and w(q) or p(μ) and m(μ) for the equilibrium quasispecies distributions and mean population fitnesses. We emphasize that in the rest of this review we talk exclusively about the stationary distributions and do not discuss any time-dependent aspects of the quasispecies evolution.

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3 Permutation Invariant Fitness Landscapes and Error Threshold We posed in the previous section the main mathematical question regarding the quasispecies theory, namely how given matrices W or M give an exact and/or approximate expressions for leading eigenvalue and, if possible, the corresponding positive eigenvector p of the eigenvalue problems (1.7) and (1.8) with the mutation matrices (2.1) and (2.2), respectively, as the functions of the mutation probability q and the mutation rate μ. One of the main reasons why this question turned out so complicated in general is the possibility of the phenomenon, which was called the error threshold. First we illustrate this phenomenon numerically, following the simplifications that were originally introduced in [37]. Note that even if the alphabet we use to compose our sequences has only two letters, the total number of different types of sequences is 2N , which becomes unrealistically large for numerical computations even for modest values of N . Swetina and Schuster [37], to overcome this difficulty, suggested to use the socalled single peak fitness landscape, in which one of the sequences, called the master sequence, is assigned the higher value of the fitness constant, and everyone else is “equally inferior,” that is w = (w + s, w, w, . . . , w),

or

m = (m + s, m, . . . , m),

where constant s > 0 is the selective advantage of the master sequence. In both cases the master sequence was chosen to be composed of all zeros. The single peak fitness landscape is an example of what we call permutation invariant fitness landscapes, for which the fitness of the given sequence is determined not by the sequence itself but by the total number of ones in this sequence (or, more mathematically rigorous, by the Hamming norm of the sequence, which, by definition, is the distance from this sequence to the zero sequence, Hi := H0i ). In this way the total population is divided now into classes of sequences, which are characterized by the Hamming distance from the zero sequence, such that j -th class contains exactly Nj types of sequences. Now, if we are able to modify our mutation matrix accordingly, then the dimension of the problem is reduced from 2N × 2N to (N +1)×(N +1) since there are exactly N +1 different classes of binary sequences of length N . The easiest way to do it is for the continuous time model (1.8), (2.2). Indeed, let νij be the rate of mutations from class j to class i. Assuming again that μ is the rate of mutations per one site per time unit, we have ⎧ ⎪ ⎪ ⎪(N − j )μ, ⎪ ⎨j μ, νij = ⎪−Nμ, ⎪ ⎪ ⎪ ⎩ 0,

i = j + 1, i = j − 1, i = j, otherwise,

Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred. . .

35

since only one elementary event is possible per small time interval, and mutation in any of the 0 sites yields a sequence in class j + 1, whereas mutation in any of the 1 sites yields a sequence in class j − 1. Therefore, for the permutation invariant fitness landscapes and Crow–Kimura quasispecies model (1.4) the eigenvalue problem (1.8), (2.2) takes the form (M + N )p = λp,

(3.1)

where, as before, M = diag(m0 , . . . , mN ) (we abuse the notations here, by using the same letter to denote now the fitness of the j -th class, a more correct, and uglier, notation would be mHj ), and N = [νij ](N +1)×(N +1) , which has an especially simple tri-diagonal form ⎡

−N 1 0 ⎢ N −N 2 ⎢ ⎢ ⎢ 0 N − 1 −N N = μS = μ ⎢ ⎢ ... ... ... ⎢ ⎣ 0 0 ... 0 0 ...

0 0 3 ... ... ...

... ... ... ... 2 0

... ... ... ... −N 1

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥. ... ⎥ ⎥ N ⎦ −N

(3.2)

For the eigenvalue model (1.7), (2.1) the bookkeeping of the mutation probabilities is slightly more involved and leads [25] to the eigenvalue problem W Rp = λp,

(3.3)

with W = diag(w0 , . . . , wN ), the fitness landscape of sequence classes, and matrix R = [rij ], where rij is the probability that a sequence from class j mutates into sequence of class i, explicitly [25] rij =

min{i,j } a=j +i−N

     j N − i N 1 − q i+j −2a q , a j −a q

i, j = 0, . . . , N.

(3.4)

Here is a numerical illustration how the equilibrium quasispecies distribution p(q) changes with respect to q for different sequence lengths and the single peak fitness landscape W = diag(10, 1, . . . , 1), see Fig. 2. It is quite clear that if the sequence length increases there appears a threshold value of the mutation probability, after which the quasispecies distribution of sequence classes does not change and remains binomial, which means that the distribution of the sequence types is (almost) uniform after this critical mutation rate. This phenomenon is called the error threshold. Right at this point we note that this particular phenomenon is landscape dependent, and at least for some fitness landscapes does not manifest itself [38].

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Fig. 2 The quasispecies vector of the eigenvalue problem (3.3), (3.4) depending on the mutation probability q per site for different sequence lengths N . The fitness landscape is W = diag(10, 1, . . . , 1). Note the different scales for different graphs

There is a heuristic way to derive the value of the error threshold for the given w, s, N. Basically one assumes that no backward mutations to the fittest class are possible, and then the first differential equation for the master sequence takes the form  p˙ 0 (t) = (w + s)r00 − w(t) p0 (t), w(t) = w + sp0 (t). We have r00 = (1 − q)N , and hence the condition (w + s)(1 − q)N = w implies “extinction” of the master sequence (we put the word “extinction” into the quotes since, strictly speaking, the quasispecies models are written for the frequencies and hence it makes no sense to discuss the phenomenon of extinction within the framework of these models). The last equality can be manipulated as $ w q∗ = 1 − N , w+s which for the example above for N = 100 gives the threshold value q ∗ ≈ 0.0227,

Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred. . .

37

which is very close to the sharp transition observed in Fig. 2. The same relation can be manipulated as follows: N∗ =

log σ , q

σ =

w+s , w

(3.5)

which is often interpreted as the critical condition on the attainable length of a polynucleotide sequence given the mutation probability. Again, jumping ahead, we note that there exists a rigorous proof of the formula (3.5). We note, however, that this formula applies only to the case of the single peak landscape, and therefore it is dangerous to make conclusions that sequence length is inversely proportional to the mutation probability. In [36] we conjectured that for the general quasispecies model (1.7), (2.1) the error threshold mutation rate q ∗ , if it exists, can be determined by the formula %  & N 2 −1 wi 1& N N w(0.5) ∗ 1−q = = ' i=0 . w(0) 2 maxj {wj } Among other things, this expression shows that the inverse relationship of the sequence length and critical mutation probability does not pertain to any possible fitness landscape, for more details see [36]. Very similar picture is observed for the Crow–Kimura quasispecies model. In particular, the formula (3.5) turns into N∗ =

s , μ

for the single peak landscape M = (m + s, m, . . . , m), s > 0. We reiterate that it is generally not applicable to other possible fitness landscapes.

4 The Only Exact Solution and Isometry Group of the Hypercube Here we return to the general eigenvalue problems (1.7), (2.1) and (1.8), (2.2), i.e., we do not assume here that the fitness landscape is permutation invariant. The discussed above phenomenon of the error threshold indirectly implies that it is quite naive to expect that even for such simple fitness landscapes, such as single peak landscape, we expect to find simple explicit formulas for the mean population fitness and quasispecies distribution. There is, however, one special case, for which exact solution can be written for a specific general fitness landscape. This observation is due to Rumschitzki [27], who noticed that in general for the mutation matrix Q can be found simple decomposition Qk = Q1 ⊗ Qk−1 ,

k = 2, . . . , N,

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where ( ) 1−q q Q1 = , q 1−q and ⊗ is the Kronecker or tensor product of matrices (e.g., [22]). This decomposition immediately implies that all the eigenvalues and eigenvector of matrix Q can be found in terms of easily analyzed matrix Q1 . To include also the fitness landscape, let us define the following matrices: Wk =

) ( 1 0 , 0 sk

k = 1, . . . , N.

If we define recursive procedure (QW )k = (Q1 W k ) ⊗ (QW )k−1 ,

k = 2, . . . , N,

where (QW )1 = Q1 W 1 , then at the N -th step we obtain that (QW )N = QW , where Q is given by (2.1) and matrix W is diagonal, with elements wii =

*

sk ,

i = 1, . . . , 2N − 1,

w00 = 0,

k : ak =1

assuming that index i has the binary representation i = [a1 , a2 , . . . , aN ],

ak ∈ {0, 1}.

Since the matrix QW is given again as N -fold Kronecker product, then all the eigenvalues and eigenvectors can be calculated using the eigenvalues and eigenvectors of 2 × 2 matrices Q1 W k . Biologically this exactly solvable case corresponds to the multiplicative fitness landscape, when the fitness of a given sequence is represented by the multiplicative (independent) contribution of all the sites with 1’s. It is customary in this case to speak of no epistasis fitness landscape, that is, the cumulative effect of all the sites is given by independent contributions of each site. Similar formulas can be written for the Crow–Kimura model (1.8) with (2.2), in this case the fitness landscape becomes additive, that is, again, fitness of a given sequence is given by independent contributions of each site. All these formulas can be directly generalized to more than two letters in the sequence alphabet and to non-uniform mutation probabilities (rates) along the sequences, providing a false sense of generality. What is more important however, and what was clearly observed in [14], the potential solvability of the quasispecies model is directly related to

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the isometry group of the underlying finite metric space, which can be identified with N -dimensional hypercube, presented in Fig. 1. It is interesting to note that the connections of the Eigen eigenvalue problem and the isometry group of the hypercube were not explored further until very recently.

5 The Ising Model, the Maximum Principle, and the Hamilton–Jacobi Equation Approach In the same year when the paper by Rumischitzki [27] was submitted for publication, it was noted in [23] that the eigenvalue problem (1.7), (2.1) was already studied for one specific fitness landscape in the disguise of the transfer matrix of two-dimensional Ising model of statistical physics [26]. On the one hand, this observation emphasized the nontriviality and complexity of analytical analysis of the quasispecies model, and on the other hand, it opened the gates for a stream of papers that used statistical physics methods to analyze different reincarnations of the quasispecies model (see, e.g., [4, 17, 24], and especially [3] and the references therein). The methods borrowed from statistical physics usually imply some infinite sequence limit under an appropriate scaling of the model parameters, and hence the results are mostly asymptotical, contrary to the exact results that can be obtained with the formulas from the previous section. In line with the tradition from statistical physics these asymptotical results are very often called “exact” in the literature, which should be kept in mind while studying such approaches. The methods of statistical physics indeed allowed significant progress in the analysis of the quasispecies model, as it is discussed at length in a very detailed and accessible paper [3]. Most importantly, they caused a number of researches to formulate and prove, by rigorous mathematical methods, what is now called the maximum principle [2, 5, 18]. The maximum principle applies to the models (3.1), (3.2) and (3.3), (3.4), that is, to the permutation invariant fitness landscapes, and allows to obtain, under some technical conditions, an approximation for the mean population fitness. For model (3.1), (3.2) it takes the following form (we note that we do not try to formulate the most general form of the maximum principle): Assume that the permutation invariant fitness landscape can be represented as mi = Nf (xi ),

xi =

i ∈ [0, 1], N

√ and define function g(x) = μ − 2μ x(1 − x). Then it can be proved that the mean population fitness m(μ) satisfies  m(μ) ≈ N sup f (x) − g(x) . x∈[0,1]

(5.1)

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For example, assume that we deal with single peak landscape of the form M = diag(N, 0, 0, . . . , 0), hence f (x) = 1 if x = 0 and f (x) = 0 if x > 0. Simple calculations imply that in this case + N(1 − μ), μ < 1, m(μ) = 0, μ ≥ 1, which also provides a proof of the threshold mutation rate for the single peak landscape discussed earlier. This approximation provides a remarkable agreement with numerical computations (see, e.g., [6]). An analogous result holds for the classical model (3.3), (3.4). It was obtained initially in [29], and rigorous proof can be found in [2]. It is important to emphasize at this point that although the maximum principle is an extremely powerful device to analyze various fitness landscapes, it only applies to permutation invariant landscapes, and also relies on several technical conditions, the most important of which is the continuity of function f : more precisely, in [18] it was assumed that f must have only finite number of discontinuities and be either left or right continuous at every point; in [2] the continuity of f was required. We are still far from understanding the realistic general fitness landscapes, but it is almost universally accepted that in many cases the evolution proceeds with huge leaps, and at least some fitness landscapes are essentially discontinuous. There are simple examples (see below) that show that a formal application of the maximum principle in the case of discontinuous fitness landscapes can lead to erroneous conclusions [6, 39]. Another mathematical approach to the quasispecies models with permutation invariant fitness landscapes is based on the limiting procedure that transform the original system of ordinary differential equations into one first order partial differential equation of Hamilton–Jacobi type. There is a recent review of results obtained with this approach [30]; therefore, here we just mention the main idea. Again, for simplicity, we only present the results for model (3.1), (3.2). In terms of total population numbers the Crow–Kimura model with tri-diagonal mutation matrix can be written as n˙ i (t) = mi ni (t)−N μni (t)+(N −i+1)μni+1 (t)+(i+1)μni−1 (t), Now we introduce the ansatz ni (t) = eN u(t,x) ,

x =1−

2i , N

mi = Nf (x),

i = 0, . . . , N.

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41

and use formal Taylor series. After dropping the terms of higher order with respect to 1/N we obtain the equation 1 − x 2 ∂u ∂u 1 + x −2 ∂u = f (x) − μ + μ e ∂x + μ e ∂x , ∂t 2 2

−1 ≤ x ≤ 1,

for some given initial condition u(0, x). We are not aware of any rigorous proof that would justify the discussed above limit; we note however that assuming that this equation indeed holds in the limit of infinite sequence length it is quite straightforward to obtain the maximum principle (5.1) [39], which implies that at least some continuity conditions of f should be required.

6 Linear Algebra of the Quasispecies Model Having discussed the maximum principle and the Hamilton–Jacobi equation, we remark that in both approaches first some infinite sequence limit is taken and after it the problem at hands is analyzed. In the papers [6, 36] we undertook a somewhat different approach, in which we first rigorously manipulate the corresponding eigenvalue problems and only after we take the limit N → ∞. It turns out that in this way it is possible to obtain new results, which cannot be derived by the maximum principle or the Hamilton–Jacobi equation. As we discussed above it is quite straightforward to calculate the eigenvalues and eigenvectors of mutation matrices (2.1), (2.2), (3.2), and (3.4). Moreover, as these calculations show, the corresponding eigenvectors always form a basis for Rl or RN +1 . Therefore, the basic idea that we used was to rewrite the full eigenvalue problem in the basis of the eigenvectors of the mutation matrices. In particular, in [6] we analyzed the problem (3.1) and found that it is possible to derive a parametric solution to this problem in the form pi (s) = Fij (s),

F (s) = mj Fjj (s),

μ=

s F (s), 2

m(s) = F (s),

(6.1)

where s is some parameter, −N

Fij (s) = 2

N  cik ckj , 1 + ks k=0

and matrix C = [cij ] is composed of the eigenvectors of matrix S defined in (3.2). It may look that this specific parametric form of the solution to the eigenvalue problem (3.1) does not simplify the situation. It turns out, however, that, given an explicit form of the fitness landscape M, these formulas allow us to make further analysis, and in particular, consider, under an appropriate scaling, the limiting procedure N → ∞. Here are two examples, the proofs can be found in [6].

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Proposition 6.1 Consider the eigenvalue problem (3.1), (3.2). If the fitness landscapes is the single peak landscape M = diag(N, 0, . . . , 0) then, for μ < 1, m(μ) = 1 − μ, N →∞ N lim

lim pi (μ) = (1 − μ)μi ,

N →∞

i = 0, . . . , N.

(6.2)

If the fitness landscape is M = diag(0, . . . , 0, N, 0, . . . , 0), where N = 2A and the only non-zero rate exactly at the position A, then m(μ) = r∞ = lim N →∞ N



 μ2

+ 1 − μ,

pA±k ≈

1 − r∞ 1 + r∞

k .

(6.3)

Several remarks are in order. First, we do not claim that the result in (6.2) is new. To the best of our knowledge, the expression for m/N was derived originally in [17] and, as we showed above, elementary follows from the maximum principle. In [28] the geometric distribution of the quasispecies vector was derived using some heuristic methods. We provided a rigorous proof illustrating our parametric solution, and also gave an estimate of speed of convergence to this distribution. Second, the example (6.3) is not a single peak landscape because here the whole class of sequences has the same fitness N , recall that we consider the case of permutation invariant fitness landscape. In Fig. 3 it can be seen that approximation is quite good even for moderate values of N . Also, as our analysis predicted, in this case there is no error threshold. Finally, this example cannot be treated by both the maximum principle and the Hamilton–Jacobi approach, because the limit fitness function is neither left or right continuous at x = 1/2. This shows, among other things, that the approach to manipulate first the eigenvalue problem and only after it take the limit N → ∞ is, at least in some situations, a more general one. We obtained some other interesting results based on parametric solution (6.1), but probably the most important consequence of its analysis was a heuristic approach

Fig. 3 Comparison of the numerical solutions (gray, dashed) of the eigenvalue problem (3.1) and approximations in (6.3) (black) for N = 100. (a) Mean population fitness, (b) the quasispecies vector of frequencies of sequence classes

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Fig. 4 A numerical illustration of Theorem 6.2: A typical picture of the qualitative behavior of w(q). Here N = 3, the fitness landscape is W = diag(10, 3, 3, 2, 3, 2,√2, 1), w(0) = 10, w(1) = 10

that allowed actually compute the quasispecies distribution for a wide variety of fitness landscapes. We discuss this approach in the next section. We undertook a similar approach in [36] to analyze model (1.7), (2.1). As a result we obtained a number of proofs for the results whose validity was generally acknowledged; however, this acknowledgement was based mostly on numerical computations. We note that q as defined in [36] is equal to 1 − q as defined in Sect. 2. In particular we prove [36]. Theorem 6.2 Consider the eigenvalue problem (1.7), (2.1). If the fitness landscape W is such that the leading eigenvalues w(0), w(1) as functions of the mutation probability q have multiplicities 1, then there exist an absolute minimum wˆ of function w(q) for 0.5 ≤ q < 1. The point of this minimum is determined by the condition w  (q) = 0. For q ≤ 0.5 function w(q) ˆ is nondecreasing and convex (Fig. 4). There are a number of other exact results in [36], but, similar to the previously discussed work, probably the most important result of this work was the general idea how to advance an analysis of non-permutation invariant fitness landscapes. We discuss these ideas in Sects. 8 and 9.

7 Formulas for the Quasispecies Distribution While the maximum principle provides a powerful tool to analyze the behavior of the mean population fitness, there are very few results of explicit computations of the quasispecies distribution, an example (6.2) notwithstanding. In [33] two of us suggested a general heuristic procedure how to derive a limiting quasispecies distribution for the model (3.1). In [33] we noticed that the matrix S in (3.2) is exactly the matrix in the standard polynomial basis of the linear differential operator S : P (s) → (1 − s 2 )P  (s) − N (1 − s)P (s),

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and hence the eigenvalue problem (3.2) can be written in the form m ◦ P (s) + μ(1 − s 2 )P  (s) − μN (1 − s)P (s) = mP (s), N i where m ◦ P (s) = i=0 mi pi s . By dividing by N and taking the formal limit N → ∞ we end up with r ◦ P (s) − μ(1 − s)P (s) = rP (s),

r=

m , N

r=

m . N

(7.1)

Now we conjecture (but did not prove rigorously) that if for some given r one can solve Eq. (7.1) with the conditions r ◦ P (1) = r, P (1) = 1, then P (s) is the probability generating function of the equilibrium quasispecies vector. For example, if we take r = (1, 0, . . . , 0), that is, the single peak landscape, then r ◦ P (s) = P (0) and hence (7.1) takes the form −μ(1 − s)P (s) + P (0) = rP (s). Plugging s = 0 into the last expression and assuming P (0) = 0 we find immediately r = 1 − μ, that is, the same expression as in (6.2). Using the condition P (1) = 1 yields immediately P (s) =

1−μ , 1 − μs

which is the probability generating function of the geometric distribution with the parameter μ (see again (6.2)). Assuming that our approach valid we can prove the following general fact: Lemma 7.1 Assume that r = (r0 , r1 , . . .) such that r0 > ri , i = 1, 2, . . .. Then r = r0 − μ,

pi = ,i

μi p0

j =1 (r0

− rj )

,

i > 0,

p0 =

1+

∞

1 μk j =1 (r0 −rj )

.

k=1 ,k

These formulas provide a solution for the quasispecies distribution in the limit case N → ∞ if and only if % &* & n ∗ n μ < μ = lim inf ' (r0 − rj ). j =1

The critical value μ∗ gives the error threshold of the mutation rate of the permutation invariant Crow–Kimura model.

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Fig. 5 A numerical illustration of Lemma 7.1: Numerical solution (gray) with N = 200 is compared with the analytical predictions for the fitness landscape m = N (2, 0, 1, 0, 1, . . .). Note the error threshold and √ its theoretical prediction 2

For example, for the fitness landscape r = (2, 0, 1, 0, 1, 0, . . .) we immediately conclude that the error threshold we have μ∗ =

√ 2,

and p0 =

2 − μ2 , 2+μ

p 1 = p0

μ , 2

...,

see the numerical illustration in Fig. 5. We finish this section noting that in a series of recent papers [9, 10, 12] Cerf and Dalmau analyzed a more complicated case of model (3.3) and (3.4). As a result they obtained, in the form of infinite series, explicit solution for the quasispecies distribution for the permutation invariant Eigen quasispecies model (3.3) and (3.4).

8 Two-Valued Fitness Landscapes and Isometry Group of the Hypercube Let us reiterate that now we have very powerful tools to analyze permutation invariant fitness landscapes with the help of, e.g., the maximum principle or the explicit formulas discussed in the previous section. Much less is available to tackle non-permutation invariant fitness landscapes, which are, of course, much more biologically realistic. In [34] two of us considered a special case of the full eigenvalue problem (1.7), (2.1) with the fitness landscape that we dubbed twovalued fitness landscape. Let A be a non-empty subset of indices A ⊆ {0, 1, . . . , l − 1}. We fix two constants w ≥ 0 and s > 0 and consider the fitness landscape of the form + wk =

w + s, k ∈ A, w,

k∈ / A.

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One of the main theoretical results can be then formulated as follows (for a proof, which is based on the analysis in [36], see [34]). Lemma 8.1 Let X be the finite metric space on indices {0, 1, . . . , 2N − 1} with the Hamming distance between the binary representations of indices (i.e., geometrically the N dimensional hypercube, see Fig. 1), let G be a group that acts on X by isometries (i.e., G ≤ Iso(X)), and let A in the definition of the two-valued fitness landscape be a G-orbit. Then the mean population fitness w(q) is a root of algebraic equation (with coefficients depending on q) of degree at most N + 1. We note for the readers that are not closely familiar with the language of a group action, in [34] there is a section with completely elementary discussion of the terminology and results. Also we remark that we give an explicit form for the algebraic equation for w(q). Here are just two examples. First, let us consider again the classical single peak landscape W = diag(w + s, w, . . . , w). The group G here is simply the trivial group. Then the equation that determines w is given by N   (2q − 1)d 1  N 1 = . N d w − w(2q − 1)2 2 s d=0

A very similar expression for a slightly different model was obtained originally in [17]. Of course, single peak landscape is an example of a permutation invariant fitness landscape. Here is an example that is not permutation invariant. Let  G = Q8 = ±1, ±i, ±j, ±k | i 2 = j 2 = k 2 = −1, ij = k, j k = i, ki = j be the classical quaternion group of order 8. Consider the embedding Q8 → S8 where we choose i → (0212)(4657), j → (0415)(2736). As a G-orbit we can, for instance, take A = {7, 11, 13, 14, 112, 176, 208, 224} ⊂ X,

N ≥ 8.

It can be seen immediately that this fitness landscape is not permutation invariant. We can show that, e.g., for N = 8, the leading eigenvalue of (1.7), (2.1) is determined by 8  d=0

Rd (2q − 1) 64 , = w − w(2q − 1)d s

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where R0 = R8 = 2, R1 = R7 = 1, R2 = R6 = 14, R3 = R5 = 15, R4 = 0. Equally easy to write an algebraic equation for an arbitrary N . In the nutshell, the geometry of the underlying hypercube turned out to be crucial in determining the cases, when significant simplification of the original 2N × 2N eigenvalue problem can be made. This fact was originally noticed in [14], but did not get subsequent development until our recent paper [34].

9 Abstract Quasispecies Model Having studied the two-valued fitness landscapes discussed in the previous section, we asked a very natural mathematical question: Why to focus all the attention on the hypercube? From mathematical point of view nothing precludes us from considering the following generalized quasispecies model [35]. Let (X, d) be a finite metric space with integer valued metric d. Consider a group  ≤ Iso(X) of isometries of X that acts transitively on X. Since it acts transitively, we can pick any point x0 ∈ X and consider the function dx0 : X −→ N such that dx0 (x) = d(x0 , x). By definition, the diameter of X is N = diam X = max{dx0 (x) | x ∈ X}. Also consider a fitness function w : X −→ R≥0 which can be represented as a vector with nonnegative components. Together with the introduced notations consider fitness matrix W = diag(w), symmetric mutation matrix Q = [q d(x,y) (1− q)N −d(x,y) ] for q ∈ [0, 1], and the distance polynomial  pX (q) = q d(x,x0 ) (1 − q)N −d(x,x0 ) , x0 ∈ X. x∈X

Now we call the problem to find the leading eigenvalue w(q) and/or the corresponding positive eigenvector p of the eigenvalue problem QWp = pX (q)wp the generalized algebraic quasispecies problem. It turns into classical Eigen’s problem (1.7), if X = {0, 1}N is the N -dimensional binary cube with the usual Hamming metric, in this case pX (q) = 1. We gave an extensive treatment of the generalized quasispecies problem in [34, 35], here we would like to state just one specific result. Namely, consider a generalization of two-valued fitness landscape in the form that the fitness function w is constant on each G-orbit, where G ≤  and has at least two values. Consider the decomposition X0 = A0 

t i=1

Ai ,

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Fig. 6 Complete graphs on 3–10 vertices, representing the mutational landscape with simplex geometry

such that A0 is the union of G-orbits on which w(A0 ) = w ≥ 0, and w(Ai ) = w + si , si > 0. Then the following theorem can be proved (see [35]). Theorem 9.1 The dominant eigenvalue of all the examples of the generalized quasispecies eigenvalue problem considered in [35] can be found as a root of an algebraic equation of degree at most t · (N + 1) = t · (diam(X) + 1). Moreover, this equation can be written down in the explicit form. Arguably the simplest possible generalized quasispecies model is generated by the geometry of simplex, which can be represented as a complete graph (see Fig. 6). In this case we deal with a finite metric space X of diameter N = 1, that is, we consider the case in which individuals of a population can mutate into any other individual with the same probability q. We note that this abstraction can be actually considered as a mathematical description of the switching of antigenic variants for some bacteria. Let A ⊆ X such that w(A) = w + s. Then, according to the theorem above, the leading eigenvalue of the generalized quasispecies problem can be found as a root of a quadratic equation, which (see [34] for a derivation) takes the form   |A| 2q − 1 |A| + 1− = 1, (n + 1)(u − u) n + 1 (q + n(1 − q))u − (2q − 1)u

u=

w , s

u=

w . s

Here |A| is the cardinality of set A, and n = |X|. That is, this case turns out to be significantly simpler than the classical quasispecies problem. Moreover, it can be proved that for this simplicial landscape the error threshold exists (see the discussion and rigorous derivations in [34]). In a similar manner other possible geometries can be analyzed, see, for instance, the analysis of regular m-gon and hyperoctahedron mutational landscapes in [35].

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10 Concluding Remarks As we discussed above, since Manfred Eigen’s original paper [15] a lot of mathematical peculiarities of the quasispecies model were rigorously analyzed. At the same time we would like to conclude our presentation with a number of still open mathematical questions. • In Sect. 7 we presented a heuristic algorithm to compute the quasispecies distribution for the permutation invariant Crow–Kimura model. A proof of validity of this approach is still missing. • There exist a few sufficient conditions on the fitness landscape for the error threshold to exist (e.g., [39]). We are not aware of any necessary and sufficient conditions of this sort. • While the abstract results on the generalized quasispecies model discussed in Sect. 9 are of significant interest, our paper [35] considers only the examples in which X = A0  A1 . It is important to consider examples with more complicated partition of X (for instance, important mesa-landscapes [39] have exactly this more complicated form). • In terms of the generalized quasispecies model it would be interesting to study the following question: What are the properties of X and the distance function d that guarantee that at least for some fitness landscapes the error threshold exists. This question also has some direct connections with various forms of the Ising model. A list of open mathematical questions about the now classical quasispecies model can be easily extended, and it is our hope that at least for some of these problems methods and approaches discussed in this review can be of some help. Acknowledgment ASB’s research is supported in part by Russian Science Foundation grant #19-11-00008.

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A Survey on Quasiperiodic Topology Roberto De Leo

To Sergey P. Novikov, in occasion of his 80th birthday, with deep respect and gratitude.

Abstract This article is a survey of the Novikov problem of the structure of leaves of the foliations induced by a collection of closed 1-forms in a compact manifold M. Equivalently, this is to the study of the level sets of multivalued functions on M. To date, this problem was thoroughly investigated only for M = Tn and multivalued maps F : Tn → Rn−1 in three different particular cases: when all components of F but one are multivalued, started by Novikov in 1982; when all components of F but one are singlevalued, started by Zorich in 1994; when none of the components is singlevalued, started by Arnold in 1991. The first two problems can be formulated as the study of the level sets of certain quasiperiodic functions, the last as level sets of pseudoperiodic functions. In this survey we present the main analytical and numerical results to date and some physical phenomena where they play a fundamental role. Keywords Quasiperiodic functions · Quasiperiodic topology · Closed 1-forms · Foliations · Multivalued functions

1 Introduction About a century ago, Morse [93] discovered a deep, beautiful, and far-reaching relation—what we now call Morse theory [92]—between topology and analysis, namely the fact that many fundamental topological information are encoded in the R. De Leo () Department of Mathematics, Howard University, Washington, DC, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_3

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set of critical points of each “generic enough” function (Morse function). In other words, given a topological space M, several of its fundamental properties—that, thanks to a celebrated result of Gelfand and Naimark [63], are known to be fully encoded inside C 0 (M)—can also be extracted from elements of the space of much more regular functions C ∞ (M). We recall that Morse theory is a much more powerful tool for the category of compact manifolds than for the one of non-compact ones, due to the much larger geometrical and topological freedom the latter ones enjoy. Nevertheless, it is reasonable to think that there exist some “well-behaving” classes of non-compact manifolds that are close enough, in some sense, to the compact case that some strong general geometrical and topological property holds for them. A trivial example of such class is the set of periodic submanifolds of Rn , since they are all regular covers of compact submanifolds of Tn and therefore can be treated with the (more powerful) tools of Morse theory for compact manifolds. At the beginning of the 1980s, Novikov [95–97] identified a class that represents a natural first step from the compact to the non-compact case, namely manifolds Mˆ that are the covering space p : Mˆ → M, with discrete group , of a compact ˆ of functions whose differential df manifold M and the linear subspace of C ∞ (M) is invariant by —a class clearly modeled on the paradigmatic case of Rn → Tn = Rn /Zn and the subspace of C ∞ (Rn ) of pseudoperiodic functions (see Sect. 3). From a more algebraic point of view, this corresponds exactly to extending Morse theory from functions to closed 1-forms. Indeed, while such a function f on Mˆ does not generically descend to a function on M (since, for a generic point x ∈ M, it has different values at the points γ · x for all γ ∈ ), the exact 1-form df on Mˆ does descend to a well-defined closed 1-form ω = p∗ (df ) on M; inversely, if ω is a closed 1-form on M, there exist some abelian covering p : Mˆ → M where p∗ ω is exact. In Novikov’s terminology, f (or, by abuse of language, ω) is a multivalued function on M and so the corresponding Novikov-Morse theory can be also simply thought as a Morse theory for multivalued functions. The Novikov-Morse theory has two main goals [98]: 1. To estimate the number of critical points mi (ω) with Morse index i for a closed 1-form ω on M. 2. To investigate the topological structure of the leaves of the foliation ω = 0 on M, i.e. the level sets of the multivalued function f . The present article is a survey of the second point, developed since the 1980s mainly by Novikov and his Moscow topology school. We refer the reader interested in the first one to the excellent monographs on the subject by Farber [53] and Pajitnov [105]. The survey is structured as follows. In Sect. 2 we present the main results on the Novikov general problem of the structure of level sets of multivalued functions on a compact manifold M. One of the main results is that such level sets are quasiperiodic manifolds, namely they are the (generally infinite) union of a finite number of compact components. From Sect. 3, and throughout the rest of the paper, we set M = Tn . Following Arnold [11], we call multivalued functions on Tn

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pseudoperiodic—note that each pseudoperiodic function is the sum of a linear function and a periodic one, see Sect. 3. Arnold’s problem, developed jointly with D.A. Panov and I.A. Dynnikov, is studying the topology of the level lines of a pseudoperiodic map f : Tn → Rn−1 with all multivalued components (namely, the linear part of this map has full rank). Section 4 is dedicated to a very rich and important particular case, namely the level lines of a pseudoperiodic map f : Tn → Rn−1 having exactly one single-valued component and all remaining pseudoperiodic components with linear part only. This case was introduced by Novikov, that extracted the case n = 3 from solid state physics literature, and studied it from several points of view jointly with his pupils A. Zorich, I.A. Dynnikov, S. Tsarev, A. Maltsev and the present author, and more recently with A. Skripchenko. Besides having a surprisingly rich and complicated topological and geometrical structure, this case is strictly related to the physics of metals and, in particular, it is essential for the theoretical first-principle prediction of some important experimental data concerning the magnetoresistance of metals. In this section we present the main analytical and numerical results and some experimental data of the 1950s that were compared with the theory only recently by the present author, thanks to those results. Finally, in Sect. 5 we consider a case somehow “dual” to the Novikov one in Sect. 4, namely the case of multivalued maps f : Tn → Rn−1 where all but one components are pseudoperiodic functions with linear part only. This case was introduced by A. Zorich in the 1990s as a generalization of the aforementioned problem of Novikov for n = 3, that corresponds to the study of foliations induced on a Riemann surface i : Mg2 → T3 by the 1-form ω = i ∗ B, where B is a constant 1-form on the torus. Clearly a closed 1-form ω built this way is very far from being generic. Increasing the dimension of the torus (with respect to the genus of the surface) makes the 1-form, loosely speaking, closer and closer to being generic, allowing the use of more powerful techniques. Even in this case, the asymptotics of the leaves turned out to be topologically very rich. The main contributions to this field are due to Zorich, M. Kontsevich, and A. Avila.

2 Quasiperiodic Manifolds One of the two main goals of the Novikov-Morse theory is studying the structure of the level sets of multivalued functions or, equivalently, of the leaves of foliations induced by closed 1-forms. Before getting to the main results of this section, we list a few important results on this topic pre-dating the Morse-Novikov theory. All these results tend to be mostly about how the leaves wind inside the space rather than on the structure of the leaves; moreover, the authors seem to be unaware of any application of the subject outside of the field of geometry. We will need throughout the article the following definition: Definition 1 The degree of a closed 1-form ω on a compact manifold

of irrationality

M is irr ω = dimQ  γ1 ω, . . . , γr ωQ , namely the dimension over Q of the vector

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space spanned in R by κi = γi ω, where the γi are a base for H1 (M, Z) and r = dimR H1 (M, R) is the first Betti number of M. Clearly the irrationality degree of ω ranges from 1 to r. We say that ω is rational if irr ω ≤ 1 and fully irrational if irr ω = rk H1 (M, Z). Remark 1 The definition of irr ω is not universally accepted. An alternate definition, e.g. used by I.A. Melnikova, is calling irrationality degree of ω the quantity that, in our terminology, is irr ω−1 (e.g., see [91]) and several authors actually switched from one definition to the other over the years. Moreover some other authors, e.g. M. Farber, rather call irr ω the rank of ω (e.g., see [54]) and use the notation rk ω. The first fundamental result in this field is perhaps the following result of Kolmogorov [71] on the 2-torus, generalized to T3 by Arnold [12] in 1992 and to any Tn by Kozlov [73] in 2006: Theorem 1 ([71]) Any closed 1-form ω without critical points on T2 is smoothly conjugate to a constant 1-form ω0 , namely there exists a diffeomorphism f of T2 in itself such that f ∗ ω = ω0 . Definition 2 A closed 1-form is Morse if it is locally the differential of a Morse function. In the 1940s Maier [83] proved the following important result on closed surfaces, generalized later to any dimension in 1998 by Farber, Katz, and Levine [54]: Theorem 2 ([83]) Any closed Morse 1-form ω on a closed surface M decomposes it in several connected components of two types: periodic components Ki (M, ω) and minimal components Mj (M, ω). Periodic components are union of compact level sets of ω, while every non-singular level set in a minimal component is dense in it. Example 1 Consider the case M = T2 and ω constant. Then, when irr ω ≤ 1, K(T2 , ω) = T2 (and so M(T2 , ω) = ∅) while, when irr ω = 2, M(T2 , ω) = T2 (and so K(T2 , ω) = ∅). At the end of the 1970s, Imanishi [68] showed an important result that somehow generalizes the basic example above: Theorem 3 ([68]) A Morse form on a closed manifold M n , n ≥ 2, defines a foliation with only two types of leaves: closed and locally dense. If the foliation has a locally dense leaf, then irr ω ≥ 2. If ω has no zeros of index 1 and n − 1, then either K(M, ω) = M (and irr ω ≤ 1) or M(M, ω) = M (and irr ω ≥ 2). This result, in particular, completely describes the topology of the foliation defined on a manifold M n by a closed Morse 1-form without zeros. According to the following important result of Tischler [115], though, the existence of such 1-form has strong consequences on the topology of M n : Theorem 4 ([115]) Suppose that M admits a closed 1-form without zeros. Then M is a fiber bundle over S1 .

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None of these results aimed to ascertain the structure of the level sets themselves, neither the authors appear aware of the many connections of this field with other parts of mathematics and other sciences. In particular, the problem of the magnetoresistance in normal metals (see Sect. 4.1.1) specifically requires the study of such structures, which is one of the reasons for Novikov’s interest in the problem. Starting from the 1980s, he and his Moscow topological school have found several fundamental results that we briefly recall in the rest of this section. Definition 3 ([96]) We say that a manifold L is periodic when it is a Zk -covering of a compact manifold for some integer k ≥ 1. Theorem 5 ([96], [117]) The non-singular leaves of a closed 1-form ω without zeros and with irr(ω) = k are periodic manifolds with monodromy group Zk−1 . Proof Outline The idea of the proof is the following. Since the (cohomology classes of) rational 1-forms are dense in H 1 (M, R), we can approximate ω with a rational one ω0 as closely as we please and, moreover, we can choose a basis in H1 (M, R) so that irr(ω − ω0 ) = k − 1 and Ann(ω0 ) = γ2 , . . . , γr , Ann(ω) = γk+1 , . . . , γr . 1 Now, the rational 1-form ω0 induces a Morse function over

the circle fω : M → S , 2π i

ω0

γx0 ,x , where γx0 ,x is any e.g. by fixing any x0 ∈ M and setting f0 (x) = e smooth curve joining x0 and x. Indeed the topology of the leaves won’t change if we multiply ω by any non-zero constant and so we can assume WLOG that all the κi are integers, so that, by choosing a different path from x0 and x, the integral of ω will change by an integer. Since f has no critical points, f : M → S1 is a fiber bundle and we can consider the pull-back bundle Mˆ = exp∗ M → R, where exp : R → S1 is the exponential map, so that exp(F (m)) ˆ = f0 (π(m)), ˆ where π : Mˆ → M is the Z-covering canonical projection. In other words, in Mˆ the 1-form ω0 is exact: π ∗ ω0 = dF . Since in Mˆ we “unbundled” the cycle γ1 , then irr π ∗ ω = k − 1 and, since ω and ω0 are “close enough,” the leaves of π ∗ ω, which are diffeomorphic to those of ω, project along the trajectories induced by any Riemannian metric onto those of π ∗ ω0 with monodromy group Zk−1 . In other words, the leaves of ω are Zk−1 -coverings of some compact manifold.  

When the closed 1-form ω has critical points, the situation becomes much more complicated and leads to the quasiperiodicity of leaves. Definition 4 ([96, 99]) Let {W1 , . . . , Wp } a finite collection of k − 1-dimensional manifolds with (k − 1)-stratified boundary, namely with maps σj : Wj → ∂I k−1 onto the boundary of the unit cube and transversal along all sub-cubes of lower dimension. A sequence j : Zk−1 → {1, . . . , p} is admissible iff we can glue the manifold Wjn1 ,...,nk−1 along all neighboring faces of the (k − 1)-lattice. The sequence j is quasiperiodic iff it is quasiperiodic as a function (in the sense of Weyl1 ). Finally, j is special-quasiperiodic if jn1 ,...,nk−1 is defined as the index of the open set Ui the  point x0 + k−1 i=1 ni ui belongs to, where {U1 , . . . , Us } are disjoint open sets such 1 See

[29] for a short survey on the several types of almost- and quasi-periodic functions.

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that Tk−1 = ∪si=1 Ui , x0 a point on the torus and u1 , . . . , uk−1 vectors such that  x0 + k−1 i=1 ni ui never falls on a boundary point of the Ui . It was conjectured by Novikov in [96] that non-singular leaves of closed 1-forms with zeros would be quasiperiodic manifolds. The works of Zorich [117], Le Tu [74], and Alanya [1] around the end of the 1980s, the first in case of small perturbations of rational closed 1-forms and the second two in the general case, proved the following stronger result: Theorem 6 (Zorich (1987), [74], [1]) The non-singular leaves of a closed Morse 1-form ω are special quasiperiodic manifolds. Proof Outline The idea for the general case is the following. Given a closed 1-form ω with irr ω = k, we can always find a basis {ω1 , . . . , ωr } of H 1 (M, Z), dual to a basis {γ1 , . . . , γr } of H1 (M, Z), such that ω = ki=1 κi ωi and ωi , γj  = 0 for j ≥ k + 1 and i ≥ k. Similarly to what done above, the forms {ω1 , . . . , ωk−1 } induce a map f : M → Tk−1 and therefore a minimal covering π : Mˆ → M where π ∗ ω is exact and a map fˆ : Mˆ → Rk−1 such that exp(fˆ(m)) ˆ = f (π(m)). Given k−1 and basis vectors {e , . . . , e any generic point x ∈ R 0 1 k−1 }, the inverse images f −1 (I0k−1 + x0 + k−1 n e ) provide exactly the finite number of manifolds the i=1 i i leaves of ω can be built with.   To the knowledge of the author no progress has been done to date in case of level sets of 2 or more closed 1-forms since the following conjecture [98, 99]: Conjecture 1 ([98]) The generic common level of any number of closed 1-forms is a quasiperiodic manifold. Before moving to expose in more detail the specific results relative to the level sets of multivalued functions on n-tori, we end this section by exposing several important results on foliations of closed Morse 1-forms found in the same years by other research groups. In the 1980s, Henˇc [67] studied foliations defined by closed Morse 1-forms cohomologous to one without zeros: Definition 5 The foliation induced on a manifold M by a closed 1-form ω is minimal when the whole M is a minimal component, namely M(M, ω) = M. Theorem 7 ([67]) Let ω0 be a closed 1-form without zeros inducing a minimal foliation on M. Then the foliation induced on M by any closed Morse 1-form ω cohomologous to ω0 has exactly one minimal component. In particular this happens when irr ω0 ≥ 2. Arnoux and Levitt [14] proved in the same years, though, that this condition is not necessary: Theorem 8 ([14]) In any cohomology class with irrationality degree at least 2, there is a closed Morse 1-form ω defining a minimal foliation. On the other hand, Levitt [75] studied a class of Morse forms, which he called weakly complete, that, similarly to closed 1-forms without zeros, satisfy the

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condition of defining a foliation without minimal components when irr ω ≤ 1 or a minimal foliation otherwise. In particular, he showed the following: Theorem 9 ([75]) Let M n be a manifold and corank(π1 (M n )) be the maximum rank of a free quotient group of its fundamental group π1 (M n ). Then, if corank(π1 (M n )) = 1, all closed Morse 1-forms on M n without zeros of index 0 or n are weakly complete. A further general result, complementing the result by Arnoux and Levitt cited above, was found in the 1990s by Melnikova [91], that showed that all closed Morse 1forms with high enough irrationality degree have a minimal component: Definition 6 We say that subgroup H ⊂ Hn−1 (M n , Z) is isotropic if the intersection pairing Hn−1 (M n , Z) × Hn−1 (M n , Z) → Hn−2 (M n , Z) is identically zero restricted to it. We denote by h(M) the largest rank of isotropic subgroups of Hn−1 (M n , Z). Example 2 If n = 2, H1 (Mg2 , Z)  Z2g and the largest subgroup over which the intersection product is zero has rank g. Theorem 10 ([91]) If ω is a closed Morse 1-form on M n and irr ω ≥ h(M n ), then ω has a non-compact leaf. In particular, any closed Morse 1-form ω on a surface Mg2 has a non-compact leaf if irr ω > g. The following interesting results were found at the end of the 1990s by Farber et al. [54]. Definition 7 (Calabi, 1969 [26]) A closed 1-form is transitive if, for any x ∈ M where ω(x) = 0, there exists a smooth loop γ : S1 → M based at p such that ω(γ˙ (t)) > 0 for all t. Theorem 11 ([54]) Let M be a compact manifold such that H1 (M, Z) has no torsion and the cup-pairing product H 1 (M, Z) × H 1 (M, Z) → H 2 (M, Z) is non-degenerate and let ω be a closed transitive Morse 1-form with maximal irrationality degree. Then M(M, ω) = M. In particular, this shows that all non-singular leaves of a fully irrational transitive Morse closed 1-form on Tn (namely, without center-type singularities) are dense in the whole space Finally, at the beginning of the new century, I. Gelbukh generalized and refined several of the previously mentioned results. In [57, 58] she generalized Theorem 2 by bounding the number k(ω) of periodic components Ki (M, ω) and the number m(ω) of minimal components Mj (M, ω) of the foliation of Morse form on a manifold of arbitrary dimension:

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Theorem 12 ([58]) For a closed Morse 1-form ω with q zeros on a manifold M, k(ω) + m(ω) ≤ b1 (M) + q − 1, k(ω) + 2m(ω) ≤ b1 (M) + q − 1, where b1 (M) = corank(π1 (M)) is the co-rank of the fundamental group, and b1 (M) is the first Betti number. She developed methods for calculating b1 (M) in [61] and showed in [62] that b1 (M) ≤ h(M) and that the inequality can be strict, e.g. in case of the Heisenberg 3-nilmanifold or the Kodaira-Thurston nilmanifold. In particular, for n-torus, b1 (Tn ) = 1, this allows to generalize Example 1 to arbitrary dimensions and arbitrary closed Morse 1-form: Example 1 Consider a closed Morse 1-form ω on Tn . Then K(Tn , ω) = Tn (and so M(Tn , ω) = ∅) if and only if irr ω ≤ 1. Moreover, in the same articles she generalized Theorem 10: Theorem 13 ([58]) If irr ω > b1 (M), then the foliation has a minimal component. The next result [59] is somehow a converse to a part of Theorem 3: Theorem 14 (Gelbukh (2011)) If the foliation of a closed Morse 1-form has no minimal components, then there exists a closed Morse 1-form ω with irr ω ≤ 1 defining the same foliation. Finally, she made considerable progress in the topology of foliations induced by closed Morse 1-forms on surfaces [60]: Theorem 15 (Gelbukh (2011)) Let ω be a closed Morse 1-form on a closed orientable surface Mg2 and let c(ω) be the maximum number of homologically independent compact non-singular leaves of ω. Then c(ω) +

 i

g(V (Li )) + g

. j

 Mj (M, ω) = g,

where V (·) is a small closed regular neighborhood and Li are the compact singular leaves of ω.

3 Pseudoperiodic Functions From this moment on, we will assume throughout the paper that M n = Tn . Following Arnold [11], we call pseudoperiodic the multivalued maps defined on Tn . By abuse of notation, we will use the same letter to indicate a pseudoperiodic map Tn → R and the corresponding singlevalued map Rn → R.

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Proposition 1 Any pseudoperiodic function f is the sum of a linear function Sf with a periodic one εf . Proof Outline Let f : Tn → R be pseudoperiodic. Then f is a singlevalued function on Rn and his differential df descends to a closed 1-form ωf on Tn . Let Bf be a constant 1-form such that [ωf ] = [Bf ] in H 1 (Tn , R). Then ωf − Bf is exact, namely ωf − Bf = dεf for some singlevalued function f : Tn → R. In Rn , Bf = dSf for a linear function Sf , so f = Sf + εf modulo constants.   Arnold refers to Pseudoperiodic Topology as the study of level sets of pseudoperiodic maps [13] and sets the beginning of this field to the studies on vector fields on tori started by H. Poincaré. For example, in this language the Kolmogorov-ArnoldKozlov theorem mentioned in the previous section reads as follows: Theorem 16 ([12, 71], Kozlov (2006)) For any pseudoperiodic function f on Tn without critical points, there is a global diffeomorphism F of Tn such that εF ∗ f = 0. The following general proposition reflects the fact that clearly every level set of a closed 1-form ω = B + d with B = 0 lies between two planes of the foliation B = 0 and that, while the topology of a pseudoperiodic manifold “at a small scale” can get quite complicated, it is nevertheless very similar to a plane from a “global” point of view: Proposition 2 Every level set of a non-degenerate pseudoperiodic function f : Rn → Rk is contained in a finite radius cylinder about the n−k-plane Sf = Sf−1 (0) and has at least one non-compact component. Since the level set of any non-zero Sf are single hyperplanes, we also have the following: Proposition 3 For a generic linear function S and periodic function ε on Rn , the level sets of the pseudoperiodic function fλ = S + λε have a single non-compact component for λ small enough. A first non-trivial result on level sets of pseudoperiodic functions that are not necessarily small perturbations of a linear function was found by Arnold in [11], as a first step studying the level lines of pseudoperiodic maps f : Tn → Rn−1 . Theorem 17 ([11]) The level set of a generic pseudoperiodic function on the plane contains precisely one unbounded component. The case n > 2 was studied by Dynnikov in [42]: Theorem 18 ([42]) Let a be a regular value of a non-degenerate pseudoperiodic map f : Tn → Rn−1 and let ρa the radius of the smallest cylinder centered at Sf = Sf−1 (a) containing f −1 (a). Then there is an odd number of non-compact component of f −1 (a) and each one is a finite deformation of Sf . Moreover, if the direction Sf is fully irrational, supa∈R ρa is finite and the number of connected components in f −1 (a) does not depend on a (Fig. 1).

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10 4 5

2

0

0

–2 –5 –4 –10 –10

–5

0

5

10

–4

–2

0

2

4

 Fig. 1 (Left) Sections of the periodic function f (x) = 5i=1 cos(x i ) with a 2-plane spanned by v = (1, 0, 0.2, 0.3, 0.15)√and w = (0, 1, 0.05, 0.1, 0.4). (Right) Level sets of the pseudoperiodic function F (x, y) = y + 2 x + cos(2π x) + cos(2πy)

Unlike what happens for n = 2, even in the generic case the number of non-compact components of the level sets of a pseudoperiodic map f : Tn → Rn−1 can be any positive odd number, as an example of Panov [106, 107] shows: Theorem 19 ([106]) For every n ≥ 3 there is an open set (in the space Rn × C ∞ (Tn ) of n-vectors and n-periodic functions) of pseudoperiodic maps f : Tn → Rn−1 having level sets with k = 2m + 1 non-compact lines for any integer m ≥ 0. Open Questions What can be said about the number and structure of the level sets of a pseudoperiodic map Tn → Rk ?

4 Quasiperiodic Functions on the Plane To date, the richest topological properties have been found when are present at the same time periodic and pseudoperiodic functions. The case when f = (ε, S1 , . . . , Sn−2 ) : Tn → Rn−1 , where ε is singlevalued and the Si are purely linear pseudoperiodic functions, was introduced by Novikov in 1982 [96] and leads to the field of quasiperiodic functions. Indeed, the image in Rn of the line sets of f coincides with the line sets of the restriction of ε to affine 2-planes defined by the equations {S1 = a1 , . . . , Sn−2 = an−2 }, namely (see the definition below) the line sets of a quasiperiodic function on the plane with n quasiperiods. Quasiperiodic functions in one variable started appearing naturally in the theory of Ordinary Differential Equations, in particular in completely integrable Hamiltonian systems, since the end of the nineteenth century while it was only in the 1970s that multivariable ones were found for the first time, by S.P. Novikov,

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in concrete applications, specifically in the theory of integrable Partial Differential Equations (see [48] for several other examples and references). They were first introduced in literature by the Latvian mathematician Bohl [23], in the context of the theory of differential equations, and by the French astronomer Esclangon [50], who introduced the terminology, but they become widely known to the mathematical community only in the 1920s through the works of Bohr [24], Besikovich [21], Bochner [22] and other authors, who further extended this class of functions to almost periodic functions: Definition 8 The set AP (Rn ) of almost periodic functions over Rn is the closure in Cb (Rn , C), the set of bounded continuous complex functions over Rn , of the set of trigonometric polynomials with respect to the supremum norm. For every f ∈ AP (Rn ), we define its frequency module M(f ) as the Z-module generated by all “frequencies vectors” ν ∈ Rn such that 1 T →∞ (2T )k lim

/ [−T ,T ]n

e−iν,x f (x)dx = 0.

We say that f is quasiperiodic if M(f ) is finitely generated and we call the generators quasiperiods of f . Equivalently, we say that f : Rn → C is quasiperiodic with m > n quasiperiods if f = ϕ ◦ πm ◦  for some ϕ ∈ C 0 (Tm , C) and some affine embedding  : Rn → Rm , namely if f is the restriction to an n-plane of a periodic function on Rm . √ Example 3 Consider the function f (x) = cos(2π x) + cos( 2 2π x). Clearly f is a non-periodic √ almost periodic function. Its frequency module is generated over Z by 1 and 1/ 2, so f is quasiperiodic with two quasiperiods. For example, f can be √ seen as the restriction of ϕ(x, y) = cos(2π x) + cos(2πy) to the affine line y = 2x. Note that, if f  = ϕ ◦ πm ◦  with  parallel to , then, for some c ∈ Rn , we have that f  (x) = f (x + c) for all x ∈ Rn . Although in general we cannot find a c such that f (x + c) = f (x) for all x, we can get as close as we please to this condition: Theorem 20 ([24]) A function f ∈ Cb (Rn , C) is almost periodic iff there exists a relatively dense set of ε-almost periods for every ε > 0, namely for every ε > 0 there is a relatively dense set Tε ⊂ Rn such that |f (x + a) − f (x)| < ε for all a ∈ Tε . The main general results on the topological structure of level sets of quasiperiodic functions are due to Gusein-Zade [65, 66] and have to do with the topological invariants for the level sets Vc = f −1 (c) and lower sets Mc = f −1 ((−∞, c]) of a generic quasiperiodic function.

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As mentioned in the introduction, for both compact and periodic manifolds are defined fundamental topological numbers such the Betti numbers and the Euler characteristic. Moreover, in case of the level and lower sets of a Morse function on a compact manifold, these numbers would be piecewise constant functions of the parameter c. On the contrary, such numbers are all infinite for a generic non-compact manifold or, correspondingly, for the level and lower levels of a generic function on a non-compact manifold. In case of quasiperiodic manifolds, though, it is still possible to give a meaning to such quantities by considering the corresponding averaged quantities in the following way, that we write in case of the Euler characteristic: Definition 9 Let R > 0, BR ⊂ Rn the ball of radius R centered at the origin and f : Rn → R a Morse function. Then the density of critical points of f of order k is Nfk (c) = lim

r→∞

Nfk (BR ∩ Vc ) V ol(BR )

and similarly are defined the density of Euler characteristic χf (c) and of Betti numbers bfk (c) (resp. χ˜ f (c) and b˜fk (c) ) of Mc (resp. Vc ). Theorem 21 ([66]) For any analytic (resp. generic smooth) quasiperiodic function f which is the restriction to an n-plane of an m-periodic function, the density Nfk (c) is a well- defined piecewise analytic (resp. piecewise smooth) function of c. Moreover, in the neighborhood of every singular point c0 , f has an asymptotic expansion of the form  m−n−1  α

Ai,α |c − c0 |α lnq |c − c0 | ,

q=0

where α ranges on some finite segment of an arithmetic progression of rational numbers with difference equal to 1. Theorem 22 ([66]) For all smooth (resp. almost all smooth) quasiperiodic functions f , χf (c) =

 (−1)k Nfk (c). k≥0

Moreover, χ˜ f (c) = 0 for n even and χ˜ f (c) = 2χf (c) for n odd. Finally, χf (c) =  k bk (c) when the bk (c) are all well defined. (−1) k≥0 f f It is still an open question whether or not the density of Betti numbers is well defined under the same conditions holding for the Euler characteristic: Conjecture 2 ([66]) For all analytical (resp. almost all smooth) quasiperiodic functions, the Betti numbers densities are well defined.

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This conjecture was proved in 2000 by Esterov [51] in the smooth case for the Betti numbers densities of Mc : Theorem 23 ([51]) For almost all smooth quasiperiodic functions, the Betti numbers densities bfk (c) are well defined and depend continuously on c.

4.1 Quasiperiodic Functions in R2 with Three Quasiperiods The case n = 3 is the one that Novikov extracted from solid state physics literature (see [96]) and that attracted his interest in the general problem of levels of quasiperiodic functions. In this section we present in detail the physics of the model from which the problem originated, in which played a relevant role the KharkovMoscow school of solid state physics (among which I. Lifshitz, M. Azbel and M. Kaganov), the analytical results, found mainly by Novikov and his topology school in Moscow, and the numerical results by the present author.

4.1.1

Physics

The solid state physics basic model for a normal metal is a lattice of ions  ⊂ R3 with rank 3 surrounded by a gas of free electrons. In order to get manageable equations for this model, electrons are considered independent from each other, so that it’s enough to study the problem for a single one. The quantum Hamiltonian operator H = − + V on L2 (R3 ), where is the Laplacian and V the potential energy, corresponding to the ions lattice commutes with the action of  on R3 , so that all of its eigenstates ψn satisfy a relation ψn (x + γ ) = exp(2π ip, γ )ψn (x) for some covector p ∈ (R3 )∗ clearly defined modulo  ∗ , the dual lattice of  (Bloch theorem, e.g., see Ch. 8 in [15]). The corresponding eigenvalues εn (p) of H (energy bands) therefore are functions over (R3 )∗ /  ∗  T3 . The electrons responsible for the conductivity of the metal correspond to a particular n = n0 and a particular energy E0 . The Riemann surface {ε(p) = E0 } → T3 , where we set ε = εn0 , is called Fermi Surface (FS) of the metal and the labels p are called quasimomenta of the electrons. Even in the free electrons approximation, studying the Schrodinger equation after adding a weak (enough not to perturb the lattice) external magnetic field is known to be a very hard problem already in two dimensions (see Novikov’s results in [94] and [104]). In particular, no exact magnetic analogue of Bloch waves is known when the magnetic flux is irrational. This hard obstacle led physicists to introduce several alternate models to study phenomena involving external fields in metals. In the late 1950s, Lifshitz et al. [80] (from the Kharkov-Moscow school of solid state

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physics), in order to explain some anisotropies found experimentally in the angular dependence of magnetoresistance on the direction of an external magnetic field, proposed using a semiclassical approximation for the system. Under this approximation, electrons are treated as “classical particles” but with T3 , rather than R3 , as phase space. Under a constant magnetic field B = B i dpi , the equations of motions are given by the classical ones: p˙ i = {pi , ε(p)}B , where {pi , pj }B = ij k B k is the “magnetic” Poisson structure on T3 and ij k is the Levi-Civita tensor. This dynamical system turns out to be quite non-trivial and related to the problem of level sets of quasiperiodic functions with three quasiperiods. The bracket {, }B , indeed, has a Casimir, namely a function commuting with all other functions, represented by the “function” S(p) = B k pk , but clearly S on the phase space T3 is multivalued. Notice that very few Hamiltonian and Poissonian systems with multivalued first integrals have been studied by the dynamical systems community to date, probably because very few of them originate from classical mechanics. Notice also that the orbits of the quasimomenta of the electrons under these equations of motion are given by the intersection of the FS by planes perpendicular to the magnetic field, namely they are the level sets of a multivalued map f = (ε, S) : T3 → R2 with the first component singlevalued. The relevance of the geometry and topology of the FS in physical phenomena is well known since the 1930s, when Justi and Scheffers showed evidences that the Fermi Surface G of gold is open [69]—namely, in more formal terms, that the rank of the rings homomorphism i∗ : H1 (G, Z) → H1 (T3 , Z) induced by the embedding i : G → T3 is larger than 0 (in fact, it is maximal). The model introduced in [80] suggested that the behavior of magnetoresistance in monocrystals at low temperatures in high magnetic fields would also be quite sensitive to the FS topology: as the intensity of the magnetic field B grows, the magnetoresistance saturates isotropically to an asymptotic value if the orbits are all closed while it grows quadratically with B if there are open orbits; moreover, in this last case the magnetoresistance is not isotropic and the conductivity tensor σ has rank 1. To these first theoretical predictions of the Lifshitz model (see Fig. 2) followed many other works studying magnetoresistance from both the experimental [2– 9, 55, 108] and theoretical [27, 28, 78–82, 109, 112, 113] point of view; in particular, Stereographic Maps (SM) were experimentally built by plotting in a stereographic projection all magnetic field directions B in which a quadratic rise of σ was observed (see Fig. 3). In those times the interest on these magnetoresistance effects was due mainly to their utility as a tool to determine FS properties rather than as phenomena in their own right, and in particular SM maps provided information

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Closed FS n1 = n2 Δρ

n1 = n2 Δρ

ωc τ = 1H 2

Open FS H = const

θ1 θ2

θ

θ = θ2 Δρ

Δρ

Δρ

ωc τ = 1H 2

θ = θ1

ωc τ = 1H 2

ωc τ = 1H 2

Fig. 2 Behavior of ρ = σ −1 in metals with closed and open FS [77]. (Closed) ρ is isotropic and saturates unless the density of electrons and holes coincide, in which case ρ ∼ B 2 . (Open) ρ is highly anisotropic and it shows qualitatively different behavior in minima and maxima (resp. θ2 and θ1 in the picture): in maxima ρ ∼ H 2 , in minima it saturates (θ is the angle between B and the crystallographic axis)

Fig. 3 (Left) Four qualitative sketches for the SM of a metal suggested by Lifshitz and Peschanskii in [76]. (Right) Experimental data about magnetoresistance in gold monocrystals obtained by Gaidukov [55]

about FS topology: e.g. clearly if σ grows quadratically for some direction of B then the FS must be open, and further analysis can lead to discover the directions of the openings. Between the 1950s and 1970s SM were experimentally found for about 30 metals including gold [55], silver [6], copper [70, 108], lead [5], cadmium [7], zinc [56], thallium [7], gallium [110], and tin [4]. Despite the theoretical efforts, though, no way was found to generate them with first-principles calculations and therefore no accurate direct verification of the Lifshitz model is available to date, except for the qualitative sketches by Lifshitz and Peschanskii [79] (see Fig. 3); in particular, it was not known till now how closely the semiclassical model is able to reproduce these complex experimental data and, therefore, whether or not further purely quantomechanical corrections to the model are needed. As the magnetoresistance methods

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were replaced by newer and more accurate tools to study FS, no new SM was experimentally produced and the problem was eventually abandoned. The results on topological structure of the level lines of f = (, S) illustrated in next section will shed full light on the relation between the direction of the magnetic field and the direction of the non-compact trajectories, when they arise.

4.1.2

Topology

The main problem for the case n = 3 can be formulated as follows: given a triply periodic function ε : R3 → R and a constant 1-form B, determine when do open level lines arise as intersections of a level surface εc = ε−1 (c) with a level plane of B and, in case they do, find their asymptotics. We call simply B-sections such level lines. Note that ε descends in T3 to a smooth function and B to a closed 1-form. By abuse of notation, we will use the same symbol for the corresponding objects in R3 and T3 . Since the B-sections only depend on the direction of B, we will sometimes consider, by a further abuse of notation, B ∈ RP2 . It will be clear by the context which object we mean by B. The following definition is fundamental for this problem: Definition 10 We call topological rank of an embedding i : Mg2 → T3 of a Riemann surface into the 3-torus the rank of the induced ring homomorphism i∗ : H1 (Mg2 , Z) → H1 (T3 , Z). Remark 2 Since H1 (Mg2 , Z)  Z2g and H1 (T3 , Z)  Z3 , in order to have an embedding with maximal rank we must have g ≥ 3 (see Fig. 4 for examples). With respect to foliations induced on Mg2 by general closed 1-forms, here we have a further structure due to the embedding of Mg2 inside T3 and to the fact that the level lines are all planar in the universal covering. This special situation allows us the following decomposition of Mg2 . Let Ci be the components of εc filled by leaves

(a)

(c)

(b)

B

(d)

Fig. 4 (Left) Surfaces embedded with rank 0, 1, 2, and 3 in T3 . (Right) A genus 4 surface embedded with rank 3 in T3 and some of its closed and open B-sections for some B nearly vertical

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homologous to zero in T3 . Then Mg2 \ ∪Ci is the union of periodic and minimal components Kj and Mk whose boundaries are all loops homotopic to zero. Definition 11 The genus of the components Kj and Mk of Mg2 is the genus of the corresponding manifold obtained by quotienting to a single point each boundary component. The following observations by Dynnikov [43] are fundamental to understand the topological structure behind this problem: Theorem 24 ([43]) If a component Kj or Mk of εc has topological rank 2, then there is no component with higher rank and all components with topological rank 2 of εc , c ∈ R, share the same integral indivisible homology class in H2 (T3 , Z) modulo sign. Theorem 25 ([43]) If a component Kj or Mk of εc has genus g0 > 1, then there is a neighborhood U of c such that all components of εc have genus smaller than g0 for all c ∈ U . Definition 12 We denote by FB (Mg2 ) the foliation induced by B on Mg2 → T3 and we say that FB (Mg2 ) is: 1. trivial if both the genus and the topological rank of all components Kj are not larger than 1 (in this case no Mk arises); 2. integrable if the genus of all its components Kj and Mk is not larger than one and there is at least one component with topological rank 2; 3. chaotic if some component Kj or Mk has genus larger than 1. In the integrable case, the indivisible 2-cycle  ∈ H2 (T3 , Z) common, modulo sign, to all rank-2 components is called the soul of FB (Mg2 ). Remark 3 Every non-zero integer 2-cycle in T3 determines a rational direction in its universal cover R3 . By abuse of notation, therefore, throughout this section we often consider  as an element of QP2 rather than H2 (T3 , Z). The structure of open B-sections in the integrable case is very simple [41]: Proposition 4 ([41]) Let N → T3 be a surface embedded with topological rank 2 and let  = [N ] ∈ H2 (T3 , Z). Then the open B-sections of N are strongly asymptotic to a straight line with direction  × B. Once a function ε has been fixed, since only the direction of B determines the foliation, the “phase space” of this problem is [a, b] × RP2 , where [a, b] = ε(T3 ). Correspondingly, there are two natural ways to tackle this problem: either fix a level for ε and study what happens by tilting B, or fix B and study what happens by changing the “energy level”. The first approach was taken by Zorich and resulted in the following theorem (see Fig. 4), based on a clever elementary topological argument [116]:

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Theorem 26 ([116]) Consider an embedding i : Mg2 → T3 and let B be a rational constant 1-form on T3 such that ω = i ∗ B is a Morse closed 1-form on Mg2 . Then FB (Mg2 ) is either trivial or integrable and FB  (Mg2 ) is, correspondingly, trivial or integrable for all B  close enough to B. Proof Outline Every rational closed 1-form on Mg2 is the differential of a map fω : Mg2 → S1 . This is a Morse map by hypothesis and so decomposes Mg2 in elementary cobordisms, the only non-trivial one being pants, whose boundary loops are Bsections. Since, by construction, two of the three boundaries of such pants lie on the same 2-torus embedded in T3 , it follows easily that at least one of the three must be homologous, and so homotopic, to zero in T3 . Hence, the genus of all periodic components Ki (if any) is equal to 1 (by Theorem 3 there are no minimal components). Under small enough perturbations, closed leaves non-homotopic to zero might become open (namely, minimal components might arise) but the leaves homotopic to zero will stay so and so all periodic and minimal components will still have genus 1, namely FB (Mg2 ) is either trivial or integrable. The second approach was taken by I.A. Dynnikov, who proved, in a series of articles [41, 43, 45] and with much more sophisticated arguments, the results below. Theorem 27 ([45]) Let ε : T3 → R be a Morse function. Then there are functions e1 , e2 : RP2 → R such that • e1 (B)  e2 (B) for all B ∈ RP2 ; • FB (εc ) is trivial if and only if c ∈ / [e1 (B), e2 (B)]; • if e1 (B) < e2 (B), then FB (εc ) is integrable for all c ∈ [e1 (B), e2 (B)], and its soul B (εc ) is independent of c. Definition 13 Given a surface Mg2 embedded in T3 and a  ∈ QP2 , we call stability zone for Mg2 relative to  the set D (Mg2 ) = {B ∈ RP2 ; Mg2 ,B = } . We denote by D(Mg2 ) the union of all stability zones and we call stereographic map of Mg2 the map Mg2 : D(Mg2 ) → QP2 associating the soul to each B that has one. Finally, we denote by E(Mg2 ) the set of directions for which FB (Mg2 ) is chaotic. Proposition 5 ([45]) Let M and N be two disjoint embedded surfaces in T3 . Then M and N are compatible on all directions on which they are both defined.

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Definition 14 Given a function f : T3 → R, and a  ∈ QP2 , we call stability zone for f relative to  the set D (f ) = ∪c D (fc ) and we denote by D(f ) the union of all stability zones. We call stereographic map of f the map f : D(f ) → QP2 associating the soul to each B that has one for some c ∈ R. Finally, we set E(f ) = ∪c E(fc ). Theorem 28 ([45]) For a generic surface Mg2 ⊂ T3 , the sets DMg2 , are disjoint closed domains with piece-wise smooth boundary. The set E(Mg2 ) is disjoint from QP2 and has zero measure. The set of directions B with trivial B-sections is open. Examples of such stability zones are shown in Fig. 10. Theorem 29 ([45]) Stability zones of a generic f ∈ C ∞ (T3 ) are closed domains with piecewise smooth boundary. If  =  then D (f ) and D (f ) can only have intersections at the boundary and the number of their common points is at most countable. Finally, either the whole RP2 is covered by a single generalized stability zone or the number of zones is countably infinite and the set E(f ) is uncountable. It follows that the interior of D(f ) is an open everywhere dense subset of RP2 and its complement E(f ) has the form of a two-dimensional cut out fractal set [32, 35]. Examples of such maps f are shown in Fig. 6. Note that several properties about the structure of the stability zones are not well understood yet: Conjecture 3 ([45]) The area of a stability region D (Mg2 ) does not exceed C/  3 for some constant C that depends only on Mg2 . The sets D (f ) are connected and simply connected. Open Questions • Is the number of sides of a stability zone finite? • Are stability zones convex? • If not, can two stability zones meet in more than one point? • Does a generic surface Mg2 have only finitely many stability zones? • Does any of these theorems still hold, possibly in a weaker form, if we consider, rather than constant 1-forms, closed 1-forms with zeros? • What can be said if we replace T3 by some other compact 3-manifold (e.g. Mg2 × S1 ) and B with a closed 1-form without zeros? The following theorem connects E(f ) with the image of the stereographic map f [32]:

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Theorem 30 ([32]) If there is more than one generalized stability zone, then the closure of E(f ) coincides with the set of accumulation points of the set f (D(f )). It is plausible, but still unknown, that E(f ) has always zero measure. The following stronger conjecture was proposed in [102]: Conjecture 4 ([102]) Whenever E(f ) is non-empty, the Hausdorff dimension [52] of E(f ) is strictly between 1 and 2 for every f . Another important but still not completely understood problem is the structure of chaotic level sets. In [45] Dynnikov presented an abstracted sophisticated construction that showed the existence of such sections whose closure is contained in a minimal component of topological rank larger than 2. In [39], in case of a polyhedral surface (see the next section), Dynnikov and the present author provided for the first time a concrete case of a chaotic section on a simple surface. Shortly afterwards, Dynnikov started using the Interval Exchange theory tools to study chaotic sections, provided new examples with similar but more generic surfaces and proposed the following conjecture [46]: Conjecture 5 ([46]) If FB (Mg2 ) is chaotic, almost all the B-sections of Mg2 consist in a single connected curve. In 2013 Skripchenko [114] supported the conjecture by building an example of surface for which there is at least a direction on which this is true. More recently, though, Dynnikov and Skripchenko [49] constructed examples of surfaces Mg2 such that FB (Mg2 ) is chaotic but on each B-section of Mg2 lie infinitely many components. Moreover, in this case, typical connected components do have an asymptotic direction, showing in particular that the corresponding foliation on Mg2 is not uniquely ergodic. The most recent result on this subject is by A. Avila, P. Hubert and S. Skripchenko, that studied the case of the μ-cube (namely, the regular skew apeirohedron {4, 6|4}, see [30] and next section) introduced by Dynnikov and the present author in [39], and proved the following important particular result [18]: Theorem 31 ([18]) For almost all B with FB (μ − cube) chaotic, every B-section has an asymptotic direction with a diffusion rate strictly between 1/2 and 1.

4.1.3

Numerical Analysis

No algorithm is known to generate analytically, in general, the stability zones relative to a given surface or function. Given a direction B inside a stability zone D (Mg2 ), in principle the analytical boundaries of D (Mg2 ) can be found by looking at the cylinders Ci of closed orbits separating the pairs of genus-1 rank components. Let us discuss in some detail the simplest non-trivial case, namely when Mg2 is a surface of genus 3. Note that, numerically, we can work only with directions in QP2 but this is not a significant restriction for this problem because QP2 is dense

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Fig. 5 A change of cylinder inside a stability zone of D(c). On the left is shown a critical leaf at the base of a cylinder, the critical point is p1  (0.035, 0.463, 0.25). At the opposite base lies the critical point p = p4 + (0, 0, 1). The middle picture shows what happens at the boundary between the two stability zones of cylinders, namely the point p1 has a saddle connection with p2 . The picture on the right shows the base of the new cylinder. At one base still lies the point p1 but at the opposite one now lies p4 + (1, 1, 0)

in D(f ) for every f . For practical reasons, we choose to represent the direction B as one of the two integer indivisible vectors in its equivalence class. Since B is rational, all B-sections will be compact, some homologous to zero in T3 and some not. In the integrable case, there will be exactly two genus-1 components K1,2 filled by B-sections non-homologous to zero and they will be separated by exactly two cylinders C1,2 of sections homologous to zero. The heights of the two cylinders is non-zero in some open topological discs D1,2 containing B and the corresponding stability zone is given by D = D1 ∩ D2 . The zone D is subdivided in open sets Ej corresponding to different pairs of critical points at the bases of the cylinders. The equation of the side of each Ej is given by B, p2 − p1  = 0, where p1,2 are the critical points at the bases of the cylinder that determines the breaking of the P1,2 (see Fig. 5 for an example). Using this procedure, Dynnikov [43] started the numerical study of this problem by finding ten stability zones of the stereographic map of one of the simplest trigonometric polynomials with level sets of topological rank equal to 3, the function c(x, y, z) = cos(2π x) + cos(2πy) + cos(2π z) , whose regular level sets cc = c−1 (c) are either spheres (for c < −1 and c > 1) or genus-3 surfaces (for −1 < c < 1). Note that the surface cc is a translate of c−c and therefore, in this case, e1 (B) = −e2 (B) for all B ∈ RP2 , so that D(c), in principle equal to the union of all the D(cc ), in this case is simply equal to D(c0 ). Unfortunately, though, the procedure above is hard to implement in a computer language. In order to produce a reasonably good approximation of a SM Mg2 we decided, therefore, to rather select a lattice of rational directions covering homogeneously RP2 or some proper subset of it (in our computations so far we used a number of points ranging from 104 to 108 ) and evaluating the soul (when defined) for each of such directions. In concrete, for each B on the lattice we evaluate numerically

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the four critical sections, namely the B-sections through the points where the plane perpendicular to B is tangent to Mg2 . When FB (Mg2 ) is integrable, all critical points at the bases of the two cylinders are homoclinic saddles with one critical loop homologous to zero and all four such loops are homologous in Mg2 to the same γ ∈ H1 (Mg2 , Z). Then we evaluate the homology class γ and the rank-2 subring Ann(γ ) ⊂ H1 (Mg2 , Z) of all (cohomology classes of) loops having zero intersection numbers with γ . The image of the annihilator of γ , Ann(γ ), into H1 (T3 , Z) is a rank-2 subring R that, in turn, determines a rank-1 subring L ⊂ H2 (T3 , Z) as the set of all 2-cycles whose intersection number with all elements of R is zero. The soul of FB (Mg2 ) is, modulo sign, any of the two indivisible elements of L. We implemented this algorithm in NTC [31] (nts.sf.net), an open source C++ library we built on top of the well-known 3D computer graphics, image processing, and visualization open source C++ library VTK (www.vtk.org) by Schroeder et al. [111]. In order to check its reliability, we successfully tested NTC’s results on the SM of eight surfaces, the zero level of the function c above and several level sets of a piecewise quadratic function—all these surfaces are genus-3, are embedded with topological rank 3, and are invariant by permutations of the coordinate axes. Their symmetry is cubic, namely their Brillouin zone is the unit cube. In Fig. 6 (left) we show c0 , the whole SM D(c), and a detail of it in the region [0, 1]2 in the chart Bz = 1. A rough numerical evaluation of its box dimension gives an estimate of about 1.83, in agreement with Novikov’s Conjecture 4. We later generalized NTC to genus-4 surfaces and used it to evaluate the SM of FS of noble metals (see the next section). Recently, in order to see which kind of SM arise from functions of higher genus and less symmetry, we started the study of the SM of the function d(x, y, z) = cos(2π x) cos(2πy) + cos(2πy) cos(2π z) + cos(2π z) cos(2π x) , whose regular level sets dc are either spheres (for c < −1 and c > 0) or genus-4 surfaces (for −1 < c < 0). Each of the genus-4 level sets has topological rank 4. Note also that d, besides being invariant by integer translations along the coordinate axes, is invariant with respect to translations by 1/2 along the cube diagonals, namely it has a bcc invariance. In Fig. 6 (right) we show the whole SM D(d) in its first Brillouin zone, the SM D(d) and a detail of it in the region [0, 1]2 in the chart Bz = 1. A rough numerical evaluation of its box dimension gives an estimate of about 1.69, again in agreement with Novikov’s Conjecture 4. In order to have a look at simpler SM of functions and, at the same time, to decrease substantially the time to generate, in general, SM of surfaces, we generalized in [36] the main results of Zorich and Dynnikov to polyhedral surfaces. The most noticeable difference with the smooth case is that in the polyhedral case saddle points with index smaller than −1 (e.g. monkey saddles) are generic. In Fig. 6 we show the results for three noteworthy ones: two of the three regular skew apeirohedra (see [30]), the muoctahedron (with Schläfli symbol {6, 4|4}) and the μ-cube (with Schläfli symbol {4, 6|4}), and the cubic polyhedra P1 , namely the

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Fig. 6 (Left) From top to bottom: the surface c0 , the SM D(c) in the square [0, 1]2 of the chart Bz = 1 and in the whole RP2 . (Right) From top to bottom: the surface d0 , the SM D(d) in the square [0, 1]2 of the chart Bz = 1 and in the whole RP2

only cubic polyhedron with all screw vertices, similarly to the μ-cube being the only cubic polyhedron with all monkey-saddle vertices (see [64]). Each one of these surfaces is triply periodic and enjoys the following property: their interior is identical, modulo translations, to their exterior, so that their open B-sections arise

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Fig. 7 (Left) From top to bottom: the μ-octahedron, its SM D(μ-octahedron) in the square [0, 1]2 of the chart Bz = 1 and in the whole RP2 . (Center) From top to bottom: the μ-cube, its SM D (μ-cube) in the square [0, 1]2 of the chart Bz = 1 and in the whole RP2 . (Right) From top to bottom: the P1 polyhedron, its SM D(P1 ) in the square [0, 1]2 of the chart Bz = 1 and in the whole RP2

for them for every B and therefore their SM coincides with the SM of some function. A rough numerical evaluation of their box dimension gives an estimate of 1.69, 1.72 and 1.68, in agreement with Novikov’s Conjecture 4 (Fig. 7). The strategy of looking for the most possible elementary cases proved to be successful and it produced a breakthrough. In case of the μ-cube, indeed, we were able to find an explicit algorithm producing the non-trivial part of D(μ-cube) and, in a joint work with Dynnikov [40], we were able to prove several fundamental properties about the structure of D(μ-cube): Theorem 32 ([40]) The following properties hold for the μ-cube:

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1. The largest SZ of its SM are the (closed) square Qz with sides [0 : 1 : 1], [1 : 0 : 1], [0 : −1 : 1] and [−1 : 0 : 1] and the two other ones Qx and Qy obtained, respectively, by switching the z axis with the x axis and with the y axis. The set RP2 \ (Qx ∪ Qy ∪ Qz ) is the disjoint union of four open triangles Ti and the restriction of D(μ-cube) to each Ti is a Levitt-Yoccoz gasket. 2. E(μ-cube) has measure zero. 3. there exist four constants A, B, C, D such that A 

3 2

 p 

B , 

C D  a  3   3

for all souls of D(μ-cube). 4. Let n any recursive ordering of the souls of D(μ-cube). Then 2 log23 (1 + 2n) + 1  n

2

 3(1 + 2n)2 log3 α ,

where α is the Tribonacci constant. The Levitt-Yoccoz gasket (see Fig. 8) is a self-projective parabolic fractal set (see [37] for details on its story and geometry) generated by the following recursive algorithm: given a triangle in RP2 with vertices [e1 ], [e2 ], and [e3 ], subtract from it the inscribed one with vertices [e1 + e2 ], [e2 + e3 ], and [e3 + e1 ] and proceed recursively with the three triangles left. The second and third statement proves Conjecture 3 and partially Conjecture 4 for the μ-cube. The fourth led the present author to the study of attractors of general self-projective semigroups

Fig. 8 (Left) Seven iterations of the Levitt-Yoccoz gasket. (Right) A B-section for the direction B = [α 2 − α − 1 : α − 1 : 1], where α  1.839 is the Tribonacci constant, giving rise to chaotic sections on the μ-cube

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based on the asymptotics of the norms of corresponding semigroups of linear transformations [38]. Moreover, we were able in this case to find explicit directions B with FB (μ-cube) chaotic, for example B = [α 2 −α−1 : α−1 : 1], where α is the unique real solution of α 3 = α 2 + α + 1 (Tribonacci constant). Previously, the only known examples were abstract combinatorial constructions by S.P. Tsarev in the 2-irrational case and Dynnikov in the fully irrational one (both published in [44]). A further step in proving the Novikov conjecture for the μ-cube is due to Avila, Hubert, and Skripchenko in [19]. Let us denote by dimH the Hausdorff dimension of a set: Theorem 33 ([19]) dimH E(μ-cube) < 2. To date no non-trivial lower bound for the Hausdorff dimension of E(μ-cube) is known but, according to a conjecture of the present author on attractors of general self-projective semigroups (see [37]), dimH E(μ-cube) ≥ 1.63. A numerical rough estimate in [40] gives dimH E(μ-cube)  1.72. Finally, we mention a rigidity result on the stability zone of foliations of the torus in μ-parallelepipeds. Definition 15 A μ-parallelepiped is a polyhedron obtained from a μ-cube after rescaling over the three coordinate axes. Theorem 34 Consider a family of μ-parallelepipeds foliating the plane and let f be any piecewise function having that family as level sets. Then D(f ) = D(μ-cube). Numerical Objects Yet Unexplored • SMs relative to surfaces of genus higher than 4. • SMs relative to functions whose level sets have genus higher than 4. • B-sections when B is a closed 1-form with zeros.

4.1.4

Back to Physics

The results of Sect. 4.1.2 shed full light on the structure of the orbits of quasielectrons under a magnetic field. Given a FS Mg2 , the set of magnetic field directions B for which the chaotic regime arise has zero measure, so it is in principle undetectable experimentally. A generic direction gives rise to either all orbits homologous to zero or to open orbits that lie on a rank-1 component of Mg2 of topological rank 1 or 2. The first and last case are stable by small perturbations of the direction of B while the second arises only across 1-parametric families of directions (see the schematic representation in Fig. 9). This picture immediately explains qualitatively the SM obtained experimentally (see Fig. 10): the dark islands, where the magnetoresistance grows quadratically with the intensity of B, are exactly the SZ of D(Mg2 ). The soul of each SZ dictates the asymptotics of the open orbits: if B ∈ D (Mg2 ), then every non-singular open

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.

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Stability Zones (stable open trajectories)

q1

B Singular closed trajectories

Carrier of open trajectories

.

. . Chaotic trajectories of Tsarev or Dynnikov type

2D Discs

. Unstable periodic trajectories

q2 Open trajectories

Fig. 9 (Left) Model of the SM of a generic surface; (Right) a genus-1 rank-2 component filled by open trajectories (from [86])

Fig. 10 (Top) FS of gold and its experimental [55] and numerical [33, 34] Stereographic Map. (Middle) FS of silver and its experimental [4] and numerical [33, 34] Stereographic Map. (Bottom) FS of copper and its experimental [70] and numerical Stereographic Map

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orbit is strongly asymptotic to the direction B × . The souls2 of the SZ are a hidden quantum number of purely topological origin of the system and that was absolutely unknown previously to the physics community. Note also that now we are able to tell that some of the proposed SM (e.g., see Fig. 3) are qualitatively wrong since two SZ with different souls cannot overlap for elementary topological reasons. As mentioned in Sect. 4.1.1, the SM was found experimentally for many metals. None of those, to the author’s knowledge, has genus 3. The smallest genus arises for the FS of noble metals, for whom g = 4. In this case, in the integral regime the FS is generically decomposed into two components of genus 1 and rank 2 separated by three cylinders of orbits homologous to zero—note that the smallest genus needed to have four pairs of genus-1 components of open orbits is g = 5. After adding support for genus-4 surfaces to the NTC library, we extracted from [25] an accurate approximation for the Fermi energy of the noble metals and finally produced [33, 34] the SM shown in Fig. 10 for gold, silver, and copper, the first metals for which a SM was produced experimentally. As pictures show, there is a striking closeness of the experimental data with the ones coming from the semiclassical approximation. The study of the physics of the magnetoresistance in a strong magnetic field has been continued in the last 20 years, at the light of the deep mathematical advances, by A. Maltsev, jointly with Novikov and Dynnikov, in a long series of papers [47, 84, 86–90, 101–103]. Among the main results of his analysis we mention the observation that, under the integrable regime, the conductivity tensor has rank 1, namely the current and flow in only one direction, and that the precise boundaries of a SZ are not detectable via standard magnetoresistance measurements but could be determined experimentally by studying oscillatory phenomena. Open Tasks • Exploring SMs relative to FSs of metals for which experimental data are already in literature. • Encouraging the solid state community to get new more detailed data.

4.2 QP Functions in R2 with 4 and More Quasiperiods In 2004, Maltsev [85] showed that quasiperiodic functions with any number of quasiperiods could play a relevant role in solid state physics. His arguments are based on a result of Beenakker [20] that, at the end on the 1980s, used a semiclassical approximation to explain an anomalous phenomenon in the magnetoresistance of a two-dimensional electron gas (2DEG) subject to a weak periodic potential. A 2DEG is a semiconductor structure where the motion of electrons in one direction is somehow constrained (and therefore quantized) so that in many phenomena only the projection of the momentum in the plane perpendicular to the constrained direction plays a role and so the system can be considered two-dimensional. The major 2 From

the point of view of physics, souls are Miller indices of the dual lattice.

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systems where such 2D behavior has been observed and studied are metal-oxidesemiconductor structures, quantum well and superlattices (see [10] for a thorough review on this topic). According to Beenakker’s analysis, when a constant magnetic field B and an electric field E = dV are applied to a 2DEG, the drift of the center of the cyclotron orbits r satisfies, in appropriate units, the equations eff

r˙i =

∂V 1 1 eff {ri , VB (r)}B = ij k B B k , 2 2 ∂rj B B

eff

where VB is an effective potential depending on V and B. Since the relevant degrees of freedom of the electron gas are in the plane perpendicular to B, we can take B = Bz dz and consider V as a function of the two coordinates (x, y) in the plane perpendicular to B. The motion of electrons, therefore, is given by the eff level lines of VB (x, y) and Maltsev’s analysis of the problem in [85, 90] shows that it is possible to generate quasi-periodic effective potentials with any number of quasiperiods by using several standard techniques and that, in turn, the topological properties of the trajectories—namely, whether they are closed or open and their asymptotic directions—can be detected experimentally through measurements of the magnetoresistance of the 2DEG, similarly to the case of metals. Note that in this case, though, we have more freedom since we cannot just change the direction of B but also the potential, that took the role of the Fermi energy. We recall that the Novikov problem for n = 4 consists in studying any of the following equivalent objects: 1. The level sets of a multivalued function F = (S1 , S2 , ε) : T4 → R3 , where S1 and S2 are pseudoperiodic functions (see the next section) and ε is singlevalued. 2. The 2-planar sections of four-periodic hypersurfaces of R4 . 3. The leaves of the foliations induced on an embedded hypersurface i : M 3 ⊂ T4 by pairs of closed 1-forms ω1,2 = i ∗ B1,2 , where B1 and B2 are non-parallel constant 1-forms on T4 . 4. The trajectories of the equations of motions corresponding to the Hamiltonian ε via the Poisson bracket  {pi , pj }B = ∗ dpi ∧ dpj ∧ B1 ∧ B2 on T4 , where ∗ is the Hodge star with respect to the Euclidean metrics and B1 and B2 non-parallel constant 1-forms on T4 . 5. The level sets of quasiperiodic functions with 4 quasiperiods. Given a fixed function ε, the phase space for this case is R × G4,2 (R). It is easy to see that, just like for n = 3, all rational directions in G4,2 (R) give rise to all closed (possibly non-homologous to zero) sections in T4 . Indeed, a rational 0 is given (in several ways) by a pair of rational directions B1 and B2 . All leaves of B1 are 3-tori Ta embedded in T4 so every level set εc = ε−1 (c) restricts on Ta to a triply periodic surface Ma ⊂ T a . The 0 -sections of ε−1 (c), therefore, are B2 -sections of

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Ma and so, by Zorich theorem, they are all closed and lie in a genus-1 component of Ma embedded with rank 1 or 2 in Ta . Moreover, this result is stable with respect to small changes in the direction of B2 , namely over all rational lines r passing through 0 it is defined an r so that, for all (rational or not)  on r closer to 0 then r , all -sections have a strong asymptotic directions. In fact, the same result clearly holds for any number n ≥ 4 of quasiperiods. The problem, though, is what happens for the directions  close to 0 outside those rational lines. So far, only two analytical works (and no numerical one) have been published on level sets of quasiperiodic functions with more than three quasiperiods, namely a first one by Novikov [100] and a follow-up jointly with Dynnikov [48]. Their main result is the following analogue of the Zorich theorem for the case n = 3: Definition 16 A level hypersurface εc = ε−1 (c) is topologically completely integrable (TCI) for the 2-plane  ∈ G4,2 if all -sections are either compact or have a strong asymptotic direction. We say that εc is stable if this property is still true after small perturbations of  and ε. Theorem 35 (Novikov and Dynnikov (2005)) There is an open dense set of smooth functions U ⊂ C ∞ (T4 ) with the property that, for every f ∈ U , there is an open dense set Vf ⊂ G4,2 (R) such that every εc is a TCI for all  ∈ Vf . Moreover, for a given c and , there is a R > 0 such that all open -sections of εc are contained inside some cylinder of radius R. Conjecture 6 (Novikov and Dynnikov (2005)) A generic level εc is TCI stable for almost all  ∈ G4,2 (R). The case n ≥ 5 is believed to be much more complicated and no Zorich-like result is believed to hold. Open Tasks Explore numerically the cases n = 4 and n = 5.

5 Level Lines of n − 2 Quasiperiodic Functions in Rn−1 with n Quasiperiods In this last section we consider yet another problem in quasiperiodic topology, this time dual in some sense to the Novikov problem discussed in Sect. 2, namely the case of multivalued maps f : Tn → Rn−1 with all but one components singlevalued. The study of this problem was started by Zorich in 1994 [118–120] as a generalization of his study of the Novikov problem for n = 3. In that case, Zorich proved that every open B-section of a surface Mg2 → T3 is strongly asymptotic to a straight line for all B close enough to rational (Theorem 26). Note that, for n = 3, the closed 1-form induced by B on Mg2 has irrationality degree not higher than 3, while a generic one has irrationality degree equal to the number of cycles 2g, namely 1-forms obtained through this construction are highly non-generic. It is natural to ask whether B-sections keep having an asymptotic directions when we make the setting more generic, namely

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by considering the embedding of Mg2 in some Tn , n > 3. In this case, for n large enough, we can get forms with full irrationality degree, which enables one to use sophisticated dynamical systems tools that are unavailable for the n = 3 case. As in the case n = 3, we call Ci (B) the components of Mg2 filled by B-sections homologous to zero in Tn , so that Mg2 \∪Ci (B) is the union of periodic and minimal components Cj (B) whose boundaries are all loops homotopic to zero. We denote by gj the genus of Cj (B) and by N the number of the Cj (B). Theorem 36 ([118]) Let Mg2 → Tn be an embedding with full topological rank and assume n ≥ 4g − 3. Then, for almost all B ∈ RP n , the following properties hold: 1. 1 ≤ N (B) ≤ g . 2. To each Cj (B) is associated a direction cj ∈ RP n such that lim

t→∞

γ˜ (t) − γ˜ (0) = cj t

for every B-section γ in Cj (B), where γ˜ = πn−1 γ . 3. lim sup t→∞

log d(γ˜ (t), (t)) = α(gj ) < 1 , log t

where (t) is the straight line parallel to cj passing through γ˜ (0) and 1+α(gj ) is the second Lyapunov exponent of the Teichmuller geodesic flow on the principal stratum of squares of holomorphic differentials on the surface of genus gj . It is noteworthy that Zorich was led by the study of this case to prove, jointly with Kontsevich [72] and Avila [16, 17] a more general, important, and far-reaching theorem on foliations induced by closed 1-forms on surfaces, of which the previous result is a corollary: Theorem 37 (Zorich et al. (2007)) For almost all abelian differential ω on Mg2 without maxima and minima in any connected component of any stratum H(k1 , · · · , ks ) of the moduli space of Abelian differentials with zeros of index k1 , . . . , ks , for the foliation of the closed 1-form ω0 = "(ω) there exists a complete flag of subspaces V1 ⊂ V2 ⊂ · · · ⊂ Vg ⊂ H1 (Mg2 , R) with the following properties: 1. For any leaf γ and point p0 ∈ γ we have lim

t→∞

cp0 (t) = c, t

where the non-zero cycle c ∈ H1 (Mg2 , R) is proportional to the Poincaré dual of the cohomology class of ω0 and V1 = span{c}.

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2. For any ψ ∈ Ann(Vj ) ⊂ H1 (Mg2 , R) \ Ann(Vj +1 ), any leaf γ and any point p0 ∈ γ , lim sup t→∞

logψ, cp0 (t) = νi+1 for all i = 1, . . . , g − 1. log t

3. For any ψ ∈ Ann(Vg ) ⊂ H1 (Mg2 , R), any leaf γ and any point p0 ∈ γ , lim sup t→∞

logψ, cp0 (t) =0 log t

4. For any ψ ∈ Ann(Vg ) ⊂ H1 (Mg2 , R), ψ = 1, any leaf γ and any point p0 ∈ γ , ψ, cp0 (t) is bounded from above by a constant depending only on the foliation and the norm chosen on the cohomology. All limits converge uniformly and the numbers 2, 1 + ν2 , . . . , 1 + νg are the top g exponents of the Teichmuller geodesic flow on the corresponding connected component of the stratum H(k1 , · · · , ks ). Open Questions What can be said about the asymptotic direction of open leaves in the intermediate case 3 < n < 4g − 3? Acknowledgements The author gladly thanks S.P. Novikov for introducing him to the subject and for his encouragement and support throughout the years and A. Zorich and I. Gelbukh for several discussions and suggestions on the present survey. All numerical calculations by the author presented here were made on the computational clusters of the National Institute for Nuclear Physics (INFN) in Cagliari (Italy) and, at Howard University in Washington, DC (USA), on those of the College of Arts and Sciences and of the Center for Computational Biology and Bioinformatics. Part of this work was supported by an Advanced Summer Faculty Research Fellowship of Howard University.

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92. J. Milnor, Morse theory, Annals of Mathematics Studies, vol. 51 (Princeton University Press, Princeton, 1963) 93. M. Morse, Relations between the critical points of a real function of n independent variables. Trans. Am. Math. Soc. 27, 345–396 (1925) 94. S.P. Novikov, Bloch functions in a magnetic field and vector bundles. Doklady AN SSSR (Russ. Math. Doklady) 257(3), 538–543 (1981) 95. S.P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory.Soviet Math. Dokl 24, 222–226 (1981) 96. S.P. Novikov, Hamiltonian formalism and a multivalued analog of Morse theory. Russ. Math. Surv. 37(5), 3–49 (1982) 97. S.P. Novikov, Bloch homology, critical points of functions and closed 1-forms. Soviet Math. Dokl 33, 551–555 (1986) 98. S.P. Novikov, Quasiperiodic structures in topology. Topological methods in modern mathematics, pp. 223–233 (Stony Brook, New York, 1991) 99. S.P. Novikov, The semiclassical electron in a magnetic field and lattice some problems of low dimensional “periodic” topology. Geom. Funct. Anal. 5(2), 434–444 (1995) 100. S.P. Novikov, The levels of quasiperiodic functions on the plane, Hamiltonian systems and topology. Russ. Math. Surv. 54(math-ph/9909032), 1031–1032 (1999) 101. S.P. Novikov, A.Ya. Maltsev, Topological phenomena in normal metals. Usp. Fiz. Nauk 41(3), 231–239 (1998) 102. S.P. Novikov, A.Ya. Maltsev, Dynamical systems, topology and conductivity in normal metals. J. Stat. Phys. 115, 31–46 (2003) 103. S.P. Novikov, A.Ya. Mal’tsev, Topological quantum characteristics observed in the investigation of the conductivity in normal metals. J. Exp. Theor. Phys. Lett. 63(10), 855–860 (1996) 104. S.P. Novikov, A.P. Veselov, Two-dimensional Schrödinger operators in periodic fields. Sovr. Probl. Math. 23, 3–23 (1983) 105. A.V. Pajitnov, Circle-Valued Morse Theory, vol. 32 (Walter de Gruyter, Berlin, 2006) 106. D.A. Panov, Multicomponent pseudo-periodic mappings. Funct. Anal. Appl. 30(1), 23–29 (1996) 107. D.A. Panov, Pseudoperiodic mappings. Trans. Am. Math. Soc. Ser. 2 197, 117–134 (1999) 108. A.B. Pippard, An experimental determination of the Fermi surface in Copper. Phil. Trans. R. Soc. A 250, 325 (1957) 109. A.B. Pippard, Magnetoresistance in Metals (Cambridge University Press, Cambridge, 1989) 110. W.A. Reed, J.A. Marcus, Topology of the Fermi surface of Gallium. Phys. Rev. 126(4), 1298 (1962) 111. W. Schroeder, K. Martin, B. Lorense, Visualization Toolkit: An Object-Oriented Approach to 3D Graphics (Kitware, New York, 2006) 112. D. Shoenberg, Proc. Mag. 5, 105 (1960) 113. D. Shoenberg, Proc. Trans. R. Soc. A 255, 85 (1962) 114. A. Skripchenko, On connectedness of chaotic sections of some 3-periodic surfaces. Ann. Glob. Anal. Geom. 43(3), 253–271 (2013) 115. D. Tischler, On fibering certain foliated manifolds over S1 . Topology 9(2), 153–154 (1970) 116. A.V. Zorich, A problem of Novikov on the semiclassical motion of electrons in a uniform almost rational magnetic field. Usp. Mat. Nauk (RMS) 39(5), 235–236 (1984) 117. A.V. Zorich, The quasiperiodic structure of level surfaces of a Morse 1-form close to a rational one—a problem of S.P. Novikov. Math. USSR Izv. 31(3), 635–655 (1988) 118. A.V. Zorich, Asymptotic flag of an orientable measured foliation on a surface. Geom. Study Folia. Tokyo 1993 479, 498 (1994) 119. A.V. Zorich, On hyperplane sections of periodic surfaces, in Solitons, Geometry, and Topology: On the Crossroad, ed. by V.M. Buchstaber, S.P. Novikov. Translations of the AMS, Ser. 2, vol. 179 (AMS, Providence, 1997) 120. A.V. Zorich, How do the leaves of a closed 1-form wind around a surface? Am. Math. Soc. Transl. Ser. 2 197, 135–178 (1999)

Combining Bifurcation Analysis and Population Heterogeneity to Ask Meaningful Questions Irina Kareva

Abstract Classical approaches to analyzing dynamical systems, such as bifurcation analysis, can provide invaluable insights into underlying structure of a mathematical model and the spectrum of all possible dynamical behaviors. However, these models frequently fail to take into account population heterogeneity, which, while critically important to understanding and predicting the behavior of any evolving system, is a common simplification that is made in the analysis of many mathematical models of ecological systems. Attempts to include population heterogeneity frequently result in expanding system dimensionality, effectively preventing qualitative analysis. Reduction theorem, or hidden keystone variable (HKV) method, allows incorporating population heterogeneity while still permitting the use of classical bifurcation analysis. A combination of these methods allows visualizing evolutionary trajectories and making meaningful predictions about system dynamics of evolving populations. Here, we discuss three examples of combination of these methods to augment understanding of evolving ecological systems. We demonstrate what new meaningful questions can be asked through this approach, and propose that application of the HKV method to the large existing literature of fully analyzed models can reveal new and meaningful dynamical behaviors, if the right questions are asked.

1 Introduction Heterogeneity is a major driving force behind the dynamics of evolving systems. When it is heritable and when it affects fitness, heterogeneity is what makes evolution possible [1–4]. This comes from the fact that the environment in which

I. Kareva () Mathematical and Computational Sciences Center, School of Human Evolution and Social Change, Arizona State University, Tempe, AZ, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_4

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the individuals interact is composed not only of the outside world (such as resources necessary for survival or members of other species) but also of individuals themselves. Therefore, selective pressures that are imposed on them come both from the environment and from each other. Furthermore, selective pressures that individuals experience from other members of the same population will be perceived differently depending on population composition, which in turn may be changing as a result of these selective pressures. In a vast majority of conceptual, and often even in descriptive mathematical models of population dynamics, whether it be models of predator–prey interactions, spread of infectious diseases or tumor growth, population homogeneity is the first simplification that is made. It is not treated as homogeneity per se; rather, one assumes that an average rate of growth or death or infectiousness is a reasonable enough approximation if the system has already reached some kind of stabilized state of evolutionary development. However, by ignoring population heterogeneity in such a way, one ends up either ignoring natural selection or assuming that it has already “done its work.” This assumption is often incorrect within the context of such models, since natural selection may be in fact a key driver behind dynamics of most systems that are of interest and importance. Equation-based models are usually avoided in questions that require modeling high levels of heterogeneity. This is a result of the inevitable increase of system dimensionality that often accompanies attempts to account for population heterogeneity, to the point at which obtaining any kind of qualitative understanding of the system becomes nearly impossible. Assuming population homogeneity, while making systems of equations computationally and sometimes even analytically manageable, causes the loss of many aspects of system dynamics that come from intra-species interactions and natural selection. Parametrically homogeneous systems can nevertheless still provide exceptionally valuable information about the structure of the system, which can be obtained through extensively developed analytical techniques, such as bifurcation analysis [5]. A skillfully constructed bifurcation diagram can both reveal various possible dynamical regimes that a system can exhibit as a result of variations in parameter values and initial conditions, and provide analytical boundaries as functions of system parameters. This information can then be used to construct a theoretical framework for understanding a biological system that could never have been obtained experimentally. Reduction theorem, also known as parameter distribution technique, or hidden keystone variable (HKV) method, is a method that allows building on insights obtained from bifurcation analysis while incorporating population heterogeneity [6– 9]. It allows investigating more fully the dynamics of an evolving system while overcoming this problem of immense system dimensionality in a wide class of mathematical models. This approach of course makes sense only when there exists a meaningful research question that a parametrically heterogeneous model can help answer (otherwise this becomes little more than a mathematical exercise).

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In what follows, we will first briefly describe this approach and the assumptions and limitations that are associated with implementation of the HKV method. We will then describe several examples that reveal how a combination of the classical bifurcation analysis techniques with the HKV method can reveal previously inaccessible dynamical behaviors. We will conclude with a brief discussion on the possibilities for rich dynamical behaviors that still remain to be revealed in the already existing body of literature of fully analyzed mathematical models.

1.1 General Strategy Assume the population of individuals is composed of clones xa , and that each individual clone xa is characterized by parameter value a, which corresponds to a measure of some intrinsic heritable trait, such as birth rate, death rate, resource consumption rate, etc. The total population size is given by N (t) = xa (t) if a

the system is discrete, and N(t) = a xa (t)da if the system is continuous. Then, since different clones can grow and die at different rates, the distribution of clones (t) within the population Pa (t) = xNa(t) can change over time due to system dynamics. Consequently, the mean value of the parameter Et [a], which now becomes a function of time, changes over time as well. Analysis of a parametrically heterogeneous system involves the following steps: 1. Analyze autonomous parametrically homogeneous system to the extent possible using well-developed analytical tools, such as bifurcation analysis. 2. Replace parameter a with its mean value Et [a], which is a function of time. 3. Introduce an auxiliary system of differential equations, which define keystone variables that actually determine the dynamics of the system. (Note: the term “keystone” is chosen here in parallel to the notion of keystone species in ecology. Just like keystone species have disproportionately large effect on their environment relative to their abundance, keystone variables determine the direction in which the system will evolve without being explicitly present in the original system). 4. Express the mean and variance of the distributed through keystone variables. The mean of the parameter, which now changes over time due to system dynamics, can now “travel” through the different domains of the phase-parameter portrait of the original parametrically homogeneous system. 5. Calculate numerical solutions. Exact formulation of the Reduction theorem and the theory underlying the method can be found in [7–9]. A summary of definitions and associated notation is provided in Table 1.

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Table 1 Definitions and notation used in the application of the HKV method Definition Selection system

Notation and explanation A mathematical model of an inhomogeneous population, where every individual is characterized by a vector-parameter a = (a1 , . . . , an ) that takes on values from set A

Clone xa Total population size N(t)

Set of all individuals that are characterized by a fixed value of parameter a  N (t) = xa if the number of possible values A

of a is finite and N (t) = A xa (t)da if it is infinite

Growth rate of a clone xa

dxa (t) dt

Fitness of an individual within the population

dxa (t) dt /xa (t)

Distribution of clones within the population

Pa (t) =

Expected value of a distributed parameter

For all expressions of the type

xa (t) N (t)

A

f (a)xa da , N (t)

standard notation Et [f ] of the expected value is used

1.2 Advantages and Drawbacks of the Reduction Theorem One of the most important properties of this method is that it allows reducing an otherwise infinitely dimensional system to low dimensionality. However, as with any method, there are limitations to the application of the Reduction theorem. Most importantly, the transformation can be done (with some  generalizations) only to Lotka–Volterra type equations of the form x(t) = x(t) t F(t, f (E [a]), where x(t) is a vector, a is a parameter or a vector of parameters that characterize individual heterogeneity within the population, and where the form of f (Et [a]) is system-specific. It can also increase the dimensionality of the original parametrically homogeneous system at a possible cost of auxiliary keystone equations (although these would typically still be on the order of only one or two extra equations, depending on the original system). Finally, the resulting system is typically non-autonomous, so one cannot perform standard bifurcation analysis and has to resort to calculating numerical solutions. When studying numerical solutions of such parametrically heterogeneous systems, one can observe trajectories that could not have been observed in a parametrically homogeneous systems. This phenomenon occurs because the expected value of the parameter “travels” through the phase-parameter portrait, causing the system to undergo qualitative phase transitions as the now dynamic parameter crosses bifurcation boundaries. Now, if there exists a complete bifurcation diagram for the specific parametrically homogeneous model, one can identify what boundaries have been crossed during system evolution.

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One can now not only track the distribution of different clones within the population as the system evolves but also observe how different initial distributions of clones in the population can lead to different trajectories. One can therefore capture effect of sensitivity to initial population composition both to changes in intrinsic properties of individuals (such as birth or death rates) or to changes in the external factors (environment). This results from the fact that different clones have different fitness depending on initial population composition, since the selective pressures that are imposed on them result not only from the external environment but from surrounding clones as well. Therefore, the HKV method allows for equation-based models to generate novel behaviors by incorporating all the properties of a complex system [2] without significantly increasing system dimensionality. Notably, unlike agent-based models, which are the standard computational tool for studying complex systems, the HKV method does not allow incorporating spatial heterogeneity. Next, we will describe several examples of when application of the two methods coupled with a meaningful research question allowed answering questions and visualizing previously unobserved evolutionary trajectories. Example 1 Sustainability: Using a parametrically heterogeneous model to study resource depletion, transitional regimes, and intervention strategies. In this first example, we focus on a model of consumer interaction with shared resources that are critical for population survival. In this model, each consumer is characterized by their own value of parameter c, which determines the degree, to which the consumer depletes or restores shared resources. The model was initially proposed in [10] in the context of niche construction, and was later expanded in [11]. It contains two coupled differential equations, written as follows: ⎛ xc (t) 0 12 3



=

consumers



z(t) 0123 shared resource

rxc (t) 0 12 3

c 0123

⎜ ⎝consumption

γ − δz(t) 0 12 3 natural resource turnover

N (t) kz(t) 0 12 3

⎞ ⎟ ⎠

dynamic carrying capacity

population growth rate

=

−

+

N (t) (1 − c) e z(t) + N (t) 12 3 0

(1.1) ,

change in resource caused by consumers (depletion if c > 1, restoration if c < 1)

 where N (t) = A xc (t) is the total population size over all possible values of parameter c. As one can see, it is assumed that the population grows according to the logistic growth function with a dynamic carrying capacity, determined by the shared resource z(t). The consumers can either deplete the shared resource, or contribute to it, depending on the value of parameter c: c > 1 results in resource depletion, while c < 1 results in its restoration. The resource z(t) also has a natural turnover rate, which can allow for sustainable coexistence of consumers with the

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resource. However, since increase in growth rate with respect to parameter c creates an incentive for consumers to maximize resource consumption in the short term, this is likely to lead to destruction of shared resources, a notion that has been known as “the tragedy of the commons” [12–14]. Situation when survival of the population depends on the over-depleted resource is known as “evolutionary suicide.” It occurs when short-term increases in fitness due to resource overconsumption lead to eventual destruction of the shared resource and the population’s extinction [15, 16]. Several questions can be asked of this model, such as: 1. How will such a system behave depending on the number of over-consumers in it? What are the possible dynamical regimes that such a system can realize as it is heading for resource exhaustion and eventual population collapse? 2. Can we identify transitional regimes that can serve as warning signals of approaching collapse? 3. What, if any, intervention measures could be implemented to prevent the tragedy of the commons and possibly even evolutionary suicide? Answering these questions requires a combination of both classical bifurcation analysis and the HKV method that allows visualizing evolutionary trajectories as the system evolves over time. Question 1 How will such a system behave depending on the number of over-consumers in it? Answering this question can be achieved through conducting stability and bifurcation analysis, as has been done in [11]. In this work, we progressively increased the value of parameter c and observed a series of dynamical regimes, ranging from sustainable coexistence with the common resource with ever decreasing domain of attraction, to sustained oscillatory regimes, to population collapse due to complete depletion of the common resource. The results are summarized in Fig. 1. In domain 1, when the parameter of resource (over)consumption is small, the shared carrying capacity remains large, successfully supporting the entire population, since no individual is taking more resource than they replenish. In domain 2, a parabolic sector appears near the origin, decreasing the domain of attraction of the nontrivial equilibrium point A. The population can still sustainably coexist with the resource even with moderate levels of overconsumption but the range of initial conditions, where it is possible, decreases. As the value of c is further increased, the range of possible initial conditions that allow sustainable coexistence with the common resource decreases and is now bounded by an unstable limit cycle, which appears around point A through a catastrophic Hopf bifurcation in domain 3, and via “generalized” Hopf bifurcation in domain 6. Finally, in domains 4 and 5, population extinction is inevitable due to extremely high overconsumption rates unsupportable by the resource.

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Fig. 1 Bifurcation diagram of System (1.1) in the (γ , c) and (N, z) phase-parameter spaces for fixed positive parameters e and δ. The nontrivial equilibrium point A is globally stable in domain 1; it shares basins of attraction with equilibrium O in domains 2 and 3. The separatrix of O and the unstable limit cycle that contains point A serve, correspondingly, as the boundaries of the basins of attraction. Only equilibrium O is globally stable in domains 4, which also contains unstable √ nontrivial A, and 5, which contains the elliptic sector. Domain 6 exists only for δ > 5 + 24, where the stable limit cycle that is in turn contained inside an unstable limit cycle shares basins of attraction with equilibrium O. Boundaries between domains K, S, H, Nul, C correspond, respectively, to appearance of an attracting sector in a neighborhood of O, appearance of unstable limit cycle containing A, change of stability of equilibrium A via Hopf bifurcations, disappearance of positive A, and saddle-node bifurcation of limit cycles. The figure is adapted from Fig. 4 in [11]

Question 2 Can we identify transitional regimes that can serve as warning signals of approaching collapse? Answering this question will require introducing population heterogeneity into the model to allow us to visualize evolutionary trajectories. As it stands, in the parametrically homogeneous case, analyzed in [11], the parameter value c is always a constant, and therefore, the population will always remain in the corresponding domain of Fig. 1. However, let us introduce a keystone variable q(t), such that q(t) =

N(t) . kz(t)

Then, xc (t) = rxc (t)(c − q(t) ).

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Consequently, xc (t) = xc (0) = xc (0)ect−q(t) , and thus / N (t) =

A

xc (t)dc = N0 e−q(t)

/ A

ect Pc (0)dc = N0 e−q(t) M0 (t),

∞ (0) and M0 (t) = 0 ect Pc (0) is the moment generating function where Pc (0) = xNc (0) (mgf) of the initial distribution of clones within the population. The expected value of parameter c can then be calculated as /

/ E [c] = t

A

cPc (t)dc =

A

cPc (0)

M0 (t) etc dc = . M0 (t) M0 (t)

The final system of equations thus becomes   N(t) N(t) = N(t) E t [c] − kz(t)  N(t) 1 − E t [c]  , z(t) = γ − δz(t) + e z(t) + N (t)

(1.2)

where Et [c] is determined by the moment generating function of the initial distribution of clones in the population, as are consequently the dynamics of the entire system. Note that in comparison to the parametrically homogeneous System (1.1), in the parametrically heterogeneous System (1.2) the fixed value of the parameter c has been replaced by its expected value at each time instant t. It is easy to verify that the rate of change of Et [c] is equal to the variance of c at each time moment t in accordance to Fisher’s fundamental theorem. Therefore, as the system evolves with time, the expected value of c will also change with each time step, causing it to “travel” through the phase-parametric portrait. A full analysis of this system, with all the derivations and proofs, was done in [11]. As an example, consider Fig. 2, where the initial distribution for this model was taken to be truncated exponential, allowing for different maximal values of parameter c. The panels on the left depict the dynamics predicted by a parametrically homogeneous system, while the panels on the right depict the dynamics of a heterogeneous system. Firstly, one can clearly observe the qualitative differences in predictions for population size and resource dynamics over time depending on the degree of population heterogeneity: a heterogeneous system survives longer, since it contains both over-consumers and under-consumers, with the latter delaying the collapse of the shared resource by “subsidizing” the former. Secondly, as one can see for the case, when the initial distribution of parameter c ∈ [0 2.33] (red dashed line), the system does in fact exhibit several transitional

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Fig. 2 Comparison of trajectories of Systems (1.1) and (1.2) for different values of parameter c or range of possible values of c, respectively. As one can see, while both parametrically homogeneous and heterogeneous populations can go extinct due to the exhaustion of common resource by over-consumers, time to extinction of a parametrically heterogeneous population is expected to be much larger. Moreover, in a parametrically heterogeneous system one can sometimes observe a transitional oscillatory regime preceding collapse, which is not observed in a parametrically homogeneous system. This figure is adapted from Fig. 5 in [11]

regimes as it goes through a period of growth through a period of seeming stability, to an oscillatory regime, which precedes population collapse. A closer look at this system in Fig. 3 reveals that during this period of apparent stability, the expected value of parameter c increases (Fig. 3c), revealing the changes in population composition that will lead to its eventual collapse. It is not always clear that system collapse is approaching, and so one has to learn to recognize early warning signals, such as increased flickering and data auto-correlation [17–19] in order to try and prevent the tragedy of the commons. Application of the HKV method to relevant systems of ODEs allows to visualize exactly how the system passes through these dynamical regimes as it evolves. One can see that while changes in population size and resource over time may seem to give no cause for alarm, the mean value of the parameter of overconsumption may signal trouble: the system will be evolving towards maximizing c, and as soon as the buffer capacity of the resource (in this case it is proportional to natural resource restoration and decay rates) is exhausted, both the population and the resource collapse. Notably, in [19] Dakos et al. analyzed eight ancient abrupt climate shifts and showed that in each case they were preceded by a characteristic slowing down of fluctuations before the actual shift, similarly to behaviors predicted in Fig. 2, suggesting that even a relatively simple parametrically heterogeneous model can provide meaningful results and even qualitative, if not quantitative, predictions.

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Fig. 3 System [9] with initial truncated exponential distribution on the interval c ∈ [0 2.5]. An example of what transitional regimes the system can go through before the population crashes, depicting (a) trajectories for the total population size N(t), (b) total amount of renewable resource z(t), (c) expected value of the parameter c, and (d) the change over time in distribution of various clone types within the population. Initial conditions fall within the parameter range of domain 6 of the phase-parameter portrait of the non-distributed system (Fig. 1). Since the rate of natural resource decay is high, it takes more time even for the most efficient consumer to “get to it,” and so the population survives longer, and the transitional regimes are more evident. This figure is adapted from Fig. 7 in [11]

Question 3 What, if any, intervention measures could be implemented to prevent the tragedy of the commons and possibly even evolutionary suicide? In order to address this question, we can introduce a punishment/reward function that can affect individuals in the population based on the value of parameter of overconsumption c. The updated system of equations would look as follows: ⎛ x (t) 0 c12 3



=

consumers

z(t) 0123 shared resource

rxc (t) 0 12 3

N(t) kz(t) 0 12 3

c − ⎜ 0123 ⎟ ⎜consumption ⎟ ⎜ ⎟+ ⎝ dynamic carrying ⎠

population growth rate

=

⎞

γ − δz(t) 0 12 3 natural resource turnover

capacity

+

xc (t)f (c) 0 12 3 punishment/reward

N (t) (1 − c) e z(t) + N (t) 12 3 0

.

change in resource caused by consumers (depletion if c > 1, restoration if c < 1)

(1.3)

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This way, depending on the form of the punishment function f (c), one can try to impose punishment on over-consumers, reward under-consumers, and hopefully be able to maintain population composition in a range where it can sustainably coexist with its dynamic resource. In [20], we investigated three types of punishment/reward functions: 1. Moderate punishment f (c) = a 1−c 1+c . 2. Severe punishment/generous reward f (c) = a(1 − c)3 , where the parameter a denotes the severity of implementation of punishment on individuals with the corresponding value of parameter c. 3. Separating punishment and reward: f (c) = ρ(1 − cη ). This functional form allows to separate the influence of reward for underconsumption, primarily accounted for with parameter ρ, and punishment for overconsumption, accounted for with parameter η. We evaluated the effectiveness of these three types of punishment/reward functions on system evolution and calculated predicted outcomes for different initial distributions of clones within the population, which were taken to be truncated exponential and Beta distributions. The initial distributions were chosen in such a way as to give significantly different shapes of the initial probability density function; in applications, they should be matched to real data, when it is available. We observed that the intensity of implementation of punishment/reward has to differ for different initial distributions if one is to successfully crub overconsumption, and so in order to be able to make any reasonable predictions one needs to know the initial composition of the affected population (see Fig. 4).

Fig. 4 The importance of evaluating the range of possible values of $c_{f}$, illustrated for different initial distribution. Punishment function is of the type f (c) = ρ(1 − cη ), where ρ = 0.6, η = 1.2. Initial distributions are taken to be truncated exponential with parameter μ = 10, and beta with parameters α = 2, β = 2 and α = 2, β = 5; ρ = 0.6, η = 1.2. The top row corresponds to c ∈ [0, 3]; the bottom row corresponds to c ∈ [0, 4]. Figure adapted from Fig. 12 in [20]. The simulations were conducted by Benjamin Morin

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We also observed that severe punishment/generous reward approach was much more effective in preventing the tragedy of the commons than the moderate punishment/reward function, particularly for the cases, when over-consumers were present at higher frequencies (such as for Beta initial distributions). This comes not only from the severity of punishment but also from the fact that moderate punishment allows more time for over-consumers to replicate, and thus by the time the punishment has an appreciable effect, the population composition had changed, and moderate punishment will no longer be effective. So, in punishment implementation one needs to take into account not only the severity of punishment but also the time window that moderate punishment may provide, to over-consumers to “develop resistance” by dominating the population. Within the frameworks of the proposed model, moderate implementation of more severe punishment/reward system is more effective than severe implementation of moderate punishment/reward. Complete analysis of System (1.3) and further simulations are reported in detail in [20]. In [20], we proposed just one way to try and modify individuals’ payoffs in order to prevent resource overconsumption—through inflicting punishment and reward that affects the growth rates of clones directly. This approach can be modified depending on different situations, inflicting punishment or reward based not just on the intrinsic value of c but on total resource currently available. To summarize, a system of two equations describing the dynamics of consumers depleting and replenishing shared resources was simple enough to allow complete analysis and generation of a comprehensive bifurcation diagram. However, application of the HKV method allowed to qualitatively expand the realm of questions that the model could answer, which could have significant practical applications, particularly in the area of sustainability. Example 2 Mixed strategies and natural selection. In this example, we will explore a reformulation of the model of consumer– resource interactions within the context of strategy selection. Specifically, we will look at a model that deals with the question of strategies of resource allocation. Broadly speaking, the two main strategies that can be adopted by different species in response to different selective pressures that come from their environment are either to invest the resources into rapid proliferation, which has been suggested to be the preferable strategy in unstable environments, or into physiological maintenance and increasing environmental carrying capacity at the expense of rapid proliferation, which would allow maximizing fitness in more stable conditions [21–23]. The main criticism of this theory came from empirical studies: however intuitive the heuristic may seem, the adaptations that were predicted by either selective strategy were rarely if ever observed in nature [24]. Nevertheless, there may be merit to this theory if one focuses not on looking for pure strategies but rather explores a continuum. In [25], we described such a situation by introducing parameter α to denote the strategy of investing available resource solely in reproduction, and (1 − α) to denote the strategy of investing the available resource primarily into increasing and maintaining the physiological carrying capacity. We considered the dynamics over time of a

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population of individuals xα characterized by their particular value of “strategy” α, which can fall anywhere within the continuum α ∈ [0, 1]. When α = 0, each  individual was assumed to grow according to the functional form r c2 N z+z − φ ,

where N(t) = A xα dα is the total population size of all individuals, and z(t) is the shared resource. As one can see, in this case shared resources z(t) are used to increase the rate of proliferation of individuals xα . When α = 1, each individual grows according to the logistic growth function with dynamic carrying capacity,  bN given by r c1 − kz . If the individual uses both strategies with probabilities α and (1 − α), respectively, i.e., uses some of the resource towards rapid proliferation and some towards physiological maintenance, then   the per capita   growth rate of each z α-clone is given by αr c1 − bN c + − α) − φ . (1 2 N +z kz Shared resource z(t) is assumed to have a natural turnover rate, replenishing naturally at some constant rate γ and decaying at a rate δz(t); it can also be consumed or restored by all the individuals. The consumption–restoration process (t)(1−c) is accounted for by the term e N z(t)+N (t) ; as the number of consumers increases, the amount of resource will increase or decrease depending on the value of parameter c1 for α-strategy or c2 for (1 − α)-strategy. Full derivation of the system is given in [25]. The final model then becomes dN(t) 12 3 0 dt population size

⎛

  bN αr c1 − kz 0 12 3



 z −φ (1 − α) c2 N +z 0 12 3

⎞

+ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ = rN (t) ⎜proportion of individuals proportion of individuals investing ⎟ , ⎟ ⎜ ⎝investing resource directly resource in physiological maintenance ⎠ in proliferation

dz(t) = 12 3 0 dt shared resource

(1.4)

γ − δz(t) 0 12 3 natural resource turnover

⎞ α (1 − c1 ) (1 − α) (1 − c2 ) + ⎟ ⎜ N(t) + z(t) N (t) + z(t) ⎟ ⎜ 12 3 12 3 0 0 ⎟ ⎜ ⎟ + eN(t) ⎜ resource consumed/restored ⎟ . ⎜ resource consumed/restored ⎟ ⎜ by individuals investing it in ⎠ ⎝ by individuals investing it in ⎛

proliferation

physiological maintenance

Several questions can now be asked of such a model, such as:

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1. If one allows for the possibility of resource overconsumption, which strategy is preferable for avoiding population collapse and consequently the tragedy of the commons? 2. Which strategy (allocating shared resources towards rapid proliferation, or towards slower proliferation but increased physiological and environmental maintenance) is more likely to become dominant as a result of natural selection? Similarly to the previous example, answering these questions will require the use both of classical analytical methods and the HKV method. Question 1 If one allows for the possibility of resource overconsumption, which strategy is preferable for avoiding population collapse as a result of the tragedy of the commons? Answering this question, as in the previous case, can be achieved through conducting stability analysis and in particular by evaluating how system behavior changes with regard to changes in strategy parameter α and concurrent changes in parameters of resource consumption c1 and c2 . The obtained bifurcation diagram (see Fig. 5) describes the possible dynamical regimes of a population that is homogeneous with respect to α. An important conclusion from the bifurcation analysis is that the main qualitative regimes of behaviors and also the sequence in which they appear as the parameters of (over-) consumption change are very similar for both extreme cases. However, wider domains of sustainable coexistence with shared resource were identified for the second strategy of allocating the resources towards physiological maintenance even under increasing values of parameters of resource (over)consumption. This suggests that at least in the case of a parametrically homogeneous system, investing in physiological maintenance might be a more sustainable strategy. Question 2 Which strategy (allocating shared resources towards rapid proliferation, or towards slower proliferation but increased physiological and environmental maintenance) is more likely to become dominant as a result of natural selection? The answer to this question required application of the HKV method to distribute parameter α (the details of this relatively complex transformation are given in [25]). The resulting parametrically heterogeneous system allowed exploring the changes in predicted evolutionary trajectories depending on the initial composition of the population with respect to different strategies. The results of these simulations revealed that in this system, the direction of population evolution is extremely sensitive to initial population composition (see Fig. 6). This suggests that even though one strategy might be preferable for a parametrically homogeneous population, in a parametrically heterogeneous case the direction of the evolutionary trajectory is determined primarily by initial distribution of clones with the population. This can be interpreted as “founder effect,” when the

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Fig. 5 Bifurcation diagram of the System (1.4). (a–c) present schematically (c1 , c2 , α) parameter portraits for fixed values of γ , δ, e = 1 and (d) represents the corresponding typical phase portraits. In domain 1 there exists a nontrivial globally attracting equilibrium point Aα . Domains 2 and 6 are the regions of bistability; in domain 2, there is a nontrivial stable node, while in domain 6 there exists a stable oscillatory regime. In these regions population survival is conditional on the initial population size and the initial amount of resource. In domain 3, an unstable limit cycle is formed around the point Aα , shrinking the range of possible initial conditions that will lead to sustainable population survival. In domain 4, point Aα is unstable, so any perturbation will lead to population collapse. In domain 5, an elliptic sector appears, which implies that a population is bound for extinction regardless of initial conditions. Finally, domain 0 corresponds to the case, when only trivial equilibrium B 0, γδ is globally attractive, which is of no biological interest

initial composition of the small population determines the subsequent evolutionary trajectory of the population over time [26]. Example 3 Oncolytic viruses In this final example, we look at a model proposed in [27], which describes the dynamics of cancer cells that can be infected by an oncolytic virus, i.e., a virus that can specifically infect and kill cancer cells but leave normal cells unharmed [28–30]. The proposed model considers two types of cancer cells, infected and uninfected, growing in a logistic fashion. The system is described by the following two equations:

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dt 0123

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where X is the size of the uninfected cancer cell population; Y is the size of the infected cancer cell population; r1 and r2 are the maximum per capita growth rates of uninfected and infected cells, respectively; K is the carrying capacity; β is the transmission coefficient, which may also include the replication rate of the virus; and δ is the rate of additional infected cell death rate as caused by the virus. The following questions can be asked and answered by this model: 1. What transitional regimes occur as the cancer cell population gains resistance to the virus? Can we use the model to infer dynamics of evolution of resistance? 2. Why are cytotoxic therapies effective in some patients and not others? Similarly to the previous cases, both approaches—classic bifurcation analysis and modeling of heterogeneity—will be necessary to answer these questions. Question 1 What are the transitional regimes that occur as the cancer cell population gains resistance to the virus? Can we use the model to infer dynamics of evolution of resistance? In order to answer this question, bifurcation analysis needs to be performed. A full bifurcation diagram can give a sense of what transitional regimes a population goes through as it moves from the area of phase-parameter space of tumor elimination to that of tumor growth, similarly to the previous examples. The complete phase-parameter portrait of System (1.5) is shown in Fig. 7. The model exhibits all possible outcomes of life cycle of infected and uninfected cells. In

Fig. 7 Bifurcation diagram of the parametrically homogeneous system reported in [27] and reproduced here in System (1.5). All possible outcomes of oncolytic virus infection are as follows: no effect on the tumor (domains I and II), stabilization or reduction of the tumor load (domains IV and V), and complete elimination of the tumor (domain VIII). Moreover there are two domains (domains III and VII) where the final outcome crucially depends on the initial conditions and can result either in failure of virus therapy or in stabilization (domain III) and elimination (domain VII) of the tumor. The figure is adapted from Fig. 1 in [27]

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domains 1 and 2, there is no effect of the viral infection on the tumor; in domains IV and V, tumor load is stabilized and even reduced. Complete elimination of the tumor can be observed in domain VIII. Furthermore, there are two domains (domains III and VII) where the final outcome crucially depends on the initial conditions and can result either in failure of virus therapy or in stabilization (domain III) and elimination (domain VII) of the tumor. Introduction of population heterogeneity with respect to parameter of viral transmission β allowed more complete visualization of possible evolutionary trajectories of the tumor. For example, in simulations in Fig. 8, parameter values were chosen in such a way as to start in domain VIII, where complete tumor elimination occurs. However, as the system evolved, the dynamics crossed from the domain of complete tumor elimination (VIII) to that of bistability (domain VII) to end up in the domain of tumor escape (domain I). Furthermore, differences in variances of initial distributions resulted in changes in predicted tumor dynamics, with lower variances corresponding to longer periods of near-negligible tumor sizes,

Fig. 8 Solutions of parametrically heterogeneous system reported in [27] with Gamma-distributed parameter of transmission of the oncolytic virus. Exact parameter values used in panels (a), (b), (c) and (d) can be found in Fig. 2 in [27]. The solutions here reflect the fact that the degree of heterogeneity plays an important role in the model dynamics. The parameter values and initial conditions are the same for all four simulations; the difference comes from different initial variances of the initial distribution; the greater the initial variance, the faster we reach the unfavorable domain I. The figure is reproduced with permission from Fig. 2 of [27]

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a dynamical regime that can be interpreted as tumor dormancy, or “cancer without disease,” when a tumor is present in the tissue but it is not growing [31–33]. More broadly, we can infer from the bifurcation analysis and subsequent simulations that as the tumor population becomes resistant, it travels through the various domains described in Fig. 7, allowing us to better understand the transitional regimes of evolution of resistance. Question 2 Why are cytotoxic therapies effective in some patients and not others? The answer to this question came from further simulations conducted by the authors, where they showed that initial composition of the population may be one of the culprits underlying emergence of resistance in some tumors but not others. Specifically, in Fig. 9 they showed that differences in variance of the initial

Fig. 9 Solutions of parametrically heterogeneous system presented in [27] with both uninfected cell specific and infected cell specific distributions of transmission coefficient. The initial conditions and parameter values are the same for both cases; the two cases differ only in the initial variance of the initial distribution of the transmission coefficient; exact parameter values used in panels (A), (b), (c) and (d) can be found in Fig. 7 in [27]. As one can see, even a small difference in the variance of the initial distribution of the cell clones may yield dramatically different results: in the first case, the tumor is cured, whereas in the second case, virus therapy fails. The figure is adapted from Fig. 7 in [27]

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distribution of cell clones within the population can lead to qualitatively different final outcomes of oncolytic therapy. These results may shed some light on the question of variability in therapeutic successes for other interventions, a topic that is of vital importance.

2 Conclusions Classic techniques for analysis of dynamical systems can provide critical insights into the possible dynamical regimes that a system can realize. Unfortunately, doing full bifurcation analysis is labor intensive and is not always possible due to increasing complexities of many models. However, there already exists a very rich body of literature of fully analyzed parametrically homogeneous models in many fields, including ecology [34, 35], epidemiology [36–38], among others. As the examples presented here demonstrate, even relatively simple two-dimensional systems can reveal rich, unexpected, and meaningful behaviors. Application of the HKV method to introduce population heterogeneity in a meaningful way and utilizing previously performed analysis can reveal a new layer of understanding of many existing models that was not accessible before. This of course is possible only if we ask the right questions. Acknowledgements The author would like to thank anonymous reviewers for helpful and insightful comments. This research received no external funding. Disclosure of Potential Conflicts of Interest IK is an employee of EMD Serono, U.S. subsidiary of Merck KGaA. Opinions expressed in this paper do not necessarily reflect the opinions of Merck KGaA.

References 1. C. Darwin, On the Origin of Species by Means of Natural Selection: Or the Preservation of Favoured Races in the Struggle for Life (Yushodo Bookseller’s, Tokyo, 1880) 2. S.E. Page, Diversity and Complexity (Princeton University Press, Princeton, 2010) 3. G. Bell, Selection: The Mechanism of Evolution (Oxford University Press on Demand, Oxford, 2008) 4. C. Johnson, Introduction to Natural Selection (University Park Press, Baltimore, 1976), pp. vii–213 5. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer Science & Business Media, London, 2013) 6. G.P. Karev, Inhomogeneous maps and mathematical theory of selection. J. Differ. Equ. Appl. 14(1), 31–58 (2008) 7. G.P. Karev, On mathematical theory of selection: continuous time population dynamics. J. Math. Biol. 60(1), 107–129 (2010)

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8. G.P. Karev, The HKV method of solving of replicator equations and models of biological populations and communities. arXiv preprint arXiv:1211.6596 (2012) 9. G. Karev, I. Kareva, Replicator equations and models of biological populations and communities. Math. Model. Nat. Phenom. 9(3), 68–95 (2014) 10. D.C. Krakauer, K.M. Page, D.H. Erwin, Diversity, dilemmas, and monopolies of niche construction. Am. Nat. 173(1), 26–40 (2008) 11. I. Kareva, F. Berezovskaya, C. Castillo-Chavez, Transitional regimes as early warning signals in resource dependent competition models. Math. Biosci. 240(2), 114–123 (2012) 12. G. Hardin, The tragedy of the commons’. Science 162(3859), 1243 (1968) 13. E. Ostrom, Coping with tragedies of the commons. Annu. Rev. Polit. Sci. 2(1), 493–535 (1999) 14. E.E. Ostrom, T.E. Dietz, N.E. Dolšak, P.C. Stern, S.E. Stonich, E.U. Weber, The Drama of the Commons (National Academy Press, Washington, 2002) 15. K. Parvinen, Evolutionary suicide. Acta Biotheor. 53(3), 241–264 (2005) 16. D. J Rankin, A. López-Sepulcre, Can adaptation lead to extinction? Oikos 111(3), 616–619 (2005) 17. M. Scheffer, J. Bascompte, W.A. Brock, V. Brovkin, S.R. Carpenter, V. Dakos, et al., Earlywarning signals for critical transitions. Nature 461(7260), 53–59 (2009) 18. M. Scheffer, S. Carpenter, J.A. Foley, C. Folke, B. Walker, Catastrophic shifts in ecosystems. Nature 413(6856), 591–596 (2001) 19. V. Dakos, M. Scheffer, N.E.H. van, V. Brovkin, V. Petoukhov, H. Held, Slowing down as an early warning signal for abrupt climate change. Proc. Natl. Acad. Sci. 105(38), 14308–14312 (2008) 20. I. Kareva, B. Morin, G. Karev, Preventing the tragedy of the commons through punishment of over-consumers and encouragement of under-consumers. Bull. Math. Biol. 75(4), 565–588 (2013) 21. B.-E. Sæther, M.E. Visser, V. Grøtan, S. Engen, Evidence for r-and K-selection in a wild bird population: a reciprocal link between ecology and evolution. Proc. R. Soc. B 283(1829), 20152411 (2016) 22. E.R. Pianka, On r-and K-selection. Am. Nat. 104(940), 592–597 (1970) 23. J.H. Andrews, R.F. Harris, r-and K-selection and microbial ecology, in Advances in Microbial Ecology (Springer, Boston, 1986), pp. 99–147 24. S.C. Stearns, Life history evolution: successes, limitations, and prospects. Naturwissenschaften 87(11), 476–486 (2000) 25. I. Kareva, F. Berezovkaya, G. Karev, Mixed strategies and natural selection in resource allocation. Math. Biosci. Eng. 10(5–6), 1561–1586 (2013) 26. J.G. Lambrinos, How interactions between ecology and evolution influence contemporary invasion dynamics. Ecology 85(8), 2061–2070 (2004) 27. G.P. Karev, A.S. Novozhilov, E.V. Koonin, Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics. Biol. Direct. 1(1), 30 (2006) 28. E.A. Chiocca, S.D. Rabkin, Oncolytic viruses and their application to cancer immunotherapy. Cancer Immunol. Res. 2(4), 295–300 (2014) 29. H.L. Kaufman, F.J. Kohlhapp, A. Zloza, Oncolytic viruses: a new class of immunotherapy drugs. Nature reviews drug discovery. Nat. Res. 14(9), 642–662 (2015) 30. D.L. Bartlett, Z. Liu, M. Sathaiah, R. Ravindranathan, Z. Guo, Y. He, et al., Oncolytic viruses as therapeutic cancer vaccines. Mol. Cancer 12(1), 103 (2013) 31. G.N. Naumov, E. Bender, D. Zurakowski, S.-Y. Kang, D. Sampson, E. Flynn, et al., A model of human tumor dormancy: an angiogenic switch from the nonangiogenic phenotype. J. Natl. Cancer Inst. 98(5), 316–325 (2006) 32. J. Folkman, R. Kalluri, Cancer without disease. Nature 427(6977), 787–787 (2004) 33. I. Kareva, Primary and metastatic tumor dormancy as a result of population heterogeneity. Biol. Direct 11(1), 37 (2016) 34. A.D. Bazykin, Nonlinear Dynamics of Interacting Populations (World Scientific, Singapore, 1998)

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35. F. Berezovskaya, G. Karev, T.W. Snell, Modeling the dynamics of natural rotifer populations: phase-parametric analysis. Ecol. Complex. 2(4), 395–409 (2005) 36. F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology (Springer, New York, 2001) 37. A.S. Novozhilov, On the spread of epidemics in a closed heterogeneous population. Math. Biosci. 215(2), 177–185 (2008) 38. A.S. Novozhilov, Epidemiological models with parametric heterogeneity: deterministic theory for closed populations. Math. Model. Nat. Phenom. 7(3), 147–167 (2012)

Polyvariant Ontogeny in Plants: When the Second Eigenvalue Plays a Primary Role Dmitrii O. Logofet

Abstract In a local population of a plant species, individual plants may follow various pathways through successive stages in their ontogenesis or/and different schedules in the succession of stages. This feature is called polyvariant ontogeny and considered to be the major mechanism of adaptation at the population level. While the field study reveals this phenomenon in nature, its quantitative characteristics are gained from what is called a matrix model of stage-structured population dynamics. The classical Perron–Frobenius theorem for nonnegative matrices provides for the existence and uniqueness of the dominant eigenvalue, λ1 (L), of the model matrix L, the asymptotic growth rate, which is considered to be the quantitative measure of adaptation. This chapter reports on a recently developed theory of rank1 corrections of nonnegative matrices, a strong extension of the classics in what concerns λ2 (L), the second positive eigenvalue, why λ2 (L) is always less than 1, and what it means for the matrix population model.

1 Introduction This chapter is devoted to matrix population models of stage-structured, singlespecies populations, the stages being supposed discrete and distinguishable in botanical studies. A profound introduction to what is called matrix population models can be found, e.g., in Caswell [1], but I confine here to what is necessary for the mathematical comprehension of the content, see also [2]. Next subsection presents a concept that underlies theoretical views on the plant development and growth. Those views stem from the fundamental idea of developmental biology that the whole course of development can be represented as a sequence of certain discrete, definable stages [3].

D. O. Logofet () A.M. Obukhov Institute of Atmospherics Physics, Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_5

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Fig. 1 Ontogeny scale of woodreed Calamagrostis epigeios. Stage notations: p plantule, j juvenile, im immature, v virginal, g1, g2, g3 generative, ss subsenile, s senile (adapted from [4] and [5])

1.1 Scale of Ontogenesis When botanists monitor a plant population, they are guided by the so-called scale of ontogenesis, a pictorial representation of the consecutive stages that an individual of a given species proceeds in the course of its development. Those stages have canonical names and notations, and they are either known from the background knowledge of species biology or defined in the current study. Figure 1 shows a sample of the ontogeny scale for a perennial grass species. We see that the stage of an individual plant can be determined from the morphology of its above-ground parts.

1.2 Life History and the Life Cycle Life history of an individual plant is a verbal description of how specifically the plant proceeds through the scale of ontogenesis, and it turns into the individual life cycle when we fix the time step as the interval between two consecutive censuses (e.g., 1 year) and consider the contribution the plant makes to the population recruitment when passing through the reproductive stage(s). Thus, the “cycle” appears due to the recruitment, which is observed either directly when the reproduction is vegetative or/and indirectly via the soil seed bank and seed germination, when it is generative.

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Fixing the time step introduces the time dimension into the study and enables monitoring the population dynamics in terms of

1.3 Population Structure, Population Vector, and the Life Cycle Graph Consecutive stages in the ontogeny scale can be naturally numbered with the natural numbers, and let xj (t), j = 1, 2, . . . , n, denote the number of the individuals belonging to the j-th1 stage at the moment, t, of observation. Expressed in the absolute or relative number, xj is called the stage-specific population size, while the population (column) vector, x(t) = [x1 (t), . . . , xn (t)]T ∈ Rn+ ,

(1)

belongs to the positive orthant of Rn and represents the population structure. The population structure is relative when normalized by the sum of vector components, N(t) = ||x(t)||1 , thus becoming a stochastic vector. Botanical studies of local populations are normally conducted on permanent sample plots. They mine data of the kind that H. Caswell called “identified individuals”: “Data in which individuals are marked and followed over time. In some cases, each individual is observed at each time (e.g., trees in a quadrate); in others an individual may disappear for a while and reappear later (e.g., birds marked by leg bands, or whales identified by color patterns)” [1, p. 134]. Therefore, the population vector (1) is reliably determined on (a limited number of) bounded sample plots at each discrete time moment, t, of observation. Determined also are all the changes in the status of each individual plant at time t that have occurred to the moment t + 1, as well as detected all the new plants that have appeared since the previous census, the population recruitment at t + 1. Summarizing this information qualitatively for all individuals of the local population results in what is called the life cycle graph (LCG): its vertices (or nodes) correspond to the stages depicted in the ontogeny scale, while the directed edges (arcs) illustrate all possible transitions between stage statuses and the recruitment by reproductive plants. Figure 2 shows the LCG for the case where both the age (in years) and stage are detectable in individual plants. The set of nodes has therefore two apparent dimensions in this special case. In other cases, LCGs are “onedimensional.” The plantule stage is absent from the LCG as the annual censuses took place at mid-August, when the plantule stage became already immature. For the same reason, the newly reproduced plants are recruited at the virginal stage. LCGs like that in Fig. 2 illustrate vividly what the next subsection is devoted to. 1 Do

not confuse with j as the notation of stage.

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Fig. 2 Life cycle graph for a mature population of woodreed C. epigeios reproducing vegetatively: solid arrows represent ontogenetic transitions, dashed arrows represent reproductions; stage notations as in Fig. 1; empty boxes designate the age-stage statuses that were not observed in the field study, the patterned ones indicate states participating in reproduction. Further explanation is given in Sect. 2 (adapted from [5])

1.4 Polyvariant Ontogeny as a Mechanism of Adaptation So, individual plants follow various, polyvariant pathways through successive stages in their ontogenies, while many stages may last for various numbers of years. This diversity of ontogenetic pathways and stage schedules within a local population of a single species is called polyvariant ontogeny in the Russian school of geobotany [6], and it is considered to be the major mechanism of adaptation at the population level (ibidem). Although the time dimension is implicit in “one-dimensional” LCGs (Fig. 3), the LCG still illustrates the irregular schedules of stages by the self-loops at certain stages (arcs va  va and g  g), meaning that some plants remain at the stage for one year more. Under favorable conditions, the virginal plants may accelerate their ontogeny and skip the first generative stage (va → gt); the unfavorable conditions result in some generative plants not flowering next season to accumulate more resources for the future flowering (va ← g). So, the LCGs illustrate effectively the idea of polyvariant ontogeny as the mechanism of adaptation, and we will see in the next section how they provide for the quantitative measure of adaptation.

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Fig. 3 Life cycle graph of Eritrichium caucasicum, an alpine herbaceous perennial: j, young individuals (seedlings and juvenile plants); va, virginal and adult vegetative plants; g, generative, and gt, terminal generative plants, which die after flowering [7]

1.5 Life Cycle Graph as the Digraph Associated to a Matrix In terms of graph theory, the digraphs depicted in Figs. 2 and 3 are weighted digraphs due the weights assigned to each arc [8]. Irrespectively of whether those weights are numeric or symbolic, they can be arranged into a square matrix, A = [aij ], in accordance with the following simple rule after the nodes having been numbered (ibidem): if, in the digraph, there is an arc i→j weighted with wij , then aij = wij ; if there is no arc i→j, then aij = 0, (i, j = 1, 2, . . . , n) .

(2)

The arc direction can be changed to the opposite one in the above rule, the ensuing matrix becoming AT , the transpose of A. Therefore, the both modes can coexist without conflicts, unlike the famous war between Lilliput and Blefuscu [9]. However, the latter “Big End” is preferable in matrix population models for the reason explained in the next section.

2 Mathematics of the Matrix Population Model The mathematics we deal with are linear algebra and matrix theory, in particular, the chapter on nonnegative matrices, yet various mathematical disciplines may also be engaged in solving the problems motivated by model applications.

2.1 Basic Model Equation, or Where the Matrix Emerge From While the LCG represents rather a qualitative view of individual life cycles, it gives an insight to how the population dynamics is expressed in quantitative terms.

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For the sake of simplicity, consider the LCG in Fig. 3, the further notions and methods being applicable to any kind of discrete-structured population with “identified individuals” data. Let the positive weights c, d, . . . , k, l represent frequencies of the corresponding transitions (observed at time moment t + 1), while a and b be the average numbers of alive young plants recruited from (the seeds produced by) the generative plant and the terminally generative plant, respectively, at t + 1. Defined in this way, these model parameters are called vital rates or demographic parameters. Following these meanings and adopting the natural order of components in the population vector, we have the following system of linear equations:

(3) System (3) takes on the following vector-matrix form: x (t + 1) = L x(t),

(4)

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Equation (4) summarizes the changes that occur in the population vector with time, and it is therefore the basic model equation, which underlies theoretical studies and practical model applications. Remark 1 The LCG in Fig. 3 coincides with the digraph associated to matrix L by the rule alternate to (2) (“Big End”). Thus, matrix L emerges from a given LCG and the meanings of vital rates by the well-known canon of linear algebra. Remark 2 The vital rates in matrix L depend, by definition, on the data gained at the years t and t +1. To indicate this dependence, it is formally sufficient to include only t into the notation, L = L(t).

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2.2 Population Projection Matrix, L = T + F Since matrix L “projects” the current population structure for one time step further, it was called the population projection matrix [10, 11]. Note that the projection matrix A has quite another definition in matrix theory: A2 = A [12], meaning that the second projection cannot change the result of the first one. This makes no sense in population models. Decomposition, L = T + F, to the parts T plus F responsible for the ontogenetic transitions and recruitments, respectively, is practically important since the matrix entries have different meanings (as c, d, . . . , k, l and a, b, respectively, in Fig. 3) and the ways they are calculated are respectively different. Matrix (5) decomposes as ⎡

00 ⎢cd L=T +F =⎢ ⎣0f 0 k

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(6)

While the T part is calculated uniquely (due to all the transitions being recorded, hence their frequencies fixed), the F part is not when the LCG contains more than one reproductive stages, hence F has more than one positive entries (two in Fig. 3, seven in Fig. 2). The progenies of different parent groups cannot be distinguished among the total population recruitment observed in the field [7, 13, 14], therefore the status-specific reproduction rates remain uncertain and confined only to the observed recruitment size. In general, this confinement takes on the form of the recruitment balance equation, x (t + 1) –T x(t) = F x(t)

(7)

([15], p. 220), with the unknowns in the right-hand side and all the coefficients known from the data. The first-component equation of system (7), the only nontrivial one, reduces by Eq. (6) to x1 (t + 1) = a x3 (t) + b x4 (t)

(8)

in the E. caucasicum case, with the unknowns a, b and the known xj s. Theoretical value of decomposition (6) will be exposed in Sect. 2.4.

2.3 Asymptotic Properties and the Measure of Adaptation The basic model Eq. (4) suggests that x(t) = L (t–1) . . . L(1)L(0)x(t)

(9)

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when matrix does depend on t, and x(t) = Lt x(0), t = 0, 1, . . . ,

(10)

when it does not and remains constant. In the latter case, under the terms formulated below, the sequence {x(t), t = 0, 1, ...} converges in the following sense: 4 t logt→∞ L x(0) λt = x ∗ , 1

(11)

where λ1 > 0 is the dominant eigenvalue of matrix L and x* > 0 is the corresponding positive eigenvector whose norm is proportional to that of x(0) [16, 17]. The fact the eigenvalue λ1 > 0 does exist such that λ1 = ρ(L), the spectral radius of L, together with a corresponding positive eigenvector, is certified by the classical Perron–Frobenius theorem for nonnegative matrices (see, e.g., [18], §II.5.5). The theorem requires the matrix to be irreducible [12], which is equivalent to its associated digraph being strongly connected, i.e., having a directed route between any pair of nodes ([8], Chapter 7). The LCG in Fig. 3 is obviously strongly connected, whereas that in Fig. 2 is obviously not. The lack of strong connectedness is normally caused by the presence of post-reproductive stages in the LCG, yet there exists a reproductive core in this case, the maximal strongly connected subgraph [19], and the theorem has to be applied to the submatrix associated with the reproductive core of the LCG (backgrounded with light gray in Fig. 2). Limit (11) features the case where λ1 = ρ(L) > |λj | for any other eigenvalue λj ∈ C in the spectrum of irreducible matrix L, and such a matrix is called primitive [12]. Otherwise, when there are more than one eigenvalues whose modulus equals ρ(L), the irreducible matrix L is called imprimitive, while the number of those eigenvalues is called the index of imprimitivity (ibidem). It is calculable via the lengths of all simple cycles in the (reproductive core of the) LCG [12, 20] and equals 1 in all practical cases [21, 22], confining the study to the primitive matrix L. So, the dynamics of population vectors are asymptotically geometric with λ1 as the exponent: Lt x(0) ∼ λ1 t x∗ ,

(12)

i.e., the growth when λ1 > 1 or decline when λ1 < 1, along the direction defined by the dominant vector. Although λ1 does show the rate at which the population vector increases along its asymptotic direction (defined by x*), λ1 (L(t)) is determined in a unique algebraic way by the matrix entries, namely, by those ontogenetic transitions and reproductions that occurred in the population during the interval of time from t to t + 1. This is why λ1 (L(t)), “in spite of its asymptotic nature, does determine the growth potential that the local population possessed in a particular environment on that very interval of time. In this sense, λ1 (L(t)) does measure how the local

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population adapts to its environment” ([21], p. 48), i.e., it serves as a quantitative measure of adaptation local in space and time.

2.4 Net Reproductive Rate, R0 Given a discrete-structured population with the population projection matrix L = T + F, the “net reproductive rate R0 is the mean number of offspring by which a newborn individual will be replaced by the end of its life and thus the rate by which the population increases from one generation to the next” [1, p. 126]. Let a new generation at the initial time moment be described by a vector g(0). As “the end of its life” may be uncertain in a nontrivial LCG, the number of offspring can be replenished at each of the infinite number of time steps, with the total recruitment being equal to  F g(0) + F T g(0) + F T 2 g(0) + F T 3 g(0) + · · · = F I + T + T 2 + T 3 + . . . g(0)

(13) when matrices T and F remain constant over time. When converging, series (13) sums up to F(I – T)–1 , resulting, by Perron–Frobenius theorem, in 6 5 R0 = ρ F (I –T )–1

(14)

[23], where I denotes the identity matrix. Then it is quite logical that R0 > 1 yields the population growth and R0 < 1 the decline. Moreover, the mathematical fact is that ⎧ ⎧ ⎫ ⎫ ⎨1

(15)

where ⇐⇒ means “if an only if” (ibidem; [24]). Remark 3 Each column j of matrix T contains the rates of transitions from the jth stage to any other, including the delay at j. If none of the stage-j individuals dies out of these transitions, the column sum equals 1 (as the full event frequency), otherwise it is less than 1. In general, all column sums are ≤ 1, hence matrix T is always substochastic, being stochastic in quite unrealistic cases only. It means that ρ(T) ≤ 1 ([18], §III.3.1), hence series (13) does converge when T is truly substochastic. The simplest illustration descends back to the Leslie [25] model of a hypothetical age-structured population and reduces, for the sake of clarity, to a particular number of age classes, say, n = 5, with the 3rd and 4th ones being reproductive. Then the Leslie matrix takes on the following form:

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D. O. Logofet

⎡

0 0 ⎢s 0 ⎢ 1 ⎢ L = T + F = ⎢ 0 s2 ⎢ ⎣0 0 0 0

0 0 0 0 0 0 s3 0 0 s4

⎤ ⎡ 0 0 ⎢ ⎥ 0⎥ ⎢0 ⎥ ⎢ 0 ⎥+⎢ 0 ⎥ ⎢ 0⎦ ⎣0 0 0

0 0 0 0 0

b3 b4 0 00 0 0 0 0 0 0 0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(16)

with the age-specific survival and birth rates. It is easy to show that T t = 0 for t ≥ 5, hence series (13) converges just to its finite sum, and we have ⎡

(b3 s1 s2 + b4 s1 s2 s3 ) (b3 s2 + b4 s2 s3 ) (b3 + b4 s3 ) b4 ⎢ 0 0 0 0 ⎢ ⎢ F (I –T )–1 = ⎢ 0 0 0 0 ⎢ ⎣ 0 0 0 0 0 0 0 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎦ 0 (17)

([26], Example 2.2), whereby 6 5 R0 (L) = ρ F (I –T )–1 = (b3 s1 s2 + b4 s1 s2 s3 )

(18)

is simple, indeed. Correspondingly simple are the conditions (15) on the vital rates under which λ1 >, =, or < 1.

3 Potential-Growth Indicators and the Merit of Indication Combined together, the three statements (15) represent what is called the indicator ability of the function R0 (L), a particular case of what is considered in this section.

3.1 General Definition Let R(L) be a scalar function of an irreducible nonnegative matrix L. Definition 1 R(L) is called a potential-growth indicator (PGI) if it possesses the indicator ability, i.e., ⎫ ⎧ ⎫ ⎧ ⎨1 [2, 27].

(19)

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What R(L) indicates is the location of λ1 (L) on the real axis relative to 1, hence whether a model population with the reproductive core submatrix L declines, remains stable, or grows in the asymptotics. Function R0 (L) is an apparent sample of PGI due to its property (15), and it is apparently not a unique sample. For instance, if a given R(L) is a PGI, then R(L)α is also a PGI for any α > 0. Other samples can be shown too as an exercise for beginners in real analysis. In the vast variety of PGIs, there is one of special importance due to its algebraic simplicity, which is considered in the next subsection.

3.2 Simple Indicator, R1 Another exercise for beginners reveals that simple expression (18) for R0 (L) in the Leslie case can be obtained in another way, as R1 (L) = 1– det (I –L) ,

(20)

the same being true for the n×n Leslie matrix. In fact, the indicator ability of R1 (L) was revealed already by P.H. Leslie for his n×n matrix [25], yet he did not use the PGI terminology. Function R1 (L) is really simpler to calculate by (20), but to prove its indicator ability (19) is not so simple in more general cases than the Leslie case. The nearest expansion to the Lefkovitch [28] matrix for stage-structured population, Lef kovitch = Leslie + diag {d1 , d2 , . . . , dn } ,

(21)

where dj ≥ 0 is a rate of delay at stage j, was obtained much later [29], yet still before the PGI terminology was coined [2]. The PGIs R0 (L) and R1 (L) are no longer equal in Lefkovitch matrices. This can be illustrated by matrix (5) with e = 0: R0 (L) = [af + b (f l + k (1–h))] c/ [(1–h) (1–d)] , R1 (L) = acf + bc (k (1–h) + f l ) , which are apparently different, yet the both being linear w.r.t. the birth rates a and b. A further generalization concerned the so-called Logofet matrices [30], Logofet = Lefkovitch +

(22)

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D. O. Logofet

where the triangle block of potentially nonzero elements ∗ locates below the first subdiagonal and corresponds to accelerated “progressive” transitions j → i, i ≥ j +2. For any strongly connected, “progressive” LCG, Klochkova’s theorem (ibidem; [2]) established the indicator ability of R1 (L) and the explicit form of (20) via the matrix entries corresponding to simple cycles in the LCG. However, Fig. 3 with e > 0 bears witness to the existence of polyvariant LCGs with the “regressive” arcs, too, signifying a trait of adaptation (see also [31], Fig. 1). Rather common are regressive links for size-structured populations of plant species, signifying shrinkage effects [32–34]. In further subsections, I present a proof that R1 (L) (20) is still the PGI for matrices L with both “progressive” and “regressive” elements.

3.3 Why λ2 Is Crucial for Indication There is a graphic explanation why at all can function R1 (L) indicate the location of λ1 (L) relative to 1. By definition (20), we have R1 (L) = 1–p (1; L) ,

(23)

where p(1; L) is the characteristic polynomial of the irreducible matrix L evaluated at λ = 1. How does the plot of p(λ; L) looks like in the plane (λ, p)? Since the coefficient at the highest power of λ equals 1 (see (20)), while the dominant root λ1 > 0 is simple by Perron–Frobenius theorem, the rightest point λ where the plot intersects the horizontal semi-axis is λ = λ1 > 0 with a positive derivative p (λ1 ) > 0, see Fig. 4 (where L is omitted to simplify the notations). Note that the indicator ability (19) to be verified for R1 (λ) can be also expressed equivalently in terms of p(1) = 1 – R1 (1): ⎧ ⎫ ⎧ ⎫ ⎨1

(24)

Fig. 4 Qualitative types of the characteristic polynomial curve around the unit value of its argument: (a) λ1 < 1, p(1) > 0; (b) λ1 >1, p(1) < 0; (c) 1 ≤ λ2 < λ1 , p(1) > 0 (adapted from [35])

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While the equality case is trivially true in (24), the nontrivial inequality ones deserve investigation in the three qualitatively different cases covering all the options (depicted as three corresponding panels in Fig. 4). Cases (a) and (b) are simply evident due the simplicity of λ1 as the dominant root of p(λ), and they illustrate the “legal” PGI property (24). Case (c) does apparently lose the property, but it can only exist if there exists the second positive λ2 ≥ 1 in the spectrum of L. If it does not exist or is less than 1, then the situation is actually covered be the “legal” case (a) or (b). So, λ2 (L) > 0 plays the crucial role indeed as it deprives R1 (L) of its PGI ability when λ2 (L) ≥ 1, and I devote the next two subsections to the proof that this case does practically not exist among (irreducible) population projection matrices.

3.4 Rank-One Corrections of Nonnegative Matrices It is clear intuitively and well established mathematically that λ1 (A) = ρ(A), the largest, or leading, or dominant eigenvalue, or the Perron root, is a monotone increasing function of any entry to the irreducible nonnegative matrix A ([12], Ch. 8). But what can we say if the argument increment represents a matrix, rather than a scalar, with the entries of different signs? In simple cases, the answer resorts to the idea of rank-one corrections for nonnegative matrices [35]. Definition 2 A matrix B is called a rank-one correction of a matrix A if rank(B − A) ≤ 1 (ibidem). We use the term rank-one correction in two related senses: it is both the addition of a rank-one matrix B − A to a given matrix A (as in Definition 1), and the result of such an addition (matrix B). Theorem 1 Suppose matrix B is a rank-one correction of a matrix A and both these matrices are nonnegative, then B has at most one real eigenvalue (counting multiplicities) strictly greater than ρ(A). The proof is given in [35]. So, if a rank-one correction B of a matrix A ≥ 0 is nonnegative, then either ρ(B) ≤ ρ(A) or the dominant eigenvalue of B is simple and unique in the interval (ρ(A), +∞). In other words, no rank-one correction can produce two real eigenvalues greater than ρ(A), provided we stay in the set of nonnegative matrices. Although ρ(B) can be made arbitrary large by a large enough correction (B – A), the second largest real eigenvalue, if it exists, always remains bounded by ρ(A).

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Fig. 5 Life cycle graph of Androsace albana: pl, seedlings; j, juvenile plants; im, immature; v, vegetative adults; g, generative plants (adapted from [36])

3.5 Population Projection Matrix as a Rank-1 Correction of Its Transition Part Now, we consider the population projection matrix, L = T + F, as a rank-one correction of its transition part, T, by its fertility part, F. Two evident cases are the single-row F when recruitment occurs in a single stage only (Figs. 2 and 3) and the single-column F when the parents at a single reproductive stage produce the newborns that may occur in several earlier stages (Fig. 5). Since F is nonnegative (and F = 0), we have ρ(L) > ρ(T), and it is the sole eigenvalue in the interval (ρ(T), +∞) by Theorem 1. The spectral radius of a substochastic matrix cannot exceed 1 ([18], §III.3.1), hence the second positive eigenvalue, λ2 (L), if it exists, always meets the condition: λ2 (L) ≤ 1.

(25)

Remind that for R1 (L) being a PGI, it is necessary and sufficient that λ2 (L) < 1 or does not exist (Sect. 3.3). Therefore, the case λ2 (L) = 1 needs a special investigation, which has been implemented as Theorem 3 by Protasov and Logofet [35]. It is only possible when ρ(T) = 1, and the Theorem has described all the cases where a truly substochastic matrix T may have ρ(T) = 1. None of those special cases can realize in modeling practice.

3.6 When R1 Is Linear, or the Merit of Indication Two evident cases of rank-one fertility matrix, namely, the single-row F and the single-column F, lead to one more remarkable property of R1 (L). Since each summand in the standard expression of the determinant contains only one element of each row/column, the power at which the reproduction rates enter expression (20) cannot be greater than 1.

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For example, Fig. 5 suggests that matrix L takes on the following form: ⎡ ⎢ ⎢ ⎢ L=T +F =⎢ ⎢ ⎣

0 0 000 d 0 000 ef h00 00k l 0 000 m0

⎤

⎡

0000a ⎥ ⎢ 0000b ⎥ ⎢ ⎥ ⎢ ⎥+⎢ 0 0 0 0 c ⎥ ⎢ ⎦ ⎣000 00 000 00

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

(26)

with positive reproduction rates a, b, c and R1 (L) = a (df k + ek) m + bf km + ckm + h + l–hl.

(27)

Similarly, for matrix L (5) with reproduction rates a and b, we have R1 (L) = acf + b (ck + cf l–chk) + d + h–dh + ef.

(28)

The merit of linearity ensues from that of indication, i.e., speculations on the population viability (λ1 1) prior to the calculation of its λ1 [27]. Of particular demand are those speculations when the data are insufficient to calibrate the matrix in a unique way. For example, when the data are of the “identified individuals with uncertain parents” type (Figs. 2 and 3), only the transition part T of matrix L = T + F can be uniquely calculated, whereas the fertility part F is only confined to the recruitment balance equation (7), see Sect. 2.2, a linear equation for several unknown reproduction rates (for seven in case of Fig. 2 and two of Fig. 3). This equation defines a hyperplane in the parametric space of reproduction rates, which cuts a polyhedron, B, out of the positive orthant. Any point b ∈ B fits the data, and the λ1 (L(b)) attains its minimal and maximal values as a continuous function on a compact domain, thus posing “the limits of adaptation in the local population of a clonal plant with polyvariant ontogeny” [37]. Those “limits” turn out calculable as the values of λ1 (L(b)) at certain vertices of the polyhedron B (Theorem 1 in [21, 27]), while the conjugate minimization/maximization problems for R1 (L(b)) are solvable by the routine methods of linear programming [38, 39]. Although the extremal points of R1 (L(b)) may not coincide with those of λ1 (L(b)) ([27], Appendix B), the linearity of R1 (L(b)) provides also for a tool to test the expert hypotheses on whether the observed population growth be compatible with certain hierarchies among stage-specific reproduction rates or/and reproduction contributions as far as those hierarchies are expressed as the additional linear inequality constraints [27], constraining λ1 (L(b)) further to a polyhedron A ⊆ B. If maxA R1 (L(b)) < 1, then the hypothesis fails; otherwise, if maxA R1 (L(b)) > 1, then it should be accepted [40].

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4 Discussion and Conclusions While the algebraic expressions of R0 (L) and R1 (L) coincide for the Leslie matrix L (the first row plus the first subdiagonal of nonzero entries), they do differ already for the Lefkovitch matrix (21), as illustrated for matrix (5) with e = 0. These PGIs differ also in the level of generality that the pattern of L may have to possess the indicator ability (19). In this regard, the “competition” between them reminds rather a “pursuit” where R1 (L) is just approaching to R0 (L) (Fig. 6). Indeed, the maximal level of generality in the pattern of L = T + F was achieved in a break-through paper by Cushing and Yicang [23] for what they call the net reproductive value: just the convergence of series (13) resulting in formula (14). The indicator ability of that value was also proved (ibidem, Theorem 3), thus showing it “both biologically meaningful and analytically useful” (ibidem, p. 298). Since that time, the function became very popular so quickly as it’s hard to say today who coined the notation R0 (L). The maximal level that R1 (L) has reached is the rank-one F, a wide, yet not universal, class of patterns among population projection matrices. Polyvariant ontogeny in plants may well generate situations where F has a higher rank (e.g., [33], Fig. 1; [13], Fig. 4). It occurs when the parents at more than one reproductive statuses (4 to 5 ibidem) produce the newborns observed at more than one recruiting statuses (2 ibidem). Two recruiting statuses generate a fertility matrix F with two nonzero rows, hence, of rank 2 (ibidem, Table 4). However, the theorem of rankone corrections cannot be expanded to rank-k (k ≥ 2) corrections ([35], Example 3). Nevertheless, calculations certify the indicator ability (19) of R1 (L) for all the

Fig. 6 Historical course of competition between the PGIs R0 (L) and R1 (L) generality in the pattern of L = T + F; other original terms are also cited

for the level of

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calibrated matrices L = T + F with rank-two Fs ([13], Table 4). An explanation can be found in a sufficient condition for R1 (L) to be the PGI, formulated below as Remark 4 Let the fertility part F in a population projection matrix L = T + F be of rank-k, k ≥ 2, and be represented as F = F 0 + F 1 , rank (F 1 ) = 1,

(29)

ρ (T + F 0 ) < 1.

(30)

and let also

Then, if there exists λ2 (L) > 0, we have λ2 (L) ≤ ρ (T + F 0 ) < 1 as an apparent consequence of Theorem 1 presented above in Sect. 3.4. Due to the crucial role the λ2 > 0 plays in the indicator ability of R1 (L) (see Fig. 4), it means that R1 (L) is still the PGI in this particular case of rank-k (k ≥ 2) matrix F. There is a mention of “scaling theorems for R0 ” in Fig. 6, with a reference to Li and Schneider [24], who proved R0 (L) to serve as a scaling factor for matrix F that ensures ρ(T + F/R0 ) = 1 (ibidem, Corollary 3.2), i.e., to stabilize an otherwise unstable population. Moreover, given an irreducible L = T + F, the authors determined, for any s > ρ(T), a scaling function q(s) such that ρ(T + F/q(s)) = s, while R0 (s) = R0 (L)/q(s) (ibidem, Theorem 4.4). Those findings revealed profound mathematics behind the net reproductive rate R0 (L), but the uniform scaling of F can hardly meet the needs of practical demography, where the reproduction rates are rather stage-specific, while the control/management efforts are rather concentrated on a few stages of interest [33, 41–43]. Rank-one corrections of population projection matrices, with the condition R1 (L) = 1, thus provide for a more versatile approach to stabilization problems. Although losing the competition for generality, the PGI R1 (L) wins that for computational reliability since the determinant is much easier to compute than the spectral radius [44, 45]. Although the task to enumerate all the vertices of a polyhedron is hard (recursively enumerable) in general [46, 47], it is still feasible in practical cases of polyvariant ontogeny under reproductive uncertainty, where the number of uncertain parameters does not exceed 5 ([21], Table 1), 7 (Fig. 2), or 8 ([13], Table 5). Therefore, solving the constraint minimization/maximization problems for R1 (b) by routine methods of linear programming really helps in finding the extreme values of λ1 (L) under reproductive uncertainty. Since Leslie [25] times, we get used to think of the second-in-modulus eigenvalue λ2 (L) ∈ C with its sole regard to the rate at which the sequence of x(t) = Lt x(0), t = 0, 1, . . . , converges to the limit (11). The second positive eigenvalue, with its crucial role in the indicator ability of R1 (L), has motivated the expansion of the

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classical Perron–Frobenius theory for nonnegative matrices with the recent theory of rank-one corrections. To conclude, this is a story how matrix population model, essentially linear systems of difference equations, still serve as the source of nontrivial mathematical problems when applied in modeling practice.

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43. A.L. Koop, C.C. Horvitz, Projection matrix analysis of the demography of an invasive nonnative shrub (Ardisia elliptica). Ecology 86(10), 2661–2672 (2005) 44. Y. Nesterov, V.Y. Protasov, Optimizing the spectral radius. SIAM J. Matrix Anal. Appl. 34(3), 999–1013 (2013) 45. V.D. Blondel, Y. Nesterov, Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices. SIAM J. Matrix Anal. Appl. 31(3), 865–876 (2010) 46. L. Khachiyan, The problem of calculating the volume of a polyhedron is enumerably hard. Russ. Math. Surv. 44(3), 199–200 (1989) 47. L. Khachiyan, E. Boros, K. Borys, K. Elbassioni, V. Gurvich, Generating all vertices of a polyhedron is hard. Discrete Comput. Geom. 39(1–3), 174–190 (2008)

Recurrence as a Basis for the Assessment of Predictability of the Irregular Population Dynamics Alexander B. Medvinsky

Abstract I give a brief overview of a number of methods that are aimed to assess predictability of population dynamics. Besides, a few examples of using the methods based on the recurrence nature of fluctuations of the population size in order to evaluate numerically the horizon of predictability time series resulted from both field observations and mathematical modeling of population dynamics are given in this paper. Keywords Predictability · Population dynamics

1 Introduction Prediction implies the estimation of future states of dynamical systems on the basis of the time series, which are resulted from mathematical modeling, experiments, and/or observations. In other words, to predict the future states of a system, it is necessary to know the past of this system. From the deterministic standpoint, the initial state of a system at time t0 completely determines the states for every time t > t0 [1]. However, it is well known now that there is a limitation of predictability of subsequent system states that is related to unavoidable uncertainty in the initial conditions provided that the system evolution exhibits rapid (but limited) growth of the initial uncertainty [2], i.e., if the system dynamics is chaotic [3]. Hand in hand with the amplification of the uncertainty, the chaotic dynamics are characterized by recurrence to initial conditions, i.e., dynamical systems invariably recur to immediate proximities to their initial state [4]. Such fuzzy recurrences of the states of the dynamical systems open the door to prediction of their future states based on their past.

A. B. Medvinsky () Institute of Theoretical and Experimental Biophysics, Pushchino, Moscow Region, Russia © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_6

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So far, however, the frequency of occurrence, mechanisms, and functional value of chaotic regimes in the dynamics of populations remain obscure [5, 6]. Nevertheless, even independently of the mechanisms underlying irregular oscillations in the population size, the problem of assessment of predictability of oscillations in the population size is still urgent [7]. The observed fluctuations in population sizes are typically very irregular [6], which is usually associated with both the inherent nonlinearity of the processes underlying population dynamics and external influences. Such irregularity causes the population dynamics to be unpredictable over the finite time that, following Lighthill [8], can be termed the horizon of predictability (Tpr ). A numerical estimation of the horizon of predictability critically depends on the measurement accuracy and length of the time series. In order to assess Tpr , a number of approaches have been proposed. A brief description of some of the approaches and their application to the evaluation of Tpr is given below.

2 Predictability of Chaotic Dynamics Firstly, let us consider as a simple example the chaotic mapping x (t + 1) = f (x(t)) , with close initial values of x(0) and y(0) located at a distance between them of |y(0) − x(0)|. In an m-dimensional phase space, the change in the distance between two close x and y trajectories during the unit time interval [n, n + 1] is described as   y (n + 1) − x (n + 1) = J (x(n)) (y(n) − x(n)) + O |y(n) − x(n)|2 , where J is a Jacobian. Let ei be the eigenvector of J, li its eigenvalue, and δ(n) = y(n)−x(n). Decomposing δ(n) into the eigenvectors with coefficients β i , one  can find δ(n + 1) = β i li ei . If the vector δ(n) is parallel to one of the eigenvectors ei , it will lengthen or shorten with time depending on the value of the factor li . In order to characterize the process as a whole, one can introduce a proper average over the different local stretching factors. Then the Lyapunov exponent (the length N of the time series is assumed infinite) is

(N ) λi = lim ln li . N →∞

If the largest of these exponents,  = max λi , is positive, the dynamics is chaotic. Since  > 0 is a signature of chaos, it is a considerable interest to determine its value for time series of a finite length.

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The time series of a finite length N, which can be represented in the following form: {x(n)} = {x(1), x(2), . . . , x(N )} , n = 1, . . . , N,

(2.1)

is assigned a vector in p-dimensional space (space of embedding): X(n) = [x(n), x (n − h) , . . . , x (n − (p − 1) h)] .

(2.2)

The vector in Eq. (2.2) defines the dynamics of the system represented by time series (2.1) in the interval from n−(p−1)h to n (h is the time lag). Then, the k(t) of X(i) vectors that falls into the ε-neighborhood, U(t), of X(i) are searched for a certain value of n = t. Then, the mean change in the norm ( X(t) − X(i) ) over time τ is tracked to calculate the function S(t), which characterizes the increase in the difference between vectors X(t) and X(i) over time τ in the overall time series (2.1) [9–11]:  M 1  1  ln S (τ ) = M k (t) t−1

 X (t) − X (i)

,

(2.3)

i∈U

where M = N−(p−1)h. The function (2.3) increases in a linear manner when the norm of the difference between vectors X(t) and X(i) becomes smaller than the size of the attractor. The inclination of the linear region of S(τ) allows the value of the dominant Lyapunov exponent  to be assessed. Using the universe dependence of the horizon of predictability, Tpr , on  [12], it is possible to estimate the degree of predictability for the studied chaotic dynamics.

3 Using Recurrence of Time Series to Assess the Dominant Lyapunov Exponent in a Microbial One-Predator-Two-Prey System Since long-term field observations of population dynamics in varying environments faces difficulties in explaining underlying ecological mechanisms [13], the laboratory experiments under controlled environmental conditions allow us to get closer to understanding these mechanisms. In order to identify the dynamical patterns, which can occur in a microbial food web, the experiments with the use of chemostat systems consisting of the ciliate Tetrahymena pyriformis as a predator and two coexisting prey bacteria, Pedobacter and Brevundimonas, were carried out [14]. The dynamical patterns in such a two-prey, one-predator system were found to include chaotic behavior, stable limit cycles as well as coexistence at equilibrium. Changes in the population dynamics were triggered by changes in the dilution rate.

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Fuzzy recurrence of the states of dynamical systems is resulted in a set of nearest neighbors of points on the phase trajectory. As a result, Eq. (2.3) can be used in order to evaluate the dominant Lyapunov exponent  numerically. The irregular time series characteristic of the predator Tetrahymena pyriformis dynamics under the dilution rate of 0.5 d−1 were found to be chaotic with  of about +0.15. Correspondingly, the horizon of predictability is approximately equal to 6.7 d while duration of the experiments (under chemostat conditions) exceeded 30 days.

4 Assessment of the Horizon of Predictability of Irregular Dynamics Without Computing the Dominant Lyapunov Exponent Along with the above approach involving estimation of the predictability of time series with the use of the dominant Lyapunov exponent, there is an algorithm for numerical estimation of the horizon of predictability, which does not directly utilize the Lyapunov exponent [15]. This algorithm implies the following successive steps for the time series u(t), where t ∈ [0,T]: 1. Construction of the vector            T T T T T , u − 1 .u − 2 ,...,u − (d − 1) = u ,u 2 2 2 2 2 where d is the embedding dimension; 2. Searching at the interval [0,T/2] the d-dimensional vectors U (ti ) = (U (ti ) , U (ti − 1) , . . . , U (ti − (d − 1))) , i = 1, 2, . . . , m, such that

 



u T − U (ti ) < ε & 1;

2 3. Prediction of the value u



 m 1  T +1 = U (ti+1 ) ; 2 m i=1

 4. Construction of the vector u T2 + 1 according to item (2.1) above and taking  into account that the value u T2 + 1 is known now; : ; 5. The next iteration at the interval 0, T2 + 1 , and following iterations until the point T is reached;

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6. Calculation of the prediction error [16] E(n) =

<  = u (t) − u (t − 1) 1  T2 +n − 1 ; t= T2 +2 n u(t) − u (t − 1)

(4.1)

7. The quantitative assessment of the horizon of predictability (Tpr ) as a function of the prediction error E(n) given by Eq. (4.1). For this purpose, a limiting value EL 1) the recruitment begins to decrease. In discrete-state Ricker map (5.1), int [x] denotes the integer part of x, σ measures the intensity of the noise, while ν is a standard normal random variable. In order to assess quantitatively the predictability of irregular oscillations, which arise in model (5.1), the algorithm proposed by Kaplan and Glass [15] was used. The prediction error was calculated with the use of Eq. (4.1). Figure 1a allows us to compare real values Nt obtained with the use of Eq. (5.1) and predicted values of this function. Here the horizon of predictability Tpr = 28 time steps. The value Tpr varied from one computer experiment to another; in the average Tpr under σ = 0.0001 was found to be equal to 20 time steps. Notice that the imposition of the same

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noise on continuous-state chaotic Ricker dynamics yielded Tpr to be, on average, equal to 15 time steps. The difference between the values of Tpr obtained for the continuous-state chaotic Ricker dynamics and their discrete-state noisy counterparts died out as the noise intensity increased. Under σ = 0.01, the averaged values of the horizon of predictability of the discrete-state and continuous-state dynamics became practically identical and equal to ten time steps [20].

6 Evaluation of Predictability of Irregular Dynamics with the Use of Reconstruction of Phase Trajectories In order to break the curse of short time series in population dynamics prediction, Turchin and Taylor [22] proposed an approach based on reconstruction of the phase trajectory, which corresponds to the studied experimental or observation data. It is essential such a reconstruction can be applied to rather short time series (of several tens of points), which sets this approach apart from the above methods, which demand time series of at least thousands of points for their efficient use. The first step in this approach is the assumption that the short time series u(t) resulted from an experiment or field monitoring can be represented by a function F: u(t) = F (u (t − 1) , u (t − 2) , . . . , u (t − p) , ε(t)) , where t is time and ε is an exogenous variable, which describes the impacts (for example, temperature variations during a season) that are not associated with endogenous processes (for example, with interactions between populations) related to the dynamic feedbacks within the system resulted in a time lag τ = 1, 2, . . . , p. The second step is the assumption that the rate of change, r(t), in a population size u∗ (t) is determined in the following way: r(t) = a0 + a1 X + a2 Y + a11 X2 + a22 Y 2 + a12 XY + ε(t),

(6.1)

where ai (i = 0, 1, 2) and aij (ij = 11, 12, 22) are constants; X = (u(t − 1))θ , ∗ (t) . Y = (u(t − 1))τ (θ and τ are constants), and r(t) = log u∗u(t−1) At the next stage, the numerical values of the parameters in Eq. (6.1) are chosen in order to maximally reproduce the comparatively short time series u(t) by the time series u∗ (t). Then, the time series u∗ (t), which now adequately reproduces the real dynamic process, can be extended beyond the time interval of observations. Such a reconstruction allows removal of the limitations on the numerical evaluation of the horizon of predictability of the process under study imposed by the length of the time series. This reconstruction technique assumes stationarity in the time series. Notice, however, that for the vast majority of environmental time series, this condition is not met.

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7 Reconstruction of Phase Trajectories as a Tool to Assess Chaoticity of Population Dynamics Turchin and Ellner [23] used the reconstruction of phase trajectories to estimate chaoticity of vole populations and found recurrent short-term chaos as the regime that characterizes vole dynamics in northern Fennoscandia. In other words, the vole dynamics occurred to be a blend of order and irregularity resulted in the dominant Lyapunov exponent very close to zero. This means that though the vole dynamics are not regular, the fluctuations of the vole population size are nevertheless highly predictable. Prevailing majority of case studies occurred to show that a most frequently found kind of population dynamics in the wild is oscillations, which are characterized by the values of the Lyapunov exponent in the [−0.1, +0.1] range [13].

8 Recurrence Quantification Analysis of the Time Series Another approach to the analysis of the population time series obtained experimentally or by field observations is based on the so-called recurrence quantification analysis. Historically, the concept of recurrence of the state of a dynamic system dates back to the work by Poincaré [4] where Poincaré demonstrated that a dynamics system very closely approaches its initial state infinitely many times. Almost 100 years later, Eckmann et al. [24] introduced a graphical tool, which they called a recurrence plot. In the framework of this tool, the repetition of the states of a studied dynamic system is represented as a recurrence matrix: > >  Rij (ε) = H ε − >X(i) − X(j ) > ; i, j = 1, . . . , N,

(1)

where X(k) (k = i, j) is the vector that defines the state of the studied system, N is the number of measurements of the system’s state, ε is a small parameter (see [25] for some rules in selecting its numerical value), and H is the Heaviside function (H(x) = 0 if x < 0 and otherwise H(x) = 1). Matrix (1) is used to visualize recurrence as a recurrence plot. Recurrence plots are sets of points in the system of coordinates i and j; usually, the black dots correspond to the coordinates (i, j) with Rij = 1 and the white dots correspond to the coordinates with Rij = 0. Non-regular dynamics typical of many environmental processes are embodied in the diagonal segments formed by the black dots that manifest recurrency of these processes [26]. The length (l) of such diagonal segments satisfies the following identity [25]:   *l−1 1 − Ri−1 j −1 1 − Ri+l j +l Ri+k j +k ≡ 1, k=0

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where Ri−1 j−1 = 0 if R1j = 1 or Ri1 = 1 and Ri+k j+k = 0 if RNj = 1 or RjN = 1 [26]. Since such diagonal segments with a length l suggest that the dynamic process follows a recurrence pattern over l time steps, the horizon of predictability for this process can be represented as an average length of the diagonal segments: N l=lmin

lP (l)

l=lmin

P (l)

Tpr = N

(2)

,

where P(l) is the histogram of diagonal segments with a length l, i.e., P (l) =

N    *l−1 1 − Ri−1 j −1 1 − Ri+l j +l Ri+k j +k ; i = j ; i,j =1

k=0

and lmin is the threshold segment length, which allows exclusion of the short segments corresponding to the stretch of time throughout which the autocorrelation goes to zero (or some arbitrary small value) [25]. The recurrence quantification analysis allows for the assessment of the horizon of predictability without theoretical reconstruction of phase trajectories but rather directly based on the specific dynamic features of the time series under study.

9 Recurrence Plots for the Analysis of Predictability of Plankton Dynamics Figure 2 demonstrates the phytoplankton time series resulted from the long-term hydrological monitoring of the system of Naroch Lakes (Belarus), which consists of three water bodies, Lake Batorino, Lake Myastro, and Lake Naroch; Lake Naroch in turn is subdivided into Large Stretch and Small Stretch. It is evident that these time series are strongly irregular; in other words, the phytoplankton biomass oscillations display no visible repetition. Nonetheless, numerical recurrence analysis detects repetition characteristic of these oscillations. Figure 3 shows the recurrence plots, which correspond to the time series in Fig. 2. Diagonal segments parallel to the main diagonal (the line of identity) occur when the corresponding phase trajectory visits the same region of the phase space at different times and a segment of the trajectory runs almost in parallel to another segment for t time units. As a result, the length of these diagonal segments in Fig. 2 is conditioned by the duration of a similar local evolution of the phytoplankton biomass. Figure 4 represents the bacterioplankton dynamics in each of the water bodies of the Naroch lakes system, while Fig. 5 shows the corresponding recurrence plots.

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Fig. 2 The dynamics of phytoplankton in the Naroch Lakes. The abscissa shows the observation data (years) and the ordinate, biomass (g/m3 ). Breaks in the time series correspond to the months when no measurements were carried out; see [27] for the measurement protocol

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The results of the recurrence quantification analysis based on the assessments of the length of diagonal segments in the recurrence plots in accordance with Eq. (2) [26, 27] are given in Table 1. One can see that the values of the horizon of predictability, Tpr , of bacterioplankton inhabiting the Naroch Lakes appeared 1.4– 2 times greater than the corresponding values for the phytoplankton. The question now arises: what are the factors affecting the predictability? Firstly, the value of Tpr can depend on trophic interactions between separate populations [13]. However, the trophic interactions are unlikely to be the only source of limitations in predictability of population dynamics. The changing environment is one more dimension of the dynamical context likely to influence fluctuations in the population size [28]. In particular, it has been shown that irregular bacterioplankton oscillations in the Naroch Lakes are phase-locked by seasonal variations in the water temperature, while the phytoplankton dynamics in Lake Naroch, the largest

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Fig. 4 The dynamics of bacterioplankton abundance (106 cells/mL) in the Naroch Lakes: (a) Small Stretch of Lake Naroch, (b) Large Stretch of Lake Naroch, (c) Lake Myastro, (d) Lake Batorino. Breaks in the time series correspond to the months when no measurements were carried out [26]

of the Naroch Lakes, and Lake Myastro are not synchronized with the temperature variations. At the same time, the temperature variations are able to phase-lock the fluctuations of the phytoplankton abundance in Lake Batorino, which is the smallest among the Naroch Lakes [27]. Hence, temperature is apparently the factor that has significant impact on the predictability of the bacterioplankton fluctuations. Notice that lack of synchronization between the phytoplankton fluctuations and oscillations of temperature does not necessary imply the absence of an impact of temperature on the phytoplankton dynamics. It just may mean that the phytoplankton dynamics are likely to be controlled by both the temperature and some other factors. It has been demonstrated that the horizon of predictability of the phytoplankton dynamics is do controlled not only by the temperature but also by trophic interactions and nutrient supply. This can lead to the worst predictability of the dynamics of phytoplankton in comparison with the dynamics of the bacterioplankton; see Table 1 [26, 27].

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Fig. 5 Bacterioplankton recurrence plots: (a) Lake Naroch, Small Stretch; (b) Lake Naroch, Large Stretch; (c) Lake Myastro; (d) Lake Batorino [27] Table 1 Assessments of the horizon of predictability, Tpr , for phytoplankton and bacterioplankton oscillations in each of the Naroch Lakes Reservoir Lake Naroch, Small Stretch Lake Naroch, Large Stretch Lake Myastro Lake Batorino

Phytoplankton Tpr (month) 2.4 2.3 2.5 2.5

Bacterioplankton Tpr (month) 4.8 4.6 4.7 3.4

10 Conclusion All the above-mentioned methods that have been designed for the analysis of nonlinear time series although can be useful (with some significant limitations; see, for example, [10]) when estimating predictability of population processes, provide few opportunities to reveal the mechanisms underlying complex changes in population size. The role of different factors in controlling predictability of population dynamics can be established by juxtaposition of the results obtained with the use of the analysis of field observations and the results of mathematical modeling [6, 29, 30].

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Acknowledgments The author is deeply grateful to anonymous reviewers. This work was partly supported by the Russian Foundation for Basic Research (grant # 17-04-00048).

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Total Analysis of Population Time Series: Estimation of Model Parameters and Identification of Population Dynamics Types Lev V. Nedorezov

Abstract Current publication is devoted to the problem in identifying the type of population dynamics (on an example of green oak tortrix (Tortrix viridana L.)). For fitting of time series of tortrix fluctuations (Korzukhin and Semevsky, Sinecology of forest. Gidrometeoizdat, Saint-Petersburg, 1992), generalized discrete logistic model was used. Results of model parameter estimations obtained with ordinary least squares (OLS) and method of extreme points (MEP) (Nedorezov. Chaos and Order in Population Dynamics: Modeling, Analysis, Forecast. LAP Lambert Academic Publishing, Saarbrucken. 2012. p. 352; Nedorezov. J. Gen. Biol. 73(2), 114–123, 2012; Nedorezov. Biophysics 61(1), 149–154, 2016) were compared. It was assumed that the model demonstrates good correspondence to time series if and only if deviations between time series and model trajectory satisfy with several statistical tests. It was shown that model with OLS estimations of parameters cannot be used for fitting of time series. Analyses of four various variants of MEP estimations were provided, and it was obtained that observed dynamic regime of population dynamics isn’t cyclic (if length of cycle is less than 1500 years). For the selected dynamic regimes, a rapid decrease in values of auto-correlation functions with further small fluctuations near zero level was observed. It means that forecasting the change in population size for short or long time periods is practically impossible. Keywords Population dynamics · Generalized discrete logistic model · Green oak tortrix fluctuations

L. V. Nedorezov () Research Center for Interdisciplinary Environmental Cooperation RAS, Saint-Petersburg, Russia © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_7

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1 Introduction One of the most important problems in modern ecology is a problem of identification of population dynamics type for the existing time series [1, 2]. Solution of this problem is scientifically based background for forecast, selection of optimal methods of population dynamics management, and some other important ecological problems. Determination of population dynamics type can be provided or with use of various biological tests [1, 3–5] or with use of mathematical models of ecosystem dynamics [6–13]. Note that in the first case, it is possible to use information of qualitatively different types; in the second case we can use existing time series only and may have serious problems when time series are rather short or when we have several correlated time series [14, 15]. In most cases the second way is preferred: it allows obtaining quantitative estimations of population dynamics characteristics. But at the same time, the second way doesn’t allow using the existing information totally: we can effectively use several existing time series only [6, 16]. For various mass species of forest insects (and, in particular, for green oak leaf roller (Tortrix viridana L.); [17–21]), we have long time series, and for analyses of these datasets it is possible to use ecological models with several (visible and invisible) variables [1, 2, 22–24]. But before applying of complex multi-component models describing population/ecosystem dynamics, we have to be sure that we cannot obtain sufficient description of population dynamics with simpler models [25]. In current publication for fitting of known time series [17] on dynamics of green oak leaf roller (Tortrix viridana L.), generalized discrete logistic model was used [26, 27]. Estimations of model parameters were provided by two various statistical approaches: ordinary least squares (OLS) [28, 29] and method of extreme points (MEP) [6, 7, 14, 30]. Provided calculations show that OLS estimations belong to “non-biological zone” of space of model parameters, and it doesn’t allow determining of population dynamics type and present a real forecast. Searching of MEP estimations of model parameters was provided within the boundaries of “biological zone” of space of model parameters. Several points with extreme properties were found; on qualitative level all extreme points correspond to one and the same dynamic regime: it isn’t a cyclic fluctuations of population size (when length of cycle is less than 1500 years), and it can be characterized by fast decreasing auto-correlation function. It means that constructing of good forecasts can be a serious problem.

2 Model Description In literature it is possible to find a rather small number of mathematical models of isolated population dynamics which have rich set of dynamic regimes and can

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be used as a “brick” for constructing complicated multi-component models of ecosystem dynamics [26, 27, 31–36]. Discrete logistic model is from that pool of models [6]: ? xk+1 =

ax k (b − xk ) , xk ≤ b . 0, xk > b

(2.1)

In (2.1) xk is a population size (or population density) at time moment k (time has a discrete nature). Parameter b is maximum value of population size; in (2.1) it is assumed that if at any time moment population size is greater than b population size of next generation is equal to zero. Product ab is maximal value of birth rate (which is defined as relation of sizes of two nearest generations). Parameters a and b, and initial value of population size x0 are non-negative amounts, a, b, x0 ≥ 0. If inequality ab ≤ 4 is truthful, trajectories of model (1) are non-negative and bounded at 0 ≤ x0 ≤ b. If ab > 4 and 0 < x0 < b, possibility for model trajectory to escape from domain {x : x < b} to {x : x > b} is appeared; in this situation, identification of population dynamics regime is practically impossible: behavior of model trajectory will correspond to regime of population extinction, but it doesn’t correspond to reality. In other words, domain {(a, b) : ab > 4} in a space of model parameters can be defined as “non-biological zone.”

3 Used Statistical Criterions Within the framework of traditional approach to model parameter estimation, it is assumed that the model allows obtaining sufficient approximation of empirical datasets if and only if deviations (between model/theoretical values and respective empirical amounts) are values of independent stochastic variables with normal distribution [28, 29]. It is also assumed that hypothesis about equivalence of deviations of averages to zero cannot be rejected for the selected significance level (it is equal to assumption that there are no regular errors in measurements). For the analysis of properties of sets of deviations, Kolmogorov–Smirnov test, Lilliefors test, Shapiro–Wilk test [37–39], and other tests are used for checking of the normality of deviations (for checking the correspondence of sample of deviations to Normal distribution). For checking the absence/existence of serial correlation in sequence of deviations, Durbin–Watson test, “jumps up–jumps down” test, and some other tests can be used [28, 29, 40–42]. Estimations of model parameters can be found, for example, with the help of the following functional form (ordinary least squares (OLS)): Q (a, b, x0 ) =

N   ∗ 2 xk − g (a, b, x0 , k) → min . k=0

a,b,x0

(3.1)

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  In (3.1) xk∗ , k = 0, . . . , N is the existing initial sample (change in population sizes from year to year), N + 1 is a sample size; g(a, b, x0 , k) is a trajectory of model (1) which can be obtained for fixed values of parameters a and b, and initial population size x0 , g(a, b, x0 , 0) = x0 . In literature, one can find other types/modifications of functional form (2). For example, if we want to take into account stronger influence of small values of initial sample onto final result (onto model parameter estimations), we can use functional form (2) with weights: Q (a, b, x0 ) =

N 

 2 wk xk∗ − g (a, b, x0 , k) → min .

k=0

a,b,x0

In this expression weights wk are non-negative values for all k, wk ≥ 0, w0 + . . . + wN = 1. But now we have no criterions for selection of weights wk . We have a recommendation only in such occasion where we need to have bigger values of weight for smaller deviation. We have also to note that weights may depend on amount of deviations. One more way for modification of functional form (2) is the following: Q (a, b, x0 ) =

N 

∗

x − g (a, b, x0 , k) γ → min . k k=0

a,b,x0

In this expression γ is positive number, and it is not obligatory that γ = 2. We can conclude that now we have no criterions for the selection of functional forms of type (2), and the functional forms correspond to our imagination about ways for obtaining best estimations of model parameters only. It is assumed that values which give global minimum for loss function Q are the best estimations of model parameters. If for these estimations of parameters, one of the used statistical criterions gives negative result (in particular, Null hypothesis about correspondence of distribution of set of deviations to Normal distribution, serial correlation is observed in sequence of residuals, etc.), and it gives a background for conclusion that the model cannot give sufficient fitting of time series. It is important to repeat it again: final conclusion about suitability or non-suitability of model for fitting of considering time series, we make using one point of a space of model parameters. Moreover, we use for final conclusion estimations of model parameters but not its real values. Use of loss functions for finding estimations of model parameters is one of the basic limitations of OLS. It becomes a more serious problem in a situation when we have to use several correlated time series [11–14, 43–45]. Use of method of extreme points (MEP) [7, 16] doesn’t assume using of any loss function. For obtaining MEP estimations of model parameters, first of all we have to construct feasible set of points ∗ in a space  = {(a, b, x0 ) : a, b, x0 ≥ 0}. Set ∗ must be constructed by the following way: at the beginning we have to choose a set of statistical criterions which must be used for checking of properties of sets

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of deviations between theoretical/model values and respective values of initial time series. We have also to fix significance level for all criterions. After that we have to find points in a space  which correspond to sets of deviations when all the used statistical criterions give desired results. If feasible set is empty, ∗ = ∅, we get a background for conclusion that model isn’t suitable for fitting of time series. When ∗ isn’t empty for approximation of time series, we have to choose points with extreme properties (for example, we can choose points with maximum p-value for one or other statistical criterion). In this paper, 5% significance level was fixed for all the used criterions. For every selected point of space of model parameters, set of respective deviations was checked on symmetry with respect to origin (it was provided with tests of homogeneity of two samples: Kolmogorov–Smirnov test, Mann–Whitney U-test, Lehmann–Rosenblatt test, and Wald–Wolfowitz test were used). Monotonic behavior of branches of density function was checked with Spearman rank correlation coefficient [37, 46]. For the analysis of absence/existence of serial correlation in sequences of deviations, Swed–Eisenhart test and “jumps up–jumps down” test were used [29, 40–42]. Note that feasible set ∗ can be defined as confidence set: for all points of ∗ statistical criterions give required results, and we have got a background for conclusion about invalidity of model for fitting of time series.

4 OLS Estimations of Model Parameters For model (1), minimizing of loss function (2) allowed obtaining the following estimations for parameters a, b, and x0 : x0 ≈ 0.086465, a ≈ 0.090102, and b ≈ 54.236778; for these estimations we have Q(a, b, x0 ) ≈ 1215.035. This point of space of model parameters belongs to zone where origin is global stable equilibrium; when time step k = 30 population size xk = 65.12392, and after that step population size becomes equal to zero (and it doesn’t correspond to reality). Analysis of deviations shows that with 5% significance level, hypothesis about equivalence of average to zero cannot be rejected. At the same time, probability that distribution of deviations corresponds to Normal distribution is following: p < 0.1 (Kolmogorov–Smirnov test), p < 0.01 (Lilliefors test), p = 0.0116 (Shapiro–Wilk test). Thus with 1% significance level, hypothesis about Normality of deviations must be rejected. Testing of symmetry of distribution of deviations gave the following results: probability of event that distribution is symmetric is equal to p = 0.584645 (Wald– Wolfowitz test), p = 0.033261 (Mann–Whitney U-test). It allows concluding that with 5% significance level, hypothesis about symmetry must be rejected, and consequently OLS estimation doesn’t belong to feasible set ∗ . Thus, obtained results allow concluding that with OLS estimations, model (1) cannot give sufficient approximation of time series. Regime of population extinction is observed in the model at 31st time step; distribution of deviation doesn’t correspond to Normal (respective hypothesis must be rejected with 1% significance

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level) and so on. In such a situation, we don’t need to use other statistical criterions: for all possible results of application of other tests, we’ll have the same final result: model (1) cannot be used for fitting of empirical datasets, and cannot be used for description of green oak leaf roller population dynamics.

5 MEP Estimations of Model Parameters In Fig. 1 there is a projection of 120,000 points of feasible set ∗ onto plane (a, b). All points were found at pure stochastic search in domain [0, 100] × [0, 5] × [0, 130]. As we can see on Fig. 1, big number of points of feasible set ∗ are within the boundaries of “biological zone” where inequalities ab ≤ 4 and x0 ≤ b are truthful. Highest concentration of points (Fig. 1) is observed near curve ab = 4. It indicates that with a big probability, population dynamics of green oak leaf roller corresponds to cyclic regime with big length. For points of set ∗ , it was obtained that minimum value for Kolmogorov– Smirnov test (d = 0.25064) was observed at x0 = 26.16194, a = 0.11327, b = 35.31373 (ab = 3.999978). For these estimations, minimum value (0.019231) was also observed for Lehmann–Rosenblatt test. For t = 0.26, probability K(t) of Kolmogorov distribution is close to zero [37], and respectively with significance level which is close to one, we cannot reject Null hypothesis about symmetry of distribution of deviations. Lehmann–Rosenblatt test shows that this hypothesis cannot be rejected with significance level 0.997. It means that Null hypothesis about symmetry of distribution must be accepted. Close result was obtained for Mann– Whitney test: U = 60 with critical level 45 when sample size is equal to 26. b 120 100 80 60 40 20 a

0 0

0.5

1

1.5

Fig. 1 Projection of 120,000 points of feasible set ∗ onto plane (a, b)

2

2.5

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Checking of monotonic behavior of branches of density function of deviations can be provided in two possible variants. If sample  size is  rather big, then we can check pointed out property for deviations ek+ and −ek− separately where ek = xk∗ − g (a, b, x0 , k) . Deviation ek+ is positive value of deviation ek , and respectively ek− is a negative   one. If sample size is small, then we can check pointed out property for set ek+ ∪  −   −ek . Let’s consider a situation when ek+ is sufficient big sample, k = 1, . . . ,  +∗  m. And let ek be a sample of ordered positive deviations: +∗ e1+∗ ≤ e2+∗ ≤ · · · ≤ em .

Monotonic decreasing of density function means that bigger values (in sample) must be observed with smaller probabilities. Respectively, for lengths of intervals : +∗ ; : +∗ ; : +∗ +∗ ; +∗ , , em 0, e1 , e1 , e2 , . . . , em−1 we need to have the:similar; order (in ideal situation). Rank 1 will correspond to the shortest interval 0, e1+∗ , while biggest rank m will correspond to the biggest : +∗ ; +∗ . Ideal case must be compared with real situation which is interval em−1 , em   determined by sample ek+∗ . For this reason, we have to calculate Spearman rank correlation coefficient ρ (and/or Kendall correlation coefficient τ ), and check Null hypothesis H0 : ρ = 0 with alternative hypothesis H1 : ρ > 0. For selected significance level, Null hypothesis must be rejected. Note we have stronger result in a case when we can reject Null hypothesis with smaller significance level. For pointed out parameters, we have p − value = 0.02052 for Spearman rank correlation coefficient, and p − value = 0.02325 for Kendall correlation coefficient τ . Thus Null hypotheses must be rejected for both coefficients with 3% significance level. Analysis of behavior of auto-correlation function r(k) shows that for 0 < k ≤ 15000 all values of this function belong to close interval [−0.02, 0.02]. It allows concluding that if observed process is cyclic, the length of cycle is bigger than 1500 years. Moreover, fast decreasing of values of this function (r(0) = 1) and further fluctuations in narrow limits near zero level is typical behavior for processes which forget their history very fast (for example, like pure stochastic processes). In Fig. 2 considering time series and model (1) trajectory obtained for pointed out parameters are presented. For points of feasible set ∗ (Fig. 1), it was obtained that maximum value for Mann–Whitney test U = 119 was observed for x0 = 0.529402, a = 0.299619, b = 13.234814 (note this point belongs to “biological zone” of space of model parameters, ab = 3.9654). For these parameters, we have p − value = 0.1838 (Kolmogorov–Smirnov test), p − value = 0.06028 (Lehmann–Rosenblatt test). Thus with 6% significance level, Null hypothesis about symmetry cannot be

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population size

50

real dataset

40

model trajectory

30 20 10 0 1960

years 1965

1970

1975

1980

1985

1990

Fig. 2 Time series of fluctuations of green oak leaf roller (solid line) and trajectory of discrete logistic model (1) (broken line) obtained for parameters when maximum amounts for p − value are observed for Kolmogorov–Smirnov and Lehmann–Rosenblatt tests

rejected, but we have to note that amount of p − value is very close to critical threshold. Spearman rank correlation coefficient ρ = 0.6178 with p − value = 0.0004933. Kendall correlation coefficient τ = 0.4277 with p − value = 0.0009166. Taking it into account, we have to accept hypothesis about monotonic behavior of branches of density function. Analysis of behavior of auto-correlation function r(k) shows that for 8 < k ≤ 15000 all values of this function belong to close interval [−0.08074, 0.0685]. Like in a previous case if observed process is cyclic, length of this cycle must be bigger than 1500 years. In Fig. 3 considering time series and model (1) trajectory obtained for pointed out parameters are presented. For set ∗ (Fig. 1), maximum value of Spearman rank correlation coefficient r = 0.888547 is observed for the following estimations of model parameters: x0 = 0.573105, a = 0.217602, b = 18.18987 (this point belongs to “biological zone” of space of model parameters, ab = 3.958159). For the obtained parameters, p − value = 0.1226 for Kolmogorov–Smirnov test, p − value = 0.20171 for Lehmann–Rosenblatt test, p − value = 0.218743 for Wald–Wolfowitz test, and p − value = 0.293265 for Mann–Whitney U-test. Thus with 12% significance level, hypothesis about symmetry of deviation’s distribution cannot be rejected. Spearman rank correlation coefficient ρ = 0.888547 with p − value = 8.367 · 10−7 . Kendall correlation coefficient τ = 0.7169231 with p − value = 5.988 · 10−9 . Taking into account the presented results for deviations, we have to accept hypothesis about monotonic behavior of branches of density function. Analysis of behavior of auto-correlation function r(k) shows that for 11 < k ≤ 15000 all values of this function belong to close interval [−0.038, 0.035].

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10 0 1960

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1970

1975

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1985

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Fig. 3 Time series of fluctuations of green oak leaf roller (solid line) and trajectory of discrete logistic model (1) (broken line) obtained for parameters when maximum amounts for p − value are observed for Mann–Whitney U-test

Like in previous cases if observed process is cyclic, length of this cycle must be bigger than 1500 years.

6 Conclusion Provided analysis of fluctuations of green oak leaf roller population [17] with generalized discrete logistic model showed that estimations of model parameters obtained with ordinary least squares method belong to “non-biological zone.” If we use this approach only we obtain a background for conclusion that model doesn’t allow obtaining sufficient approximation for considering time series. Deviations between theoretical/model values and empirical numbers don’t correspond to several common requirements which must be observed if we have “good” correspondence between the model and the existing dataset. For example, hypothesis about Normality of set of deviations must be rejected with 1% significance level. Moreover, for estimated parameters model predict population extinction to 1993 year that doesn’t correspond to reality. Approach to estimation of model parameters based on method of extreme points (MEP) allowed presenting several most suitable points for fitting of space of model parameters. All presented points are from “biological zone,” and deviations between theoretical/model trajectories and real dataset are satisfied to set of statistical criterions. In other words, analysis of deviations doesn’t allow concluding that model isn’t suitable for fitting of considering time series.

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It is interesting to note that all variants of dynamic regimes which are observed for MEP estimations of model parameters correspond (on a qualitative level) to one and the same population size behavior. This is not a cyclic regime with cycle length in 1500 years or less. Moreover, in all situations, a rapid decrease in values of auto-correlation function (calculated for model trajectories) with further small fluctuations near zero level is observed. It is a typical behavior for processes which “forget their history” very fast.

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Modelling Cancer Dynamics Using Cellular Automata Álvaro G. López, Jesús M. Seoane, and Miguel A. F. Sanjuán

Abstract Cellular automata and agent-based models have become the cornerstone of the simulation of many complex biological phenomena. More specifically, they are making major breakthroughs in the understanding of cancer development. Besides, these discrete spatio-temporal models can be hybridized with more traditional models based on differential equations, allowing to faithfully represent multiscale open systems. These systems typically consist of many entities that can perform a vast repertoire of actions, which depend on the concentration of substances diffused in their environments, as well as their mutual interaction through different coupling mechanisms. In the present chapter, we use a hybrid cellular automaton model to explore the dynamics of tumor growth in the presence of an immunological response. A mathematical expression is derived, which describes the speed at which a tumor is erased by a population of immune cytotoxic cells, depending on the morphology of the tumors and the intrinsic capacity of the immune cells to detect and destroy their adversaries. Finally, the coevolution of tumor– immune aggregates is simulated and the likelihood of a prolonged tumor mass dormancy mediated by the immune system is discussed. Keywords Cancer dynamics · Cellular automata · Tumour growth · Immune cells · Mathematical modelling

Á. G. López · J. M. Seoane Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain e-mail: [email protected] M. A. F. Sanjuán () Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain Department of Applied Informatics, Kaunas University of Technology, Kaunas, Lithuania Institute for Physical Science and Technology, University of Maryland, College Park, MD, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_8

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PACS 05.45.Ac, 05.45.Df, 05.45.Pq

1 Introduction The oversimplification of cancer as the growth of an independent subset of rebel mutated cells within a tissue presents great difficulties explaining tumor development [1, 2]. The relative importance of the dynamics at the tissue level during carcinogenesis, represented by the interactions of the tumor cells with their environment, compared to the role played by mutations, or in connection with them, is still a subject of intense debate [3, 4]. Perhaps, in order to unveil the origin of cancer, a fundamental question that needs to be addressed first is why healthy somatic cells, as part of a tissue, do not grow unlimitedly. It is hard to believe that eukaryotic cells have lost their ability to reproduce in the absence of growth factors, since autopoiesis pervades life at all scales. Assuming this fact, we would then have to understand how a tissue as a whole self-organizes contributing to this suppression and control of cell growth. Undoubtedly, chemical and physical interactions between both similar and different types of cells within the tissue should play a key role in differentiation and tumorigenesis. The tumor microenvironment includes stromal cells (e.g., immune cells, fibroblasts, or endothelial cells), the extracellular matrix, and signalling molecules such as cytokines or growth factors. The particular cellular and molecular mechanisms, as well as their role in tumor development, are complex and not sufficiently well understood [5]. Even though all of them might prove to be important in the fight against cancer, immunotherapy is lately focusing great attention. Probably, this is because the immune system is better known and has evolved for centuries to neatly destroy threatening foreign organisms in our body. As it occurs with any other evolutionary entity, when a tumor forms, it should develop its own biochemical imprints (antigens), which would allow for its recognition by the immune system. Therefore, there is evidence and hope that it can be trained to effectively destroy tumor cells, which originate in the body, as well. The history of immunotherapy for cancer dates back to the beginning of the twentieth century, when the physician Paul Ehrlich suggested that the immune system might protect an organism from the development of cancer [6]. Around 50 years later, this proposition was more formally reintroduced by Macfarlane Burnet [7, 8] and, later on, by Lewis Thomas [9]. After suffering major setbacks [10, 11], the immunosurveillance theory gained renewed consistence close to 20 years ago, thanks to several experimental works with genetically altered mice [12, 13]. Currently, the immunosurveillance of tumors is more properly referred as cancer immunoediting. Given the genetic heterogeneity of tumors, this control system coevolves with them and seems to act as a natural selective force, editing its phenotype by selecting those cells that are unresponsive to immune detection. Adoptive cell transfer using chimeric antigen receptors [14, 15], the modulation of CTLA-4 activity by means of monoclonal antibodies [16], or the blocking of the PD-1 receptor [17] are a few outstanding examples of the increasing importance that

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immunotherapy is gaining. Nevertheless, and despite some dramatic cases of cure, the advantages of immunotherapy are still modest in general, and only some cancers (less than 10%) might benefit from immunotherapy nowadays [18]. Therefore, there is still a long way to go in the investigation of immunotherapy for cancer, as the many ongoing clinical trials indicate [19]. The progress of tumor immunotherapy with T lymphocytes mainly relies on our capacity to uncover and understand the molecular and cellular basis of the T-cell-mediated antitumor response. However, due to the highly complex regulatory mechanisms that control both cell growth and the immune system, this task can be hardly achieved without the use of mathematical models. From a theoretical point of view, these models provide an analytical framework in which fundamental questions concerning cancer dynamics can be addressed in a rigorous fashion. The practical reason for their development is to make quantitative predictions that permit the refinement of the existing therapies or even the design of new ones. Because cancer is a biological phenomenon occurring at multiple scales, mathematical models of tumor growth are becoming increasingly sophisticated. In particular, agent-based modelling and cellular automata are the groundbreaking instruments of contemporary research in the study of cancer dynamics [20–25]. These models allow to accurately represent the cell heterogeneity within a tissue, and can be hybridized with more traditional models based on differential equations, which allow to represent the substances that diffuse through the tissue and the intracellular dynamics as well. More particularly, mathematical models describing a growing tumor that interacts with the cellular arm of the immune system have demonstrated their potential to explain different properties of tumor–immune interactions [26]. In the present work, we adopt the view of enzyme kinetics to describe tumor– immune interactions at the cellular scale [27, 28]. Enzymatic reactions can be viewed in an abstract manner as an asymmetric interaction between two entities, one being rather passive (the substrate) and the other being rather active (the enzyme). When these two entities make contact, the latter affects the former transforming it into some other entity (the product). Thus, an enzymatic reaction can be casted in three steps: the formation of a complex from the two parts, a subsequent transformation of the passive part by its active counterpart and their final dissociation. As long as these conditions are fulfilled, there is no general reason preventing us to use this conceptual framework not only at the chemical scale, but also at the cellular scale and, perhaps, even at higher scales. For example, the growth of microorganisms in the presence of a limited substrate obeys the Michaelis– Menten kinetics [29]. In ecology, the intake rate of a consumer as a function of the density of preys is also a kinetics of this type [30]. In all these cases, whenever there is a considerable imbalance between the number of active and passive elements, saturation occurs. This is due to the limited capacity of the active part to interact with a sufficiently high number of elements of the passive counterpart. Note that, in part, this is also true in the reverse direction, since the passive elements cannot interact with an enormous number of active elements for short times. Nevertheless, the situation is not completely symmetrical, since an active element can interact

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with many passive elements, while a passive element usually interacts with one or a few active elements, before it is transformed. In summary, interactions occur locally and require some time. Inspired by this reasoning, a mathematical model describing tumor–immune interactions was designed by Kuznetsov et al. [31] to explore a possible dynamical origin of the dormancy and the sneaking through of tumors. In their original model, the rate at which a tumor is lysed increases linearly with the number of immune cells, just as in an ordinary Lotka–Volterra model [32, 33]. Simply put, the velocity at which a tumor is destroyed can be increased without bounds by simply adding more immune cells. Nevertheless, their work served as a foundation for other works concerning the interactions between immune and tumor cells [34, 35]. Among these works, a mathematical model was validated using experiments from mice [36] and men [37]. To reproduce the experimental data, these authors proposed a new fractional cell kill for the lysis of tumor cells by CD8+ lymphocytes. The fractional cell kill is a key concept in the study of tumor lysis, which has sometimes been confused with the rate of tumor cell lysis, because the notion of rate is used in a loose sense [35, 38]. However, strictly speaking, these concepts are different. The rate represents the speed at which the tumor is lysed, while the fractional cell kill is defined as the speed at which the logarithm of the tumor size is reduced. Well, these authors noticed that the lysis curves seen in experimental settings exhibited saturation. Briefly, the fraction of lysed tumor cells after a certain time (usually a few hours in chromium release assays) versus different values of the initial effectorto-target ratio saturates for increasing values of the latter. Therefore, they proposed a Hill function [38, 39] depending on the effector-to-target ratio as the mathematical function describing the rate at which a tumor is lysed. Their brilliant achievement notwithstanding, little theoretical explanation was given to this function and the original proposal [31] was partly forgotten. In this chapter, we use several cellular automata models to characterize more rigorously the nature of the mathematical expression that governs the lysis of tumor cells by cytotoxic cells. Our study indicates that this mathematical function emerges from spatial and geometrical restraints. Interestingly, simulations are provided in the limit of immunodeficient environments, where saturation becomes less evident. We demonstrate that the current mathematical function works bad for such environments, and retake the conceptual framework of enzyme kinetics to propose another fractional cell kill. We show that this new function behaves better in the limit in which the immune cell population is small compared to the tumor size, and that the parameters appearing in it have a clear physical and biological interpretation. Then, we investigate the kinetics of tumor lysis in different limiting situations. This second analysis allows us to further explore the mathematical expression. As we will see, to reproduce also the time series as well as the lysis curves, one last rearrangement must be introduced, which we believe makes it theoretically more conspicuous. To conclude, we explore the transient and asymptotic dynamics that results from the coevolution of a growing tumor and the cell-mediated immune response. A cellular automaton is used to analyze the correspondence between this dynamics and the three phases of the theory of immunoedition: elimination,

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equilibrium, and escape. The exploration of different immunological scenarios enables the discussion of a possible dynamical origin of tumor dormancy and the sneaking through of tumors, as originally proposed by Kuznetsov et al. [31]. Our results demonstrate that the immune system can keep a tumor dormant for long periods of time, but that this dormancy is based on a frail equilibrium between the mechanisms that spur the immune response and the growth of the tumor. Thus, we question the capacity of the cell-mediated immune response to sustain long periods of dormancy, as those appearing in recurrent disease. We suggest that its role might be rather to synergize with other types of tumor dormancy.

2 Model Description 2.1 A Hybrid Cellular Automaton Model The simulations are accomplished by means of a cellular automaton (CA) model developed in [40] to study the interactions between tumor and immune effector cells. This model was built on a previously CA model designed to study the effects of competition for nutrients and growth factors in avascular tumors [41]. It is hybrid because the cells are treated discretely, allowing them to occupy several grid points in a particular spatial domain, and evolve according to probabilistic and direct rules. On the other hand, the diffusion of nutrients (such as glucose and oxygen) or growth factors from the vessels into such spatial region is represented through linear reaction–diffusion equations, which are continuous and deterministic. We expose separately the equations governing the diffusion of substances and the rules describing the behavior of cells.

2.1.1

Diffusion of Nutrients

Two types of nutrients are utilized in this model, making a distinction between those which are specific for cell division N(x, y, t), and others M(x, y, t) that are related to the remaining cellular activities. The partial differential equations for the diffusion of nutrients are ∂N = DN ∇ 2 N − k1 T N − k2 H N − k3 EN ∂t ∂M = DM ∇ 2 M − k4 T M − k5 H M − k6 EM, ∂t

(1) (2)

where T (x, y, t), H (x, y, t), and E(x, y, t) are functions representing the number of tumor, healthy, and immune cells at time t and position (x, y). For simplicity, we assume that both types of nutrients have the same diffusion coefficient DN = DM = D. Following [40], we consider that the competition parameters are equal k2 = k3 =

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k5 = k6 = k, except for the tumor cells, which compete more aggressively. We set k1 = λN k and k4 = λM k, with λM and λN greater than one. An adiabatic limit is considered, assuming that the solutions are stationary. This approximation holds because the time it takes a tumor cell to complete its cell cycle, which is of the order of days [42], is much longer than that of the diffusion of nutrients. A quadrilateral domain  = [0, L] × [0, L] is considered and Dirichlet boundary conditions are imposed on the vertical sides of the domain, where the vessels are placed, assigning N(0, y) = N (L, y) = N0 and = M(0, y) = M(L, y) = M0 . For simplicity, the horizontal upper and lower bounds of the domain obey periodic boundary conditions N(x, 0) = N (x, L) and M(x, 0) = M(x, L), wrapping them together to form a cylinder. Finally, the diffusion equations are nondimensionalized as explained in [41], and the equations are numerically solved by using finite-difference methods with successive overrelaxation. The resolution of the grid n equals 300 pixels in all our simulations. We describe these two steps before enumerating the CA rules.

2.1.2

Cellular Automata Rules

Since the CA used in the last part of the present work is just a variation of the one used in the first analysis, here we simply present the CA rules and the algorithm for this first study. The modifications latter required are introduced along the way. The study of the lysis of tumors with different morphologies is carried out in two successive steps. The first is devoted to the growth of the tumors, while the second focuses on their lysis by the CTLs. 1. We generate distinct solid tumors as monoclonal growths, arising after many iterations of the cellular automaton. At each CA iteration the tumor cells can divide, move, or die attending to certain probabilistic rules that depend on the nutrient concentration per tumor cell and some specific parameters. Each of these parameters θa represent the intrinsic capacity of the tumor cells to carry out a particular action a. The precise probabilistic laws and the corresponding actions are described ahead in detail. Attending to morphology, diverse types of tumors can be generated, depending on the nutrient competition parameters among tumor cells α, λN . We simulate four types of geometries (spherical, papillary, filamentary, and disconnected), and inspect four tumors of different sizes for each shape. 2. The lysis of tumor cells is a hand-to-hand struggle comprising several processes. After recognition of these cells through antigen presentation via MHC class I molecules, the CD8+ T cells proceed to induce apoptosis. The principal mechanism involves the injection of proteases through pores on the cell membrane that have been previously opened by polymerization of perforins. Even though death may take about an hour to become evident, it takes minutes for a T cell to program antigen-specific target cells to die [43]. We assign a time of 10 min for each iteration of the CA, and other choices can be made. Therefore, twenty-four iterations of the CA equal the 4 h after which the lysis of tumor cells is measured

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in the experiments [36]. Since the cell cycle time of a tumor cell is generally a few times longer, we assume a second adiabatic approximation and suspend the tumor cell dynamics during T cell lysis. The rules governing the effector cells evolution are as follows. At each iteration, those immune cells that are in contact with at least one tumor cell might lyse them with certain probability. The intrinsic cytotoxic capability, which in the model also accounts for the capacity of T cells to recognize tumor cells [44], is related to the parameter θlys . If a T cell destroys a tumor cell, recruitment might be induced in its neighboring CA elements. When immune cells are not in direct contact with a tumor cell, they can either move or become inactivated. Thus, the present CA model does not represent T cell infiltration into the tumor mass, which is discussed somewhere else [45]. We consider that a single T cell cannot lyse more than three times, leaving the region of interest when this occurs [40]. The precise probabilistic laws and the corresponding actions are again thoroughly described ahead. Each of the sixteen solid tumors is co-cultivated with different effector-to-target ratios as initial conditions (see Fig. 1) and the lysis is computed 4 h later. Because our study mainly focuses on how fast lymphocytes lyse a tumor, an important simplification between our cellular automaton and the one presented in [40] deserves notification. We have excluded a constant source of NK cells from

Fig. 1 (a) Schematic representation of the cellular automaton grid in a square domain, with some tumor cells (pink) growing from its center, and some necrotic cells (gray) at its core. Two vertical vessels on the boundary supply the nutrients required for cell division and other cellular activities. The upper and lower bounds are identified, forming a cylinder. (b) To study the lysis of the tumors, the initial conditions are always prepared by randomly placing the effector cells in a rectangular region outside the tumor. The size of this domain is selected so that for the maximum values of the effector-to-target ratio the region is almost filled with effector cells

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the model. The CA rules are now described for the two steps, one corresponding to the development of the tumors, and the other related to the lysis of the tumor cells by the cytotoxic T cells. They are almost the same as those used in [40], and any difference will be explicitly remarked. In what follows, T (( x ) and E(( x ) are the tumor and immune cells at position x(, while N(( x ) and M(( x ) are the concentration of nutrients in nondimensional variables at position x(. N (( x ) represents those nutrients required for cell division, and M(( x ) those required for other cellular activities. The role of the healthy cells is simplified to passive competitors for nutrients that allow the tumor cells to freely divide or migrate. For the first step, corresponding to the growth of the tumors, the following rules apply. At each CA iteration the tumor cells are randomly selected one by one, and a dice is rolled to choose whether each of these cell divides (1), migrates (2), or dies (3). 1. A tumor cell divides with probability 

Pdiv

(N/T )2 = 1 − exp − 2 θdiv

 .

(3)

This probability is compared to the probability that results from applying this same formula to a randomly generated√number using a normal distribution and the same standard deviation θdiv / 2. If the former is greater than the last, division takes place. The higher the value of θdiv , the more metabolic requirements for a cell to proliferate. When a cell at position x( = (x, y) divides, if there are neighboring CA elements that are not currently occupied by tumor cells, we randomly select one x( = (x  , y  ) and place there the newborn cell, thus making T (x( ) = 1 and H (x( ) = 0 or D(x( ) = 0, where D(( x ) is the function representing the necrotic cells at position x(. However, if all the neighboring elements are occupied, we let the cells pile up, making T (( x ) → T (( x ) + 1. Concerning the computation of probabilities, some discussion is here deserved. Firstly, we recall that a much more reasonable and simple way that gives very similar results is to generate a random number with uniform distribution between 0 and 1, and to compare Pdiv to the value of that number. This is the standard procedure. The reason why we proceed otherwise in [45–47] is because in [41] it was suggested that the distribution was Gaussian, making us think that N/T had to be considered the random variable. However, the random variable corresponding to every action in a cellular automaton obeys in fact a Bernoulli distribution, since the action takes place or it does not. Then, the particular probability that decides whether this occurs or not depends on the concentration of diffused substances through some function. In particular, here a sigmoid function is used, defined by means of a Gaussian profile. As far as we have investigated, using more simple profiles, as long as they are monotonic, gives very similar results.

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2. A tumor cell migrates with probability 

Pmig

 √ ( T M)2 = 1 − exp − . 2 θmig

(4)

If Pmig is greater than the probability of a randomly generated number, migration proceeds, otherwise it does not. The higher the value of θmig , the more metabolic requirements for a cell to migrate, unless there are too many tumor cells. When a cell at position x( moves, if there are neighboring CA elements that are not currently occupied by tumor cells, we randomly select one at x( and place the cell there. If there is more than one cell in the original position, the moving cell simply replaces the healthy or the necrotic cell, thus making the transformation T (( x ) → T (( x ) − 1, T (x( ) = 1 and H (x( ) = 0 or D(x( ) = 0. 3. On the other hand, if there is only one tumor cell at x(, then it interchanges its position with the healthy or necrotic cell at x( . If all the neighboring elements are occupied, we displace the cell to a randomly selected neighboring element. 4. A tumor cell dies with probability Pnec

  (M/T )2 . = exp − 2 θnec

(5)

If Pnec is higher than the probability of a randomly generated number, necrosis proceeds, otherwise it does not. The higher the value of θnec , the greater the probability for a cell to die. When a cell at position x( dies, we make T (( x) → T (( x ) − 1. If this is the only cell at x(, then D(( x ) = 1. We now describe the CA rules for the second step, corresponding to the lysis of the tumors. At each CA iteration the immune cells that have one or more tumor cells as first neighbors carry out an attempt to lyse a randomly chosen surrounding tumor cell. This process occurs with probability ⎛

⎛



⎜ 1 Plys = 1 − exp ⎝− 2 ⎝ θlys i∈η

⎞2 ⎞ ⎟ Ei ⎠ ⎠ ,

(6)

1

where ηn indicates summation up to the n-th nearest neighbors. If Plys is higher than the probability of a randomly generated number, then the selected tumor cell dies. Therefore, T (x( ) = 0, D(x( ) = 1, and the immune cell counter decreases by a unit. If the counter reaches a value of zero, it dies and it is replaced by a healthy cell. The smaller the value of θlys , the greater the probability for an effector cell to lyse a tumor cell. This parameter was not present in [40] and is introduced here to model the intrinsic cytotoxicity of T cells. When a tumor cell is destroyed by an immune cell, the first neighboring cells are flagged for recruitment. For each CA element without tumor cells a new immune cell is born with probability

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⎞−2 ⎞  ⎟ ⎜ 1 = exp ⎝− 2 ⎝ Ti ⎠ ⎠ . θrec ⎛

Prec

⎛

(7)

i∈η1

If Prec is higher than the probability of a randomly generated number, recruitment proceeds. The higher the value of θrec , the less surrounding tumor cells that are required for T cell recruitment to success. When a cell is recruited at position x( , we make D(x( ) = 0 or H (x( ) = 0, and E(x( ) = 1. Those effector cells whose immediate neighborhood is not occupied by tumor cells either migrate or become inactivated. To decide which of these two processes is carried out, a coin is flipped. If the output is migration, it occurs for sure. In the opposite case, inactivation occurs with probability ⎞−2 ⎞  ⎟ ⎜ 1 = 1 − exp ⎝− 2 ⎝ Ti ⎠ ⎠ . θinc i∈η ⎛

Pinc

⎛

(8)

3

If Pinc is higher than the probability of a randomly generated number, inactivation proceeds. The smaller the value of θinc , the less surrounding tumor cells that are required for a T cell to become inactivated. When a cell disappears from position x(, we simply make H (( x ) = 1 and E(( x ) = 0.

2.1.3

The Algorithm

The algorithm starts with a domain full of healthy cells, except for a single tumor cell placed at the center of the domain. Firstly, during this period of growth, each CA step corresponds to 1 day. Every iteration begins with the integration of the reaction– diffusion equations, using a finite-difference scheme and a successive overrelaxation method. Then all the tumor cells are randomly selected with equal probability, and the CA rules are applied. As in previous works [41], every time an action takes place, the reaction–diffusion equations are locally solved in a neighborhood with size 20 × 20 grid points. Once the tumors have been grown, their dynamics is halted. The immune cells are randomly placed in the vicinity of the tumor and start to evolve. Now the CA step corresponds to 10 min. Firstly, the reaction–diffusion equations are solved and all the immune cells are randomly selected. Then every immune cell is randomly selected and the CA rules are applied. For each immune cell, after applying the CA rules, the nutrients are computed in a local region, in exactly the same manner as before. The algorithm stops when a maximum number of twentyfour steps have been reached, or when the tumor has disappeared.

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2.2 An Ordinary Differential Equation Model In the present investigation the results of the in silico experiments performed with the cellular automaton model are fitted by means of a least-squares fitting method to a Lotka–Volterra type model. The continuous model of cell-mediated immune response to tumor growth consists of three interacting cell populations: the tumor cells T (t), the host healthy cells H (t), and the immune effector cells E(t). Our study focuses mainly on CD8+ T lymphocytes, but the model can be easily modified to reproduce NK cell dynamics. The system of differential equations [35] reads   T − a12 H T − K(E, T )T 1− K1   H dH − a21 T H = r2 H 1 − dt K2 dT = r1 T dt

dE K 2 (E, T )T 2 = σ − d3 E + g E − a31 T E, dt h + K 2 (E, T )T 2

(9) (10) (11)

with K(E, T ) = d

(E/T )λ . s + (E/T )λ

(12)

The tumor cells and the host healthy cells grow logistically with growth rates and carrying capacities r1 , K1 and r2 , K2 , respectively. The terms a12 and a21 model the competition for nutrients and space among tumor and healthy cells. A term representing the fractional cell kill of tumor cells by CTLs is given by the nonlinear function K(E, T ), which constitutes the main topic of the present work. Here the parameter σ incorporates a constant input of lymphocytes into the tissue where the tumor develops, but it can be related to a background of NK cells as well [38]. The inactivation of the effector cells and their migration from the tumor area is given by the term d3 E, whereas the parameters g, h stand for the recruitment of immune cells to the tumor domain mediated by cytokines, such as IFN-γ or TNF-α, after the tumor and the immune cells interact. Finally, the competition between the tumor and the T cells for resources is given by a31 . These differential equations are solved using a fourth order Runge–Kutta integrator. This continuous model has been validated [35] using experiments from [36] and the parameter values are listed in Table 1. In the present work only those parameters appearing in the fractional cell kill (d, λ, and s) are inspected. Accordingly to the CA model, we have set σ = 0 in the ODE model since the CA does not include a constant input of effector cells. We have also selected a value g = 0.15, which is very close to one of the values appearing in Table 1. Importantly the CA model and the ODE model include the same type of processes. The logistic growth of tumor cells in the CA model arises as a consequence of competition for nutrients [41].

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Table 1 The values of the parameters in the ordinary differential equation model used to fit experiments from [36] Parameter r1 K1 a12 d(nn) d(nl) d(ln) d(ll) s(nn) s(nl) s(ln) s(ll) λ(nn) λ(nl) λ(ln) λ(ll) r2 K2 a21 σ d3 g(nn) g(nl) g(ln) g(ll) h a31

Units day−1 cell cell −1 day −1 day−1

None

None

day−1 cell cell −1 day −1 cells day−1 day−1 day−1

cell2 cell −1 day −1

Value 5.14 × 10−1 9.8 × 108 1.1 × 10−10 2.20 3.47 2.60 7.86 1.6 2.5 1.4 × 10−1 4.0 × 10−1 1.2 × 10−1 2.1 × 10−1 7.0 × 10−1 7.0 × 10−1 1.80 × 10−1 1.0 × 109 4.8 × 10−10 7.5 × 104 6.12 × 10−2 3.75 × 10−2 3.75 × 10−2 1.13 × 10−1 3.00 × 10−1 2.02 × 107 2.8 × 10−9

Description Tumor cells growth rate Tumor carrying capacity Competition of host cells with tumor cells Saturation level of fractional tumor cell kill

Steepness coefficient of fractional tumor cell kill

Exponent of fractional tumor cell kill

Host cells growth rate Host cells carrying capacity Competition of tumor cells with host cells Constant source of effector cells Inactivation rate of effector cells Maximum recruitment rate

Steepness coefficient for the recruitment Immune–tumor competition

The parenthesis represents four different cases: a primary challenge with control-transduced cells followed by a secondary one with ligand (nl) or control (nn) cells, and a primary interaction with ligand-transduced cells followed again by ligand (ll) or ligand-negative (ln) rechallenges

There is also competition among healthy cells and tumor cells for nutrients, which in the ODE model is represented by the competing Lotka–Volterra terms between healthy and tumor cells. T cell lysis, inactivation, and recruitment are also present in both models. Only the competition term between tumor and immune cells a31 is different. Although we keep this parameter as shown in Table 1, if desired, it can be made equal to zero. As far as we have investigated, reducing the value of this parameter produces no appreciable consequences in our study. Notwithstanding this correspondence, we recall that during the second step of our CA simulations, the tumor dynamics is suspended. Accordingly, the parameter r1 should be made equal to zero. Again, we keep this parameter as shown in Table 1. Reducing the value of this parameter produces no significant consequences in our study when the T cells

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are effective, because the time scale of T cell lysis (less than 1 h) is considerably smaller that the time scale of cell division (around 1 day). For immunodeficient scenarios the effects are more sensitive, but still small. In other words, T cell dynamics dominates during the first 4 h.

3 The Lysis of Tumors in the Absence of Growth 3.1 Tumors In Fig. 2 we depict the simulated solid tumors with four distinct morphologies, depending on the nutrient competition among tumor cells. The apparent threedimensionality is an artifact resulting from the fact that we let cells pile up at the CA grid points. This piling mechanism was assumed in [41] for computational simplicity, and does not have any consequence in our study, since once the tumors Fig. 2 Tumors generated using the cellular automaton model. Tumors become increasingly branchy as the competition for nutrients increases. Colors go from dark purple (one cell) to light pink (highest number of cells in a grid point for each tumor). We set the parameters λM = 10 and θnec = 0 in all the cases, disregarding necrosis. (a)–(d) Spherical tumors with increasing size and parameters α = 2/n, λN = 25, θdiv = 0.3, and θmig = ∞. (e)–(h) Papillary tumors with increasing size and parameters α = 4/n, λN = 200, θdiv = 0.3, and θmig = ∞. (i)–(l) Filamentary tumors with increasing size and parameters α = 8/n, λN = 270, θdiv = 0.3, and θmig = ∞. (m)–(p) Disconnected tumors with increasing size and parameters α = 3/n, λN = 200, θdiv = 0.75, and θmig = 0.02

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are grown, we project them to study their lysis. High values of α and λN lead to more branchy tumors, gradually changing from spherical to filamentary. This break of the spherical symmetry of the tumors is explained if we consider that when some nearby neoplastic cells on the boundary of a tumor compete aggressively for nutrients, those cells that divide and take ahead at some step preserve this advantage at the next step, stealing the nutrients to those cells left behind. The four geometries are comparable to a variety of histologies [41], such as a basal cell carcinoma, a squamous papilloma, a trichoblastoma, and a plasmacytoma. Note that the necrosis of tumor cells due to the scarcity of nutrients in the core of the masses has been neglected, since it has no relevance in our study. In the CA this is achieved by setting θnec = 0.01 for all our simulations. Except for the disconnected patterns appearing in the last row in Fig. 2, motility has been also disregarded, considering sufficiently high values of θmig .

3.2 Effective Immune Response In the model given by Eqs. (9)–(11), the fractional cell kill of tumor cells by CTLs is given by the function K(E, T ). In [35] we opted for expressing this function in the form K(E, T ) = d

Eλ , h(T ) + E λ

(13)

with h(T ) = sT λ . Written this way, the fractional cell kill clearly states that the more the effector cells, the greater the fractional cell kill, but bearing in mind the saturation of antigen-mediated immune response, which depends on the tumor burden. We propose that the saturation is due to the crowding of immune effector cells, which is evident if we recall that these cells need to be in contact with tumor cells to exterminate them. In a solid tumor, once all the tumor cells on its surface are in contact with a first line of immune cells, the remaining effector cells are not lysing, although the adjacent lines behind probably contribute to immune stimulation through several feedback mechanisms. Therefore, at a certain point, no matter how many more immune cells are present in the region of interest, the rate at which the tumor is lysed remains practically unaltered. Before saturation appears, if two tumors of the same nature and different size at a certain time instant are lysed at the same rate by the immune system, the bigger tumor will require more effector cells. Put more simply, if two tumors of different size are reduced to a particular fraction of its size after a certain period of time, the bigger tumor will require more effector cells. The number of effector cells E for which the fractional tumor cell kill is half of its maximum d increases monotonically with the tumor size h(T ).

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We use simulations to demonstrate that these assertions are sufficient to explain the fractional cell kill law, even though there might be others. With this purpose, for every tumor pictured in the previous section, we prepare co-cultures with different effector-to-target ratios. Then, we let the CA evolve and measure the lysis 4 h later (see Fig. 3). As previously explained, the tumors have been projected before the lysis starts, to better correlate the geometry and the parameters in the fractional cell kill. Otherwise, we would have two-dimensionally distributed lymphocytes fighting three-dimensional-like tumors, since in our CA we do not let the immune cells pile up. We do it this way to avoid the unfair situation in which just a few immune cells

Fig. 3 Lysed tumors after 4 h for different effector-to-target ratios. The effector cells (green) form satellites that advance destroying their neoplastic enemies (violet) and leave apoptotic bodies (light gray) behind them. The parameter values of the CA are θlys = 0.3, θrec = 1.0, θinc = 0.5, λM = 10, λN = 25, and α = 2/L. (a)–(d) Spherical tumor in Fig. 2b with E0 /T0 taking values on the set {0.0025, 0.005, 0.05, 0.75}, respectively. (e)–(h) Papillary tumor in Fig. 2f with E0 /T0 taking values on the set {0.005, 0.0075, 0.025, 0.25}, respectively. (i)–(l) Filamentary tumor in Fig. 2l with E0 /T0 taking values on the set {0.0025, 0.0075, 0.017, 0.025}, respectively. (m)–(p) Disconnected tumor in Fig. 2n with E0 /T0 taking values on the set {0.005, 0.0075, 0.015, 0.025}, respectively

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are facing a big pile of tumor cells, and vice versa. Finally, the results are fitted to the ODE model using a least-squares fitting method. We recall that such model was validated using as initial conditions typical cell populations of 106 cells, while the CA automaton grid used can harbor at most 9 × 104 cells. However, this is not a hurdle at all, since if desired, the cell populations in the ODE model can be renormalized and its parameters redefined so as the cell numbers coincide. The resulting lysis curves are depicted in Fig. 4 and the values of the parameters d, λ, and s in Eq. (13) are listed in Table 2, together with the fractal dimension DF of the boundary of the initial tumors. Satellitosis is clearly appreciated as a consequence of T cell recruitment, and the resulting clusters of cells act like wave fronts that advance lysing the tumor. Note that the immune cells that are far enough from the tumor become inactivated after several iterations of the CA. Consequently, only the T cells that are able to make contact with the tumor, gain traction in killing and subsequent recruitment, appear in the figures. There is a correlation between the box-counting dimension and the parameters d and λ for the connected tumors examined, but this is not case for the disconnected one. The disconnected tumors shown in Fig. 2 display the highest box-counting dimension, because they are very drilled, so that most of the tumor cells are on its boundary. However, they are rather spherical, and for this reason the part of the boundary that is in the center of the mass is not initially accessible to the immune cells. These facts explain the low values of d and λ for such tumors, which are comparable to the spherical ones. Therefore, in our model, those tumors with a bigger surface of contact are lysed faster. Indeed, what matters to the cytotoxic cells is how accessible their enemies are. The more the tumor cells there are between an immune cell and some other tumor cell, the Fig. 4 The lysis of tumor cells after 4 h versus the effector-to-target ratio E0 /T0 in immunocompetent environments. The parameter values of the CA related to the lysis, recruitment, and inactivation are θlys = 0.3, θrec = 1.0, and θinc = 0.5, respectively. The solid curve corresponds to the ODE model, while the points correspond to the cellular automaton results. (a) The spherical tumor in Fig. 2b. (b) The papillary tumor in Fig. 2f. (c) The filamentary tumor in Fig. 2l. (d) The disconnected tumor in Fig. 2n

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Table 2 The parameter values modified in the model shown in Eqs. (9)–(11) corresponding to an effective immune response Parameter d(s) d(p) d(f) d(d) λ(s) λ(p) λ(f) λ(d) s DF (s) DF (p) DF (f) DF (d)

Units day−1

None

None None

Value 9±4 20 ± 1 32 ± 2 13 ± 3 0.61 ± 0.07 0.87 ± 0.04 0.89 ± 0.03 0.63 ± 0.03 0.15 1.09 ± 0.02

Description Saturation level of fractional tumor cell kill

Exponent of fractional tumor cell kill

Steepness coefficient of fractional tumor cell kill Box-counting dimension of the boundary before the lysis starts

1.21 ± 0.04 1.36 ± 0.02 1.72 ± 0.04

The parameters λ and d are obtained through a least-square fitting of the lysis of tumor cells between the CA simulations and the ODE model. The mean value and standard deviations are computed for each morphology using four different tumors sizes: spherical (s), papillary (p), filamentary (f), and disconnected (d)

lower the rate at which the effector cells kill their victims. This is starkly evident for the spherical tumors, which correspond to the smallest values of d and λ. Thus, according to our model, Eq. (13) is a robust emergent property of the tumor–immune interaction depending on the spatial distribution of the tumor cells. It reflects the saturation of an effective immune system, which depends on the tumor size. This saturation is fruit of the crowding of the effector cells and the arduousness to establish contact with their adversaries. Nevertheless, it takes hours for the effector cells to fully lyse the tumors so far investigated, which denotes that this extrinsic limitation to the lytic capacity of the immune system is barely important compared to the immunoevasive maneuvers that tumor cells commonly orchestrate [1].

3.3 Ineffective Immune Response Tumor cells find ways to evade the immune surveillance through a broad range of mechanisms [48]. They can acquire the ability to repress tumor antigens, MHC class I proteins, or NKG2D ligands. They may also learn to destroy receptors or to saturate them, induce suppressor T cells formation, launch counterattacks against immunocytes by releasing cytokines, avoid apoptosis, etc. It is therefore pertinent

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to ask ourselves if the fractional cell kill can cover situations in which the tumor microenvironment is immunodeficient. In [38] the authors show that the lysis curves corresponding to NK cells in the experiments borrowed from [36] do not show saturation, and that a fractional cell kill given by a simple power law cE ν works to fit such data. Because much higher values of the effector-to-target ratio are required to obtain similar values for the lysis compared to the CTLs curves, it was suggested that when the effector cells are less effective, saturation is not observed. Mathematical arguments have been given [35] to explain this lack of saturation. Briefly, when the cytotoxic cells are less effective, only a fraction f of the effector cells are interacting with the tumor. Thus we can replace E by f E in the fractional cell kill. Now, defining s˜ = s/f λ , the fractional cell kill law remains unchanged. This suggests that the parameter s is related to the effectiveness of the cytotoxic cells, being this parameter inversely proportional to the effectiveness of such cells. On the other hand, if the effectiveness is small enough (f & 1), then h(T ) dominates over E λ in Eq. (13), as long as E is not too high. The resulting lysis term becomes df λ E λ T 1−λ /s. This facts legitimize the estimation cE ν T that has been used in other works [35, 38] to reproduce the fractional cell kill of tumor cells. Nevertheless, here we do not want to introduce phenomenological functions of this type, but rather concentrate our efforts on the significance of s. To this end, we diminish the intrinsic cytotoxic capacity of the immune cells, which is encoded in the parameter θlys in our cellular automaton. Higher values of this parameter represent more ineffective T cells. The results can be seen in Fig. 5 and the values of the parameters are listed in Table 3. As we increase the parameter θlys , the saturation appearing in the lysis curves becomes less evident, and at a certain point it disappears. When θlys = 10, the ODE model can be adjusted to the CA results. However, increasing s is not sufficient to reproduce this data, and considerable variations of

Fig. 5 The lysis of tumor cells after 4 h versus the effector-to-target ratio E0 /T0 in immunosuppressed environments. The spherical tumor represented in Fig. 2b is studied, with recruitment and inactivation CA parameters θrec = 1.0 and θinc = 0.5. The solid curve corresponds to the ODE model, while the points correspond to the cellular automaton results. (a) A more ineffective, but still effective, adaptive response is here represented, with θlys = 10. (b) A value of the intrinsic cytotoxic capacity θlys = 100 is set for the most ineffective immune system

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Table 3 The parameter values of the fractional cell kill given by Eq. (13) Parameter d(s) d(i) λ(s) λ(i) s(s) s(i)

Units day−1 None None

Value 3.80 1.56 0.62 0.17 0.50 1.10

Description Saturation level of the fractional tumor cell kill Exponent of the fractional tumor cell kill Steepness coefficient of the fractional tumor cell kill

These parameters are obtained through a least-square fitting of the lysis of tumor cells between the CA simulations and the ODE model (see Fig. 5). Two cases are represented: a very ineffective (i) and a semi-effective (s) immune responses

the remaining parameters d and λ are required. A much more dramatic case arises when θlys = 100. In this case we have not been able to find any values of the parameters that represent faithfully the CA results. The best fitting provided by the ODE model exhibits considerable saturation. The conclusion is that the fractional cell kill represented by Eq. (13) works bad for immunodeficient environments and also confuses the geometrical effects and the intrinsic cytotoxic capacity of the immune cells. In the next section, we propose a new fractional cell kill that allows to fit the results more accurately by simply reducing the value of s.

3.4 Modification of the Fractional Cell Kill In [35], the particular nature of the function h(T ) appearing in Eq. (13) was also discussed, proving that if instead of h(T ) = sT λ , h(T ) = sT λ+ λ is used, the empirical results can also be validated by simply decreasing the value of s, even for values λ/λ greater that one. This means that the original proposal of a saturating fractional cell kill depending on the quotient E/T cannot be guaranteed. Furthermore, from a theoretical point of view, the function h(T ) = sT λ makes the model ill-defined in the limit of very big tumors (T → ∞) facing a comparably small fixed number of immune cells. The reason is that in this limit we get unbounded velocity for the lysis (K(E, T )T → ∞). We demonstrate that h(T ) = sT is a much better choice. It has been shown [44, 49] that for a fixed number of effector cells E0 , the Michaelis–Menten kinetics govern the lysis of tumor cells. The value of the lytic velocity at tumor saturation, i.e., when T → ∞, is reported in such works as a measure of the intrinsic cytotoxic capability of a particular number of effector cells. A Michaelis–Menten decay in Eq. (13) is obtained for a constant value of effector cells as long as h(T ) = sT is used. The value at saturation for a fixed number of effector cells is then dE0λ /s. An argument supporting saturation comes from the following fact. If the number of tumor cells is much higher than

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a fixed number of effector cells, the velocity at which the tumor cells are lysed cannot be enhanced by increasing the number of the neoplastic cells. This occurs because T cells kill tumor cells one by one, and for such ratios all the effector cells are already busy fighting other cells. In a similar fashion, for an enzymatic reaction, one cannot increase arbitrarily the velocity at which the products are formed by simply adding more substrate. Precisely, this reasoning is reminiscent of the original formulation proposed by [31], in which the cell populations are regarded as chemical species obeying enzymatic kinetics in the quasi-steady state regime. In such work, the tumor cells are the substrate, the effector cells are the enzyme, and the products are the dead cells. Indeed, in the following section we use enzyme kinetics as a metalanguage to provide an analytical derivation of the fractional cell kill. A fractional cell kill function that yields bounded velocity for the lysis of tumor cells when any of these two cell populations is sufficiently high compared to the other is represented by K(E, T ) = d

Eλ . sT + E λ

(14)

If we focus only on the lysis of tumor cells, the velocity at which the tumor is reduced can be represented by the following nonlinear differential equation: T˙ = −d

Eλ T. sT + E λ

(15)

Following the point of view of [31], this mathematical expression can be regarded as a Michaelis–Menten kinetics where the rate constants of the formation of the “enzyme–substrate" conjugates, their dissociation and their conversion to product depend nonlinearly (as power laws) on the enzyme concentration. It establishes the saturation of the velocity of the lysis of tumor cells for both the tumor and the immune cell populations. In Fig. 6a we first reproduce the experiments of the spherical tumor shown in Fig. 2b for θlys = 0.3. This allows us to obtain the parameter values of the modified fractional cell kill shown in Eq. (14). Then we carry out the simulations of the preceding section for immunodeficient environments and see how, mainly by increasing the value of s, the CA results are reproduced (see Fig. 6b, c). The parameter values are listed in Table 4. This sheds light into the significance of this parameter, which is now manifestly related to the intrinsic cytotoxic potential of the T cells. Moreover, this implies that the limit T → ∞, for which the quantity dE λ /s is obtained, is not a good measure of lymphocyte cytotoxicity, as suggested in [44, 49]. This limit, which for a constant value of the T cells implies a linear decay of the tumor, involves geometry as well. Ideally, if we consider that there is just one immune cell, and it takes this cell 1 h to lyse one tumor cell, then a spherical tumor would be reduced at approximately one cell per hour (assuming that this immune cell does not become inactivated at some step). However, the geometry of the tumor, which is coded in the parameters d and λ, clearly affects how fast this single cell can erase it.

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Fig. 6 The lysis of tumor cells after 4 h versus the effector-to-target ratio E0 /T0 for increasing ineffectiveness of the lymphocytes. The spherical tumor represented in Fig. 2b is studied, with recruitment and inactivation CA parameters θrec = 1.0 and θinc = 0.5. The solid curve corresponds to the ODE model, while the points correspond to the cellular automaton results. (a) An effective immune response for θlys = 0.3. (b) A more ineffective, but still effective, adaptive response is here represented, with θlys = 10. (c) A value of the intrinsic cytotoxic capacity θlys = 100 is set for the most ineffective immune system. As shown in Table 4, the intrinsic cytotoxic potential of the T cells is chiefly represented by parameter s in Eq. (14)

Even though the reduction of saturation for ordinary values of the effector-totarget ratio can be justified mathematically and numerically, the change in curvature for the CA results appearing in Fig. 6c requires a positive feedback mechanism. Certainly, the mechanism responsible for this phenomenon is the recruitment of immune cells, which becomes increasingly important as the effectiveness of the T cells decreases.

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Table 4 The parameter values of the fractional cell kill appearing in Eq. (14) Parameter d(e) d(s) d(i) λ(e) λ(s) λ(i) s(e) s(s) s(i)

Units day−1

None

cellsλ−1

Value 9.22 9.62 9.52 0.50 0.51 0.55 1.0 × 10−5 1.4 × 10−4 9.5 × 10−4

Description Saturation level of the fractional tumor cell kill

Exponent of the fractional tumor cell kill

Steepness coefficient of the fractional tumor cell kill

These parameters are obtained through a least-square fitting of the lysis of tumor cells between the CA simulations and the ODE model (see Fig. 6). Three cases are represented: an effective (e), a semi-effective (s), and ineffective immune responses (i). Note that it is only the parameter s, which is related to the intrinsic cytotoxic capacity, that varies substantially. It increases as the immune cells become less effective

4 The Fractional Cell Kill as a Michaelis–Menten Kinetics The fractional cell kill represented by Eq. (14) can be derived from the Michaelis– Menten kinetics [27, 28] assuming that the rate constants of the reaction depend on the enzyme concentration. During the process of lysis, the effector cells E bound to the tumor cells T forming complexes C, and dead tumor cells T ∗ result from this interaction. Therefore, the tumor cells play the role of the substrate and the effector cells act as the enzyme. This cellular reaction can be written in the form k1

k2

k−1

k−2

E + T  C  T ∗ + E.

(16)

Once a tumor cell is induced to apoptosis it cannot resurrect, so we must set k−2 = 0. Generally, also the backward reaction represented by k−1 should be disregarded, since after tumor cell recognition and complex formation, destruction proceeds. However, we keep this term for reasons explained below. Assuming that the law of mass action holds, the system of differential equations governing the reactions is dE dt dT dt dC dt dT ∗ dt

= −k1 ET + (k−1 + k2 )C

(17)

= −k1 ET + k−1 C

(18)

= k1 ET − (k−1 + k2 )C

(19)

= k2 C.

(20)

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The Briggs–Haldane [50] quasi-steady state approximation C˙ = 0 was assumed in [31] because in their model the time scale of the process of lysis is considerably smaller than that of cell growth. However, if we focus on the process of lysis only, the quasi-steady state approximation requires E0 & 1, T0 + KM

(21)

where KM = (k−1 + k2 )/k1 is the Michaelis constant, and E0 and T0 are the initial concentrations of the effector and the tumor cells, respectively. Because we are dealing with situations in which the substrate concentration can be smaller than the enzyme, the quasi-steady state approximation implies KM ) E0 . Since this condition cannot be generally guaranteed, instead, we consider Michaelis and Menten original formulation, and suppose that the substrate is in instantaneous equilibrium with the complex. We believe this is more reasonable, because it takes about 1 h for a cytotoxic T cell to fully lyse one tumor cell and, if the cells are effective, the recognition and complex formation should occur quite fast when brought together. In this manner, we have k1 ET = k−1 C. From Eqs. (17) and (19) we get the conservation law E +C = E0 . These two equations put together and substituted in Eq. (20) yield T dT ∗ = k2 k1 E0 . dt k1 T + k−1

(22)

So far, this is nothing else but the Michaelis–Menten kinetics. It is at this point that we have to consider a dependence of the rate constants of the reaction on the concentration of the effector cells. The mathematical relations are derived heuristically, based on the idea that for higher concentrations of the immune cells the rate constants vary in such a manner that the whole process is pushed backwards. Since saturation is due to the crowding of T cells, and this depends on the geometry of the tumor, it seems a natural choice to use power laws. Once the first lines of effector cells cover the surface of a solid tumor, the remaining immune cells are not in contact with it. Alternatively, an equivalent argument is attained if we suppose that the non-interacting effector cells do interact with some tumor cells unsuccessfully (say ghost tumor cells), so that the complexes are dissociated without lysis. The more the effector cells, the higher the rate of dissociation, and when the number of effector cells is small compared to the number of tumor cells, the dissociation should vanish. Therefore, we consider a power law dependence k−1 (E0 ) = κ−1 E0α , with 0 < α < 1, as suggested from the experiments. Substitution in Eq. (22) yields dT ∗ E0 = k2 k1 T. dt k1 T + κ−1 E0α

(23)

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The fractional cell production of dead cells in this equation already resembles very much to Eq. (14). To obtain the exact result we have to consider dependence of k1 and k2 on the effector concentration as well. Note that for the inverse reaction to take place complexes have to be formed first, and this requires some time. Therefore, saying that complexes dissociate without lysis is not exactly equivalent to stating that the complexes are not formed. These rates should decay for increasing concentrations of the effector cells, diminishing the rate of formation of complexes and products. Once again, we postulate power law relations in the form k1 (E0 ) = −γ −β κ1 E0 and k2 (E0 ) = κ2 E0 , where again 0 < β < 1 and 0 < γ < 1. It might result surprising that in the limit E0 → ∞ these functional relations tend to zero, suggesting that the reaction stops. However, this is not the case, because when substituted in Eqs. (17)–(20), k1 (E)E and k2 (E)C both increase with the number of effector cells. Replacing the rate functions in Eq. (23) we obtain 1−γ

E0 dT ∗ κ1 κ2 = T. dt κ−1 κ1 T + E α+β 0 κ−1

(24)

We now rename the constants λ = α + β, s = κ1 /κ−1 , d = κ1 κ2 /κ−1 , and remember that the velocity for the lysis must remain bounded for E0 → ∞, which imposes the constraint α + β + γ = 1. Thus, the velocity at which dead tumor cells accumulate is given by the nonlinear function E0λ dT ∗ =d T. dt sT + E0λ

(25)

5 Decay Laws in Tumor Cell Lysis Using the Michaelis–Menten kinetics as the modelling framework describing tumor–immune interactions at the cellular scale [31], a mathematical expression describing the velocity at which a population of cytotoxic cells lyse a tumor has been derived in the previous sections. A schematic representation of such a cellular reaction is seen in Fig. 7. When a T cell identifies a tumor cell through the recognition of antigens, these two cells form complexes. As a result, apoptosis is induced and a dead tumor cell is produced. However, some of the assumptions that lead to the Michaelis–Menten kinetics, such as a high substrate concentration compared to the enzyme concentration, or high values of the Michaelis constant compared to the enzyme concentration, are not met in the present case. To reproduce experiments, the constant rates of the reaction require dependence on the number of effector cells, in such a manner that saturation of the velocity is also found for increasing numbers of the effector cells. As previously stated, saturation occurs in both directions. Disregarding other processes, the differential equation [46]

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Fig. 7 The cell-mediated immune response as an enzymatic reaction. An interaction between an activated lymphocyte E, colored in green, and a tumor cell T , painted in red. When the lymphocyte identifies the tumor cell these two cells form a complex. The result of the interaction is the initial T cell and an apoptotic tumor cell T ∗ . This cellular interaction is similar to an enzymatic chemical reaction, where the tumor cell plays the role of the substrate and the T cell acts as an enzyme (figure obtained from Ref. [45])

describing the velocity at which the tumor cells are destroyed is T˙ = −K(E, T )T , with K(E, T ) the fractional cell kill, which can be written as K(E, T ) = d

Eλ , sT + E λ

(26)

where T and E represent the number of tumor cells and immune cells, respectively. The parameters d and λ depend on the tumor geometry. Less spherical tumors lead to higher values of these parameters. On the other hand, the parameter s is related to the intrinsic ability of the cytotoxic cells to recognize and destroy their adversaries. Smaller values of this parameter are related to more effective immune cells. Thus, the velocity at which a tumor is lysed is given by T˙ = −d

Eλ T. sT + E λ

(27)

In the present section we delve deeper into the significance of this mathematical expression by examining the different limits that it provides. To reproduce also the time series as well as the lysis curves, we introduce one last rearrangement.

5.1 The Limits of the Fractional Cell Kill We begin by carefully examining the different limits that this equation possesses (see Fig. 8). For a fixed number of immune cells E0 , when the immune cell population is small compared to the tumor size (E0λ & sT ), the tumor cell population is reduced at a constant velocity T˙ = −dE0λ /s.

(28)

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Fig. 8 The limits of the fractional cell kill in the absence of T-cell infiltration. (a) A small immune cell population facing a big tumor. In this limiting situation the decay of the tumor is rather linear, as shown in Eq. (28). (b) The intermediate case in which a considerable part of the tumor is covered with immune cells. (c) A tumor, in which surface is totally covered with immune cells. In this extreme case the velocity of the decay can be approximated by a parabolic decay, as shown in Eq. (31) (figure obtained from Ref. [45])

Fig. 9 A single cell lysing a tumor. (a) The linear decay of a tumor in the limit in which there is only one immune cell. (b) The path (green) of a lymphocyte after a certain time, modelled by an unbiased random walk in a square domain, which is occupied by tumor cells (red). The initial condition is set on the left bottom corner (figure obtained from Ref. [45])

This linear decay makes perfect sense if we bear in mind the extreme situation in which there is only one lymphocyte fighting a tumor of a certain size. Ideally, if it takes the immune cell approximately 1 h to lyse a tumor cell, then the velocity of the decay is simply one tumor cell per hour. Even though this is fairly obvious, in Fig. 9 we show the random walk of a lymphocyte lysing a tumor that occupies a square domain, at one cell per hour. In practice, the velocity clearly depends on the intrinsic ability of the cytotoxic cell s to lyse the tumor cells and also on the tumor morphology λ and d. On the other hand, when the immune cell population is

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high enough compared to the tumor cell population (E0λ ) sT ), Eq. (27) yields an exponential decay T˙ = −dT .

(29)

Now, the scenario corresponds to the case in which the tumor is totally covered with effector cells. For the sake of simplicity, we consider a tumor spheroid [51]. At each step the immune cells lyse a layer of tumor cells, and the radius of the spheroid decreases. In the next round another layer is eliminated but, since the tumor has smaller radius, so it does the length of this second layer. Therefore, the velocity decreases as the tumor is gradually erased. Nevertheless, note that for a three-dimensional solid tumor the reduction occurs in surface, while the tumor is distributed in volume, suggesting that the decay should be slower than exponential. It has been demonstrated [46] that Eq. (27) reproduces accurately the values of the lysis after some fixed time versus different values of the effector-to-target ratio as initial conditions. However, here we show that it is unable to reproduce the time series of the tumor decay faithfully. A more general mathematical function which is better at reproducing the time series of the tumor decay can be derived in the following manner. Assume that a two-dimensional tumor with the shape of a disk is plainly covered with immune cells. As shown in Fig. 10a, a layer of tumor cells is erased by the immune cells at each step, like peeling an onion. If we write the radius of the disk at the n-th step as Rn , and the diameter of a cell as R, the dynamics of the tumor can be represented by a very simple map in the form Rn+1 = Rn − R. Since the area of a disk is related to the radius through A = π R 2 , a direct 1/2 substitution yields the map An+1 = An + π R 2 − 2π 1/2 RAn , where An is the area of the disk at the n-th step. If we consider that the immune cells lyse at a constant rate, then R = c t, and we obtain

An 1/2 = π c2 t − 2π 1/2 cAn .

t

(30)

Fig. 10 Two tumors with a destroyed layer. (a) A tumor with the shape of a disk and initial radius R0 . At each step the immune system erases a layer (light red), reducing its radius by an amount

R. (b) Again a tumor with a destroyed layer, but exhibiting a more complex geometry (figure obtained from Ref. [45])

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We assume that the superficial cell density σ of the tumor is approximately constant, which was missing in previous works. Finally, if the tumor is big enough so that the time intervals can be considered infinitesimal and defining a decay constant as d = 2π 1/2 σ 1/2 c, we obtain the differential equation T˙ = −dT 1/2 .

(31)

More simply, if we consider a disk of area A = π R 2 and assume that the velocity at which the radius decreases is constant R˙ = −c, with c > 0, we can write dR dA = 2π R = −2π 1/2 cA1/2 . dt dt

(32)

If the tumor has a more sophisticated geometry, we can still apply Eq. (31) under appropriate assumptions. Things get even more complicated if we take an initial tumor which is not a convex set, as the one depicted in Fig. 10b. Even in the case in which all the immune cells act synchronously and are equally effective, the topology of the tumor might change during the process of lysis, becoming disconnected. Assuming equal decay rates d and using Eq. (31), it is straightforward to verify that the total area of two tumors with the shape of a disk does not decay as a whole with the same velocity than that of a single tumor with such shape and equal total area. The two small tumors decay faster, because the ratio between the perimeter and the enclosed area is larger. Analytically, this is simply a consequence of the nonlinear nature of Eq. (31). Therefore, we designate the mean value of the variations of the radius of such sequence of disks as R. Then, we write the variation of the radius as δn R, where δn accounts for the deviations with respect to the mean value that must be bounded. The map is now Rn+1 = Rn − δn R and the area goes as 1/2 An+1 = An + π δn2 R 2 − 2π 1/2 δn RAn . Making the same assumptions as in the previous case, the final result is T˙ = −d(t)T 1/2 ,

(33)

where d(t) = 2π 1/2 σ 1/2 cδ(t), and δ(t) a function which takes into account the deviations from Eq. (31) due to the change in morphology and connectedness at each step. In the results we show that these deviations due to a complex morphology are small for the connected tumors here examined. Therefore, the parabolic decay represented in Eq. (31) works well at reproducing the decay of the tumors in the limit in which they are completely surrounded by immune cells, as long as they are not formed by disconnected pieces and their shape does not differ too much from a spherical shape. An explicit relation between δ(t) and the geometrical properties of the tumor can be derived. It is given by the expression  L2 (t) δ(t) = , (34) 4π A(t) where L(t) is the length of the boundary of the tumor, while A(t) is the total area occupied. If the value of δ(t) does not change substantially along the process of lysis, we can approximate the parameter as d = 2π 1/2 σ 1/2 cδ0 .

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5.2 The Effects of Morphology on the Maximum Fractional Cell Kill We use the same cellular automaton model to inspect three different morphologies of two-dimensional tumors: a spherical tumor, a papillary tumor, and a filamentary tumor. The tumors generated with the cellular automaton are shown in Fig. 11. We place these three tumors inside a circumference and, for each of them, we repeat the experiments for several initial conditions. To this end, we fill with immune

Fig. 11 Three tumors grown by iteration of the cellular automaton. A grid of n × n cells, with n = 300 has been used. We disregard necrosis and motility of tumor cells by setting the parameters, θnec = 0 and θmig = ∞. In all the three cases λM = 10. (a) A spherical tumor obtained for parameter values α = 2/n, λN = 25, and θdiv = 0.3. (b) A papillary tumor obtained for parameter values α = 4/n, λN = 200, and θdiv = 0.3. (c) A filamentary tumor obtained for parameter values α = 8/n, λN = 270, and θdiv = 0.3. These three tumors have grown up to approximately 9100 cells (figure obtained from Ref. [45])

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Fig. 12 How the initial conditions are set to investigate the lysis of the different tumors. The tumors are inscribed in a circumference and immune cells are placed in the surroundings for different angles γ . Since those cells that are not close to the tumor outer layer become inactivated during the first steps of the CA, small values of the angle correspond to the case shown in Fig. 8a, while the case γ = 2π is related to Fig. 8c (figure obtained from Ref. [45])

cells the remaining space of the circumference for increasing angles, as depicted in Fig. 12. The time series representing the decay of the tumors are shown in Fig. 13. As explained in previous sections, we see a tendency towards linearity as the tumor is initially less covered with immune cells. Even the curvature is inverted for such small values of the initial angle, but this is surely a consequence of recruitment in the cellular automaton. Note also that the stochastic effects are more noticeable when the number of initial effector cells is low. The cases in which the tumors are totally covered with immune cells as initial conditions (γ = 2π ) are fitted to the equation T˙ = −dT ν and also to T˙ = −dT , to elucidate which type of decay represents better the tumor cell lysis. The parameters are obtained through a least-square fitting method, and are listed in Table 5. As it can be seen in Fig. 14, the exponential decay is much worse at describing the time evolution of this dynamical system. Moreover, the value of ν that gives the best fit to the power-law function is equal to one half for the papillary and the filamentary tumors, and practically one half for the spherical case. The agreement is striking and, as previously predicted, the fluctuations are higher when the tumors exhibit a more complex geometry. Concerning the parameter d, we see that more branchy tumors display higher values. The explanation for this behavior is evident, since the higher it is the contact surface of a tumor, the more cells that can interact with it and the faster the speed at which it is lysed. This is in conformity with results obtained in the previous sections, where it was claimed that tumors with a spherical symmetry are harder to lyse. The crucial concept here is the accessibility that the immune cells have to the tumor cells. Thus, we have demonstrated that in the limit in which a solid tumor is totally covered with immune cells, the velocity at which it decays is slower than exponential. This fact requires modifying Eq. (27) so that such limit is attained. The mathematical arguments previously employed can be perfectly extended to tumors

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Fig. 13 The decay of the three tumors for different initial conditions. The immune cells are placed in the neighborhood of the tumors for values of the angles γ = {π/6, π/2, π, 3π/2, 2π } and we iterate the CA. The CA actions corresponding to the lymphocytes have parameter values θlys = 0.3, θrec = 0.5, and θinc = 0.5. The tumor cells dynamics has been frozen, and the parameters related to the diffusion of nutrients are the same as those appearing in previous figures. (a) The decay of the spherical tumor for the different initial conditions. (b) The decay of the papillary tumor for the different initial conditions. (c) The decay of the filamentary tumor for the different initial conditions. As less immune cells are placed in the vicinity of the tumors as initial conditions (from γ = 2π to γ = π/6), the parabolic decay transforms into a more or less linear type of decay (figure obtained from Ref. [45])

that live in a three-dimensional space. If we recall that saturation of the velocity must be attained in the limit of infinitely big tumors, we propose that the kinetics of tumor lysis in the cell-mediated immune response to tumor growth is given by T˙ = −d

Eλ T ν, sT ν + E λ

(35)

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Table 5 The parameter values of the power-law decay T˙ = −dT ν and the exponential decay T˙ = −dT , to which the data of the cellular automaton are fitted by means of a least-squares fitting method Power-law decay Parameter d(s) d(p) d(f) ν(s) ν(p) ν(f)

Units cell1/2 h−1 cell1/2 h−1 cell1/2 h−1

Value 1.34 3.36 7.31 0.49 0.50 0.50

Exponential decay Units Value h−1 0.04 h−1 0.10 h−1 0.21

Description Rate of decay Rate of decay Rate of decay Exponent Exponent Exponent

We see that as the geometry of the tumor changes from spherical (s) through papillary (p) to filamentary (f), the parameter d increases. However, the value of ν is almost the same for the three geometries (table obtained from Ref. [45])

where the exponent ν depends on the dimension of the space, the morphology of the tumor and its connectedness. For realistic, connected, and rather spherical solid tumors we have ν = 2/3, with the 2 standing for surface, and the 3 for volume. However, in those cases in which the tumor is very disconnected and the immune cells are well mixed with the tumor cells, as, for instance, in hematological cancers or solid tumors profusely infiltrated with lymphocytes, ν = 1 should be used. The exponential decay arising in the limit E0λ ) sT would be then interpreted from a stochastic point of view, regarding the process as a Poisson process. Indeed, not all the immune cells have the same capacity to recognize a tumor cell, neither they act synchronously. In this case, the decay of a tumor does not differ substantially from other types of decay phenomena, as, for example, one-decay processes in radioactivity. For intermediate situations, the exponent ν will take a value between 2/3 and 1.

6 Dynamics of Tumor–Immune Aggregates To conclude this study we use a hybrid cellular automaton to investigate the dormancy of a tumor mass, mediated by the cellular immune response. Even though an interesting work has been previously carried out in this context [40], the present study includes new features, which we believe makes it more realistic, permitting a correlation between the results and the theory of immunoedition. Mainly, the time scale of the cytotoxic cell action (about 1 h) differs from the time scale of tumor cell proliferation (about 1 day). Secondly, our cellular automaton includes a new parameter that allows us to represent immunosuppressed environments.

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Fig. 14 The decay of the three tumors for γ = 2π . We have iterated the cellular automaton in the limit in which the tumors are totally covered with immune cells. The results are fitted to a power-law function T˙ = −dT ν , shown in red, and an exponential decay T˙ = −dT , shown in blue, to elucidate which type of decay represents better the velocity with which the tumors shrink. (a) The decay of the spherical tumor. (b) The decay of the papillary tumor. (c) The decay of the filamentary tumor. In all the cases a power-law function with an approximate value of ν = 1/2 fits much better the results of the CA. Therefore, the decay is parabolic. The exact values are listed in Table 5 (figure obtained from Ref. [45])

The exploration of different immunological scenarios enables the discussion of a possible dynamical origin of tumor dormancy and the sneaking through of tumors, as originally proposed by Kuznetsov et al. [31]. Before embarking on this study, some information on the immunoediting of tumors is deserved.

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6.1 Cancer Immunoedition Cancer immunoediting can be described by three phases: elimination, equilibrium, and escape. The first of these three Es [52] corresponds to what has traditionally been termed immunosurveillance [53], and involves the innate and the adaptive immune responses. During this phase, the immune system keeps in check a tumor cell population, successfully recognizing and destroying the majority of its cells. However, some residual tumor cells might remain unnoticed and asymptomatic for a long period of time, which can range from 5 years to more than 20 years. This period of time defines a second stage, in which a small cell population is kept at equilibrium. Finally, the phase of escape is led by some tumor cells that might present a priori or have acquired along their evolutionary process, a nonimmunogenic phenotype. The mechanisms through which a tumor can be maintained at low cell numbers (i.e., dormant) are diverse. In a first approach, cancer dormancy can be generally classified into two categories: tumor mass dormancy and cellular dormancy [54]. In the former case, the equilibrium of a tumor is the result of a balance between cell growth and cell death. In the latter, the cells arrest and survive in a quiescent state until more benevolent conditions are provided by their environment. The occurrence of tumor mass dormancy is commonly associated with two different mechanisms [55]. The first is angiogenic dormancy, which occurs when the cells are unable to induce angiogenesis, and therefore to recruit oxygen and other nutrients to their location. In this manner, the proliferation rate is counterweighted by elevated rates of apoptosis. The second mechanism is the immune system response. This response is very complex and involves many types of cells and molecules [43]. There is evidence that the cell-mediated immune response collaborates with the humoral immune response to promote the dormancy of tumors, and that CD8+ lymphocytes and IFN-γ play a transcendental function in its maintenance [56].

6.2 Cellular Automata Rules Revisited Most of the CA rules for the tumor and the cytotoxic cells are the same as before. However, one more action concerning the immune cells is included and the algorithm is modified to allow for the coevolution of both cell populations. Such an action corresponds to a constant input of cytotoxic cells into the domain (Fig. 15). Even though we do not make a distinction between the innate and the adaptive immune responses, this constant source of immune cells allows to model the presence of NK cells in a tacit manner. These cells are placed at random in the domain, at points that are not occupied by tumor cells. Every such grid point is examined and, if a probabilistic condition holds, the healthy or dead cells that might occupy it are replaced with an immune cell. An immune cell is placed in the background with probability

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Fig. 15 The cellular automaton. A grid representing the cellular automaton during the growth of a tumor in the presence of immune effector cells. The tumor cells are shown in red and the immune cells appear in blue. The remaining spots are occupied by healthy or dead cells. The vertical black stripes in the boundary of the square domain represent the vessels from which nutrients diffuse. Periodic boundary conditions are considered in the remaining part of the boundary. Some immune cells are scattered in the region, and some other form clusters that advance reducing the tumor (figure obtained from Ref. [47])

Pbkg = f −

1  Ei , n2

(36)

i∈CA

where f is a number between 0 and 1 that represents the intensity of the input of immune cells into the tissue. If Pbkg is greater than a randomly generated number between zero and one, then an immune cell appears in the corresponding grid point. Again, the algorithm starts with a domain full of healthy cells, except for a single tumor cell placed at the center of the domain. Firstly, we let the tumor grow until it is detected by the immune system, when it has reached some specific size Tdet . During this period of growth, each CA step corresponds to 1 day. Each iteration begins with the integration of the reaction–diffusion equations, using a finite-difference scheme and a successive overrelaxation method. Then all the tumor cells are randomly selected with equal probability, and the CA rules are applied. As in previous works [41], every time an action takes place, the reaction–diffusion equations are locally solved in a neighborhood with size 20 × 20 grid points. When the time of detection is reached, the immune cells start to evolve. Now the CA step corresponds to 1 h, and during the next twenty-three steps, only the immune cells are computed. First, the background source of immune cells is executed. Then, the reaction–diffusion equations are solved and all the immune cells are randomly selected. For each immune cell, after applying the CA rules, the nutrients are computed in a local region, in exactly the same manner as before. Every twenty-three iterations, the tumor cells are checked and the tumor cell rules are applied as previously described, before immune detection. The algorithm stops when a maximum number of steps

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after the elapse of the immune response have been reached or when a tumor cell is at a distance of two grid points from its boundary.

6.3 Simulations and Results We study the evolution of the tumor and the immune system for three different scenarios. The first scenario is used as a reference, and it is characterized by high levels of immune cell recruitment and negligible necrosis due to the scarcity of nutrients in the core of the tumor masses. In the second scenario, the recruitment levels are reduced, while the necrosis of tumor cells is enhanced in the third. Unless specified, the remaining parameters are all the same in every case. Beginning with one tumor cell, the tumors grow up to Tdet = 5 × 103 cells, and at this moment the immune response triggers. In order to elucidate the effects of tumor immunogenicity, we devise what shall be called a transient bifurcation diagram. Given a dynamical system, a bifurcation diagram is a plot of the asymptotic values of a particular variable against a set of values of some relevant parameter. However, in many situations there might exist very long transients before the asymptotic state is established. These transients are of great importance in our context, since tumors may exhibit long periods of latency before the development of recurrence. Therefore, we compute the number of tumor cells for the last 100 iterations of a trajectory comprising 24,000 iterations of the CA from immune detection. Then, these 100 points are represented on the vertical axis for different values of the parameter θlys , which lies on the horizontal axis. If we assign to each of these iterations a time of 1 h, we are registering the size of the tumor for approximately the last 4 days of a period of 33 months from immune detection. We recall that the parameter θlys codes the intrinsic ability of the immune cells to recognize and lyse their adversaries. Higher values of this parameter correspond to more immunodeficient environments.

6.3.1

Reference Scenario

The set of parameters for this scenario is chosen similar to previous works, in which it has been demonstrated that they generate reasonable tumor dynamics [41, 46]. The specific values are θdiv = 0.3, θnec = 0.05, θmig = ∞, θrec = 1.0, θinc = 0.1, λM = 10, λN = 25, and α = 2/n. Regarding the natural flow of immune cells into the tissue, two situations are inspected for each scenario. The first corresponds to a high input of immune cells into the tumor area. In this case a value f = 0.10 is set, which means that approximately 10% of the background is occupied by immune cells, if there are not too many immune cells piled up. The other has a lower input of 5%, thus f = 0.05. In the absence of immune response, the tumors display a rather spherical shape. As we can see from the transition bifurcation diagrams shown in Fig. 16, three different regions are clearly distinguished. In the first region, when

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Fig. 16 Transient bifurcation diagrams. Two transient bifurcation diagrams for the reference scenario. The size of the tumor T for the last 100 h of a trajectory comprising 1000 days is plotted against the parameter that models the immunogenicity of the tumor θlys . The size of the tumor has been “normalized,” dividing it by the number of total grid points n2 . Tumors having escaped the region are assigned a value of T = 2.2, which is over the maximum obtained in all our simulations. (a) A transient bifurcation diagram for a constant input of tumor cells into the domain given by f = 0.1. (b) A transient bifurcation diagram for a constant input of tumor cells into the domain given by f = 0.05. Three different regimes are clearly discerned. The first (1) corresponds to the elimination of the tumors, the second (2) to abiding small tumors kept in equilibrium by the immune cells, and the third (3) to fast growing tumors that escape the domain (figure obtained from Ref. [47])

the immune system is effective, the tumors are completely eliminated. The second is related to an equilibrium phase, for which tumors spend very long transients oscillating at low cell numbers. Finally, tumors with increasing size, eventually leaving the domain through the vessels, appear in the third region. Thus, here we see how immunogenicity affects the fate of tumors, in accordance with the theory of immunoedition. To give insight into the second and the third regions, time series have been computed (see Figs. 17 and 18), until the tumor escapes. Initially, the tumors grow in the absence of immune response, and then the immune cells start to reduce them or, in the worst case, delay their growth. Depending on how effective the immune cells are, longer or shorter transients follow this reduction phase. The asymptotic dynamics is always the same: if the tumors are not totally eliminated by an efficient immune system, they eventually escape from the region. These two attractors are reminiscent of those appearing in reference [31]. As shown in Fig. 17a, the length of the transients in the second region, which are of around 12 years, clearly indicate a phase of prolonged tumor mass dormancy. During the period of dormancy the immune system keeps the tumor at low cell numbers and randomly displaces its disconnected pieces until one of them reaches the vessels. In the third region,

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Fig. 17 Asymptotic dynamics and tumor escape. Three time series of the tumor size for the reference scenario are plotted. The constant input of immune cells to the domain is f = 0.1. The size of the tumors is registered until they escape the domain through the vessels. The corresponding tumors at escape are shown below. The color bar represents the number of tumor cells at a grid point. For clearness, the immune cells at a grid point are simply colored in dark blue. The dead cells are represented in light blue. (a) A long-lived tumor is kept at equilibrium for θlys = 90. This is an example of immune-mediated tumor mass dormancy. (b) The corresponding small tumor at escape. (c) A less immunogenic tumor θlys = 106 is kept at equilibrium, but for a considerably shorter time. (d) The corresponding tumor at escape, which is noticeably bigger compared to the previous case. (e) A tumor that is barely immunogenic for θlys = 140. Now the tumor escapes very rapidly and exhibits the largest size, although the immune system delays its growth. (f) The corresponding tumor at escape (figure obtained from Ref. [47])

transients are found again, but they are shorter (less than 4 years) and the tumors at escape have bigger sizes. As predicted by Kuznetsov et al. [31], the duration of the transients is stochastic. This randomness is evident from the transient bifurcation diagrams, since after 33 months of tumor–immune struggle, some tumors have escaped and some others have not, disregarding how immunogenic they are. When the immune system barely responds to the tumor, we see very big tumors occupying the domain and escaping rapidly, as depicted in Fig. 17e. Interestingly, the equilibrium region shrinks as the normal input of cells into the tissue reduces from f = 0.1 to f = 0.05. As it is shown in Fig. 18, the oscillations during the equilibrium phase are more pronounced. This makes the equilibrium more unstable and suggests that having cells scattered all over the

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Fig. 18 Asymptotic dynamics and tumor escape. Two time series of the tumor size and the corresponding tumors at escape are plotted for the reference scenario. The constant input of immune cells to the domain is now smaller f = 0.05. (a) A long-lived tumor is kept at equilibrium for θlys = 67. Now the oscillations of the tumor size during the equilibrium are higher. (b) The corresponding tumor at escape, which again is small. (c) Another tumor θlys = 89 that is slightly reduced and kept at a constant size for a year, but that soon after escapes. (d) The corresponding tumor at escape (figure obtained from Ref. [47])

domain is important for the maintenance of dormancy. Probably, the reason is that these spread immune cells keep the tumor at a small size, not allowing its overgrowth in any specific direction. We have also explored the importance of the tumor size at detection by reducing this size to 5 × 102 cells. The results are depicted in Fig. 19 and resemble very much those shown in Fig. 16. There is no hint of a sneaking through mechanism in our model. According to the definition given by Gatenby et al. [57], sneaking through is the preferential take of tumors after small size inocula to a similar degree with that seen with large size inocula, compared to the rejection of medium sized inocula. More clearly put, small and big tumors escape immune surveillance, while intermediate do not. Such phenomenon has not been observed in the present case for other values of the tumor size at detection. However, we do not discard it, since

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Fig. 19 Transient bifurcation diagrams. Two transient bifurcation diagrams for the reference scenario. Now a smaller tumor size at detection Tdet = 500 has been considered. (a) A transient bifurcation diagram for a constant input of tumor cells into the domain given by f = 0.1. (b) A transient bifurcation diagram for a constant input of tumor cells into the domain given by f = 0.05. The effects of tumor size at detection does not introduce significant changes in the dynamics (figure obtained from Ref. [47])

motility of tumor cells has not been included in this first investigation, and might be crucial for these cells to escape. Finally, even though the tumors here inspected are genetically homogeneous and no evolutionary process is really taking place in our model, the transient bifurcation diagrams insinuate how the sculpting of the phenotype occurs, moving from the first region to the second, and then to the third. In fact, a similar cellular automaton can be used to explore the impact of heterogeneity and how the process of immunoedition takes place. It suffices to consider that the immune cells intrinsic cytotoxicity, represented by the parameter θlys , depends on the tumor cell.

6.3.2

Low Recruitment Scenario

We now evaluate the impact of the recruitment of immune cells to the domain of the tumor. For this purpose, we reduce the value of θrec from 1 to 0.35. Our interest in this parameter is due to the fact that, in many occasions, the recruitment of cells to the site of the tumor might be very complicated. The recruitment of immune cells is a very complex process, at least from a physical point of view. The extravasation of leukocytes requires an initial contact between these cells and the endothelial cells, which depends on adhesion molecules. After adhesion to the walls of the vessels, the immune cells traverse them through diapedesis, which again relies on several cytokines. Finally, chemokines bias their random walks to the tumor location [58].

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Fig. 20 Transient bifurcation diagrams. Two transient bifurcation diagrams for the low recruitment scenario θrec = 0.35. (a) A transient bifurcation diagram for a constant input of tumor cells into the domain given by f = 0.1. (b) A transient bifurcation diagram for a constant input of tumor cells into the domain given by f = 0.05. A decrease of the immune cell recruitment value reduces the window of equilibrium. Thus large periods of dormancy require significant levels of immune cell recruitment (figure obtained from Ref. [47])

Thus we expect this parameter to exhibit great fluctuations, depending on the tissue location and other factors, as, for instance, the degree of inflammation. The effects of decreasing the recruitment parameter are shown in Fig. 20. As expected, the elimination region shrinks, while the escape region widens. A dramatic reduction of the dormancy window is observed in both plots. When f = 0.1, the window still exists, but for f = 0.05 it has even disappeared. These results suggest that a relatively tight balance between lysis and growth is required to maintain the dynamical equilibrium of the tumor. Note that, as previously proposed, the equilibrium of the tumor implies that reduction must occur in an isotropic manner. If a region of the tumor grows over the immune system capacity, then a soon overgrowth and a consequent escape would be expected. In the present model, this relies on a positive feedback mechanism between the natural input of immune cells and their recruitment. The more cells there are spread in the domain, the more chances for an immune cell to lethally hit a tumor cell. When this occurs, recruitment proceeds, favoring the local aggregation of immune cells at this site of the tumor and giving rise to satellites [40]. This isotropy can be appreciated in the equilibrium represented in Figs. 17b and 18b, as opposed to those situations that lie in the third region, represented in Figs. 17d, f and 18d.

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Fig. 21 Transient bifurcation diagrams. Two transient bifurcation diagrams for the high necrosis scenario θnec = 1.0. (a) A transient bifurcation diagram for a constant input of tumor cells into the domain given by f = 0.1. (b) A transient bifurcation diagram for a constant input of tumor cells into the domain given by f = 0.05. The window of equilibrium has been reduced again, which suggests that long-lived periods of dormancy are based on a delicate equilibrium between the proliferation rate of the tumor and its lysis by the immune system (figure obtained from Ref. [47])

6.3.3

High Necrosis Scenario

Solid tumors exhibit sometimes necrotic cores due to the scarcity of nutrients. Other chemical species can be represented with the present model (e.g., growth factors) and, if desired, necrosis can be regarded as apoptosis, at least to some extent. Therefore, we now inspect the effects of cell death in the model. To this end, we increase the value of θnec from almost zero to 0.5. Obviously, the increase of necrosis facilitates the labor of the immune system. As shown in Fig. 21, the elimination region enlarges substantially, compared to the reference case. Also in the equilibrium region, lower tumor cell numbers are seen before the escape of the tumor. More importantly, the equilibrium window, which has been associated with large periods of tumor mass latency, is practically imperceptible for f = 1.0 and has completely vanished for f = 0.05. We have again computed time series, showing that transients occur in the equilibrium region, sometimes as long as those appearing before in the equilibrium, but generally shorter (see Fig. 22). In fact, the equilibrium window and the escape zone drawn in Fig. 21a overlap. It seems that the equilibrium region appearing in the reference scenario has been swept under the elimination region. Once more, the results confirm the requisite of a relatively delicate balance between the mechanisms that maintain the cytotoxic destruction of the immune system and the growth of the tumor, in order to keep it at low cell numbers for long periods of time.

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Fig. 22 Asymptotic dynamics and tumor escape. Two time series of the tumor size for the reference scenario are plotted. The constant input of immune cells to the domain is now smaller f = 0.05. The size of the tumors is registered until they escape the domain. The tumors at escape are shown beside. Again, the immune cells appear in dark blue, while the tumor cells range from red to white. The dead cells, which also appear inside the tumor, are now represented in green. (a) A quite long-lived tumor is kept at equilibrium for θlys = 92. (b) The corresponding tumor at escape. (c) Another tumor θlys = 118 that is barely reduced and kept at a constant size for less than half a year, and then escapes rapidly. (d) The corresponding tumor at escape (figure obtained from Ref. [47])

7 Conclusions In the present chapter we have explored the dynamics of tumor growth in the presence of an immunological response. This formidable task would have been much more difficult without the aid of cellular automata, which are discrete spatiotemporal models that allow to represent complex biological systems. The main idea is to use this sophisticated modelling framework to perform in silico experiments, which allow to reproduce the tissue environment as an open system. Of course, and just as it occurs with in vitro experiments, the design of these models relies upon several hypotheses, which must be thoroughly debated.

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Using a hybrid cellular automaton, we have shown that cell crowding is a plausible candidate that can explain the saturation of the fractional cell kill of tumor cells by their cytotoxic opponents observed in immunological assays on the lysis of tumors. This limitation depends on the morphology of the tumor, insofar as geometry restricts the access of effector cells to tumor cells. In theory, those tumors growing with “spherical symmetry" will be the harder to lyse, because more layers of tumor cells have to be erased to reach the cells at the center. Nevertheless, we recall that the process of T cell recruitment from circulation to the tumor site is complex, involving several steps [59]. This implies that the crowding might happen before contact with the tumor occurs, as, for example, during adhesion to the endothelium. In such a case, a relation between the parameters in the fractional cell kill and the shape of the tumor cannot be established. Furthermore, we have explored the decay laws that govern the destruction of the tumor for the extreme situations in which the immune response is too weak or very strong. We have observed that, when there is no infiltration, the decay ranges from a linear decay to a parabolic decay. The linear decay corresponds to small values of the effector-to-target ratio as initial conditions, while the parabolic decay represents a tumor that is widely surrounded by immune cells. Intermediate situations are governed by Eq. (35). The significance of this new mathematical function can be described as follows. The rate at which a tumor as a whole is destroyed by a population of immune cells increases as the both cell populations increase. However, at some point, saturation is attained. The particular functional response is given by a Hill function depending on both cell populations, in a quite symmetrical way. In the case of the effector cells, this extrinsic barrier to the lytic capacity is reflected in the parameters d and λ. These parameters depend on the geometry of the tumor. Less spherical tumors correspond to higher values of both parameters. Interestingly, the values of λ are expected to be between zero and one, as suggested by the experiments and the simulations. From the enzymatic kinetics point of view, this can be interpreted as non-cooperative binding. Certainly, if we pay attention to the process of lysis only, the best that an immune cell can do to another is not to interpose between itself and their adversaries. Of course, cooperative effects exist, as the recruitment term exemplifies. Quite the opposite, as the intrinsic lytic capacity of cytotoxic cells is decreased, saturation gradually vanishes. This capability is inversely proportional to the parameter s. On the other hand, the saturation of the lytic velocity due to a tumor of increasing size is reflected in the parameters ν, which depends on the degree of infiltration or, if desired, on its topology. More connected tumors have smaller values of this parameter, ranging from one to two-thirds (in the case of three-dimensional tumors). Finally, we have studied the transient and asymptotic dynamics of a cellular automaton model for tumor–immune interactions. We have shown that, depending on the immunogenicity of the tumor, the model furnishes three main types of dynamics, which are in close relationship with the three phases of the theory of immunoedition. Importantly, we have shown that a dynamical equilibrium between the tumor can occur for long periods of time, as proposed by Kuznetsov et al. [31]. However, after inspection of the parameter space, our model suggests that

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this equilibrium is quite fragile, since it is based on an adjusted balance between the mechanisms that stimulate the immune response and tumor cell proliferation. This also occurs in the model presented by these authors [31], since considerable levels of recruitment are required to sustain dormancy. Furthermore, the infiltration of the immune cells into the tumor mass has been neglected in the present work. We also recall that the piling of immune cells in this study has been restricted, to speed the extensive computations. Presumably, these effects would make the equilibrium much more delicate. Nevertheless, both models clearly demonstrate that a state of tumor mass dormancy mediated by the immune system is possible. It is the length of this dormant period that can be safely questioned. Thus, we conclude that, even though tumor mass dormancy can result from the cell-mediated immune response to tumor growth, long periods of dormancy, as commonly found in recurrent metastatic tumors [54, 55], are not likely to arise by this single mechanism. It is therefore pertinent to ask ourselves if the role of the cell-mediated immune response in the promotion of the dormancy of a tumor mass is rather to synergize with other types of more efficient mechanisms, as, for example, cellular dormancy. Acknowledgements This work has been supported by the Spanish Ministry of Economy and Competitiveness under Project No. FIS2013-40653-P and by the Spanish State Research Agency (AEI) and the European Regional Development Fund (FEDER) under Project No. FIS2016-76883P. M.A.F.S. acknowledges the jointly sponsored financial support by the Fulbright Program and the Spanish Ministry of Education (Program No. FMECD-ST-2016).

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Analytical Solutions for Traveling Pulses and Wave Trains in Neural Models: Excitable and Oscillatory Regimes Evgeny P. Zemskov and Mikhail A. Tsyganov

Abstract We consider a piecewise linear approximation of the diffusive MorrisLecar model of neuronal activity, the Tonnelier-Gerstner model. Exact analytical solutions for one-dimensional excitation waves are derived. The dynamics of traveling waves is related to two basic regimes of wave propagation: excitable and oscillatory cases. In the first case we describe mathematically the structure of a solitary pulse and in the second case—the form of a periodic sequence of pulses (a periodic wave train). Keywords Reaction-diffusion equations · Piecewise linear models · Traveling waves PACS 82.40.Bj, 05.45.−a, 82.40.Ck, 87.10.Ed

1 Introduction Mathematical modeling of the neuronal activity is a common problem in the theoretical biophysics and neuroscience. The neurosystems show the complex spatiotemporal behavior of extended objects from diverse interacting elements. To describe such a behavior, the formalism of reaction-diffusion wave processes is usually applied. The reaction-diffusion wave processes present self-sustaining wave

E. P. Zemskov () Federal Research Center for Computer Science and Control, Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] M. A. Tsyganov Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_9

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processes when there appear excitation traveling waves with constant shape and speed of propagation in an excitable (active) nonlinear medium. In the active medium may occur three fundamental types of the reaction-diffusion wave behavior: bistable, excitable, and oscillatory regimes. The bistable regime is characterized by the appearance of a switching wave (a front). The excitable regime exhibits a single pulse wave. In the oscillatory regime there exists a periodic pulse sequence (a wave train). All these types of nonlinear traveling waves (fronts, pulses, and wave trains) have been intensely studied. Such nonlinear waves arise, propagate, and interact each other and with boundary in active media and are described mathematically with related nonlinear equations in partial derivatives of the reaction-diffusion type. Diverse wave patterns may appear due to spontaneous formation of wave sources as well as dissipative structures under conditions when traveling waves do not collide but move away from each other. The developing pattern depends on conditions and may exist in different states characterized by the presence of one as well as several reaction-diffusion wave processes. One of the well-known models of the reaction-diffusion systems includes two equations, with a cubic nonlinearity in the first one and a linear reaction term in the second one. The equations of this system were proposed by FitzHugh [1] and Nagumo et al. [2]. This FitzHugh-Nagumo (FHN) model is also referred to as the Bonhoeffer-van der Pol model. It was originally presented as a simplification of the Hodgkin-Huxley equations describing the propagation of an action potential along nerve fibers. The FHN model is described by the reaction-diffusion equations ∂u ∂ 2u = u(1 − u)(u − a) − v + Du 2 , ∂t ∂x

(1)

∂v ∂ 2v = ε(u − v) + Dv 2 . ∂t ∂x

(2)

The positive parameters a and ε are the excitation threshold and the ratio of time scales. The constants Du,v are diffusion coefficients. The variable u represents the “activator” or potential variable. It corresponds to the potential across the membrane of the nerve fiber in the original application to the Hodgkin-Huxley model. The variable v represents the “inhibitor” or recovery variable. Depending on the parameter values, all three regimes are realized in such a system. Rinzel and Keller [3] applied a piecewise linear approximation of the McKeantype [4] for the nonlinear reaction term in the first equation and deduced an example of an analytically solvable model qualitatively well reproducing the FitzHughNagumo dynamics. They wrote ∂u ∂ 2u = −u − v + H (u − a) + Du 2 , ∂t ∂x

(3)

∂v ∂ 2v = ε(u − v) + Dv 2 , ∂t ∂x

(4)

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209

where H (u − a) is the Heaviside step function. Such approach allows to solve the equations analytically and obtain the wave solutions for u and v. Tonnelier and Gerstner [5] considered the piecewise linear version of the FHN model with the nonlinear inhibitor function of a sigmoidal type as a simplification of the MorrisLecar [6] equations related to neuron models and proposed the corresponding modification of the Rinzel-Keller model with the Heaviside step function in both equations. This model will be used in the present research. We wish to solve the reaction-diffusion equations analytically and find the exact solutions in the form of the traveling pulses and wave trains. To the best of our knowledge, before the present study, no fully analytical solutions for the traveling pulses and wave trains in the two-component reaction-diffusion systems with piecewise linear reaction functions in both equations were available. The method of the piecewise linear approximation has general applicability and is often the only way to study a variety of nonlinear problems analytically in an approximate fashion [7]. Piecewise linear models have been employed to use the translational invariance of equations as a speed selection mechanism [8, 9], to study the effect of transport memory [10–13] and the wave propagation in discrete [14– 16] and inhomogeneous [17–19] media and to consider a forcing [20, 21]. Rinzel and Keller [3] described the pulses and the wave trains, they calculated the wave speeds and performed the stability analysis in a reaction-diffusion model with non-diffusing inhibitor. Koga [22] described wave solutions of the bistable doublediffusive piecewise linear model. He considered only the case where the activator (the first variable) diffuses faster than the inhibitor (the second variable). The linear stability analysis was performed both by Rinzel and Keller and by Koga. Ito and Ohta [23] derived a motionless localized solution and a propagatingpulse solution. The research was focused on the effect of the inhibitor diffusion. In most papers related to the piecewise linear reaction-diffusion equations, onecomponent [8, 10, 12, 20, 24, 25] and two-component [22, 23, 26–29] systems are investigated, i.e., such an analytical approach is important despite the existence of many numerical or seminumerical results.

2 Tonnelier-Gerstner Model There are three wave phenomena related to traveling waves: wave formation, propagation, and interaction. Wave formation and propagation are simple processes, whereas wave interaction shows complex dynamics. In many reaction-diffusion systems, it leads to wave annihilation or wave reflection [30, 31]. These phenomena occur usually after a collision of a pair of counter-propagating waves or after a collision of a wave with no-flux boundaries of the medium. Recently [32] we have found more complex behavior at such collisions: wave reflection at a growing distance (remote reflection). The present research continues our preceding work and extend the investigations of traveling front dynamics [33] to the solitary pulses and the periodic wave trains. Here we develop the analytical description for the

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1.6

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Fig. 1 Null-clines of the sigmoidal reaction-diffusion system with piecewise linear functions in (a) bistable, (b) excitable, and (c) oscillatory regimes. Null-cline related to the activator (the first variable) reaction function is shown by thin line, whereas to the inhibitor (the second variable) function by thick line

traveling waves in the piecewise linear reaction-diffusion system, the TonnelierGerstner [5] model, which is referred also as the sigmoidal model. Wave solutions in this reaction-diffusion model depend on the intersection of null-clines, i.e., the curves plotting the equations of zero-valued reaction functions f (u, v) = 0 and g(u, v) = 0. The first regime (Fig. 1a) is bistable and the corresponding solution is a front wave (heteroclinic). The second one (Fig. 1b) is excitable and the corresponding solution is a pulse (homoclinic). In the last case (Fig. 1c), the system is in oscillatory regime and demonstrates the sequences of pulses or periodic wave trains. Reaction-diffusion system considered here incorporates the Tonnelier-Gerstner kinetics [5] and a spatial coupling via diffusion that allows traveling wave propagation. The model is described by equations ∂u ∂ 2u = −u − v + H (u − a) + Du 2 , ∂t ∂x

(5)

∂v ∂ 2v = −εv + αH (u − a) + Dv 2 , ∂t ∂x

(6)

where ε, α, a, and Du,v are positive constants. General traveling wave (ξ = x − ct is the traveling-frame coordinate and c is the wave speed) solution reads u(ξ ) = A1 eλ1 ξ + A2 eλ2 ξ +

B3 λ3 ξ B4 λ4 ξ e + e + u∗ , μ3 μ4

v(ξ ) = B3 eλ3 ξ + B4 eλ4 ξ + v ∗ ,

(7) (8)

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where  @ 1  −c ± c2 + 4Du , 2Du  @ 1  −c ± c2 + 4Dv ε = 2Dv

λ1,2 = λ3,4

(9) (10)

are the eigenvalues of the characteristic equation,     Du Du λ3,4 − 1 − ε μ3,4 = c 1 − Dv Dv

(11)

and u∗ , v ∗ are constants. We consider here two types of solutions: solitary pulses and sequences of pulses (periodic wave trains) [34].

2.1 Solitary Pulses Solitary pulses occur in the excitable regime of the active media. Calculations show that there are two pulse waves at the fixed parameter values, the fast and slow waves, as obtained elsewhere [3], where it was found that the fast pulse is a stable solution, whereas the slow wave is unstable. The case of the pulse with oscillatory tails reproduces such a situation with non-oscillatory waves. The pulse solution in this piecewise linear model consists of three segments, first of which vanishes as ξ → −∞ and the third too as ξ → +∞, i.e., the boundary conditions for the pulse solutions are as follows: u1 (ξ → −∞) = 0,

u3 (ξ → +∞) = 0,

(12)

v1 (ξ → −∞) = 0,

v3 (ξ → +∞) = 0.

(13)

Since Du,v and ε are positive, λ1,3 > 0 and λ2,4 < 0, and the pulse solution has the form ⎧ B31 λ3 ξ ⎪ ⎪ A11 eλ1 ξ + e , ξ ≤ ξ0 , ⎪ ⎪ μ3 ⎪ ⎪ ⎪ λ1 ξ λ2 ξ ⎪ ⎪ ⎨A12 e + A22 e + u(ξ ) = B32 λ ξ B42 λ ξ (14) ⎪ e 3 + e 4 + 1 − α/ε, ξ0 ≤ ξ ≤ ξ0∗ , ⎪ ⎪ μ3 μ4 ⎪ ⎪ ⎪ ⎪ B ⎪ ⎪ ⎩A23 eλ2 ξ + 43 eλ4 ξ , ξ ≥ ξ0∗ μ4

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and

v(ξ ) =

⎧ λ3 ξ ⎪ ⎪ ⎨B31 e ,

ξ ≤ ξ0 ,

B32 + B42 ⎪ ⎪ ⎩B eλ4 ξ , eλ3 ξ

eλ4 ξ

ξ0 ≤ ξ ≤ ξ0∗ ,

+ α/ε,

ξ≥

43

(15)

ξ0∗ .

From the continuity of the solutions and its derivative at the matching points ξ0 = 0 and ξ0∗ we find matching conditions u1 (ξ0 ) = u2 (ξ0 ),

du1 (ξ )

du2 (ξ )

= , dξ dξ

u2 (ξ0∗ ) = u3 (ξ0∗ ),

du2 (ξ )

du3 (ξ )

= , dξ ξ ∗ dξ ξ ∗

v1 (ξ0 ) = v2 (ξ0 ),

dv1 (ξ )

dv2 (ξ )

= , dξ ξ0 dξ ξ0

v2 (ξ0∗ ) = v3 (ξ0∗ ),

dv2 (ξ )

dv3 (ξ )

= , dξ ξ ∗ dξ ξ ∗

ξ0

ξ0

u1 (ξ0 ) = a,

0

0

0

(16)

0

u3 (ξ0∗ ) = a.

There are 10 equations for 10 unknowns: 4 constants A, 4 constants B, the coordinate ξ0∗ of the second matching point and the speed c; the first matching point ξ0 may be chosen arbitrarily, usually as zero, due to the translational invariance of the model equations. An example of traveling solitary pulses at different values of the excitation threshold a is shown in Fig. 2. We see that the pulse wave profile consists of two parts: front and back. The activator u and the inhibitor v waves have different front and back parts. The front part of activator is always monotonic, whereas the back part has a well. For the inhibitor, the front and back parts are both monotonic. When the speed of the pulse wave takes positive value, the pulse propagates from left to right, when the speed is negative, the wave travels from right to left.

2.2 Periodic Wave Trains Periodic sequences of pulses can be found in the same active media where the solitary pulses occur. Contrary to the pulse case, the number of wave trains may change depending on parameter values. When the period of the wave train is varied, there appear one or several wave trains with different speeds. The diagrams for the wave speed vs. period are called dispersion relations. For wave trains with a standard

Analytical Solutions for Traveling Pulses and Wave Trains in Neural Models:. . . 1

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Fig. 2 An example of traveling pulses at Du = 0.5, Dv = 0.1, ε = 0.1 and α = 0.1: (a, c) u = u(ξ ) (thick line) and v = v(ξ ) (thin line) profiles and (b, d) u − v diagrams. Pulse propagates with (a, b) positive (c ≈ 0.311 at a = 0.25) and with (c, d) negative (c ≈ −0.201 at a = 0.1) speeds. The second (intermediate) piece of each wave is marked by gray color

shape, the dispersion relation curves are monotonic, whereas for wave trains with oscillations in profile the dispersion relations are anomalous. The periodic wave train is a two-piece solution of the form + u(ξ ) =

u1 (ξ ), ξ00 ≤ ξ ≤ ξ0 , u2 (ξ ), ξ0 ≤ ξ ≤ ξ0∗

(17)

for u variable and + v(ξ ) =

v1 (ξ ), ξ00 ≤ ξ ≤ ξ0 ,

v2 (ξ ), ξ0 ≤ ξ ≤ ξ0∗

(18)

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for v variable, where B31 λ3 ξ B41 λ4 ξ e + e , μ3 μ4

(19)

B32 λ3 ξ B42 λ4 ξ e + e + u∗ , μ3 μ4

(20)

v1 (ξ ) = B31 eλ3 ξ + B41 eλ4 ξ ,

(21)

v2 (ξ ) = B32 eλ3 ξ + B42 eλ4 ξ + v ∗

(22)

u1 (ξ ) = A11 eλ1 ξ + A21 eλ2 ξ + u2 (ξ ) = A12 eλ1 ξ + A22 eλ2 ξ +

with u∗ = 1 − α/ε and v ∗ = α/ε. The trajectory of the periodic wave train on the u − v plane is a closed curve, so that there are two matching points. The curve starts from point u = a at ξ = ξ00 , passes this point at ξ = ξ0 , and ends in this point at ξ = ξ0∗ . At the matching point we have the conditions of continuity for functions, their derivatives, and an equation u(ξ00 ) = u(ξ0 ) = u(ξ0∗ ) = a of the fixed (matching) boundary. The value L = ξ0∗ − ξ00 corresponds to the period of wave and is the new external parameter for the solutions. Thus, the matching conditions read u1 (ξ0 ) = u2 (ξ0 ),

du1 (ξ )

du2 (ξ )

= , dξ ξ0 dξ ξ0

u2 (ξ0∗ ) = u1 (ξ00 ),

du2 (ξ )

du1 (ξ )

= , dξ ξ ∗ dξ ξ 0

v1 (ξ0 ) = v2 (ξ0 ),

dv1 (ξ )

dv2 (ξ )

= , dξ ξ0 dξ ξ0

v2 (ξ0∗ ) = v1 (ξ00 ),

dv2 (ξ )

dv1 (ξ )

= , dξ ξ ∗ dξ ξ 0

u1 (ξ0 ) = a,

0

0

0

(23)

0

u1 (ξ00 ) = a.

Here is again 10 matching equations, but there is a new varied parameter: the period L of wave train. An example of periodic wave trains at different values of the period is shown in Fig. 3. We see that the difference between wave trains with positive and negative speeds reflects in both u and v profiles [Fig. 3a, c]: the wave train is steeper in the direction of wave propagation. There is no difference between wave trains with positive and negative speeds in the u − v diagrams [Fig. 3b, d]. The only difference in the size of the closed trajectory in the u − v diagrams reflects the difference in the absolute value of the wave speed. The situation with the direction of wave propagation remains the same as for the solitary pulses: when the speed of the wave

Analytical Solutions for Traveling Pulses and Wave Trains in Neural Models:. . .

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Fig. 3 An example of periodic wave trains at Du = 1, Dv = 1, a = −0.05, ε = 0.9 and α = 1: (a, c) u = u(ξ ) (thick line) and v = v(ξ ) (thin line) profiles and (b, d) u − v diagrams. Wave train propagates with (a, b) negative (c ≈ −1.41 at L = 10) and with (c, d) positive (c ≈ 2.109 at L = 15) speeds. The second piece is marked by gray color

train has positive value, the wave train propagates from left to right, when the speed is negative, the direction of the wave propagation is opposite.

3 Morris-Lecar Model Most models of excitation wave dynamics in mathematical neuroscience correspond to the Hodgkin-Huxley mechanism. The FHN model is a two-variable simplification of the Hodgkin-Huxley system, where the membrane potential and the recovery variable reflect the dynamics of transmembrane currents. The modification of the FHN model to more realistic systems with nonlinear inhibitor functions is the Tonnelier-Gerstner [5] caricature of the Morris-Lecar [6] model. Caricatures of nonlinear reaction functions by the Heaviside functions have much in their favor.

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The piecewise linear approximation used in this paper provides insights into the most basic properties of traveling waves and can in many contexts be considered as an adequate approximation for the more complicated nonlinear reaction functions in most real models of the neural activity. The analytical description for traveling waves may be developed for more general piecewise linear reaction-diffusion models of the activator-inhibitor type with nonlinear inhibitor [35] or in the piecewise linear approximation for the MorrisLecar [6] model. Such a system is constructed from three pieces and is described by equations ∂u ∂ 2u = f (u, v) + 2 , ∂t ∂x

(24)

∂ 2v ∂v = εg(u, v) + 2 , ∂t ∂x

(25)

where the reaction functions are ⎧ ⎪ ⎪ ⎨−α1 u − v, f (u, v) = α2 u − v − ρ2 ⎪ ⎪ ⎩−α u − v + ρ , 3 3

u ≤ a, a < u < b,

(26)

u≥b

and ⎧ ⎪ ⎪ ⎨β1 u − v − σ1 , u ≤ a, g(u, v) = β2 u − v − σ2 a < u < b, ⎪ ⎪ ⎩β u − v + σ , u ≥ b. 3 3

(27)

The constants are ρ2 = a(α1 + α2 ) > 0,

(28)

ρ3 = b(α2 + α3 ) − ρ2

(29)

σ1 = 0,

(30)

σ2 = a(β2 − β1 ) > 0

(31)

σ1 > a(α1 + β1 ),

(32)

σ2 = a(β2 − β1 ) + σ1

(33)

and

in the excitable regime,

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217

in the oscillatory regime and σ1 = 0,

(34)

σ2 = b∗ (β2 − β1 ) > 0

(35)

σ3 = b(β2 − β3 ) − σ2

(36)

in the bistable regime;

is the same in all three cases. General traveling wave solution reads  u(ξ ) = An eλn ξ + u∗ ,

(37)

n

v(ξ ) =



Bm eλm ξ + v ∗ ,

(38)

m

where the eigenvalues are %  & & ε + αn (ε + αn )2 c ' c2 + ± − ε(αn + βn ). λn = − ± 2 4 2 4

(39)

The specific feature is the case when the eigenvalues are complex, i.e., when (ε + αn )2 − ε(αn + βn ) < 0. 4

(40)

In this case the traveling waves have cosine and sine terms and demonstrate the oscillations in the wave profile. The analytical solutions for the specific types of the waves (fronts, pulses, and wave trains) in the piecewise linear Morris-Lecar model will be presented in detail elsewhere.

4 Discussion The analytical approach to the solution of the reaction-diffusion equations that we consider here is much simpler than the standard solutions of the related nonlinear systems using numerical simulations. Moreover, such method allows to perform a linear stability analysis of the traveling waves. Exact results can be derived for the growth rates of disturbances. The advantage of the presented approach is that it can also be extended to the reaction-diffusion systems with inclusion of the perturbative effects. The perturbative factors, such as external fields (described by advection terms in equations), can produce a formation of the complex spatiotemporal waves

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and patterns in the reaction-diffusion systems. Such perturbations lead to a wave transition when the traveling wave changes the propagation direction, i.e., wave propagation can be effectively controlled by application of these external fields. Another effect that can be treated perturbatively is an external forcing. The forcing can be prescribed a priori, i.e., as a modulation of excitability. There exist different types of the forcing: controlling by initial conditions, global feedback, periodic forcing, and traveling-wave modulation. The last type of forcing presents a perturbation on excitable medium through a moving mask (external potential) so that the type of induced pattern or wave depends on the width and velocity of the mask. The simplest situation occurs when the mask speed is equal to the velocity of propagating excitation waves. The source of such forcing is connected with the current position of the waves so that the origin of that modulation of an excitable medium may be found in that medium inside, i.e., this is a type of “automodulation" like the autocatalysis in chemical reactions. The generalization of the problem using the above-described perturbative factors has not been explored in detail here, but we expect qualitatively similar behavior.

5 Conclusion At the moment there exist two basic approaches to mathematical modelling of spatiotemporal phenomena in neuroscience: axiomatic and dynamic. The aim of the axiomatic approach involved the qualitative characterization of the reactiondiffusion wave evolution in neuronal systems, such as solitary pulse waves in nerve tissues. Moreover, the axiomatic approach does not require any additional information on the kinetics of operating processes, which allows solving a problem in its general formulation. However, there is an essential disadvantage of this approach that it is difficult to observe complex phenomena and to attain quantitative fit to experimental data. The dynamic approach postulates that an excitable medium may be adequately described using the evolution equations in partial derivatives, the nonlinear reactiondiffusion equations. Such nonlinear equations are very complicated for analytical calculations. At present no exact solutions for traveling waves in general form have been found. Alternatively, approximate calculation methods are used, such as a kinematic approach. In the framework of this approach one can mathematically describe many processes and structures in active media. In conclusion, we would like to emphasize that our presented results are expected from corresponding previous works [33, 34], of course. It is appropriate at this point to recall that a complete analytical derivation of solitary pulses and periodic wave trains in a two-variable reaction-diffusion system with piecewise linear activator and inhibitor functions has not been performed before. Another important point is that we now have the machinery at our disposal to generalize to the case with an added external field and forcing without the need to enter into the numerical details immediately.

Analytical Solutions for Traveling Pulses and Wave Trains in Neural Models:. . .

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Numerical Study of Bifurcations Occurring at Fast Timescale in a Predator–Prey Model with Inertial Prey-Taxis Yuri V. Tyutyunov, Anna D. Zagrebneva, Vasiliy N. Govorukhin, and Lyudmila I. Titova

Abstract Bifurcations occurring in a system of partial differential equations (PDEs) describing spatiotemporal dynamics of predator and prey populations with prey-taxis have been studied numerically. The model of the local kinetics of the system assumes logistic reproduction of the prey and a simplest Lotka–Volterra functional response of the predator. Since the model ignores relatively slow and rare demographic processes of birth and death in the population of predator, the predator abundance is kept constant under the considered zero-flux boundary conditions. The abundance of predator populations together with the predator taxis coefficient were used as bifurcation parameters in the numerical study that have been made with help of two qualitatively different techniques of discretization: the Bubnov–Galerkin method and grid method of lines. It has been shown that the considered simple model of prey-taxis in predator–prey system demonstrates complex bifurcation transitions leading to periodic, quasi-periodic and chaotic spatiotemporal dynamics. Keywords Population dynamics · Bifurcations · Numerical analysis · Taxis · Spatially heterogeneous periodic dynamics · Quasi-periodic regime · Spatiotemporal chaos · Multistability

Y. V. Tyutyunov () Federal Research Center The Southern Scientific Centre of the Russian Academy of Sciences (SSC RAS), Rostov-on-Don, Russia Southern Federal University, Rostov-on-Don, Russia e-mail: [email protected] A. D. Zagrebneva Department of Computer and Computer-Based System Software, Faculty of IT Systems and Technologies, Don State Technical University, Rostov-on-Don, Russia V. N. Govorukhin · L. I. Titova Vorovich Institute of Mathematics, Mechanics and Computer Sciences, Southern Federal University, Rostov-on-Don, Russia e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_10

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1 Introduction Natural ecosystems are compound objects having complex dynamics which is determined by various internal and external, biotic and abiotic factors. It is well known that under certain conditions, organisms perform spontaneous movements in response to various stimuli, forming at the level of population both stationary and dynamic spatial structures [1–5]. Various spatial structures were experimentally observed in bacteria communities, in particular, in laboratory cultures of Escherichia coli which demonstrate chemotaxis towards attractant such as αmethyl-D,L aspartate. Direct observation of animals’ spatial activity in living natural ecosystems is much more difficult and requests long-lasting and labour-intensive field works. Although such large-scale observations in situ are rare but truly useful and wanted (see, e.g., [6–10]), inevitable measurement errors and irregularity of sampling protocol lead to inaccurate evaluation of the population abundance and does not allow rigorous distinguishing between periodic, quasi-periodic and chaotic dynamics in fields [11, 12]. At the same time, dynamic features of many complex processes in living systems, including both laboratory microcosms and large-scale ecosystems, can be described and studied in silico with the help of mathematical modelling. For example, Dolak and Hillen [13] have reproduced Berg’s experiments with bacteria E. coli [14, 15]; spatial patterns observed in experiments on E. coli and S. Typhimurium [16] have been modelled and analysed by Tyson et al. [17]. During several years with different co-authors we were studying the role of active directed movement of consumers responding to spatial heterogeneity of the food resource, in emergence of complex spatiotemporal dynamics, explaining pattern formation in aquatics and terrestrial communities [18–22]. However, even for simplest models of predator–prey system with inertial taxis [22–25] exhaustive study of possible dynamic regimes and bifurcation scenarios has not been made so far. In the present work we intend to investigate numerically a minimal prey-taxis (trophotaxis) model that was suggested and analysed earlier, and which is known to demonstrate a multifarious spectrum of spatially heterogeneous dynamics including various periodic, quasi-periodic and chaotic heterogeneous regimes generated by the prey-taxis activity of the predator [26–28]. The considered model is based on the following hypotheses: – the prey population reproduces logistically and disperses diffusively; – the average individual ration of the predator is described by the Lotka–Volterra functional response [29, 30]; – the acceleration of the predator population density depends linearly on the gradient of prey density; – the birth/death processes of the predator population are ignored by the model, assuming that predator demography is much slower than the prey’s; – the habitat domain is considered to be closed with zero-flux boundary conditions.

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The linear stability analysis of the model [26–28, 31] showed that the non-trivial homogeneous stationary regime of the system becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of the model parameters. Our recent study has extended these results to the more general case of the Gauze–Kolmogorov predator–prey model describing the local kinetics of interacting species in a spatial model with indirect prey-taxis [22]. The emerging spatiotemporal heterogeneous dynamics can naturally be interpreted as the formation of persistent groups (swarms, schools, packs) of predators that hunt and actively pursue their prey, aggregating in locations with high prey abundances. Such spatial patterns of dynamically moving, spontaneously emerging and disappearing patches of population densities can be observed in various natural trophic communities. In particular, we used the considered model of prey-taxis to explain small-scale spatial heterogeneous in a trophic system consisting of populations of marine benthic harpacticoid copepods (Crustacea) and diatomic microalgae [24]. The model belongs to the class of PDE systems with cross-diffusion, wave solutions of which principally differ from the wave dynamics demonstrated by the “reaction–diffusion” type systems (see [32]). The model describes a kind of tactic interactions between species, being intensively studied during last decades [22, 33–42]. We will study possible scenarios of bifurcation transitions caused by two parameters characterizing the predator population: the prey-taxis coefficient and total abundance of predators for the one-dimensional case of the minimal model of trophotaxis [26, 28, 31]. Bifurcation analysis of such systems requires application of numerical simulation technique because analytical methods are helpful for solving particular problems only, as the above-mentioned stability study of stationary homogeneous states [26, 28, 31]. It is well known that numerical results of a bifurcation analysis of PDE systems depend on the methods of temporal and spatial discretization and its dimensionality. Usually this manifests itself as dependence of bifurcation parameter values on the approximation scheme [43]. An inadequate order of finite-dimensional analogs of the original continuous differential equations can lead to spurious numerical solutions distorting the bifurcation picture completely [43, 44]. In order to check the numerical results obtained by simulations we use two qualitatively different methods of discretization of the original continuous PDEs: the finite-difference method of lines and the Bubnov–Galerkin method, varying the order of approximations and comparing the outcomes of computations.

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2 The Model of Inertial Prey-Taxis and Its Finite-Dimensional Approximations 2.1 The Taxis–Diffusion–Reaction System The minimal prey-taxis model describing spatiotemporal dynamics of a predator– prey system within a closed one-dimensional habitat is represented as the following system of PDEs [26, 28]: ⎧ ∂N ∂2N ⎪ ⎨ ∂t = N (1 − N − P ) + δN ∂x 2 , ∂(P v) ∂P ∂2P ∂t = − ∂x + δP ∂x 2 , ⎪ ⎩ ∂v ∂N ∂2v ∂t = κ ∂x + δV ∂x 2 ,

(1)

with zero-flux boundary conditions at the edges of the domain [0, L]

∂P

∂N

= = v|x=0,L = 0. ∂x x=0,L ∂x x=0,L

(2)

Here N(x, t), P(x, t) are the populations densities of prey and predator, respectively; v(x, t) is the taxis velocity of predators; κ is the prey-taxis coefficient characterizing the sensitivity of predator to local heterogeneity of prey distribution; δ N , δ P , δ V are the diffusion coefficients. The boundary conditions (2) reflect the closeness of the habitat. The balance equation of predator population density in system (1) ignores the birth/death processes, assuming that predator demography acts at much slower timescale than prey demography and animal movement behaviour. Thus, owing the boundary conditions (2), the average density of the predator population P  =

1 L L 0 P dx is kept constant and can be interpreted as the model parameter: 1 d d P  = dt L dt

/L P dx = 0

  ∂P

1 −P v + δP = 0. L ∂x x=0,L

Note that the model of inertial prey-taxis (1)–(2) can be rewritten in an equivalent cross-diffusion form of the indirect prey-taxis (for details see [20, 22, 24, 42]).

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2.2 Stability of Uniform Stationary States System (1)–(2) has two homogeneous stationary states: (N1 , P1 , v1 ) = (0, P  , 0) ; (N2 , P2 , v2 ) = (1 − P  , P  , 0) .

(3)

The first (trivial) stationary state (N1 , P1 , v1 ) corresponds to extinction of the prey population, while the second one (N2 , P2 , v2 ) corresponds to homogeneous distribution of both coexisting populations. The trivial stationary state (N1 , P1 , v1 ) is locally asymptotically stable if the average predator density exceeds the unity, i.e., if P > 1. The homogeneous solution (N2 , P2 , v2 ) is, on the contrary, stable with respect to homogeneous perturbations when 0 < P < 1. In view of the boundary conditions (2), considering spatially heterogeneous perturbations of (N2 , P2 , v2 ), one should present them in the form of the Fourier series:   n˜ (x, t) = nk cos(kx)eλ(k)t ; p˜ (x, t) = pk cos(kx)eλ(k)t ; v˜ (x, t) = k k  vk sin(kx)eλ(k)t ,where k = nπ /L denotes the wavenumber corresponding to k

mode number n (n = 0, ± 1, ±2, . . . ). The linear stability analysis of the non-trivial homogeneous solution (N2 , P2 , v2 ) (for details see [26]) has shown that for 0 < P < 1, homogeneous stationary state (N2 , P2 , v2 ) becomes oscillatory unstable with respect to the mode of heterogeneous perturbation with the wavenumber k, when taxis coefficient κ exceeds its critical value κ cr (k2 ) that exists for all admissible values of model parameters:      (1 − P ) (δN + δP ) + k 2 (δN δP + δN δv + δP δv ) 1 − P  + k 2 (δN + δP + δv ) κcr k 2 = . P  (1 − P )

The critical curves for first three modes (n = ± 1, ±2, ±3) computed for δ N = 0.05, δ P = 0.005, δ V = 0.0001, L = 1, are presented in Fig. 1 on the plane of parameters (P, κ). The domain of parameter values below the critical curves corresponds to stability of the uniform distribution of populations densities (N2 , P2 , v2 ) (painted by grey colour in Fig. 1). The first leaving the stability domain across the critical curve (violation of the stability condition κ < κ cr (k2 ) for some wavenumber k) corresponds to the Andronov–Hopf bifurcation of the stationary state (N2 , P2 , S2 ) and emergence in its vicinity of spatially heterogeneous periodic dynamics.

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Fig. 1 The critical curves for first three modes (n = ± 1, ±2, ±3), computed for model (1)–(2) with parameter values δ N = 0.05, δ P = 0.005, δ V = 0.0001, L = 1, on the plane of parameters (P, κ). The domain below the critical curves corresponds to stability of the uniform distribution of populations densities (N2 , P2 , v2 ) (painted by grey colour)

5

4

3 κ 2 3 1

2 1

0 0

0,2

0,4

0,6

0,8

1,0

2.3 Numerical Methods Stability analysis of the non-stationary heterogeneous regime and bifurcations caused by further increase of the prey-taxis activity in model (1)–(2) cannot be achieved analytically; we shall apply methods of numerical investigations. For spatial approximation of the initial boundary problem (1)–(2) two principally different methods have been used: the projection (Bubnov–Galerkin) method and the grid method of lines. For the both methods approximations of various orders were built in order to control the results of the simulations. In the case of the Bubnov–Galerkin method, to satisfy the boundary conditions (2) we were looking for the solution of (1)–(2) in the form of finite series as follows: N = N0 (t) +

m  k=1



π kx Nk (t) cos L

 ;

P = P  +

m  k=1

v=

m  k=1



π kx Pk (t) cos L 

vk (t) sin

π kx L

 ;

 .

The unknown coefficients of the decomposition N0 (t), . . . Nm (t), P1 (t), . . . Pm (t), v1 (t), . . . vm (t) were determined from the requirement of orthogonality of discrepancy to the basis functions. The correctness of application of the Bubnov–Galerkin method to parabolic cross-diffusion population model has been discussed by Chen and Jüngel [45].

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As a result, we obtain finite approximation of the original continuous model (1)– (2) by the following system of 3m + 1 ordinary differential equations (ODEs): ⎧ m  ⎪ dN0 ⎪ = N0 (1 − N0 − P ) + 12 Nk (Nk + Pk ) , ⎪ dt ⎪ ⎪ k=1( ⎪ ⎪   ⎪ m−i ⎪ dNi ⎪ 1  π2 ⎪ P  1 − − δ − = N Nk+i (2Nk + Pk ) + Nk Pi+k + i N ⎪ 2 dt 2 L ⎪ ⎪ k=1 ⎪ A ⎨ i−1  + Nk (Pi−k + Ni−k ) + 3Ni N0 + N0 Pi , ⎪ ⎪ k=0 ⎪ ⎪ B A ⎪ ⎪ m−i i−1 ⎪   2π 2 ⎪ dP iπ iπ i i ⎪ vk Pk+i − vk+i Pk − vk Pi−k , ⎪ ⎪ dt = − P  L vi − δP L2 Pi + 2L ⎪ k=1 k=1 ⎪ ⎪ ⎪ ⎩ dvi = −κ iπ N − δ i 2 π 2 v , i = 1 . . . m, V L2 i dt L i (4) where m is the order of approximation (the number of basic functions). In practice m is supposed to be big enough, if its doubling does not change the result of computations significantly. Note that approximation (4) inherits the conservation property of the original model (1)–(2), keeping the average density of the predator population P constant. This can easily be checked by direct substitution, taking into account (4): ⎞ ⎛   /L m d P  π kx d ⎝1  dx ⎠ = 0. Pk cos = dt dt L L k=1

0

With the method of lines we define the solution of the system (1)–(2) at knots of the regular grid {xi = ih, i = 0 . . . M, h = L/M} in the domain x ∈ [0, L]. Differentiating the nonlinear term in the second equation of (1) and using central difference approximations for the spatial derivatives we obtain the following finitedimensional analog of the original PDE system: ⎧ dN Ni−1 −2Ni +Ni+1 i ⎪ , i = 1 . . . M − 1, ⎨ dt = Ni (1 − Pi − Ni ) + δN h2 vi+1 −vi−1 Pi+1 −Pi−1 Pi−1 −2Pi +Pi+1 dPi − vi + δP , dt = −Pi 2h 2h h2 ⎪ ⎩ dvi Ni+1 −Ni−1 vi−1 −2vi +vi+1 = κ + δ , i = 2 . . . M − 2, V dt 2h h2

(5)

where Ni (t) = N(xi , t), Pi (t) = P(xi , t), vi (t) = v(xi , t) are the values of the model variable at internal knots of the grid. In the equations corresponding to the edge knots spatial derivatives of the predator taxis velocity v(t) are approximated using the forward–backward difference scheme, while all other spatial derivatives are approximated by central differences. Finally, we obtain a system of 3M + 1 ODEs, where M + 1 is the number of the grid points.

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The finite-dimensional model (5) preserves the conservative property of the original model (1)–(2) with regard to P value, which can be demonstrated by i summation of the expressions for dP dt from (5): 1 d d P  = dt L dt

 M M−1    Pi + Pi−1 h  dPi 1 dP0 dPM h = + + = 0. 2 L dt 2 dt dt i=1

i=1

Systems (4) and (5) were integrated by the Runge–Kutta method of the fourth order with precision control and automatic time step selection. For the Bubnov– Galerkin method there were 3m + 1 equations, where the number of basis functions m was varied m = 30 . . . 70. For the method of lines there were 3M + 1 equations, where the number of knots M + 1 was also varied by taking M = 100 . . . 300. The interval for the independent variable, time t ∈ [0, T] was long enough to achieve stabilization of an attractor with high precision. The construction of the Poincaré map defined by a hyperplane f (N, P, v) = 0 was done by computation of the system trajectory and the bisection method. The algorithms of simulation and numerical methods were implemented in C++. Graphical visualization and analyses of the solutions were done by means of MATLAB package. The time–frequency analysis of the variable dynamics was performed with help of the fast Fourier transform function (FFT) in MATLAB. The Lyapunov exponents were computed according to method suggested by Wolf et al. [46], and implemented in MATLAB software package MATDS by Govorukhin [47].

3 Results of Numerical Bifurcation Analysis 3.1 Bifurcations of Heterogeneous Spatiotemporal Regimes According to the numerical continuation technique of the bifurcation analysis, gradually increasing the bifurcation parameter value, a sequence of the Cauchy problems was being solved consecutively, taking the previous stabilized attractor as an initial condition for the next-step simulation. Rigorous distinguishing between periodic, quasi-periodic and chaotic regimes was based on the computation of the Fourier spectrum of the variable dynamics, calculation of the Lyapunov exponent, construction of the Poincaré map and analysis of the plane projections of the model phase trajectories. Visualizing the results we considered projections on the plane of two variables: spatially averaged

L consumption of prey by all predators NP  = L1 0 N P dx; and spatially averaged

L density of the prey population N  = L1 0 Ndx.

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Simulations were performed for t ≥ 1000. Such time horizon ensures stabilization of attractors with the following set of model parameters: δN = 0.05, δP = 0.005, δV = 0.0001, L = 1.

(6)

We considered the comparing results obtained by different methods, as being correct if both methods gave qualitatively and quantitatively close solutions (i.e., if difference between bifurcation values and plane projections did not exceed 5%). Figure 2 represents an example of such validation of simulations obtained by the Bubnov–Galerkin method (4) with m = 45 basis functions (right column of Fig. 2), and by the method of lines (5) with number of grid points M = 100 (left column of Fig. 2). It is easily seen that projections constructed with two qualitatively different approximations are practically identical in the case of deterministic dynamics (limit cycles), and lie in the same domain of the phase space in the case of chaotic dynamics. Such validation was done for all numerical results presented below. Additionally, we computed the Lyapunov exponents for the grid discretization (5). However since the determination of the all 3M + 1 Lyapunov exponents corresponding to the ODE system (5) with M = 100 requires heavy computations, estimation of just the three highest values is enough to diagnose the dynamic behaviour of the model. For the results shown in Fig. 2, with P = 0.13 we have got λ1 = 0.00001, λ2 = − 0.0048, λ3 = − 0.0083, and with P = 0.8 we have got λ1 = 0.085, λ2 = 0.0249, λ3 = − 0.0021. Thus, in the first case λ1 ≈ 0, λ2 < 0, λ3 < 0 which corresponds to periodic oscillation (Fig. 2a), while in the second case λ1 > 0, λ2 > 0 indicate chaotic dynamics (Fig. 2b). These results accord well with other criteria used for distinguishing between different types of the system dynamics such as construction of the Poincare map and computation of the Fourier spectrum. Relatively small values of positive exponents λ1 , λ2 in the latter case can be explained by small supercriticality of the model parameters with respect to the bifurcation threshold that causes transition to chaos. Another explanation of this observation can be a characteristic dynamic pattern referred as quasi-chaotic oscillations [48]. Such dynamics consist of irregularly alternating periods of convergence and divergence of closely passing trajectories, manifesting itself in a lowering of the Lyapunov exponent values. As already said, two parameters, the prey-taxis coefficient κ and spatially averaged density of the predator population P, were selected as bifurcation parameters. This choice is conditioned on the results of analysis revealing the properties of the trophotaxis model (1)–(2) [26, 28, 31]. Numerical study has shown that model (1)–(2) demonstrate various regimes: from homogeneous stationary distribution of population densities (with small value of coefficient κ and abundance of predators close to 0 or to 1) to chaotic dynamics (with big κ and P moved towards the interior of domain ]0, 1[). The diagram

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Fig. 2 The phase trajectory projection on plane (NP,N) and the Fourier spectrum of N (FFT coefficients A vs. frequency f ), obtained by two qualitatively different methods with parameter values (6), κ = 2: (a) P = 0.13; (b) P = 0.8

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Fig. 3 The diversity of dynamics in model (1)–(2) with parameter values (6) on the parameter plane (P, κ):  —stationary stable uniform distribution,  —stable spatially heterogeneous periodic dynamics,  —quasi-periodic regime,  —chaos

in Fig. 3 represents a variety of spatially heterogeneous dynamics in dependence on values of bifurcation parameters. The diagram was constructed with use of randomly noised initial conditions of the Cauchy problems that were being solved numerically up to stabilization of realizable attractors. The types of dynamics were determined by computation of the Lyapunov exponent. As one can see, with increase of the preytaxis activity (coefficient κ) the region of chaos on the parameter plane widens and chaotization arises with lower (and higher) predator abundance P. Note also that the parameter domain of stabilization of the stationary stable uniform distribution (points depicted by  in Fig. 3) corresponds to the respective domain (painted by grey colour) in Fig. 1 presenting the critical curves for the analytically obtained stability condition κ < κ cr (k2 ), ∀ k. We have analysed numerically bifurcations caused by continuation of the solutions in parameter P, with different fixed values of the taxis coefficient κ. The most interesting scenario observed in simulations was the following one obtained with κ = 2: spatially homogeneous stationary state → spatially heterogeneous periodic oscillations → period-doubled heterogeneous oscillations → quasi-periodic heterogeneous dynamics → spatiotemporal chaos → heterogeneous periodic oscillations → quasi-periodic heterogeneous dynamics → chaotic dynamics → quasi-periodic dynamics → periodic dynamics. Table 1 represents the domains of parameter P values corresponding to different dynamic regimes. These computations were

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Table 1 Domains of parameter P values with qualitatively equal dynamics of the system for the set of parameter values (6) and κ = 2 No. 1 2 3 4

The Bubnov–Galerkin method P 0.000–0.040 0.041–0.123 0.124–0.309 0.310–0.337

Method of lines P 0.000–0.040 0.041–0.123 0.124–0.307 0.308–0.334

5 6 7

0.338–0.353 0.354–0.497 0.498–0.542

0.335–0.349 0.350–0.493 0.494–0.531

8 9

0.543–0.920 0.921–0.924

0.532–0.928 0.929–0.932

10

0.925–1.000

0.933–1.000

Observed dynamics Steady state (1 − P, P, 0) Limit cycle Period-doubled limit cycle Quasi-periodic oscillations and resonance cycles Chaos Periodic oscillation Quasi-periodic oscillations and resonance cycles Chaos Quasi-periodic oscillations and resonance cycles Limit cycle

performed by the Bubnov–Galerkin method with 45 basis functions, and by the method of lines with M = 100 knots. Figures 4 and 5 represent the dynamics stabilized with P belonging to different domains given in Table 1. Figure 4 shows projections of the stabilized attractors on plane (NP,N). The Poincaré maps generated by these attractors on the hyperplane v(0.5, t) = 0 in the same projections are depicted in Fig. 5. The figures illustrate the bifurcation transitions to deterministic (periodic and quasiperiodic) oscillations and to chaotic dynamics of model (1)–(2). Complication of the dynamics can result from a period doubling bifurcation (see Figs. 4 and 5 with P = 0.11 and P = 0.13), from emergence of a stable invariant torus (see Figs. 4 and 5 with P = 0.32 and P = 0.53), from the rise of chaotic dynamics, resulting from disintegration of an invariant torus (see Figs. 4 and 5 with P = 0.345 and P = 0.8).

3.2 Coexistence of Qualitatively Different Attractors Diversity of dynamic behaviour of the predator–prey system with inertial prey-taxis is not limited to the above listed regimes. Numerical simulations have revealed the possibility of coexistence of qualitatively different attractors in model (1)– (2) with the same parameter values. The model trajectory evolves to one of the simultaneously existing multiple attractors, depending on the initial conditions. The

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Fig. 4 The phase trajectory projection on plane (NP,N) with parameters (6) and κ = 2 computed for different values of P

initial distribution of population densities and taxis velocity in simulations was taken as follows: N (x, 0) = N2 + εx + δ, P (x, 0) = 0, v (x, 0) = 0.

(7)

Here ε ∈ (0, 1) is the coefficient characterizing the initial deviation of prey population density from the uniform equilibrium distribution N2 . Parameter δ ensures physical meaning of the initial distribution, i.e., if N2 + ε > 1, then δ = 1 − (N2 + ε), otherwise δ = 0. Simulation was done with parameter values (6), κ = 2, P = 0.35.

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0.895

= 0.110

= 0.130

0.875

0.688

0.865

0.885 0.085

0.09 0.095

0.095

0.105

0.115

0.63 0.67

= 0.320

0.69

0.87

0.89

0.692

0.18

0.504

= 0.390

0.184 0.188

= 0.530

0.503

0.625 0.665

0.502 0.62 0.501

0.66

= 0.345 0.18

0.19

0.2

= 0.800

0.4

0.205 0.21 0.215

0.218

0.226

0.28

= 0.924

0.25

0.26

0.222

0.35

= 0.960

0.245 0.24

0.24

0.3 0.22

0.235

0.25 0.1

0.2

0.3

0.18 0.2 0.22 0.24

0.18 0.17 0.18 0.19

Fig. 5 Projections of the Poincaré map defined by the hyperplane v(0.5, t) = 0, on the phase plane (NP,N), constructed for different values of P with parameters (6) and κ = 2

Figure 6 shows the distributions of observed realizations of periodic and chaotic regimes in relation to coefficient ε, along with the projections of these two attractors simultaneously coexisting in system (1)–(2). Our results, in particular, the fact that the type of the stabilizing attractor does not depend on the magnitude of ε, determining the initial perturbation of stationary distribution of population density, are in good accordance with the earlier observations of other authors [49, 50].

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Fig. 6 Sensitivity of model (1)–(2) to initial distributions (7):  —spatially heterogeneous periodic dynamics (ε = 0.01);  —chaotic regime (ε = 0.3). Trajectories computed by the method of lines with M = 70 knots. See text for details

4 Discussion and Conclusion In the present work we did not attempt to reproduce dynamics of some particular natural system. The objective of the paper was to suggest a simple model that could explain complex dynamics observed in real trophic communities. We have demonstrated that inclusion into the model of inertial prey-taxis allows obtaining a variety of complex spatiotemporal regimes, including periodic, quasi-periodic and chaotic oscillations. These dynamic regimes emerge due to spatial behaviour of predators in the simplest one-dimensional version of the model that ignores local kinetics of predator populations. For that reason model (1)–(2) can be considered as a minimal prey-taxis model capable of reproducing a broad spectrum of spatiotemporal dynamics. The minimal model can be further developed by inclusion of various details, including considering of higher-dimensional (2D, 3D) spatial domains, friction forces in the velocity equation, non-linarites of the functional response, terms of reproduction and mortality in the predator equations, or even by adding a top-predator population into the trophic system.

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Our results agree well with the research of Chakraborty et al. [51–53] who used the same concept for studying population dynamics of trophic system consisting of Didinium nasutum as predator and Paramecium caudatum as prey [51] and of another trophic system involving the predatory mite Phytoseiulus persimilis and the two-spotted spider mite Tetranychus urticae as a prey [52, 53]. In particular, Chakraborty and her co-authors [51–53] have shown that observed spatial patterns depend on functional response and on initial conditions, and that strong increase of the prey-taxis coefficient causes rising of spatiotemporal chaotic dynamics. The studied phenomenon of complex spatiotemporal dynamics in model (1)– (2) is primarily determined by two parameters: the prey-taxis coefficient and the abundance of predators. Thus, we can guess that these two factors determine the type of spatial patterns in natural predator–prey systems, and that complex dynamics of natural ecosystems is driven by intrinsic mechanisms related to active spatial behaviour and trophic relations between species rather than by external forces and environmental factors. According to our analysis, the dynamics of model (1)–(2) is sensitive to the initial distribution and multiple attractors can potentially exist even in very simple systems. In other words, the system possesses the multistability property: coexistence of different stable attractors for the same values of the model parameters. Generally speaking, such coexistence of qualitatively different attractors (e.g., of periodic and chaotic oscillations) substantially complicates the task of ecosystem management, making the dynamics unpredictable and therefore uncontrollable [54]. Regarding this property and high sensitivity of the system to the initial conditions (see Fig. 6), it would be interesting to study the dependence of the attractor selection on the initial distribution, looking for possible ways to control the process. We suggest considering these problems in future work. In our numerical study we validated all simulations by double computations with qualitatively different approaches for construction of finite approximation of the initially continuous model (1)–(2): the Bubnov–Galerkin method and the method of lines. Being quite reliable, the obtained results allow us to conclude that even relatively low orders of approximations (tens of the basis functions or of the grid knots) provide with adequate description of bifurcations in the considered taxis– reaction–diffusion model, being accurate enough for obtaining qualitatively correct results. Acknowledgments The research was funded by the project 0259-2014-0004 (state reg.no. 01201363188) of SSC RAS “Development of GIS-based methods of modelling marine and terrestrial ecosystems” (Tyutyunov), by the basic part of the state assignment research, project 1.5169.2017/8.9 of the Southern Federal University “Fundamental and applied problems of mathematical modelling” (Titova), and RFBR grant 18-01-00453 “Multistable spatiotemporal scenarios for population models” (Tyutyunov, Zagrebneva, Govorukhin).

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Within Host Dynamical Immune Response to Co-infection with Malaria and Tuberculosis Edme Soho and Stephen Wirkus

Abstract Diseases have been part of human life for generations and evolve within the population, sometimes dying out while other times becoming endemic or the cause of recurrent outbreaks. Co-infection with different pathogens is common, yet little is known about how infection with one pathogen affects the host’s immunological response to another. Immunology-based models of malaria and tuberculosis (TB) are constructed by extending and modifying existing mathematical models in the literature. The two are then combined to give a single nine-variable model of co-infection with malaria and TB. The immunology-based models of malaria and TB give numerical results that agree with the biological observations. The malaria– TB co-infection model gives reasonable results and these suggest that the order in which the two diseases are introduced have an impact on the behavior of both. Keywords Tuberculosis · Malaria · Math models · Co-infection

To elaborate and explore different hypotheses, we need to give a clear understanding of the different elements or pathogens causing the different diseases, their evolution and interaction. The ultimate goal is to use the assumptions and definitions to derive realistic models for malaria, tuberculosis, and ultimately co-infection with both. In the future, the hope is that relevant data could be incorporated into this work to examine the applicability of the models to real-life situations.

E. Soho Department of Mathematics, Hostos Community College, Bronx, NY, USA S. Wirkus () School of Mathematical & Natural Sciences, Arizona State University, Glendale, AZ, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9_11

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1 Malaria and the Immune Response A major cause of disease and morbidity in Africa is from parasitic infections. Malaria is one of the most dangerous human infections and is caused by one of the four protozoan parasites of the genus plasmodium: Plasmodium falciparum, Plasmodium vivax, Plasmodium malariae, and Plasmodium ovale. Of these, P. falciparum is of greatest risk to non-immune humans and inflicts the largest burden. Early treatment of malaria will shorten its duration, prevent complications, and avoid a majority of deaths [3]. It is estimated that 40% of the world’s population is at risk of malaria, and about 90% of the populations infected with malaria live in sub-Sahara Africa [7]. Malaria is a major cause of morbidity and mortality. It ranks alongside acute respiratory infections, measles, and diarrheal diseases as a major cause of mortality worldwide. Unlike other acute diseases which produce life-long resistance to reinfection, malaria only elicits partial immunity after several years of continuous exposure during which time recurring infections and illness occur which could be fatal, especially in children. This immunity is only partially effective unless reinforced through frequent reinfection. Malaria is transmitted by the female mosquito (genus anopheles) to the human host. According to Aron and May (1982), the biology of the four species of plasmodium is generally the same and consists of two distinct phases: sexual and asexual. The asexual phase consists of sporozoites, merozoites, and trophozoites. The sexual phase consists of gametocytes. Infection of the human host begins with the bite of a female anopheline mosquito vector, and the injection of sporozoites into the blood stream. The infectious sporozoites enter the liver parenchyma cells. Here, they replicate, giving rise to the merozoite form at about the time the human host’s natural defense begins to attack the infected cells. After an incubation period of about 7 days which is not accompanied by illness, about 30,000 merozoites are released from each infected liver cell into the blood stream. The merozoites attack and invade the red blood cells (erythrocytes) where upon they change into the trophozoite form. P. falciparum, in particular, attacks all ages of erythrocytes. This form undergoes asexual division and in approximately 48 h (depending on species) the infected erythrocyte ruptures releasing about 8–32 merozoites [8] which invade other erythrocytes (see erythrocytic cycle, Fig. 1). This process is responsible for the clinical symptoms of the disease. The erythrocytic cycle usually continues until controlled by the immune system response or chemotherapy or until the patient dies (in the case of P. falciparum). Some merozoites differentiate into the sexual forms of the parasites called gametocytes (microgamete and macrogamete). Gametocytes are transmitted to a mosquito during the blood meal of an infected person. The female gametes are fertilized and develop into oocysts on the walls of the mosquito gut. Each oocyst gives rise to about 1000 immature sporozoites [8] which migrate to the mosquito salivary gland and mature to repeat the cycle.

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Fig. 1 The process of red blood cell invasion and the egress of the parasites from the host cell. See Sect. 1. Adapted from http://www. uni-kiel.de/zoologie/ zoophysiologie/MainProjects. htm

1.1 Variables Definition and Immune Response Model for Malaria We consider the following variables and parameters in Table 1 that will be used to construct our ODE model describing the immune response to malaria. Erythrocyte red blood cell (RBC) dynamics. At a given time t, the naive or healthy erythrocytes X(t) are assumed to be produced from the source (bone marrow, known as “hematopoietic stem cell”) at a constant recruitment rate sR , and die naturally at a per capita rate dR . During the infection, on contact with merozoites parasite F (t), erythrocytes get infected at a constant rate β and become the infected Y (t) using the law of mass action. Infected erythrocytes die rapidly at the constant per capita death rate dI or mature at rate α. At maturation, given an average of intracellular carrying capacity p, new merozoites, pαY , burst from the host cell to invade new erythrocytes, beginning another round of infection [1]. At the same time the infected RBCs activate the immune response, they are also being killed at the net rate k1 c1Y+Y T , where k1 is the successful removal rate of infected RBCs by the immune effector and c1 is the halfsaturation constant for the infected cells in parallel to k1 Y T used in [13] when assumed that the immune response functions are unbounded. T cells. They are recruited for the source at a constant rate sT and their proliferation is induced by the infection at rate a1 and the presence of the merozoites at the rate a2 . They die at the per capita rate dT . Without loss of generality, the T cells, as immune effector agents, will be stimulated by the parasites (immunostimulation factor) which is followed by the additional processes of autocatalytic and/or cooperative reinforcement through the function g(T ), where a3 is the immunostimulation strength for immune effector and b the halfsaturation constant for immune effector. The T cells can also clear, respectively, the merozoites and infected erythrocytes at constant rates k2 and k1 .

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Table 1 Brief description of state variables for malaria within host; see Model (1.1) Variables X Y F T sR dR β dI k2 α k1 p dF sT dT c1 a2 a1 a3 b

Description Concentration of uninfected (naive) RBC Concentration of infected RBC P. falciparum malaria parasite load (merozoites) Concentration of T cells Recruitment rate of red blood cell Natural death rate of naive RBC Infection rate of naive RBC Death rate of infected RBC Rate immune effectors clear P. falciparum Maturation rate of infected RBC Rate immune effectors (T cells) clear infected RBC Carrying capacity of infected RBC Natural death rate of P. falciparum Recruitment rate of T cells Natural death rate of T cells Half-saturation constant for infected RBC Immunostimulation strength for P. falciparum Immunostimulation strength for infected RBC Immunostimulation strength for immune effector Half-saturation constant for immune effector

Merozoites. With a finite lifetime, P. falciparum parasites die at rate dF . Mature infected erythrocytes Y (t) produce pαY merozoites per unit of time by lysin, which can again infect the naive erythrocytes (see Fig. 1).

1.2 Within Host Model for Malaria Based on all the assumptions, we propose a simple immunological model for malaria: dX = sR − dR X − βXF, dt dY Y = βXF − dI Y − k1 T, dt c1 + Y dF F = pαY − βXF − dF F − k2 T, dt c2 + F

(1.1)

dT F Y T2 . = sT − dT T − k1 Y T − k2 F T + a2 T + a1 T + a3 dt c2 + F c1 + Y b+T2

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1.3 Analysis of the Model The domain D is valid epidemiologically because the populations X, Y, F , and T are all nonnegative. We denote points in D by x = (X, Y, F, T ). The nonnegative orthant R4+ = {x ∈ R4 |x ≥ 0} is a positively invariant region if any trajectory that starts in the nonnegative quadrant remains in the same quadrant forever. It can be shown using standard techniques described in [19, 20] that if initial conditions are specified for each of the states variables at time t = 0, then there exists a unique solution satisfying these initial conditions for all time t ≥ 0. We need to show that solutions of the model are nonnegative with an initial condition X(0), Y (0), F (0) and, T (0) ≥ 0. Lemma 1 The closed positive orthant is positively invariant for Model (1.1). Proof Since Y = 0 and F = 0 are invariant planes for the model, we only need to prove that X(t) ≥ 0, and T (t) ≥ 0 for t ≥ 0 if the initial conditions are in the positive octant. Assume there exists t1 > 0 such that X(t1 ) = 0, T (t1 ) = 0, and 0 < t1 < t. Then dX(t1 ) = sR > 0, dt

and

dT (t1 ) = sT > 0, dt

which imply that X(t) ≥ 0 and T (t) ≥ 0 for t ≥ t1 . Thus, X(t) ≥ 0, and T (t) ≥ 0 for all t ≥ 0.  

1.4 Local Stability Let P = (X∗ , Y ∗ , F ∗ , T ∗ ) be steady state. Whether the human host is infected or not, there are always RBCs (erythrocytes) and T cells in the human body. This implies simply that there is no trivial equilibrium, i.e., we cannot have (X∗ , Y ∗ , F ∗ , T ∗ ) = (0, 0, 0, 0). Mathematically, we see that setting the right-hand sides of Eqs. (1.1) to zero gives the following equilibrium solutions: At the disease-free equilibrium, F ∗ = 0, we have Y ∗ = 0, X∗ = dsRR , and sT + a3

T2 − dT T = 0. b+T2

(1.2)

Solving Eq. (1.2) for nonnegative real solutions T ∗ implies that the disease-free equilibrium (DFE) exists and is ( dsRR , 0, 0, T ∗ ). If one varies the parameters of Eq. (1.2) within a positive domain of the parameters values, the long-term behavior may change since we can have one or three DFE using the “Descartes’ rule of signs”; the “Sturm chain or sequence method” (cf. Beaumont and Pierce, 1963) provides more precise conditions for the number of steady states. The appearance or disappearance of equilibria produce topological changes in the system and are examples of bifurcations.

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The Jacobian (J ) evaluated at the DFE gives ⎡

−dR 0 −βX∗ ⎢ 0 −d − k1 T ∗ βX∗ I ⎢ c1 ⎢ J =⎢ 0 pα −βX∗ − dF − ⎣ a1 ∗ − ac22 T ∗ 0 − c1 T

k2 ∗ c2 T

⎤

0 0 0 ∗

2  a3 bT∗ 2 2 − dT

⎥ ⎥ ⎥. ⎥ ⎦

b+T

1.5 Simulation It is almost impossible to find explicitly the solutions of equations (1.1), and since the parameters that govern the rates and behavior of interactions in the model may change from individual to individual and over time, we then simulate the model by solving the differential equations using an appropriate numerical method. We discuss below the results of the computational experiments within an individual; see Figs. 2 and 3.

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40 Time

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Fig. 2 Time series of mathematical model of parasite P. falciparum, red blood cells, and the immune effector at the beginning of the interaction within host; see Fig. 3 for the long-term behavior

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Fig. 3 Time series results of mathematical model, approaching steady state, of parasite P. falciparum, red blood cells, and the immune effector; see Fig. 2 for the transient solution up to t = 50. Depending on the initial conditions, we see an overall rise of the infected red blood cells with a slow decrease of the naive RBC. Following the activation of the naive immune effector cells, we see a decline of the parasite and the infected RBC, respectively

Through this simple mathematical model, we can understand the dynamics of the parasite and the immune effector (T ). Depending on the initial conditions, specifically the initial amount of merozoites released into the blood stream, we see the rise of the infected red blood cells with a slow decrease of the naive RBC. Following the activation of the naive immune effector cells, we see a decline of the parasite and the infected RBC, respectively. This type of behavior can be observed in the initial stages of malaria. In an endemic region, a prolonged exposure of the host to a parasite due to a constant presence of mosquitos can lead to a tolerant immunological state of the individual immune system.

2 Tuberculosis Within the Individual Tuberculosis (TB) has been a leading cause of death in the world for centuries. During the period after the second world war, because of the medical improvement for treatment and hygiene practices, the number of cases steadily declined in the

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world to a level that could be controlled [5, 6]. However, in one-third of the world, TB still is a big threat and much of the population harbors the latent form of tuberculosis infection [14]. Today, within the pathogen-induced diseases worldwide, TB counts more that 1.5 million deaths [4]. During its latency period, the bacteria causing TB, Mycobacterium tuberculosis (Mtb), is postulated to exist in a dormant state where the host can effectively contain the pathogen. Mtb is an obligatory aerobic-intracellular pathogen, which has a predilection for the lung tissues rich in oxygen supply [16]. The tubercle bacilli enter the host via the respiratory route. In certain situations, the bacilli spread from the site of the infection in the lung to other parts of the body through the lymphatics or blood. In the lung, the alveolar macrophages represent the first line of immune defense in the host–pathogen relationship with the phagocytosis of mycobacterium tuberculosis. This first line of defense is followed by the cell-mediated immunity with an influx of lymphocytes (T cells) and activated macrophages/monocytes into the lesion resulting in granuloma formation. One of three things may happen to the bacilli. They may remain forever within the granuloma causing no/little harm, get activated later, or may get discharged into the airways after an enormous increase in number, necrosis of bronchi, and cavitation [16]. At the site of bacilli multiplication, neutrophils (monocytes) are the first cells to arrive followed by natural killer (NK) cells, a type of lymphocyte (a white blood cell), T cells, γ /δ cells, and α/β cells. It has been noticed that a significant reduction in NK activity is associated with multidrug-resistant TB (MDR-TB) [2]. The tuberculous granuloma contain T cells (both CD4+ and CD8+ ) that contain the infection within the granuloma and prevent reactivation. T cell (CD4+ ) depletion causes a rapid activation of the infection. In humans, the pathogenesis of HIV infection which causes the loss of CD4+ T cells greatly increases susceptibility to both acute and re-activated TB [17]. In spite of this persistence over many centuries, TB can be controlled. Treatment of TB is well-known and developed in the case of non-resistant strains. To treat active TB, it is necessary to take several antibiotics at the same time. If not treated properly, TB can be fatal. The common regimen of treatment is the combination of isoniazid, rifampicin, and pyrazinamide for 2 months followed by isoniazid and rifampicin for at least 4–7 months, if the organism is known to be sensitive, until all the bacteria have been completely cleared [10, 18]. The different system models of Wigginton and Kirschner [21] on TB and host immune response, Perelson on the dynamic of T cells [15], Artavanis-Tsakonas on malaria and immunopathology [1], Mackinnon [11], serve as background models for the immune response in the environment of co-infection of TB and malaria. In regions endemic with malaria and TB such as sub-Sahara Africa, it will be interesting to track the progression of both diseases in the human host and the outcome of the interactions and respective impact of the pathogens and to investigate the possibility of repeated malaria infection promoting an active TB within the individual host.

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2.1 Variables Definition and Immune Response Model for Tuberculosis We have the following variables and parameters in Table 2 that will be used in our ODE model: Macrophage dynamics. The rate of change of the macrophages, especially the resting or naive monocytes, includes the recruitment term (sM ) from the source and a natural death term (μM MR ). This is natural assuming that the macrophages have a finite life span, and in the absence of infection the monocytes undergo a renewed process or constant turnover to maintain an equilibrium. In the close proximity of infection, the naive monocytes are infected at the maximum rate of β, which depends on the extracellular bacterial load. Here we use the sigmoidal saturation function based on a form of the Hill equation [9]. The infected macrophages can be cleared by maturation at rate α, where given the intracellular

Table 2 Brief description of state variables for TB within host; see Model (2.1) Variables T MR MI BI BE α sT dT sM β μR μI γM γB αI N αE k4 c0 c1 a2 a1 a3 b

Description Concentration of T cells Resting macrophages Infected macrophages by Mtb Intracellular bacteria (TB) Extracellular bacteria (TB) Rate of busting of chronically infected macrophages Recruitment rate of T cells Natural death rate of T cells Recruitment rate of macrophages Resting macrophages infection rate Natural death rate of resting macrophages Natural death rate of infected macrophages Rate immune effectors (T cells) clear infected macrophages Rate immune effectors (T cells) clear extracellular bacteria Intracellular bacterial growth rate Carrying capacity of infected macrophages Extracellular bacterial growth rate Extracellular bacterial rate of phagocytosis Half-saturation constant for extracellular bacteria Half-saturation constant for infected macrophages Immunostimulation strength for extracellular bacteria Immunostimulation strength for infected macrophages Immunostimulation strength for immune effector Half-saturation constant for immune effector

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carrying capacity they burst, or the immune response through the T cells, or by natural death at the constant rate μI . Bacteria. The interactions and growth of Mtb are described by extracellular and intracellular bacteria. Extracellular bacteria (BE ) grow at a maximum rate αE and are killed by macrophages at rate k4 . The intracellular bacteria grow within the macrophages at a maximum rate of αI with Hill kinetics accounting for the carrying capacity [9]. They become extracellular when the host macrophage bursts when it becomes chronically infected at an assumed threshold of half of the carrying capacity N . T cells. They are recruited for the source at a constant rate sT and their proliferation is induced by the infected macrophages at rate a1 and extracellular bacteria at the rate a2 . They have a finite lifetime and die at rate dT . Without loss of generality, the T cells, as immune effector agents, will be stimulated by the bacteria (immunostimulation factor) which is followed by the additional processes of autocatalytic and/or cooperative reinforcement through the function g(T ), where a3 is the immunostimulation strength for immune effector and b the half-saturation constant for immune effector. The T cells can also clear infected macrophages at a constant rate γ .

2.2 Within Host Model for TB Based on all the assumptions, we propose a simple immunological model for TB:   BE dMR = sM − βMR − μR MR , dt BE + c0     BI2 BE dMI = βMR − μI MI − αMI − γM MI T , dt BE + c0 BI2 + (N MI )2   BI2 dBE = αE BE − k4 MR BE + αN MI − γB BE T dt BI2 + (N MI )2     BE N MR + μI BI , −β 2 BE + c0     BI2 BI2 dBI = αI BI 1 − 2 − αN MI dt BI + (N MI )2 BI2 + (N MI )2     BE N MR − μI BI +β 2 BE + c0

(2.1)

dT BE MI T2 = sT − dT T − γ MI T − γB BE T + a2 T + a1 T + a3 . dt c0 + BE c4 + MI b+T2

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2.3 Analysis of the Model The domain D is valid epidemiologically because the populations MR , MI , BE , BI , and T are all nonnegative. We denote points in D by x = (MR , MI , BE , BI , T ). The nonnegative orthant R5+ = {x ∈ R5 |x ≥ 0} is a positively invariant region if any trajectory that starts in the nonnegative orthant remains in the same orthant forever. Parallel to the malaria model, we can use standard techniques described in [19, 20]: if initial conditions are specified for each of the state variables at time t = 0, then there exists a unique solution satisfying these initial conditions for all time t ≥ 0. We need to show that solutions of the model are nonnegative with an initial condition MR (0), MI (0), BE (0), BI (0) and, T (0) ≥ 0. Lemma 2 The closed positive orthant is positively invariant for Model (2.1). Proof Since MI = 0, BE = 0 and BI = 0 are invariant planes for the model, we only need to prove that MR (t) ≥ 0, and T (t) ≥ 0 for t ≥ 0 if the initial conditions are in the positive orthant. Assume there exists t1 > 0 such that MR (t1 ) = 0, T (t1 ) = 0, and 0 < t1 < t. Then dMR (t1 ) = sM > 0, dt

and

dT (t1 ) = sT > 0, dt

which imply that MR (t) ≥ 0 and T (t) ≥ 0 for t ≥ t1 . Thus, MR (t) ≥ 0, and T (t) ≥ 0 for all t ≥ 0.  

2.4 Local Stability Let P = (MR∗ , MI∗ , BE∗ , BI∗ , T ∗ ) be steady state. Whether the human host is infected or not, there are always macrophages and T cells in the human body. This implies simply that we cannot have (MR∗ , MI∗ , BE∗ , BI∗ , T ∗ ) = (0, 0, 0, 0, 0), i.e., that there is no trivial equilibrium. Mathematically, we see that setting the right-hand sides of equations (2.1) to zero gives the equilibrium solutions. At the disease-free equilibrium, BE∗ = 0, we have MI∗ = 0, BI∗ = 0, MR∗ = sM /μR , and sT + a3

T2 − dT T = 0. b+T2

(2.2)

Solving Eq. (2.2) for positive real solutions T ∗ implies that the disease-free equilibrium (DFE) exists and is ( μsMR , 0, 0, 0, T ∗ ). We then analyze equations (2.1) by solving the differential equations using an appropriate numerical method. We discuss below the results of these computational experiments within an individual.

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BI

MI

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MR 400

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× 10–3

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Fig. 4 The TB infection time series for Model (2.1). The transient solutions of the concentrations of macrophages, bacteria and effector, MR , MI , BE , BI and effector (T ) are shown; see Fig. 5 for the long-term behavior. Observe the decrease of intracellular bacteria while the concentration of extracellular bacteria increases

2.5 Simulation For the parameters found in the literature [9, 12, 21], we see from the simulation (Figs. 4 and 5) that at the initial stages, the intracellular bacteria give rise to extracellular bacteria that ultimately reach the carrying capacity before the intracellular bacteria die off. The effector population also approaches a carrying capacity locally around the granuloma which contains a small population of extracellular bacteria and infected macrophages, as can be seen from Fig. 5 in which the final time is 100. This type of behavior can be observed biologically in the initial stages of TB.

3 Co-infection with Malaria and TB Within the Individual The human immune system has two main responses to the introduction of foreign antigen into the body: a cellular-mediated response and a humoral response (antibodies). Both TB and malaria primarily affect a cellular-mediated immune response for which we gave a brief description in the previous sections. In this

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Fig. 5 Time series of Model (2.1) showing the long-term behavior of the concentration of macrophages, bacteria and effector, MR , MI , BE , BI and effector (T ). Figure 4 shows the initial transient solutions up to t = 5 Table 3 Brief description of state variables for Model (3.1); cf. Models (1.1) and (2.1) with respective Tables 1 and 2 with u = v

Variables X Y F T MR MI BI BE

Description Concentration of uninfected RBC Concentration of infected RBC P. falciparum malaria parasite load Concentration of T cells Resting macrophages Infected macrophages by Mtb Intracellular bacteria (TB) Extracellular bacteria (TB)

section we focus mainly on the effect that TB or malaria has on each other but also comment on how the model may explain the activation of latent TB (Table 3). We consider the following variables and parameters in Table 4 that will be used to construct our ODE model describing the immune response to malaria and TB coinfection. Based on the assumptions from the previous sections, we propose a simple

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Table 4 Parameters—symbols and descriptions Symbol sR dR β dI k2 α1 k1 p dF sT dT γM λM α2 sM k3 μR μI αI N αE k4 γB c0 = λ B c1 c3 c4 a1 a2 a3 a4 a5 b

Explanation Recruitment rate of red blood cell Natural death rate of naive RBC Infection rate of naive RBC Death rate of infected RBC Rate immune effectors C clear P. falciparum Maturation rate of infected RBC Rate immune effectors (T cells) clear infected RBC Carrying capacity of infected RBC Natural death rate of P. falciparum Recruitment rate of T cells Natural death rate of T cells Rate immune effectors (T cells) clear infected macrophages Half-saturation constant for infected macrophages Rate of busting of chronically infected macrophages Recruitment rate of macrophages Infection rate of resting macrophages Natural death rate of resting macrophages Natural death rate of infected macrophages Intracellular bacterial growth rate Carrying capacity of infected macrophages Extracellular bacterial growth rate Extracellular bacterial rate of phagocytosis Rate immune effectors (T cells) clear extracellular bacteria Half-saturation constant for extracellular bacteria Half-saturation constant for P. falciparum Half-saturation constant for infected RBC Half-saturation constant for infected macrophages Immunostimulation strength for P. falciparum Immunostimulation strength for extracellular bacteria Immunostimulation strength for infected RBC Immunostimulation strength for infected macrophages Immunostimulation strength for immune effector Half-saturation constant for immune effector

Brief description of parameters for Model (3.1); cf. Models (1.1) and (2.1) with respective Tables 1 and 2

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immunological model for TB and malaria co-infection that combines Models (1.1) and (2.1) where u = v = 1 in those models: dX = sR − dR X − βXF, dt dY Y = βXF − dI Y − k1 T, dt c3 + Y dF F = pα1 Y − βXF − dF F − k2 T, dt c1 + F dT Fu Yu = sT − dT T − k1 Y T − k2 F T + a1 T + a T 3 dt c1 + F v c3 + Y v BEu MIu T2 T + a4 T + a5 − γM MI T − γB B E T , v v c0 + BE c4 + MI b+T2   BE − μR MR , = s M − k 3 MR (3.1) BE + c0     BI2 BE − μI MI − α2 MI = k 3 MR − γ M MI T , BE + c0 BI2 + (N MI )2   BI2 = αE BE − k4 MR BE + α2 NMI − γB BE T BI2 + (N MI )2     N BE + μI BI , MR −k3 2 BE + c0     BI2 BI2 = αI B I 1 − 2 − α2 NMI BI + (N MI )2 BI2 + (N MI )2     N BE − μI BI . MR +k3 2 BE + c0 +a2

dMR dt dMI dt dBE dt

dBI dt

4 Analysis of the Model The domain D is valid epidemiologically because the populations X, Y , F , T , MR , MI , BE , and BI are all nonnegative. We let x = (X, Y, F, T , MR , MI , BE , BI ) denote the points in D. The nonnegative orthant R8+ = {x ∈ R8 |x ≥ 0} is a positively invariant region if any trajectory that starts in the nonnegative quadrant remains in the same orthant forever. We can show that if initial conditions are specified for each of the states variables at time t = 0, then there exists a unique solution satisfying these initial conditions for all time t ≥ 0. We need

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to show that solutions of the model are nonnegative with an initial condition X(0), Y (0), F (0), T (0), MR (0), MI (0), BE (0) and, BI (0) ≥ 0. Lemma 3 The closed positive orthant is positively invariant for Model (3.1). Proof Since Y = 0, F = 0, MI = 0, BE = 0, and BI = 0 are invariant planes for the model, we only need to prove that X(t) ≥ 0, T (t) ≥ 0, and MR (t) ≥ 0 for t ≥ 0 if the initial conditions are in the positive orthant. Assume there exists t1 > 0 such that X(t1 ) = 0, T (t1 ) = 0, and MR (t1 ) = 0, and 0 < t1 < t. Then dX(t1 ) = sR > 0, dt

dT (t1 ) = sT > 0, dt

and

dMR (t1 ) = sM > 0, dt

which imply that X(t) ≥ 0, T (t) ≥ 0, and MR (t) ≥ 0 for t ≥ t1 . Thus, X(t) ≥ 0, T (t) ≥ 0, and MR (t) ≥ 0 for all t ≥ 0.  

4.1 Equilibria and Reproductive Number Let P = (X∗ , Y ∗ , F ∗ , T ∗ , MR∗ , MI∗ , BE∗ , BI∗ ) be steady state. Whether the human host is infected or not, there are always RBCs (erythrocytes) and T cells in the human body. This implies simply that there is no trivial equilibrium, i.e., we cannot have (X∗ , Y ∗ , F ∗ , T ∗ , MR∗ , MI∗ , BE∗ , BI∗ ) = (0, 0, 0, 0, 0, 0, 0, 0). Mathematically we can see this, as with the previous two models, by setting the right-hand sides of the equations to zero and attempting to solve for equilibrium solutions.

4.1.1

Existence of the Disease-Free Equilibrium Point

Disease-free equilibrium points are steady state solutions where there is no disease. This is the state in which an individual has no parasites (P. falciparum) or bacteria (Mtb) in the body. Thus, we take Y ∗ = F ∗ = MI∗ = BE∗ = BI∗ = 0. From equations (3.1) we get X∗ = X0 = dsRR , where X0 is the equilibrium density of all ages of erythrocytes in the absence of pathogens; MR∗ = M0 = μsMM the equilibrium R density of macrophages; and for the T cells, T0 will depend on average degree of stimulation of leukocyte and no matter what the number of T cells in human body remains bounded, and the dynamics of (3.1) reduce to examine the dynamics of sT + a5

T2 − dT T = 0. b+T2

(4.1)

By solving Eq. (4.1) in the appropriate domain for the positive solution that represents the biologically relevant steady state of the system, the pathogen-free

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equilibrium (PFE) is given by  P0 =

 sR sM , 0, 0, T ∗ , , 0, 0, 0 . dR μR

It can easily be shown that the PFE is unstable. Finding the rest of the steady states of Model (3.1) analytically is very complicated and involved; therefore, we use numerical simulation for understanding the dynamics of the TB and malaria coinfection and interaction on the immune response.

5 Results of Simulation and Discussion One numerical observation we make before moving on is that an increase in the death rate from d = .007 to d = .03 has a noticeable impact on the effector levels as they drop when malaria is introduced. This drop can be exaggerated significantly by changing the initial conditions of the various quantities. This suggests that the order of the infections may potentially have a significant impact on the solutions and longterm behavior of the results. Thus even changing one parameter may have significant influence on the transient and/or long-term behavior of the system (Figs. 6, 7, 8, 9, 10, 11).

2

0

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600 400 200 0

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4 Time

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Fig. 6 Co-infection Model (3.1) with malaria introduced first, then TB at t = 1.5 with tf = 8 and u=v=1

× 106

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400

200

0

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60

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Fig. 7 Co-infection Model (3.1) with malaria introduced first, then TB at t = 1.5 with tf = 60 and u = v = 1

RBC Inf RBC F

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500

0

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1000

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Fig. 8 Co-infection Model (3.1) with TB introduced first, then malaria at t = 1.5 with tf = 8 and u=v=1

× 106

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Fig. 9 Co-infection Model (3.1) with TB introduced first, then malaria at t = 1.5 with tf = 60 and u = v = 1

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Fig. 10 Co-infection Model (3.1) with TB introduced first, then malaria at t = 1.5 with tf = 10. The natural death rate of effectors is increased here compared with the previous figures, from d = 0.007 to d = 0.03. This gives a noticeable drop of the effector levels when malaria is introduced

260

E. Soho and S. Wirkus × 106

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250

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Fig. 11 Co-infection Model (3.1) with TB introduced first, then malaria at t = 1.5 with tf = 60. The natural death rate of effectors is increased here compared with the previous figures, from d = 0.007 to d = 0.03. This gives a noticeable drop of the effector levels when malaria is introduced

Through the nonlinear mathematical models of malaria, TB, and their coinfection, we introduce mathematical models of the immune response in order to identify and qualitatively assess the phenomenological time evolution of the distribution of the interacting populations. The mathematical models presented here simplify the complex dynamics of the immune response and include several assumptions about the known biology. Our analysis suggests that a full knowledge of the dynamics is essential with a detailed bifurcation analysis likely shedding additional light on the long-term behavior. Moreover, having a combined model may give us insight into the role that co-infection plays in the evolution of a given infection within an individual. Additionally, a thorough analysis of the impact of the parameters via a sensitivity and uncertainty analysis may also give insight. The findings suggest the importance of incorporating the state of the health of the individual in the mathematical modeling of the immune response with respect to the location (endemic regions).

Within Host Dynamical Immune Response to Co-infection with Malaria and. . .

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Index

A Agent-based model, 93, 161 Analysis of deviation, 151, 155 Arnold’s weak resonance, 1–25 Asymptotic growth, 121 Autocorrelation function, 148, 153, 154, 156

B Bifurcation analysis, 89–108, 223, 228–235, 260 Bifurcations, 2, 6, 10–15, 89–108, 194–196, 198–200, 221–236, 245, 260

C Cancer dynamics, 159–203 Cellular automata, 159–203 Chaotic regimes, 78, 132, 228, 235 Coevolution of tumor-immune aggregates, 162 Coupling mechanisms, 210 Crow-Kimura model, 30, 38, 40, 44, 49

D Diffusive Morris-Lecar model, 216 Discrete logistic model, 148, 149, 154, 155 Discrete spatio-temporal model, 201 Dominant eigenvalue, 48, 118, 123 Dynamics indeterminacy, 4, 16 Dynamics of travelling waves, 209 Dynamics of tumor growth, 201

E Error threshold, 34-37, 42, 44, 45, 48, 49 Excitable and oscillatory regimes, 207–218

F FitzHugh-Nagumo (FHN) model, 208, 209, 215 Fuzzy recurrence, 131, 134

H Hamiltonian model, 7–14, 21–24 Hidden keystone variable (HKV) method, 90–94, 97, 100, 102, 108 Hodgkin-Huxley model, 208 Horizon of predictability, 132–137, 139, 142, 143 Host dynamical immune response, 241–260

I Immunological response, 201 Immunology-based models of malaria and tuberculosis, 242–247 Irregular population dynamics, 131–143

L Limit cycles, 3–7, 15, 24, 25, 94, 95, 103, 232 Linear piecewise approximation, 208, 209, 216

© Springer Nature Switzerland AG 2019 F. Berezovskaya, B. Toni (eds.), Advanced Mathematical Methods in Biosciences and Applications, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-15715-9

263

264 M Matrix model, 115 Matrix population model, 115–120, 128 Mean population fitness, 29, 31–33, 37, 39, 42, 43, 46 Method of extreme points (MEP), 148, 150, 152–156 Model parameters, 17, 39, 116, 135, 147–156, 223–225, 229, 236 Multistability, 236

N Neural models, 207–218 Nonnegative matrix, 120, 123 Novikov problem, 81, 82 Numerical analysis, 72–78, 222, 223, 226, 228

Index Q Quasi-equilibrium, 6, 7 Quasiperiodic functions, 57, 62–84 Quasi-periodic regime, 231 Quasispecies model, 27–49

R Rank-1 corrections, 124 Recurrence, 28, 131–143, 194 Reduction theorem, 90–108

S Solitary pulse, 209, 211–212, 214, 218 Spatially heterogeneous periodic dynamics, 231, 235 Spatiotemporal chaos, 231, 236 Stage-structured population dynamics, 121

O Ordinary least squares (OLS), 148–152, 155

P Parameter estimation, 149, 150 Perron-Frobenius theorem, 118, 119, 122 Polyvariant ontogeny, 111–128 Population dynamics, 90, 113, 115, 131–143, 147–156, 236 Population heterogeneity, 89–108 Population time series, 138, 147–156 Predator prey interactions, 90 Predictability of population dynamics, 141 Protozoan parasites, 242 Pseudoperiodic functions, 54, 55, 60–62, 81

T Taxis, 222–225, 227, 229, 231, 233 Tonnelier-Gerstner model, 209–215 Tortrix fluctuations, 148 Travelling pulses, 207–218

U Uncertainty, 127, 131, 260

W Wave trains, 207–218