Table Of ContentNONLINEAR DVNRNICS OF
INTERACTING POPULATIONS
WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE
Editor: Leon O. Chua
University of California, Berkeley
Series A. MONOGRAPHS AND TREATISES
Published Titles
Volume 9: Frequency-Domain Methods for Nonlinear Analysis: Theory and Applications
G. A. Leonov, D. V. Ponomarenko, and V. B. Smirnova
Volume 11: Nonlinear Dynamics of Interacting Populations
A. D. Bazykin
Volume 12: Attractors of Quasiperiodically Forced Systems
T. Kapitaniak and J. Wojewoda
Volume 13: Chaos in Nonlinear Oscillations: Controlling and Synchronization
M. Lakshmanan and K. Murali
Volume 14: Impulsive Differential Equations
A. M. Samoilenko and N. A. Perestyuk
Volume 15: One-Dimensional Cellular Automata
B. Voorhees
Volume 16: Turbulence, Strange Attractors and Chaos
D. Ruelle
Volume 17: The Analysis of ComplexNonlinear Mechanical Systems: A Computer
Algebra Assisted Approach
M. Lesser
Volume 19: Continuum Mechanics via Problems and Exercises
Edited by M. E. Eglit and D. H. Hodges
Volume 20: Chaotic Dynamics
C. Mira, L. Gardini, A. Barugola and J.-C. Cathala
Volume 21: Hopf Bifurcation Analysis: A Frequency Domain Approach
G. Chen and J. L. Moiola
Volume 22: Chaos and Complexity in Nonlinear Electronic Circuits
M. J. Ogorzalek
Volume 23: Nonlinear Dynamics in Particle Accelerators
R. Dilao and R. Alves-Pires
Volume 25: Chaotic Dynamics in Hamiltonian Systems
H. Dankowicz
Forthcoming Titles
Volume 4: Methods of Qualitative Theory in NonlinearDynamics (Part I)
L. Shilnikov, A. Shilnikov, D. Turaev and L. O. Chua
Volume 18: Wave Propagation in Hydrodynamic Flows
A. L. Fabrikant and Y. A. Stepanyants
Volume 24: From Chaos to Order
G. Chen and X Dong
Volume 27: Thermomechanics of Nonlinear Irreversible Behaviours
G. A. Maugin
Volume 30: Quasi-Conservative Systems: Cycles, Resonancesand Chaos
A. D. Morozov
Volume 31: CNN: A Paradigm for Complexity
L. O. Chua
Volume 32: From Order to Chaos II
L. P. Kadanoff
■L I WORLD SCIENTIFIC SERIES ON r"% See^r,-es^s AA VWo~lU -114
NONLINEAR SCIENCE %
Series Editor: Leon 0. Chua
NOHLINERR DYNAMICS OF
INTERACTING POPULATIONS
Alexander D. Bazykin
institute of Mathematical Problems in Biology
Russian Academy of Sciences
edited by
Alexander I. Khibnik
Cornell university
Bernd Krauskopf
vrije universitelt Amsterdam
NTIFICS World Scientific
ARS
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Bazykin, A. D.
Nonlinear dynamics of interacting populations / Alexander D.
Bazykin ; edited by Alexander I. Khibnik, Bernd Krauskopf.
p. cm. — (World Scientific series on non-linear science, Series A ; vol. 22)
Includes bibliographical references.
ISBN 9810216858 (alk. paper)
1. Population biology - Mathematical models. 2. Biotic
communities - Mathematical models. 3. Bifurcation theory.
I. Khibnik, Alexander I. II. Krauskopf, Bernd.
III. Title. IV. Series: World Scientific series on nonlinear
science. Series A, Monographs and treatises ; v. 22.
QH352.B39 1998
577.8'8'0151-dc21 98-11768
CIP
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to be
invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc.,
222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the
publisher.
Printed in Singapore by Uto-Print
Contents
Foreword ix
Biography of A. D. Bazykin xiii
Preface xvii
1. Ideas and Methods of Modeling Populations 1
2. Dynamics of Isolated Populations 7
2.1. Free Population 7
2.2. Population with External Resources 11
2.3. Harvested Populations 14
3. Predator-Prey Interactions 18
3.1. Volterra's Model and its Modifications 18
3.2. Elementary Factors of Interactions 21
3.2.1. Predation 21
3.2.2. Reproduction and Mortality of the Predator 25
3.2.3. List of Elementary Factors 26
3.3. One-Factor Modifications of the Volterra Model 26
3.3.1. Nonlinear Reproduction, Competition and
Mortality of Prey 27
3.3.2. Predator Saturation (Type II Trophic Function) 28
3.3.3. Nonlinear Predation at Small Prey Population Density 30
3.3.4. Predator Competition for Prey and Other Resources 30
3.3.5. Nonlinear Reproduction of the Predator at
Small Population Densities 31
3.3.6. Classification of Elementary Factors 32
3.4. Two-Factor Modifications of The Volterra Model 32
3.4.1. Prey Competition and Predator Saturation 33
3.4.2. Prey Competition and Nonlinear Reproduction of
Prey at Small Population Densities 35
V
vi Contents
3.4.3. Nonlinear Predation at Small Prey Population Density
and Predator Saturation (Type III Trophic Function) 38
3.4.4. Predator Competition for Other Resources and
Predator Saturation 40
3.4.5. Predator Competition for Prey and Predator Saturation 46
3.4.6. Nonlinear Predator Reproduction and Prey Competition 50
3.4.7. Other Two-Factor Modifications 63
3.4.8. Lower Critical Prey Density 64
3.5. Three-Factor Modifications of the Volterra Model 66
3.5.1. Predator Saturation, Nonlinear Predation
(Type III Trophic Function) and Competition among Prey 66
3.5.2. Predator Saturation, Predator Competition for
Resources Other than Prey, and Competition among Prey 67
3.5.3. Predator Saturation, Predator Competition for
Prey and Competition among Prey 81
3.5.4. Prey Competition and Competition among Predators
for Resources Other than Prey (Type III Trophic Function) 84
3.5.5. Lower Critical Prey Density and Competition among Prey 85
Appendix 91
4. Competition and Symbiosis 101
4.1 Competition 101
4.1.1. Two Logistic Populations 101
4.1.2. One of the Populations has a Lower Threshold Size 103
4.1.3. Two Populations with Lower Threshold Sizes 105
4.2. Symbiosis 107
4.2.1. Protocooperation 107
4.2.2. Mutualism 111
Appendix 113
5. Local Systems of Three Populations 117
5.1. Classification of Trophic Structures 118
5.2. Competition-Free Communities 122
5.2.1. One-Predator-Two-Preys and Two-Predators-One-Prey 122
5.2.2. System of Three Trophic Levels 124
5.2.3. Cell of a Trophic Net 125
5.3. Competing Producers in a Three-Population Community
with Trophic Relations 129
5.3.1. Community of Three Trophic Levels 129
5.3.2. Two-Predators-One-Prey 130
5.3.3. Trophic Cell 132
5.3.4. One-Predator-Two-Preys 137
5.4. Lower Critical Density of the Producer in a System of
Three Trophic Levels 153
Contents vii
Appendix 160
6. DDi ssipative Structures in Predator-Prey Systems 166
6.1. Bilocal System 167
6.2. Annular Habitat 173
6.3. Evolutionary Appearance of Dissipative Structures 176
Appendix 182
Bibliography 183
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Foreword
This book could be called Mathematical ecology beyond the Lotka-Volterra model
and we might add: far beyond. In theoretical biology and applied mathematics, the
logistic and Lotka-Volterra models have long been considered as seminal examples
of modeling and dynamics. However, it was understood only recently how different
forms of regulatory mechanisms, like birth and death, competition, consumption
and the like, result in changes of the stability and dynamics of ecological systems.
The present book brings this understanding to the attention of a broad biological
and nonlinear dynamics audience. It does so with a deep and unique insight into
the mathematical richness of basic ecological models and how this richness emerges
as the number of competing mechanisms or factors (reflected in the number of
parameters, not state variables) increases.
The main topics of the book are:
• the dynamics of elementary ecological communities, consisting of two or three
trophic levels;
• the stabilizing and destabilizing role of various regulatory mechanisms deter
mining the outcome of ecological interaction within the community;
• "dangerous boundaries" for the stability of the ecosystem and criteria for
approaching them.
• mechanisms of spatial inhomogeneity and their relationship to non-equilibrium
dynamics of ecosystems.
The strength of this book is that it systematically builds a sequence of well-
motivated ecological models of increasing difficulty and classifies them with methods
from bifurcation theory. To this end, the author emphasizes the use of higher order
degeneracies. This makes this book quite unique and interesting not only for a
biological audience, but also for the applied dynamical systems community. In fact,
this text can be used as a guided tour to bifurcation theory from the applied point
of view. The interested reader will find a wealth of intriguing examples of how
known bifurcations occur in (biological) applications.
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