Mathematical Methods Of Population Biology

CAMBRIDGE STUDIES IN MATHEMATICAL BIOLOGY: 4 Editors C. CANNINGS Department of Probability and Statistics, University of Sheffield F. HOPPENSTEADT Department of Mathematics, University of Utah MATHEMATICAL METHODS OF POPULATION BIOLOGY FRANK C. HOPPENSTEADT Professor of Mathematics, University of Utah Mathematical methods of population biology CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia © Cambridge University Press 1982 First published 1982 Library of Congress Cataloging in Publication Data Hoppensteadt, F. C. Mathematical methods of population biology. (Cambridge studies in mathematical biology ; 4) Bibliography: p. Includes indexes. 1. Population biology - Mathematical models. 1. Title. II. Series. [DNLM: 1. Biometry. 2. Models, Biological. 3. Population. QH 323.5 H798m] QH352.H66 1981 574.5'248'0724 81-9977 ISBN 0 521 23846 3 hard covers ISBN 0 521 28256 X paperback Transferred to digital printing 2004 AACR2 CONTENTS Preface page v 1 Population dynamics 1.1 Iteration and parameter estimation 1.2 Synchronization of populations 1.3 Exploitation of biological populations: fisheries Appendix: Least-squares approximation 1 1 9 16 28 2 Renewal theory and reproduction matrices 2.1 Fibonacci sequence 2.2 Renewal equation 2.3 Reproduction matrix: honest matrices 2.4 Population waves: dishonest matrices 31 31 37 40 43 3 3.1 3.2 3.3 3.4 47 47 60 69 72 Markov chains Bacterial genetics Human genetics: Mendelian traits in diploid organisms Contagion Summary 4 Perturbation methods 4.1 Approximations to dynamic processes: multiple-time-scale methods 4.2 Static states: bifurcation methods 5 Dispersal processes 5.1 Integer dispersal 5.2 Diffusion approximations 5.3 Transform methods: linear stability theory for diffusion equations 5.4 Pattern formation 5.5 Wave propagation: dynamic patterns Appendix: Derivation of the diffusion approximation Solutions to selected exercises References Author index Subject index 75 75 82 92 92 95 97 101 114 128 135 142 145 147 PREFACE This book is intended as an introduction to methods that are useful for studying population phenomena. The models are presented in terms of difference equations. Experience has shown that this approach facilitates communicating the derivation of models and statements of results about them to scientists who do not have a strong mathematical background. However, in most cases of difference equations the mathematician must exert greater analytical effort because many of the features of calculus are not available in this setting. Important models that do involve extensive use of calculus are presented in exercises. In most cases, the exercises are fronts for presenting models more detailed than those derived and studied in the text. The material is graded in terms of mathematical difficulty. The earlier chapters involve elementary difference equations, and later chapters involve topics requiring more mathematical preparation. First, models of total population and population age structure are derived and studied. Next, models of random population events are presented in terms of Markov chains. The final two chapters deal with mathematical methods used to uncover qualitative behavior of more complicated difference equations. For example, the material on geographically distributed populations eventually involves nonlinear diffusion equations. In each case, the chapter begins with a simple model, usually of some historical interest, that motivates the prima