MANAGEMENT AND ANALYSIS OF BIOLOGICAL POPULATIONS
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1 ',' Developments in Agricultural and Managed-Forest Ecology, 8 MANAGEMENT AND ANALYSIS OF BIOLOGICAL POPULATIONS by BEAN-SAN GOH Department ofmathematics, University of Western Australia, Nedlands, W.A. 6009, Australia ELSEVIER SCIENTIFIC PUBLISHING COMPANY Amsterdam - Oxfurd - New York 1980
2 ELSEVIER SCIENTIFIC PUBLISHING COMPANY 335 Jan van Galenstraat P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER NORTH-HOLLAND INC. 52, Vanderbilt Avenue New York, N.Y Library cf Congress Cataloging in Publication Data Goh, Bean-8an. Management and an&l.ysis of biological populations.. I (Developments in agricultural and managed-forest ~cology ; 8) Bibliography: p. Includes indexes. 1. Population biology--mathematical models. 2. Wildlife management--mathems.tical models. 1. Title. II. Series. QH352.G63 ISBN ' ISBN (Vol. 8) ISBN (Series) Elsevier Scientific Publishing Company, 1980 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Scientific Publishing Company, P.O. Box 330, 1000 AH Amsterdam, The Netherlands Peinted in The Netherlands
3 CONTENTS Preface.. ix Chapter 1. Introduction Models in ecology Stability concepts in ecology Ecological engineering Liapunov functions Differential equations and difference equations in ecosystem modelling Selected references Chapter 2. Optimization techniques Introduction Static optimization problems Relaxed optimization problem method Multiple objectives decision problems Continuous time optimal control Discrete time optimal control Selected references. 43 Chapter 3. Stability and nonvulnerability Introduction Local stability Finite and global stability , Single-species models '" Exploited single-species models. " Models with unspecified parameters Two-species Lotka-V01 terra models Gilpin and Ayala's competition model Prey-predator system with Type 2 functional response Nonlinear two-species models Region of ultimate confinement Nonvulnerability Discrete time models: local stability Finite and global stability in discrete time models Region of ultimate confinement and nonvulnerability A discrete time model of two competing species..118 Selected references Chapter 4. Fisheries Introduction The logistic model Optimal control of the logistic model Stability of bionomic equilibrium Stock recruitment model Stability of a stock recruitment model Global stability of the Ricker model The Beverton-Holt model: Optimal size limit Optimal control of the Beverton-Holt model Fishery with a limited fishing season Fishery with delayed recruitment..158
4 4.12. The Antarctic fin whale population Harvesting a multispecies community Selected references Chapter 5. Complex ecosystems Introduction Mathematical relationships between stability and complexity Stability of nonlinear multispecies models Stability of multispecies Lotka-Volterra models Stability of a class of nonlinear models Connective stability Nonvulnerability of nonlinear multispecies models Sector stability Population models with time delays Spatial heterogeneity in population dynamics 216 Selected references 224 Chapter 6. Management ofpests and greenhouse crops Introduction Some applications of stability concepts in pest control Graphical analysis of a control two-species system... "., Optimal control of a prey-predator system Optimal control of epidemics for pest coctrol Optimal management of greenhouse crops..262 Selected references..265 References Author Index..281 Subject Index
5 PREFACE In the real world, an ecosystem is subjected to large perturbations of its initial state and continual disturbances on its dynamics. Therefore it is important to establish that an ecosystem model is stable relative to finite perturbations of its initial state and its dynamics. One object of this book is to demonstrate that Liapunov and Liapunov-like functions can be successfully used to establish, in a nonlinear population model, stability relative to realistic perturbations. It is shown that a complex ecosystem is stable relative to realistic perturbations if it is a collection of self-regulating species such that the sum total of the stabilizing intraspecific interactions dominates the interspecific interactions. A nontrivial criterion for a complex ecosystem model to have this property is given. For a complex ecosystem model to satisfy this criterion, it is desirable that it has many null interactions between pairs of species. Another object of this book is to demonstrate the usefulness of optimal control theory in the management of biological populations. Management policies should always be implemented in a feedback manner. This would reduce some of the adverse effects of uncertainties in an ecosystem in the real world. After all, Nature itself uses feedback to cope with uncertainties in many processes. Chapter 2 contains a brief survey of static optimization techniques and optimal control theory for systems which are modelled by differential and difference equations. Chapter 3 contains methods which use Liapunov and Liapunov-like functions to establish that a given population model is stable relative to finite perturbations of its initial state and that it is nonvulnerable relative to large continual disturbances. These methods are applied to single species and two-species populations which are modelled by differential and difference equations. Fisheries provide some of the best examples of the usefulness of mathematics in the management of biological populations. A few management problems in fisheries are examined in Chapter 4. They utilize stability and optimal control theories. Chapter 5 considers stability in an ecosystem model with complexities due to species richness, nonlinearities, time delays and spatial heterogeneity. The very in teresting and important problem of pest management is considered in Chapter 6. I have greatly benefited from discussions with T.T. Agnew, C.W. Clark, M.E. Fisher, G.W. Harrison, A. Hastings, C.S. Hsu, L.S. Jennings, G. Leitmann, D. Ludwig, T.L. Vincent and C.J. Walters. Many of the results in this book were obtained in collaboration with Tom Vincent. This book was completed during my sabbatical leave which was supported by NRC grant No I thank Colin Clark and members of the Mathematics Department of the University of British Columbia for their hospitality.
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