Global Behavior of Nonlinear Difference Equations of Higher Order with Applications

Global Behavior of Nonlinear Difference Equations of Higher Order with Applications

Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 256

Global Behavior of Nonlinear Difference Equations of Higher Order with Applications by V.L. Kocic and G. Ladas Department of Mathematics, University of Rhode Island, Kingston, Rhode Island, U.S.A.... " SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data KOClc, V. L., 1953- Global behavior of nonlinear difference equations of higher order with applications I by V.L. Koclc and G. Ladas. p. c~. -- (MathematICs and Its applications; v. 256) Includes blbllographlcal references and lndexes. ISBN 978-90-481-4273-6 ISBN 978-94-017-1703-8 (ebook) 00110.1007/978-94-017-1703-8 1. NonlInear difference equatl0ns--numerlcal solutions. I. Ladas. G. E. II. TItle. III. SerIes' MathematICs and Its appllcatlons (Kluwer Academlc PublIshers) ; v. 256. OA431. K66 1993 515.625--dc20 93-7605 ISBN 978-90-481-4273-6 Printed on acid-free paper All Rights Reserved 1993 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Contents Preface Acknowledgements ix xi 1 Introduction and Preliminaries 1 1.1 An Overview... 1 1.2 Oscillation and General Background. 4 1.3 Stability and General Background.. 8 1.4 Periodicity and General Background 15 1.5 Global Attractivity for First-Order Equations 17 1.6 Some Basic Lemmas... 19 1. 7 Some Useful Theorems from Analysis 24 1.8 Notes... 25 2 Global Stability Results 27 2.1 Global Asymptotic Stability of Xn+l = xnf(xn, xn-d 27 2.2 Permanence of Xn+l = xnf(xn, Xn-k 1,, Xn-k r ) 35 2.3 Global Attractivity of Xn+I = xnf(xn, Xn-kll"" Xn-k r ) 39 2.4 Global Attractivity of Xn+I = O:Xn + F(Xn-k)...... 46 2.5 Global Stability of Xn+I = L:7=o aixn-i + (1 - A)F(L:;':o b,xn_i) 48 2.6 Notes... 53 3 Rational Recursive Sequences 55 3.1 The Rational Recursive Sequence Xn+I = (a + bxn)/(l + L:7=o aixn_;) 56 3.2 The Rational Recursive Sequence Xn+l = (a + L:7=o aixn-i)/(b + L:7=o bixn-i)... 59 3.3 Global Stability of Xn+l = l/(l:~o bixn-i)... 64 3.4 Global Stability of Xn+l = (a + bxn)/(a + Xn-k) 67 3.5 Notes... 74

vi 4 Applications 4.1 A Discrete Delay Logistic Model.... 4.2 A Simple Genotype Selection Model........ 4.3 Periodicity in a Simple Genotype Selection Model 4.4 A Model of the Spread of an Epidemic...... 4.5 Nicholson's Blowflies.... 4.6 A Discrete Analogue of a Model of Haematopoiesis 4.7 A Discrete Baleen Whale Model........... 4.8 A Semidiscretization of a Delay Logistic Model... 4.9 A Discrete Analogue of the Emden-Fowler Equation 4.10 Notes... 5 Periodic Cycles 5.1 An Invariance for X n+l = a+xn;~~::~_k±2. 5.2 The Five-Cycle: X n +2 = l+xn±! Xn.. 5.3 The Eight-Cycle: X n+3 = l+xn±:n+xn±! 5.4 Symmetric Periodic Sequences 5.5 Notes.... 6 Open Problems and Conjectures 6.1 The Rational Recursive Sequence Xn+l = (a + bxn)/(a + Xn-l) 6.2 The Rational Recursive Sequence Xn+l = (a + bx~)/(c + x~_i) 6.3 A Model for an Annual Plant.... Th D. f A -2-1/2 6.4 e ynamlcs 0 X n+l = Xn + x n _ 1 6.5 A Discrete Model with Quadratic Nonlinearity.... 6.6 A Logistic Equation with Piecewise Constant Argument 6.7 Discrete Epidemic Models.............. 6.8 Volterra Difference Equations.... 6.9 Global Attractivity of Xn+l - Xn + pxn-k = f(xn-m) 6.10 Neural Networks.... 6.11 The Fibonacci Sequence Modulo 7r 6.12 Notes. Appendix A The Riccati Difference Equation B A Generalized Contraction Principle Contents 75 75 80 87 99 105 III 119 123 125 132 133 135 138 141 144 152 153 153 155 159 162 166 168 169 169 170 172 174 176 177 177 189

C Global Behavior of Systems of Nonlinear Difference Equations 195 C.1 A Discrete Epidemic Model. 195 C.2 A Plant-Herbivore System.. 197 C.3 Discrete Competitive Systems 199 Bibliography 205 Subject Index 223 Author Index 225

PREFACE Nonlinear difference equations of order greater than one are of paramount importance in applications where the (n + 1)st generation (or state) of the system depends on the previous k generations (or states). Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering and economics. Our aim in this monograph is to initiate a systematic study of the global behavior of solutions of nonlinear scalar difference equations of order greater than one. Our primary concern is to study the global asymptotic stability of the equilibrium solution. We are also interested in whether the solutions are bounded away from zero and infinity, in the description of the semi cycles of the solutions, and in the existence of periodic solutions. This monograph contains some recent important developments in this area together with some applications to mathematical biology. Our intention is to expose the reader to the frontiers of the subject and to formulate some important open problems that require our immediate attention. There are many books in the literature dealing with the theory of linear difference equations and with the local stability of nonlinear equations. There are also several books dealing with one dimensional dynamical systems. However, there are no books dealing systematically with the global behavior of solutions of nonlinear difference equations of order greater than one. It is our hope that this book will stimulate interest among mathematicians to develop further basic results on the global behavior of such equations. This book may also encourage people in applications to develop more realistic models which involve nonlinear difference equations of order greater than one. With the use of a computer one can easily experiment with difference equations and one can easily discover that such equations possess fascinating properties with a great deal of structure and regularity. Of course all computer observations and predictions must also be proven analytically. Therefore this is a fertile area of research, still in its infancy, with deep and important results. Chapter 1 contains some basic definitions and results which are used throughout the book. In this sense, this is a self-contained monograph and the main prerequisite that the reader needs to understand the material in this book and to be able to attack the open problems and conjectures is a good foundation in analysis. In Chapter 2 we present some general results on the global asymptotic stability and global attractivity of quite general nonlinear difference equations of order greater than one. ix

x Preface Our goal in Chapter 3 is to gain some understanding of the dynamics of rational recursive sequences. Chapter 4 deals with the global asymptotic stability and the oscillatory character of some discrete models, as well as some discrete analogues of continuous models taken from Mathematical Biology and Physics. Chapter 5 deals with periodic cycles and with the general properties of rational recursive sequences with one term in the denominator. In Chapter 6 we present some conjectures and open problems about some interesting types of difference equations. The appendices contain material which although not in the mainstream of scalar nonlinear difference equations of higher order, are presented either for the sake of completeness or with the hope that they may inspire some useful generalizations. At the end of every chapter we have included some notes and references about the material presented. In addition to Chapter 6 which is entirely devoted to conjectures and open problems, in Chapters 2-6 we have included numerous research projects which, we hope, will stimulate a lot of interest and enthusiasm towards the development of a general theory with realistic applications. Kingston, 1993 v. 1. Kocic G. Ladas

ACKNOWLEDGEMENTS This monograph is the outgrowth of lecture notes and seminars which were given at the University of Rhode Island during the last five years. We are thankful to Professors R. D. Driver, E. A. Grove, J. Hoag, A. Ivanov, J. Jaros, G. A. Kamenskii, R. Levins, J. Schinas, S. Schultz, Z. Wang, and S. Zhang, and to our graduate students M. Arciero, W. Briden, E. Camollzis, R. C. DeVault, J. H. Jaroma, C. Kent, S. Kufuklis, C. Qian, I. W. Rodrigues, and P. N. Vlahos for their enthusiastic participation and for offering useful suggestions which helped to improve the exposition. Special thanks are due to Professor E. A. Grove for proofreading in great detail the entire manuscript. Finally we wish to express our warmest thanks to Dr. David Larner, head of Science and Technology Division of Kluwer Academic Publishers, for his enthusiastic support of our work. xi