Preface

TY - JOUR

T1 - Preface

AU - Kuang, Yang

N1 - Funding Information: As always, it is impossible to cover all of the relevant results in one book. Naturally, the selection of the material is largely influenced by my personal interests. In fact, the content of this book is predominantly of my own and my collaborator’s recent work, excluding those well-known basic results on DDEs. The basic mathematical prerequisite for most part of this book is a thorough understanding of Hirsch and Smale’s book (1974) Differential Equations, Dynamical Systems, and Linear Algebra, published by Academic Press, or equivalent texts (such as Waltman’s book (1986) A Second Course in Elementary Differential Equations, also published by Academic Press). The book of Freedman (1980), Deterministic Mathematical Mod- els in Population Ecology, published by Marcel Dekker, may serve as an excellent mathematical ecology background reference. Other than these, this book is largely self-contained. It thus should be useful for those who are interested in learning or applying the theory of delay differential equations in their studies regardless of their specific subject areas. In addition, this book can be used as text for graduate courses and seminars in applied mat hematics programs. In this book, the global analysis of the full nonlinear equations or systems are emphasized. Both autonomous and nonautonomous systems with various delays are treated. Specifically, I am interested in the possible influence of delays on the dynamics of the system, including such topics as stability switching for increasing delays, the long time coexistence of populations, and oscillatory aspects (such as the existence of periodic solutions and chaotic behavior) of the dynamics. It is hoped and expected that the analyses presented in this monograph will also be useful in the study of other types of DDEs in applications outside of the population dynamics area. A brief description of the organization of the book is as follows. The book is divided into two main parts encompassing nine chapters. The first three chapters constitute Part I, Delay Differential Equations. In the first chapter, various examples of delay differential equations arising in real life problems are presented. Naturally, the emphasis here is the delay models that have previously appeared in the population dynamics literature. The possible effects of time delays on some of these systems are briefly discussed. In particular, I would like to point out here that, contrary to intuition, small delays can have large influences. Chapter 2 contains the basic theory of DDEs, which is a small subset of J. Hale’s book (1977) Theory of Functional Differential Equations. The local stability of a steady state in a DDE is determined by the locations of the roots of the corresponding characteristic equation. Such a characteristic equation is generally transcendental. In Chapter 3, I present several methods that are commonly used by researchers in their studies of various characteristic equations. These methods cover both discrete and distributed delays. Part I1 consists of the remaining six chapters. It deals with the analyses of various delayed population models. Chapter 4 focuses on the global stability analyses of positive steady states. Chapter 5 mainly concerns the oscillatory aspects of the dynamics. Other kinds of single species models are also discussed. Chapter 6 contains the global stability analyses of some multi-species models. Chapter 7 covers the oscillatory aspects of the dynamics in these models. Global existence of periodic solutions in both autonomous and periodic systems are established. Chapter 8 presents some recent results on permanence theory of delayed systems. Chapter 9 documents some initial attempts on the study of several neutral delay models. The book ends with an extensive bibliography and an effective algorithm written by Mr. Lo and Professor Jackiewicz at ASU for performing simulations of any delayed models. This book uses double enumeration for theorems, expressions, and so on, within the same chapter and triple enumeration when they are referred to in other chapters. It is impossible to thank everyone who has helped me over the years to understand differential equations and mathematical biology. However, two people must be singled out. I am forever grateful to my thesis advisor Herbert I. Freedman for his constant encouragement and invaluable suggestions, to my colleague and close collaborator Hal L. Smith for numerous discussions that produced many results, some of which are included here. Moreover, the presentation of this book has been improved considerably due to their many detailed and constructive comments. I would also like to thank my students and collaborators Baorong Tang and Tao Zhao for many fruitful interactions inside and outside classrooms that yielded some results presented in this book. I am deeply indebted to Edisanter Lo for performing simulations for me on several occasions and the writing of the appendix. My deepest thanks are due to Linda Arneson for her masterful job of typing most of my research papers and this manuscript. Special thanks are also due to Bruce Long for his work on all the drawings and to the Department of Mathematics at ASU and the National Science Foundation for their support of my research. I should also mention here that it was a great pleasure to work with the professional staff of Academic Press, especially Vice President and Publisher David F. Pallai, Executive Editor in Mathematics Charles B. Glaser, and Production Editor Nancy Priest. Copyright: Copyright 2010 Elsevier B.V., All rights reserved.

PY - 1993

Y1 - 1993

UR - http://www.scopus.com/inward/record.url?scp=77956864594&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956864594&partnerID=8YFLogxK

U2 - 10.1016/S0076-5392(08)62861-1

DO - 10.1016/S0076-5392(08)62861-1

M3 - Editorial

AN - SCOPUS:77956864594

SN - 0076-5392

VL - 191

SP - ix-xii

JO - Mathematics in Science and Engineering

JF - Mathematics in Science and Engineering

IS - C

ER -