**Mathematical Surveys and Monographs**

Volume: 41;
1995;
174 pp;
Softcover

MSC: Primary 34; 35;
Secondary 44; 53; 58

**Print ISBN: 978-0-8218-4487-8
Product Code: SURV/41.S**

List Price: $64.00

AMS Member Price: $51.20

MAA Member Price: $57.60

**Electronic ISBN: 978-1-4704-1272-2
Product Code: SURV/41.S.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

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# Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems

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*Hal L. Smith*

This book presents the first comprehensive treatment of a rapidly developing area with many potential applications: the theory of monotone dynamical systems and the theory of competitive and cooperative differential equations. The primary aim is to provide potential users of the theory with techniques, results, and ideas useful in applications, while at the same time providing rigorous proofs. The main result of the first two chapters, which treat continuous-time monotone dynamical systems, is that the generic orbit converges to an equilibrium. The next two chapters deal with autonomous, competitive and cooperative, ordinary differential equations: every solution in the plane has eventually monotone components, and the Poincaré-Bendixson theory in three dimensions is discussed. Two chapters examine quasimonotone and nonquasimonotone delay differential equations, and the book closes with a discussion of applications to quasimonotone systems of reaction-diffusion type. Throughout, Smith discusses applications of the theory to many mathematical models arising in biology. An extensive guide to the literature is provided at the end of each chapter. Requiring a background in dynamical systems at the level of a first graduate course, this book would be suitable as a graduate text for a topics course.

#### Readership

Students and researchers in dynamical systems theory, applied mathematicians, and scientists.

#### Reviews & Endorsements

Written in a clear and attractive style, and errors are infrequent. Complete proofs are provided… The exposition is … at a level which should be accessible to advanced graduate students … the content is sufficiently up to date and substantial that even experts are likely to gain new insights. This book will be useful to students and researchers in dynamical systems, differential equations, and mathematical biology.

-- Bulletin of the AMS

The book … written by one of the leading experts in the field, provides a comprehensive and lucid introduction to the theory of monotone dynamical systems with continuous time … each chapter is complemented by examples illustrating the application of the theory to biological problems … highly recommended as a graduate text as well as a reference for researchers working both in the theory and in applications of monotone dynamical systems.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems

- Contents v6 free
- Preface vii8 free
- Chapter 1. Monotone Dynamical Systems 112 free
- Chapter 2. Stability and Convergence 1526
- Chapter 3. Competitive and Cooperative Differential Equations 3142
- Chapter 4. Irreducible Cooperative Systems 5566
- Chapter 5. Cooperative Systems of Delay Differential Equations 7586
- 0. Introduction 7586
- 1. The Quasimonotone condition 7889
- 2. Positively Invariant Sets, Monotone Solutions, and Contracting Rectangles 8192
- 3. Eventual Strong Monotonicity 8596
- 4. Generic Convergence for Cooperative and Irreducible Systems 89100
- 5. Stability of Equilibria 91102
- 6. A Biochemical Control Circuit with Delays 93104
- 7. Competition with Time Delays 94105
- 8. Remarks and Discussion 98109

- Chapter 6. Nonquasimonotone Delay Differential Equations 101112
- Chapter 7. Quasimonotone Systems of Parabolic Equations 119130
- 0. Introduction 119130
- 1. Parabolic Systems: The Basic Setup 120131
- 2. Maximum Principles 123134
- 3. Positively Invariant Sets, Comparison and Monotonicity 127138
- 4. The Strong Order Preserving Property 132143
- 5. Generic Convergence for Cooperative and Irreducible Systems 134145
- 6. Stability of Equilibria 136147
- 7. The Biochemical Control Circuit with Diffusion 139150
- 8. Remarks and Discussion 141152

- Chapter 8. A Competition Model 145156
- Appendix. Chain Recurrence 163174
- Bibliography 167178
- Index 173184