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Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems
Hal L. Smith, Arizona State University, Tempe

Mathematical Surveys and Monographs
1995; 174 pp; softcover
Volume: 41
ISBN-10: 0-8218-4487-3
ISBN-13: 978-0-8218-4487-8
List Price: US$60
Member Price: US$48
Order Code: SURV/41.S
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See also:

Dynamical Systems and Population Persistence - Hal L Smith and Horst R Thieme

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.


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


"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

  • Monotone dynamical systems
  • Stability and convergence
  • Competitive and cooperative differential equations
  • Irreducible cooperative systems
  • Cooperative systems of delay differential equations
  • Nonquasimonotone delay differential equations
  • Quasimonotone systems of parabolic equations
  • A competition model
  • Appendix
  • Bibliography
  • Index
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