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

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 Poincare-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.