UK Nonlinear News , February 1997.

Monotone Dynamical Systems: An introduction to the theory of Competitive and Cooperative Systems

Hal L. Smith

Mathematical Surveys and Monographs Vol 41

American Mathematical Society, ISBN 0-8218-0393-X, 1995

Reviewed by D. Knapp

Introductory courses in autonomous systems of ordinary differential equations (continuous dynamical systems) prove that flows in one dimension eventually reach steady states or become unbounded. Two dimensional flows, if bounded, can only approach periodic invariant sets (the Poincare Bendixson theorem). Flows in higher dimensions may possess exotic invariant sets, strange sets with chaotic dynamics are of particular interest.

Restrictions on the form of the equations may modify the dynamical modes which are possible. Gradient systems are constrained to flow along the local potential gradient and periodic attracting flows are not possible. Over many years the study of particular systems in chemistry and biology have revealed another broad category of systems with constrained dynamics, those described by monotone dynamical systems. The operator defined by $\dot{x} = F(x)$ is monotone if it has the following property. Given two vectors x and y we say y leads x if y-x has components which are positive. Thus y-x points into the positive "octant". A sufficient condition for F to be monotone is that F(y) leads F(x) when y leads x. This generalises the idea of an increasing function to higher dimensions.

Three dimensional systems may, if monotone, be subject to the Poincare Bendixson theorem. In other conditions it is possible to prove that the flow must approach a steady state. Such global properties are difficult to establish in other ways and allow the investigator of a system to be confident that all the dynamics are understood.

This monograph presents a very clear and well organised account of these matters with their development to partial differential equations and delay differential equations. Each chapter opens with brief description of its content and ends with references for further study. The pace is brisk, results are stated clearly but the arguments require close attention. This is an admirable text which demonstrates that mathematical study can provide real insight and will reward those who study it. The mathematical theory is presented and then applied to non trivial systems from chemistry and biology.

D. Knapp, Department of Applied Mathematics, Leeds University.

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