`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.

email: amt6dk@amsta.leeds.ac.uk.

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