You are here

Dynamical Systems and Population Persistence

Hal L. Smith and Horst R. Thieme
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 118
[Reviewed by
William J. Satzer
, on

The mathematical theory of persistence deals with the life and death of species. In particular, it is aimed at determining, based on mathematical models, which among interacting species will survive over the long term. For example, in a viral epidemic will the virus drive the host species to extinction or will the host persist? When is it possible for the virus to remain endemic in the host population?

Persistence theory offers a mathematically rigorous way to answer questions of persistence by determining positive lower bounds for the long-term component of a dynamical system such as population size or prevalence of a disease. Furthermore, such lower bounds are independent of initial conditions. The key tool is the persistence function, a kind of measure of the distance from a state of the dynamical system to the boundary of the state space. In broad terms the persistence function measures how far a species is from the brink of extinction.

This book is essentially a monograph written for graduate students in mathematics. It provides the first self-contained and integrated treatment of dynamical systems, the theory of persistence, and applications. Although it reviews basic elements of dynamical systems theory, students would be advised to complete a graduate level course in ordinary differential equations first, before tackling the material here. The text has a hint of an applied flavor (there are, for example, chapters on microbial growth in a bioreactor, a model of variable infectivity, and dividing cells in a chemostat), but the focus is entirely mathematical. It is a solid and comprehensive treatment of the subject aimed at mathematically sophisticated readers. However, I suspect that the text would be largely incomprehensible to most biologists.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

  • Introduction
  • Semiflows on metric spaces
  • Compact attractors
  • Uniform weak persistence
  • Uniform persistence
  • The interplay of attractors, repellers, and persistence
  • Existence of nontrivial fixed points via persistence
  • Nonlinear matrix models: Main act
  • Topological approaches to persistence
  • An SI endemic model with variable infectivity
  • Semiflows induced by semilinear Cauchy problems
  • Microbial growth in a tubular bioreactor
  • Dividing cells in a chemostat
  • Persistence for nonautonomous dynamical systems
  • Forced persistence in linear Cauchy problems
  • Persistence via average Lyapunov functions
  • Tools from analysis and differential equations
  • Tools from functional analysis and integral equations
  • Bibliography
  • Index