**Graduate Studies in Mathematics**

Volume: 118;
2011;
405 pp;
Hardcover

MSC: Primary 37; 92;

Print ISBN: 978-0-8218-4945-3

Product Code: GSM/118

List Price: $79.00

Individual Member Price: $63.20

**Electronic ISBN: 978-1-4704-1180-0
Product Code: GSM/118.E**

List Price: $79.00

Individual Member Price: $63.20

#### You may also like

#### Supplemental Materials

# Dynamical Systems and Population Persistence

Share this page
*Hal L. Smith; Horst R. Thieme*

The mathematical theory of persistence answers questions such as which
species, in a mathematical model of interacting species, will survive over
the long term. It applies to infinite-dimensional as well as to
finite-dimensional dynamical systems, and to discrete-time as well as to
continuous-time semiflows.

This monograph provides a self-contained treatment of persistence theory
that is accessible to graduate students. The key results for deterministic
autonomous systems are proved in full detail such as the acyclicity
theorem and the tripartition of a global compact attractor. Suitable
conditions are given for persistence to imply strong persistence even for
nonautonomous semiflows, and time-heterogeneous persistence results are
developed using so-called “average Lyapunov functions”.

Applications play a large role in the monograph from the
beginning. These include ODE models such as an SEIRS infectious
disease in a meta-population and discrete-time nonlinear matrix models
of demographic dynamics. Entire chapters are devoted to
infinite-dimensional examples including an SI epidemic model with
variable infectivity, microbial growth in a tubular bioreactor, and
an age-structured model of cells growing in a chemostat.

#### Table of Contents

# Table of Contents

## Dynamical Systems and Population Persistence

- Cover Cover11 free
- Title page i2 free
- Contents v6 free
- Preface ix10 free
- Introduction 120 free
- Semiflows on metric spaces 928 free
- Compact attractors 2948
- Uniform weak persistence 6180
- Uniform persistence 87106
- The interplay of attractors, repellers, and persistence 125144
- Existence of nontrivial fixed points via persistence 157176
- Nonlinear matrix models: Main act 163182
- Topological approaches to persistence 177196
- An SI endemic model with variable infectivity 231250
- Semiflows induced by semilinear Cauchy problems 261280
- Microbial growth in a tubular bioreactor 283302
- Dividing cells in a chemostat 307326
- Persistence for nonautonomous dynamical systems 327346
- Forced persistence in linear Cauchy problems 341360
- Persistence via average Lyapunov functions 349368
- Tools from analysis and differential equations 363382
- Tools from functional analysis and integral equations 377396
- Bibliography 391410
- Index 403422 free
- Back Cover Back Cover1426

#### Readership

Graduate students and research mathematicians interested in dynamical systems and mathematical biology.