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.
Readership: Graduate students and research mathematicians interested in dynamical systems and mathematical biology.
Author(s): Hal L. Smith, Horst R. Thieme
Series: Graduate Studies in Mathematics 118
Publisher: American Mathematical Society
Year: 2010
Language: English
Pages: xviii+405
Preface
Introduction
From uniform weak to uniform persistence
How to get uniform weak persistence.
Chapter 1 Semiflows on Metric Spaces
1.1. Metric spaces
1.2. Semiflows
1.3. Invariant sets
1.4. Exercises
Chapter 2 Compact Attractors
2.1. Compact attractors of individual sets
2.2. Compact attractors of classes of sets
2.2.1. Compact attractors of compact sets.
2.2.2. Compact attractors of neighborhoods of compact sets
2.2.3. Compact attractors of bounded sets.
2.2.4. Elementary examples.
2.2.5. Compact attractors and stability.
2.3. A sufficient condition for asymptotic smoothness
2.4. a-limit sets of total trajectories
2.5. Invariant sets identified through Lyapunov functions
2.6. Discrete semiflows induced by weak contractions
2.7. Exercises
Chapter 3 Uniform Weak Persistence
3.1. Persistence definitions
3.1.1. An SI endemic model for a fertility reducing infectious disease.
3.2. An SEIRS epidemic model in patchy host populations
3.2.1. Stability of the disease-free state.
3.2.2. Weak uniform persistence of the disease.
3.3. Nonlinear matrix models: Prolog
3.3.1. Stability of the extinction equilibrium
3.3.2. Uniform weak persistence
3.4. The May-Leonard example of cyclic competition
3.5. Exercises
Chapter 4 Uniform Persistence
4.1. From uniform weak to uniform persistence
4.1.1. A persistence result for general time-sets.
4.1.2. Application to the SEIRS epidemic model in a patchy environment.
4.2. From uniform weak to uniform persistence: Discrete case
4.3. Application to a metered endemic model of SIR type
4.3.1. Uniform persistence of the host.
4.3.2. Uniform weak persistence of the parasite
4.4. From uniform weak to uniform persistence for time-set R+
4.5. Persistence a la Baron von Münchhausen
4.5.1. Uniform parasite persistence in the SI model with fertility reduction.
4.5.2. Uniform parasite persistence in the metered SIRS model.
4.5.3. Incorporating an exposed class into the metered endemic model.
4.6. Navigating between alternative persistence functions
4.6.1. The SEIRS epidemic model for patchy host populations revisited.
4.7. A fertility reducing endemic with two stages of infection
4.7.1. The model.
4.7.2. Endemic equilibrium and its stability.
4.7.3. Reformulation of the model.
4.7.4. Persistence of the host.
4.7.5. Persistence of the disease.
4.7.6. Uniform eventual boundedness of the host
4.7.7. Persistence of the susceptible and first-stage infected part of the host population
4.7.8. A compact attractor of points.
4.8. Exercises
Chapter 5 The Interplay of Attractors, Repellers, and Persistence
5.1. An attractor of points facilitates persistence
5.2. Partition of the global attractor under uniform persistence
5.2.1. Persistence a la Caesar
5.2.2. An elementary example: scalar difference equations
5.3. Repellers and dual attractors
5.4. The cyclic competition model of May and Leonard revisited
5.5. Attractors at the brink of extinction
5.6. An attractor under two persistence functions
5.7. Persistence of bacteria and phages in a chemostat
5.8. Exercises
Chapter 6 Existence of Nontrivial Fixed Points via Persistence
6.1. Nontrivial fixed points in the global compact attractor
6.2. Periodic solutions of the Lotka-Volterra predator-prey model
6.3. Exercises
Chapter 7 Nonlinear Matrix Models: Main Act
7.1. Forward invariant balls and compact attractors of bounded sets
7.2. Existence of nontrivial fixed points
7.3. Uniform persistence and persistence attractors
7.4. Stage persistence
7.5. Exercises
Chapter 8 Topological Approaches to Persistence
8.1. Attractors and repellers
8.2. Chain transitivity and the Butler-McGehee lemma
8.3. Acyclicity implies uniform weak persistence
8.4. Uniform persistence in a food chain
8.5. The metered endemic model revisited
8.6. Nonlinear matrix models (epilog): Biennials
8.6.1. A generalized Beverton-Holt model.
8.6.2. A simple Ricker type model.
8.7. An endemic with vaccination and temporary immunity
8.7.1. Disease persistence.
8.7.2. Description of the global compact attractor
8.8. Lyapunov exponents and persistence for ODEs and maps
8.8.1. Co-cycle over a compact boundary invariant set.
8.8.2. Normal Lyapunov exponents
8.8.3. Uniformly weakly repelling sets via Lyapunov exponents.
8.8.4. Host-parasite model
8.9. Exercises
Chapter 9 An SI Endemic Model with Variable Infectivity
9.1. The model
9.1.1. Reformulation in the spirit of Lotka
9.1.2. Existence and boundedness of solutions
9.2. Host persistence and disease extinction
9.3. Uniform weak disease persistence
9.4. The semiflow
9.5. Existence of a global compact attractor
9.6. Uniform disease persistence
9.7. Disease extinction and the disease-free equilibrium
9.8. The endemic equilibrium
9.9. Persistence as a crossroad to global stability
9.10. Measure-valued distributions of infection-age
Chapter 10 Semiflows Induced by Semilinear Cauchy Problems
10.1. Classical, integral, and mild solutions
10.2. Semiflow via Lipschitz condition and contraction principle
10.3. Compactness all the way
10.4. Total trajectories
10.5. Positive solutions: The low road
10.6. Heterogeneous time-autonomous boundary conditions
Chapter 11 Microbial Growth in a Tubular Bioreactor
11.1. Model description
11.2. The no-bacteria invariant set
11.3. The solution semiflow
11.4. Bounds on solutions and the global attractor
11.5. Stability of the washout equilibrium
11.5.1. The basic reproduction number.
11.5.2. Global stability of the washout equilibrium
11.6. Persistence of the microbial population
11.7. Exercises
Chapter 12 Dividing Cells in a Chemostat
12.1. An integral equation
12.2. A Co-semigroup
12.3. A semilinear Cauchy problem
12.4. Extinction and weak persistence via Laplace transform
12.5. Exercises
Chapter 13 Persistence for Nonautonomous Dynamical Systems
13.1. The simple chemostat with time-dependent washout rate
13.2. General time-heterogeneity
13.3. Periodic and asymptotically periodic semiflows
13.4. Uniform persistence of the cell population
13.5. Exercises
Chapter 14 Forced Persistence in Linear Cauchy Problems
14.1. Uniform weak persistence and asymptotic Abel-averages
14.2. A compact attracting set
14.3. Uniform persistence in ordered Banach space
Chapter 15 Persistence via Average Lyapunov Functions
15.1. Weak average Lyapunov functions
15.2. Strong average Lyapunov functions
15.3. The time-heterogeneous hypercycle equation
15.4. Exercises
Appendix A Tools from Analysis and Differential Equations
A.1. Lower one-sided derivatives
A.2. Absolutely continuous functions
A.3. The method of fluctuation
A.4. Differential inequalities and positivity of solutions
A.4.1. ODEs.
A.4.2. PDEs.
A.5. Perron-Frobenius theory
A.6. Exercises
Appendix B Tools from Functional Analysis and Integral Equations
B.1. Compact sets in Lp(R+)
B.2. Volterra integral equations
B.3. Fourier transform methods for integro-differential equations
B.4. Closed linear operators
B.4.1. Duality
B.4.2. Inhomeogeneous Cauchy problems.
B.5. Exercises
Bibliography
Index
