Dynamical systems and population persistence

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