Dynamical Systems and Population Persistence (Graduate Studies in Mathematics, 118)

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. 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 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.--publisher's Description. 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. Hal L. Smith, Horst R. Thieme. Includes Bibliographical References (p. 391-402) And Index. 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 (résumé de l'éditeur)