Dynamical Systems and Population Persistence

Abstract

Preface ix §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 §2.3. A sufficient condition for asymptotic smoothness §2.4. α-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 §3.2. An SEIRS epidemic model in patchy host populations §3.3. Nonlinear matrix models: Prolog §3.4. The May-Leonard example of cyclic competition §3.5. Exercises v vi Contents Chapter 4. Uniform Persistence §4.1. From uniform weak to uniform persistence §4.2. From uniform weak to uniform persistence: Discrete case §4.3. Application to a metered endemic model of SIR type §4.4. From uniform weak to uniform persistence for time-set R + §4.5. Persistenceà la Baron von Münchhausen §4.6. Navigating between alternative persistence functions §4.7. A fertility reducing endemic with two stages of infection §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 §5.3. 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 Contents vii §8.5. The metered endemic model revisited §8.6. Nonlinear matrix models (epilog): Biennials §8.7. An endemic with vaccination and temporary immunity §8.8. Lyapunov exponents and persistence for ODEs and maps §8.9. Exercises Chapter 9. An SI Endemic Model with Variable Infectivity §9.1. The model §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 p...

Stochastic analysis of biochemical reaction networks with absolute concentration robustness

“…First, finding more general conditions for which the conclusions of theorem 3.2 hold will be a focus. In particular, weakening the requirement that the system possesses a conservation relation will allow the results to be applicable to more models arising in ecology and population processes, which typically do not satisfy such an assumption, though do often possess low numbers of the constituent species [42]. Second, the question of when equilibria exhibiting ACR in the deterministic modelling context are stable or unstable is, to the best of the authors' knowledge, currently open and has to date received surprisingly little consideration in the literature.…”

The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes

“…Moreover, from Lemmas 4.2-4.3 and Proposition 4.1 and the Lipschitz continuity ofê (which immediately follows from Proposition 2.3), we can apply Theorem 5.2 of Smith and Thieme [21] to conclude that the uniform weak ρ-persistence of semi-flow implies the uniform (strong) ρ-persistence. In conclusion, we obtain the following theorem.…”

“…The asymptotic smoothness of was shown by Theorem 3.3. Thus, Theorem 2.33 of Simith and Thieme [21] can be applied to complete the proof.…”

“…Therefore, we can apply Corollary B.6 of Smith and Thieme [21] to conclude that there exists a number t * > 0, which only depends onk(s), such thatê j (t) > 0 for all t > t * . From the definition ofê j , this implies thatê(t) > 0 for all t > t * + r j .…”

How Infectious Was #Deflategate?

“…Since the R(t) population doesn't interact with the rest of the population at any given time, we can omit it from our analysis. We assume that the parameters β, γ, S 0 and I 0 are all positive, which implies that the solution vector [S(t) I(t)] T exists and is positive for all t > 0 [23]. The derivation of the basic reproduction number R 0 for this SIR model follows from determining the stability threshold for the "disease-free" state [S 0 0] T , where the susceptible population has yet to be exposed to the news story.…”