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Sharp threshold dynamics for a bistable age-structured population model

Quentin Griette, Franco Herrera

TL;DR

This work analyzes sharp threshold dynamics for a bistable, age-structured population in the Gurtin–MacCamy framework. It develops a rigorous monotone-dynamics approach, combining integrated semigroups and a Volterra integral formulation to prove the existence of a unique threshold parameter that separates extinction from persistence to the high-density equilibrium when the reproductive-age support is compact. It further handles a noncompact birth-rate case by recasting the problem as a coupled integro-differential system, showing that a similar sharp-threshold behavior holds in a particular asymptotic regime (eventually constant birth and mortality rates). Overall, the paper provides a detailed, mathematically rigorous account of threshold phenomena in bistable age-structured populations with Allee-type birth functions, including precise descriptions of separatrices and basins of attraction.

Abstract

This paper is devoted to the long-term dynamics of solutions to the Gurtin-MacCamy population model with a bistable birth function. We consider a one-parameter monotone family of initial distributions for the population such that for small values of the parameter, the corresponding population density gets extinct as time passes, whereas for large values of them, the solutions exhibit a different behavior. We are interested in the intermediate set of values for the parameters, which are called threshold parameters. We prove the existence of a sharp transition between these two asymptotic dynamics; that is, there exists exactly one threshold value when the age-dependent birth rate of the population has compact support, utilizing the theory of monotone dynamical systems. The case when the birth rate is non-compactly supported is more intricate to deal with, as has been observed in several works, even if the nonlinear birth function is monostable. Nevertheless, the approach used in the present work turns out to be effective to handle a particular birth rate with noncompact support by translating the dynamics of the age-structured model into an integro-differential system.

Sharp threshold dynamics for a bistable age-structured population model

TL;DR

This work analyzes sharp threshold dynamics for a bistable, age-structured population in the Gurtin–MacCamy framework. It develops a rigorous monotone-dynamics approach, combining integrated semigroups and a Volterra integral formulation to prove the existence of a unique threshold parameter that separates extinction from persistence to the high-density equilibrium when the reproductive-age support is compact. It further handles a noncompact birth-rate case by recasting the problem as a coupled integro-differential system, showing that a similar sharp-threshold behavior holds in a particular asymptotic regime (eventually constant birth and mortality rates). Overall, the paper provides a detailed, mathematically rigorous account of threshold phenomena in bistable age-structured populations with Allee-type birth functions, including precise descriptions of separatrices and basins of attraction.

Abstract

This paper is devoted to the long-term dynamics of solutions to the Gurtin-MacCamy population model with a bistable birth function. We consider a one-parameter monotone family of initial distributions for the population such that for small values of the parameter, the corresponding population density gets extinct as time passes, whereas for large values of them, the solutions exhibit a different behavior. We are interested in the intermediate set of values for the parameters, which are called threshold parameters. We prove the existence of a sharp transition between these two asymptotic dynamics; that is, there exists exactly one threshold value when the age-dependent birth rate of the population has compact support, utilizing the theory of monotone dynamical systems. The case when the birth rate is non-compactly supported is more intricate to deal with, as has been observed in several works, even if the nonlinear birth function is monostable. Nevertheless, the approach used in the present work turns out to be effective to handle a particular birth rate with noncompact support by translating the dynamics of the age-structured model into an integro-differential system.
Paper Structure (15 sections, 22 theorems, 141 equations, 1 figure)

This paper contains 15 sections, 22 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Preliminaries
  4. Abstract setting of the problem
  5. Existence and local stability of steady states
  6. Volterra formulation
  7. Comparison principle
  8. The threshold set
  9. The separatrix
  10. The sharp threshold property
  11. Proof of the main result
  12. A non-compactly supported case
  13. The dynamical system formulation
  14. Return to the age-structured model
  15. The strongly order-preserving property

Key Result

Proposition 1.1

The Gurtin-MacCamy model possesses exactly three equilibria, say 0, $\overline{\varphi}_1$ and $\overline{\varphi}_2$, where the last are given by

Figures (1)

  • Figure 1: Graph of the nonlinearity $f\colon \mathbb R_+ \to \mathbb R_+$.

Theorems & Definitions (38)

  • Proposition 1.1
  • proof
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Proposition 2.5
  • ...and 28 more