Sharp threshold dynamics for a bistable age-structured population model

TL;DR

This paper investigates the long-term behavior of an age-structured population model with a bistable birth function, establishing a sharp threshold for population extinction or persistence based on initial conditions.

Contribution

It proves the existence of a unique threshold parameter for population dynamics in a bistable age-structured model, including cases with non-compactly supported birth rates.

Findings

Existence of a sharp transition between extinction and persistence.

Unique threshold value when birth rate has compact support.

Effective approach for non-compact support birth rates via integro-differential systems.

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.