Sharp Threshold Dynamics for a Bistable Age-Structured Population Model

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 converge to a positive equilibrium. We are interested in the intermediate values of the parameters, which are called threshold parameters. We prove the existence of a sharp transition between these two asymptotic dynamics when the age-dependent birth rate of the population has compact support; that is, there exists exactly one threshold value, by utilizing the theory of monotone dynamical systems. The case of a non-compactly supported birth rate is more intricate, 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.