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 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.

💡 Research Summary

This paper provides a rigorous mathematical analysis of sharp threshold dynamics in an age-structured population model exhibiting a strong Allee effect. The authors study the Gurtin-MacCamy model, a nonlinear partial differential equation representing the density of a population structured by age. The model’s key feature is a bistable birth function f, which has three fixed points: a trivial extinction state (0), an unstable intermediate equilibrium (κ1), and a stable high-population equilibrium (κ2). This structure induces two possible long-term fates for the population: convergence to extinction (0) for small initial sizes, or convergence to the sustained state (ϕ2) for large enough initial sizes.

The central investigation revolves around the transition between these two outcomes. The authors consider a one-parameter, monotone family of initial population distributions {u_λ}. They prove that under certain conditions, there exists a unique critical parameter value λ* that sharply separates the two asymptotic regimes. For λ < λ*, the solution u_λ(t,·) converges to 0 (extinction) as time tends to infinity. For λ > λ*, it converges to ϕ2 (persistence). The solution corresponding to the threshold λ* itself does not converge to either stable equilibrium and exhibits oscillatory or other non-convergent behavior, remaining bounded away from both states by a positive distance.

The proof strategy leverages the theory of monotone dynamical systems, made possible by the assumption that the birth function f is strictly increasing. The analysis is split into two main scenarios based on the fertility rate β(a).

1. Compact Support Case (a < ∞):* When the maximum reproductive age a* is finite (β has compact support), the problem can be naturally formulated as a semi-flow on the space of continuous functions over the compact interval