A Global Attractivity in a Nonmonotone Age-Structured Model with Age Dependent Diffusion and Death Rates

TL;DR

This paper studies the long-term behavior of an age-structured population model with age-dependent diffusion and death rates, focusing on the conditions for the global attractivity of the positive steady state despite nonmonotone birth functions.

Contribution

It establishes the global attractivity of the positive steady state in a complex age-structured PDE model with nonmonotone birth functions and age-dependent parameters, providing new theoretical insights.

Findings

Proved conditions for global attractivity of steady state.

Analyzed effects of age-dependent diffusion and death rates.

Presented illustrative examples demonstrating theoretical results.

Abstract

In this paper, we investigated the global attractivity of the positive constant steady state solution of the mature population $w(t,x)$ governed by the age-structured model: \begin{equation*} \left\{\begin{array}{ll} \frac{\partial u}{\partial t}+\frac{\partial u}{\partial a}=D(a)\frac{\partial ^2 u}{\partial x^2} - d(a)u, & t\geq t_0\geq A_l,\;a\geq 0,\; 0< x< \pi,\\ w(t,x)=\int_r^{A_l}u(t,a,x)da,& t\geq t_0\geq A_l,\; 0<x<\pi,\\ u(t,0,x)=f(w(t,x)), & t\geq t_0\geq A_l,\; 0<x<\pi,\\ u_x(t,a,0)=u_x(t,a,\pi)=0,\;& t\geq t_0\geq A_l,\; a \geq 0, \end{array} \right. \end{equation*} when the diffusion rate $D(a)$ and the death rate $d(a)$ are age dependent, and when the birth function $f(w)$ is nonmonotone. We also presented some illustrative examples.