Stability analysis and long-time convergence of a partial differential equation model of two-phase ageing

TL;DR

This paper introduces a two-phase age-structured PDE model for wild populations, analyzing its stability and long-term behavior, with implications for ecological health assessment.

Contribution

It develops a novel two-phase ageing PDE model, proving existence, uniqueness, and stability of solutions, and explores simplified models for ecological insights.

Findings

Existence and uniqueness of weak solutions for the PDE system

Global asymptotic stability of the steady state in simplified models

Proportion of individuals in each phase is uniquely determined at equilibrium

Abstract

Recent biological evidence suggests the presence of a two-phase ageing process in several species. We introduce a system of two age-structured partial differential equations (PDE) representing two phases of ageing of a wild population. The model includes a coupling of both equations through birth and transition between phases and non-linearities due to competition. We show the existence, positivity and uniqueness of weak solutions in a general setting. For a simplified system of ordinary differential equations (ODE), we show existence and uniqueness of a strictly positive steady state attracting all trajectories. We study another simplification, a coupled PDE-ODE model, for which we prove existence, uniqueness and local asymptotic stability of a strictly positive steady state. Under further assumptions, but without assuming weak non-linearities, we show the global asymptotic stability of that steady state. The uniqueness of steady states and absence of oscillations in these systems show that the proportion of individuals in each phase at equilibrium is a unique feature of the model. This paves the way to ecological applications as the experimental measure of such a proportion could help gain some insight on the health of a wild population.