Global dynamics of a two-stage structured diffusive population model in time-periodic and spatially heterogeneous environments

TL;DR

This paper analyzes the long-term behavior of a two-stage population model with diffusion in environments that change over time and space, identifying conditions for species persistence and stability of solutions.

Contribution

It provides new criteria for species persistence, uniqueness, and stability of positive solutions in a complex, time-periodic, spatially heterogeneous setting.

Findings

Principal eigenvalue determines species persistence.

Existence of at least one positive time-periodic solution when eigenvalue is positive.

Conditions for stability and asymptotic profiles of steady states.

Abstract

This work examines the global dynamics of classical solutions of a two-stage (juvenile-adult) reaction-diffusion population model in time-periodic and spatially heterogeneous environments. It is shown that the sign of the principal eigenvalue $\lambda_*$ of the time-periodic linearized system at the trivial solution completely determines the persistence of the species. Moreover, when $\lambda_*>0$, there is at least one time-periodic positive entire solution. A fairly general sufficient condition ensuring the uniqueness and global stability of the positive time-periodic solution is obtained. In particular, classical solutions eventually stabilize at the unique time-periodic positive solutions if either each subgroup's intra-stage growth and inter-stage competition rates are proportional, or the environment is temporally homogeneous and both subgroups diffuse slowly. In the later scenario, the asymptotic profile of steady states with respect to small diffusion rates is established.