Permanency of the age-structured population model on several temporally variable patches

TL;DR

This paper analyzes an age-structured population model across multiple patches with variable environments, proving conditions for persistence or extinction based on the net reproductive rate.

Contribution

It introduces a novel approach to large-time stability analysis and establishes the role of the net reproductive operator in determining population permanency.

Findings

Permanency is linked to the net reproductive rate of the entire system.

Extinction occurs if the net reproductive rate is ≤ 1.

Permanency is guaranteed if the net reproductive rate > 1.

Abstract

We consider a system of nonlinear partial differential equations that describes an age-structured population inhabiting several temporally varying patches. We prove existence and uniqueness of solution and analyze its large-time behavior in cases when the environment is constant and when it changes periodically. A pivotal assumption is that individuals can disperse and that each patch can be reached from every other patch, directly or through several intermediary patches. We introduce the net reproductive operator and characteristic equations for time-independent and periodical models and prove that permanency is defined by the net reproductive rate for the whole system. If the net reproductive rate is less or equal to one, extinction on all patches is imminent. Otherwise, permanency on all patches is guaranteed. The proof is based on a new approach to analysis of large-time stability.