Heterogeneous carrying capacities and global extinction in metapopulations

TL;DR

This paper analyzes a two-patch reaction-diffusion model with strong Allee effect and heterogeneous carrying capacities, establishing conditions for global extinction and comparing analytical results with numerical simulations.

Contribution

It provides new analytical conditions ensuring global extinction in heterogeneous two-patch models, extending previous homogeneous case results.

Findings

Extinction point is globally stable under certain conditions.

Heterogeneous capacities influence the number of stationary solutions.

Numerical simulations support analytical conditions.

Abstract

In this paper we consider a simple two patch reaction diffusion model with strong Allee effect, sufficiently distinct carrying capacities, similar reaction strengths, and strong diffusion. In the homogeneous case, i.e., in in the case of equal or similar capacities and reaction strengths, it is well known that the number of stationary solutions ranges from three (strong diffusion) to nine (weak diffusion). We provide sufficient conditions which includes the diffusion strength and reaction parameters that ensure that the extinction point is the unique and globally asymptotically stable equilibrium in the case of heterogeneous capacities. For the sake of robustness we consider several bistable reaction functions, compare our analytical result with numerical simulations, and conclude the paper with a short discussion on global extinction literature (which has provided mostly numerical results so far), as well as other related phenomena, e.g., fragmentation, the perfect mixing paradox, and the natural form the reaction diffusion patch models.