Existence of two thresholds in a bistable equation with nonlocal competition

TL;DR

This paper studies a nonlocal bistable reaction-diffusion model for populations, revealing the existence of two thresholds in initial data that determine extinction or persistence, contrasting with local models.

Contribution

It demonstrates the existence of at least two thresholds in initial data size affecting long-term outcomes in a nonlocal bistable equation, a novel insight.

Findings

Small initial data lead to extinction.

Large initial data can also cause extinction under weak selection.

Intermediate initial data can result in persistence.

Abstract

We consider a nonlocal bistable reaction-diffusion equation, which serves as a model for a population structured by a phenotypic trait, subject to mutation, trait-dependent fitness, and nonlocal competition. Within this replicator-mutator framework, we further incorporate a ''pseudo-Allee effect'' so that the long time behavior (extinction vs. survival) depends on the size of the initial data. After proving the well-posedness of the associated Cauchy problem, we investigate its long-time behavior. We first show that small initial data lead to extinction. More surprisingly, we then prove that that extinction may also occur for too large initial data, in particular when selection is not strong enough. Finally, we exhibit situations where intermediate initial data lead to persistence, thereby revealing the existence of (at least) two thresholds. These results stand in sharp contrast with the behavior observed in local bistable equations.