When the Allee threshold is an evolutionary trait: persistence vs. extinction

TL;DR

This paper investigates a complex population model with an evolving Allee threshold, analyzing how different growth dynamics influence long-term persistence or extinction, revealing diverse outcomes due to nonlocal interactions and trait evolution.

Contribution

It introduces a nonlocal parabolic model with an evolving Allee threshold and analyzes its well-posedness and long-term behavior, highlighting novel extinction and persistence scenarios.

Findings

Multiple long-term behaviors including extinction and persistence.

Rich dynamics due to nonlocal growth and trait evolution.

Contrasts with classical local reaction-diffusion results.

Abstract

We consider a nonlocal parabolic equation describing the dynamics of a population structured by a spatial position and a phenotypic trait, submitted to dispersion , mutations and growth. The growth term may be of the Fisher-KPP type but may also be subject to an Allee effect which can be weak (non-KPP monostable nonlinearity, possibly degenerate) or strong (bistable nonlinearity). The type of growth depends on the value of a variable $\theta$ : the Allee threshold, which is considered here as an evolutionary trait. After proving the well-posedness of the Cauchy problem, we study the long time behavior of the solutions. Due to the richness of the model and the interplay between the various phenomena and the nonlocality of the growth term, the outcomes (extinction vs. persistence) are various and in sharp contrast with earlier results of the existing literature on local reaction-diffusion equations.