Persistence criteria for populations with non-local dispersion

TL;DR

This paper establishes criteria for population persistence in non-local dispersal models with heterogeneous environments, linking persistence to a spectral quantity and analyzing the effects of dispersal kernel support size and tail behavior.

Contribution

It introduces an optimal persistence criterion based on a generalized principal eigenvalue for non-local dispersal equations with heterogeneous environments.

Findings

Persistence occurs if and only if the principal eigenvalue is negative.

Dispersal kernel support size influences persistence criteria.

Slow dispersal may not always be an ecological stable strategy.

Abstract

In this article, we analyse the non-local model : $$ \frac{\partial u}{\partial t}=J\star u -u + f(x,u) \quad \text{ with }\quad x \in \R^N, $$ where $J$ is a positive continuous dispersal kernel and $f(x,u)$ is a heterogeneous KPP type non-linearity describing the growth rate of the population. The ecological niche of the population is assumed to be bounded (i.e. outside a compact set, the environment is assumed to be lethal for the population). For compactly supported dispersal kernels $J$, we derive an optimal persistence criteria. We prove that a positive stationary solution exists if and only if the generalised principal eigenvalue $\lambda_p$ of the linear problem $$ J\star \varphi(x) -\varphi(x) + \partial_sf(x,0)\varphi(x)+\lambda_p\varphi(x)=0 \quad \text{ in }\quad \R^N,$$ is negative. $\lambda_p$ is a spectral quantity that we defined in the spirit of the generalised first eigenvalue of an elliptic operator. In addition, for any continuous non-negative initial data that is bounded or integrable, we establish the long time behaviour of the solution $u(t,x)$. We also analyse the impact of the size of the support of the dispersal kernel on the persistence criteria. We exhibit situations where the dispersal strategy has "no impact" on the persistence of the species and other ones where the slowest dispersal strategy is not any more an "Ecological Stable Strategy". We also discuss persistence criteria for fat-tailed kernels.