Persistence versus extinction under a climate change in mixed environments

TL;DR

This paper investigates the conditions under which species persist or go extinct in reaction-diffusion models influenced by climate change, focusing on traveling fronts in complex environments with varying favorability.

Contribution

It extends previous models to higher dimensions with environments having favorable pockets and analyzes traveling fronts under both time-independent and periodic conditions.

Findings

Traveling fronts characterize species persistence or extinction.

Environmental unfavorable regions influence species concentration and front symmetry.

New conditions for species survival in complex, partially favorable environments.

Abstract

This paper is devoted to the study of the persistence versus extinction of species in the reaction-diffusion equation: \begin{equation} u_t-\Delta u=f(t,x_1-ct,y,u) \quad\quad t>0,\ x\in\Omega,\nonumber \end{equation} where $\Omega$ is of cylindrical type or partially periodic domain, $f$ is of Fisher-KPP type and the scalar $c>0$ is a given forced speed. This type of equation originally comes from a model in population dynamics (see \cite{BDNZ},\cite{PL},\cite{SK}) to study the impact of climate change on the persistence versus extinction of species. From these works, we know that the dynamics is governed by the traveling fronts $u(t,x_1,y)=U(x_1-ct,y)$, thus characterizing the set of traveling fronts plays a major role. In this paper, we first consider a more general model than the model of \cite{BDNZ} in higher dimensional space, where the environment is only assumed to be globally unfavorable with favorable pockets extending to infinity. We consider in two frameworks: the reaction term is time-independent or time-periodic dependent. For the latter, we study the concentration of the species when the environment outside $\Omega$ becomes extremely unfavorable and further prove a symmetry breaking property of the fronts.