Persistence versus extinction under a climate change in mixed environments

This paper is devoted to the study of the persistence versus extinction of species in the reaction–diffusion equation:ut−Δu=f(t,x1−ct,y,u)t>0,x∈Ω, where Ω is of cylindrical type or partially periodic domain, f   is of Fisher-KPP type and the scalar c>0c>0 is a given forced speed. This type of equation originally comes from a model in population dynamics (see [3], [17] and [18]) 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,x1,y)=U(x1−ct,y)u(t,x1,y)=U(x1−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 [3] 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 Ω becomes extremely unfavorable and further prove a symmetry breaking property of the fronts.