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: u(t) - Delta u = f (t, x(1) - ct, y, u) t > 0, x is an element of Omega, 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 [3,17,18]) 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, 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 [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 becomes extremely unfavorable and further prove a symmetry breaking property of the fronts. (C) 2015 Elsevier Inc. All rights reserved.