In this article, we analyse the non-local model :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 λ p of the linear problemis negative. λ 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.

In this article we study some spectral properties of the linear operator L Ω + a defined on the space C(Ω) by :

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This paper is devoted to the study of the persistence versus extinction of species in the reaction-diffusion equation:where Ω 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 Ω becomes extremely unfavorable and further prove a symmetry breaking property of the fronts.Mathematical Subject Classification (2010): 35C07, 35J15, 35B09, 35P20, 92D25.