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.

This article deals with the existence and non-existence of positive solutions for the eigenvalue problem driven by nonhomogeneous fractional p&q Laplacian operator with indefinite weightsapproach replies strongly on variational analysis, in which the Mountain pass theorem plays the key role.The main difficulty in this study is that how to establish the Palais-Smale conditions. In particular, in R N , due to the lack of spatial compactness and the embedding W α,p R N ֒→ W β,q R N , we must employ the concentration-compactness principle of P.L. Lions [25] to overcome the difficulty.

This paper is devoted to providing a simple condition, in term of spectral theory, that characterizes existence/nonexistence and uniqueness of positive bounded solution towhere f is of monostable KPP type nonlinearity and periodic in y. Our contribution answers a conjecture raised by Prof. H. Berestycki: which suitable assumption can impose at infinity that characterizes existence/nonexistence and uniqueness of (0.1) instead of the followings lim inf |z|→∞ ∂ u f (z, 0) > 0 as in Berestycki et al. (Ann Mat Pura Appl 186(4):469-507, 2007) and lim sup |z|→∞ ∂ u f (z, 0) < 0 as in Berestycki et al. (Bull Math Biol 71:399, 2008) and Berestycki and Rossi (Discret Contin Dyn Syst Ser B 21:41-67, 2008) but allow ∂ u f (z, 0) to change sign all the way as |z| → ∞? Our result is simply based on maximum principle and complements to those in

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