Can a population survive in a shifting environment using non-local dispersion

Abstract

In this article, we analyse the non-local model : $\partial$ t U (t, x) = J $\star$ U (t, x) -- U (t, x) + f (x -- ct, U (t, x)) for t > 0, and x $\in$ R, where J is a positive continuous dispersal kernel and f (x, s) 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) and shifted through time at a constant speed c. For compactly supported dispersal kernels J, assuming that for c = 0 the population survive, we prove that there exists a critical speeds c * ,$\pm$ and c * * ,$\pm$ such that for all --c * ,-- < c < c * ,+ then the population will survive and will perish when c $\ge$ c * * ,+ or c $\le$ --c * * ,--. To derive this results we first obtain an optimal persistence criteria depending of the speed c for non local problem with a drift term. Namely, we prove that for a positive speed c the population persists if and only if the generalized principal eigenvalue $\lambda$ p of the linear problem cD x [$\Phi$] + J $\star$ $\Phi$ -- $\Phi$ + $\partial$ s f (x, 0)$\Phi$ + $\lambda$ p $\Phi$ = 0 in R, is negative. $\lambda$ p is a spectral quantity that we defined in the spirit of the generalized first eigenvalue of an elliptic operator. The speeds c * ,$\pm$ and c * * ,pm are then obtained through a fine analysis of the properties of $\lambda$ p with respect to c. In particular, we establish its continuity with respect to the speed c. In addition, for any continuous bounded non-negative initial data, we establish the long time behaviour of the solution U (t, x).