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

In this article, we analyse the non-local model: partial derivative U-t(t, x) = J * U(t, x) - U(t, x) + f (x - ct,U (t, x)) fort > 0, and x is an element of 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 exist critical speeds c*(,+/-) and c**(,+/-), such that for all -c*(,-) < c < c*(,+) then the population will survive and will perish when c > c**(,+) or c <= c**(,-). To derive these 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 generalised principal eigenvalue lambda(p) of the linear problem cD(x)[rho] + J * rho - rho + partial derivative(s)f (x, 0)rho lambda(p)rho = 0 in R, is negative. lambda(p) is a spectral quantity that we defined in the spirit of the generalised first eigenvalue of an elliptic operator. The speeds c(*,+/-) and c**(,+/-) 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). In the specific situation, partial derivative(s)f(x,o) > 1 and J symmetric we also investigate the behaviour of the critical speeds c* and c** with respect to the tail of the kernel J. We show in particular that even for very fat tailed kernel these two critical speeds exist. (C) 2021 Elsevier Ltd. All rights reserved.