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

International audienceIn 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**(,+) or c 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