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

In this article, we analyse the non-local model: ∂tU(t,x)=J⋆U(t,x)−U(t,x)+f(x−ct,U(t,x))fort>0, and x∈R, where J is a positive continuous dispersal kernel f(x,s) heterogeneous KPP type non-linearity describing growth rate of population. The ecological niche population assumed to be bounded (i.e. outside compact set, environment lethal for population) shifted through time at constant speed c. For compactly supported kernels J, assuming that c=0 survive, prove there exist critical speeds c∗,± c∗∗,± such all −c∗,−1 symmetric also investigate c∗ c∗∗ tail J. We show particular even very fat tailed two exist.