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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 aheterogeneous KPP type non-linearity describing the growth rate of thepopulation. 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 thepopulation) and shifted through time at a constant speed c. For compactlysupported dispersal kernels J, assuming that for c = 0 the population survive,we prove that there exists a critical speeds c * ,$\pm$ and c * * ,$\pm$ suchthat for all --c * ,-- < c < c * ,+ then the population will survive and willperish when c $\ge$ c * * ,+ or c $\le$ --c * * ,--. To derive this results wefirst obtain an optimal persistence criteria depending of the speed c for nonlocal problem with a drift term. Namely, we prove that for a positive speed cthe 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 thegeneralized first eigenvalue of an elliptic operator. The speeds c * ,$\pm$ andc * * ,pm are then obtained through a fine analysis of the properties of$\lambda$ p with respect to c. In particular, we establish its continuity withrespect to the speed c. In addition, for any continuous bounded non-negativeinitial data, we establish the long time behaviour of the solution U (t, x).

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