  In this article, we analyse the non-local model : $$ \frac{\partial
u}{\partial t}=J\star u -u + f(x,u) \quad \text{ with }\quad x \in \R^N, $$
where $J$ is a positive continuous dispersal kernel and $f(x,u)$ 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). For compactly supported dispersal kernels $J$, we derive an
optimal persistence criteria. We prove that a positive stationary solution
exists if and only if the generalised principal eigenvalue $\lambda_p$ of the
linear problem $$ J\star \varphi(x) -\varphi(x) +
\partial_sf(x,0)\varphi(x)+\lambda_p\varphi(x)=0 \quad \text{ in }\quad \R^N,$$
is negative. $\lambda_p$ is a spectral quantity that we defined in the spirit
of the generalised first eigenvalue of an elliptic operator. In addition, for
any continuous non-negative initial data that is bounded or integrable, we
establish the long time behaviour of the solution $u(t,x)$. We also analyse the
impact of the size of the support of the dispersal kernel on the persistence
criteria. We exhibit situations where the dispersal strategy has "no impact" on
the persistence of the species and other ones where the slowest dispersal
strategy is not any more an "Ecological Stable Strategy". We also discuss
persistence criteria for fat-tailed kernels.
