We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms.
More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case.
As ambient space, we can consider:
$ \bullet $bounded domains,
$ \bullet $periodic environments,
$ \bullet $transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one.
In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.
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