A logistic equation with nonlocal interactions

Luis Caffarelli; Serena Dipierro; Enrico Valdinoci

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arXiv · 1601.05552

A logistic equation with nonlocal interactions

Luis Caffarelli·Serena Dipierro·Enrico Valdinoci

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Abstract

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 L\'evy 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: bounded domains, periodic environments, and 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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Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. 2016-01-21. A logistic equation with nonlocal interactions. https://arxiv.org/abs/1601.05552

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