A logistic equation with nonlocal interactions
  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. 
Luis Caffarelli, Serena Dipierro and Enrico Valdinoci
math.AP