Feller Property and Absorption of Diffusions for Multi-Species Metacommunities

Abstract

We consider individuals of two species distributed over m patches, each with a hosting capacity $d_{i}N$ , where $d_i\in (0,1]$. We assume that all the patches are linked by the dispersal of individuals. This work examines how the metacommunity evolves in these patches. The model incorporates Wright-Fisher intra-patch reproduction and a general exchange function representing dispersal. Under minimal assumptions, we demonstrate that as $N$ approaches infinity, the processes converge to a diffusion process for which we establish the Feller property. We prove that the limiting process almost surely reaches the absorbing states in finite time.