Feller Property and Absorption of Diffusions for Multi-Species Metacommunities

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.

💡 Research Summary

The paper studies a stochastic metacommunity consisting of two species distributed across m patches, each patch i having a carrying capacity d_i N with 0 < d_i ≤ 1. Individuals reproduce according to a neutral Wright‑Fisher model within each patch, and after reproduction they migrate according to a deterministic exchange map Φ_N: K → K, where K =