Feller Property and Absorption of Diffusions for Multi-Species Metacommunities

TL;DR

This paper studies the evolution of multi-species metacommunities across interconnected patches, demonstrating convergence to a diffusion process with the Feller property and finite-time absorption, under minimal assumptions.

Contribution

It introduces a diffusion approximation for multi-species metacommunities with dispersal, establishing the Feller property and absorption behavior.

Findings

Processes converge to a diffusion as N approaches infinity.

The limiting diffusion process has the Feller property.

The process reaches absorbing states in finite time almost surely.

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.