Feller Property and Absorption of Diffusions for Multi-Species Metacommunities

TL;DR

This work derives a diffusion approximation for a two-species metacommunity distributed over $m$ patches with size-conserving migration. By decomposing the diffusion generator into diffusion and drift parts and employing a Trotter–Kato framework, the authors show that the finite-population Wright-Fisher metacommunity converges to a degenerate diffusion on $K=[0,1]^m$ with a Feller semigroup, whose core includes $C^2(K)$. They prove absorption at the corners $\{0_m,1_m\}$ is almost surely finite under a boundary-positivity drift condition, and they establish a Skorokhod-convergence result for the diffusion limit from the discrete model. The analysis also extends to a finite number of species and provides extinction/ fixation time results essential for understanding large-population metacommunity dynamics under dispersal.

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.

Figures (1)

• Figure 1: An Example of $\Delta_{\alpha}^0$ and $\Delta_{\alpha}^1$ ($m=2$)

Theorems & Definitions (23)

• theorem 1
• theorem 2: Convergence in distribution
• theorem 3: Absorption
• remark 1
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof