Probabilistic Foundations of Spatial Mean-field Models in Ecology and Applications

TL;DR

This paper develops a mathematical framework linking microscopic stochastic ecological models to their macroscopic mean-field limits, capturing spatial interactions and complex dynamics like oscillations and invasion waves.

Contribution

It introduces a general approach to derive spatially extended non-Markovian mean-field models from stochastic processes in ecology, with rigorous convergence proofs and practical applications.

Findings

Spatial mean-field models accurately approximate large stochastic ecological systems.

Finite-size effects and extinction rates are explained by bifurcations in the mean-field equations.

The framework captures complex phenomena like oscillations and invasion waves.

Abstract

Deterministic models of vegetation often summarize, at a macroscopic scale, a multitude of intrinsically random events occurring at a microscopic scale. We bridge the gap between these scales by demonstrating convergence to a mean-field limit for a general class of stochastic models representing each individual ecological event in the limit of large system size. The proof relies on classical stochastic coupling techniques that we generalize to cover spatially extended interactions. The mean-field limit is a spatially extended non-Markovian process characterized by nonlocal integro-differential equations describing the evolution of the probability for a patch of land to be in a given state (the generalized Kolmogorov equations of the process, GKEs). We thus provide an accessible general framework for spatially extending many classical finite-state models from ecology and population dynamics. We demonstrate the practical effectiveness of our approach through a detailed comparison of our limiting spatial model and the finite-size version of a specific savanna-forest model, the so-called Staver-Levin model. There is remarkable dynamic consistency between the GKEs and the finite-size system, in spite of almost sure forest extinction in the finite-size system. To resolve this apparent paradox, we show that the extinction rate drops sharply when nontrivial equilibria emerge in the GKEs, and that the finite-size system's quasi-stationary distribution (stationary distribution conditional on non-extinction) closely matches the bifurcation diagram of the GKEs. Furthermore, the limit process can support periodic oscillations of the probability distribution, thus providing an elementary example of a jump process that does not converge to a stationary distribution. In spatially extended settings, environmental heterogeneity can lead to waves of invasion and front-pinning phenomena.