  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.
