Stochastic spatial model of producer-consumer systems on the lattice

TL;DR

This paper rigorously analyzes a stochastic spatial model of producer-consumer systems on a lattice, revealing how space influences ecological dynamics and showing that spatial interactions can lead to dominance by one type and reduce coexistence.

Contribution

It provides a complete qualitative analysis of the phase diagram and demonstrates that nonspatial mean-field models fail to capture key spatial effects.

Findings

Dominant type wins regardless of initial density in certain parameter regions.

Spatial interactions reduce the coexistence region.

Nonspatial models do not accurately predict the behavior of the spatial system.

Abstract

The objective of this paper is to give a rigorous analysis of a stochastic spatial model of producer-consumer systems that has been recently introduced by Kang and the author to understand the role of space in ecological communities in which individuals compete for resources. Each point of the square lattice is occupied by an individual which is characterized by one of two possible types, and updates its type in continuous time at rate one. Each individual being thought of as a producer and consumer of resources, the new type at each update is chosen at random from a certain interaction neighborhood according to probabilities proportional to the ability of the neighbors to consume the resource produced by the individual to be updated. In addition to giving a complete qualitative picture of the phase diagram of the spatial model, our results indicate that the nonspatial deterministic mean-field approximation of the stochastic process fails to describe the behavior of the system in the presence of local interactions. In particular, we prove that, in the parameter region where the nonspatial model displays bistability, there is a dominant type that wins regardless of its initial density in the spatial model, and that the inclusion of space also translates into a significant reduction of the parameter region where both types coexist.