Coexistence and duality in competing species models

TL;DR

This paper explores stochastic spatial models of competing species based on Lotka-Volterra dynamics, using duality methods to analyze coexistence and survival in generalized spin systems and diffusions.

Contribution

It introduces duality-based techniques for proving coexistence in stochastic spatial models of competing species, extending classical Lotka-Volterra models.

Findings

Duality relates coexistence to survival of dual processes

Methods for proving coexistence in generalized models

Open questions on long-term behavior of models

Abstract

We discuss some stochastic spatial generalizations of the Lotka--Volterra model for competing species. The generalizations take the forms of spin systems on general discrete sets and interacting diffusions on integer lattices. Methods for proving coexistence in these generalizations and some related open questions are discussed. We use duality as the central point of view. It relates coexistence of the models to survival of their dual processes.