Voter Model Perturbations and Reaction Diffusion Equations

TL;DR

This paper studies particle systems that are perturbations of the voter model, demonstrating their convergence to reaction diffusion equations in higher dimensions, and explores conditions for coexistence and extinction in related models.

Contribution

It establishes a general framework connecting voter model perturbations to reaction diffusion equations and applies this to various biological and social models, confirming several conjectures.

Findings

Convergence of perturbed voter models to reaction diffusion equations in dimensions d ≥ 3.

Conditions for existence of non-trivial stationary distributions and extinction.

Application to biological and social models confirming prior conjectures.

Abstract

We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins and the second confirms a conjecture of Ohtsuki et al in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.