Voter Model Perturbations and Reaction Diffusion Equations

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 ≥ 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 [8] and the second confirms a conjecture of Ohtsuki et al [38] 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.

Motion by mean curvature in interacting particle systems

“…Theorem 1.2 shows that the probabilities P (ξ ǫ t (x) = 1) converge to a u(t, x) that satisfies motion by mean curvature. As in Theorem 1.2 in [3] one can also prove that that the rescaled particle system which takes values in {0, 1} on a fine grid also converges to u(t, x). See the discussion before Theorem 1.2 in [3] for the necessary definition.…”

“…As in Theorem 1.2 in [3] one can also prove that that the rescaled particle system which takes values in {0, 1} on a fine grid also converges to u(t, x). See the discussion before Theorem 1.2 in [3] for the necessary definition. This remark also applies to the next two examples.…”

“…Convergence results for the PDE [14] and block constructions were used to show that • When β > 4.5 and ǫ < ǫ 0 (β) there is a nontrivial stationary distribution with a density close to ρ 2 . The second part of the conclusion is an improvement due to Cox, Durrett, and Perkins [3].…”

“…Cox, Durrett and Perkins [3] introduced a class of interacting particle systems called voter model perturbations. For simplicity we will restrict our attention to processes with two states.…”

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There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term, see e.g., [10,7,8,3]. These RDEs have traveling wave solutions.

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