Voter Model Perturbations and Reaction Diffusion Equations
Ted Cox, Rick Durrett, and Ed Perkins
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 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.
Preprint
Back to Durrett's home page