# Voter Model Perturbations and Reaction Diffusion Equations

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*J. Theodore Cox; Richard Durrett; Edwin A. Perkins*

A publication of the Société Mathématique de France

The authors 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 P.D.E., some methods arising from a low density super-Brownian
limit theorem, and a block construction, the authors give general, and
often asymptotically sharp, conditions for the existence of
non-trivial stationary distributions, and for extinction of one
type.

As applications, the authors describe the phase diagrams of four
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, (iii) a continuous time version of the non-linear voter
model of Molofsky, Durrett, Dushoff, Griffeath, and Levin, and (iv) a
voter model in which opinion changes are followed by an exponentially
distributed latent period during which voters will not change
again.

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 the
authors' general results is that they do not require the process to be
attractive.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in voter model perturbations and reaction diffusion equations.