Voter model perturbations and reaction diffusion equations
By: J. Theodore Cox, Richard Durrett , Edwin A. Perkins
Contributors: Durrett, Richard, 1951- author; Perkins, Edwin Arend, 1953- author; Société mathématique de France
Series: Astérisque -- 349
Publisher: Societé mathématique de France
Published: 2013
Place of publication: Paris
Language: eng
Number of pages: 113
ISBN-13: 9782856293553
Subjects: Stochastic processes; Reaction-diffusion equations; Percolation (Statistical physics)
Description: "Keywords and phrases: Interacting particle systems, voter model, reaction diffusion equation, evolutionary game theory, Lotka-Volterra model"--Title page verso.

"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[greater than or equal to]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, 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 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, (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 ("Survival and coexistence in stochastic spatial Lotka-Volterra models", 2007) and the second confirms a conjecture of Ohtsuki et al. ("A simple rule for the evolution of cooperation on graphs and social networks", 2006) 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."--Page [4] of cover.
Notes: "Publié avec le concours du Centre national de la recherche scientifique"--Title page.

Includes bibliographical references (pages 111-113).

Includes bibliographical references.

Text in English; abstract in both English and French.