### 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 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 [8] and the second confirms a conjecture of Ohtsuki et al. [34] 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.

Original language | English (US) |
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Pages (from-to) | 1-127 |

Number of pages | 127 |

Journal | Asterisque |

Issue number | 349 |

State | Published - 2013 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Asterisque*, (349), 1-127.