%0 Generic %1 cox2011voter %A Cox, J. Theodore %A Durrett, Richard %A Perkins, Edwin %D 2011 %K Lotka_Volterra density_dependence limit_theorems particle_systems reaction-diffusion spatial_coalescent voter_model %T Voter Model Perturbations and Reaction Diffusion Equations %U http://arxiv.org/abs/1103.1676 %X 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.

@misc{cox2011voter, 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 $d \ge 3$. 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.}, added-at = {2016-04-26T07:42:41.000+0200}, author = {Cox, J. Theodore and Durrett, Richard and Perkins, Edwin}, biburl = {https://www.bibsonomy.org/bibtex/2cf60aa2792061502d9c4bfe2ca69c4a8/peter.ralph}, interhash = {e63562df91fa2d22965cc29a1da1b6d2}, intrahash = {cf60aa2792061502d9c4bfe2ca69c4a8}, keywords = {Lotka_Volterra density_dependence limit_theorems particle_systems reaction-diffusion spatial_coalescent voter_model}, note = {cite arxiv:1103.1676}, timestamp = {2016-04-26T07:42:41.000+0200}, title = {Voter Model Perturbations and Reaction Diffusion Equations}, url = {http://arxiv.org/abs/1103.1676}, year = 2011 }