Astérisque 2013; 113 pp; softcover Number: 349 ISBN13: 9782856293553 List Price: US$52 Member Price: US$41.60 Order Code: AST/349
 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 superBrownian limit theorem, and a block construction, the authors give general, and often asymptotically sharp, conditions for the existence of nontrivial 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 LotkaVolterra 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 nonlinear 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. Table of Contents  Introduction and statement of results
 Construction, duality and coupling
 Proofs of Theorems 1.2 and 1.3
 Achieving low density
 Percolation results
 Existence of stationary distributions
 Extinction of the process
 Bibliography
