Convergence of Rescaled Competing Species Processes to a Class of SPDEs

Abstract

One can construct a sequence of rescaled perturbations of voter processes in dimension $d=1$ whose approximate densities are tight. By combining both long-range models and fixed kernel models in the perturbations and considering the critical long-range case, results of Cox and Perkins (2005) are refined. As a special case we are able to consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition. In the case of long-range interactions only, the approximate densities converge to continuous space time densities which solve a class of SPDEs (stochastic partial differential equations), namely the heat equation with a class of drifts, driven by Fisher-Wright noise. If the initial condition of the limiting SPDE is integrable, weak uniqueness of the limits follows. The results obtained extend the results of Mueller and Tribe (1995) for the voter model by including perturbations. In particular, spatial versions of the Lotka-Volterra model as introduced in Neuhauser and Pacala (1999) are covered for parameters approaching one. Their model incorporates a fecundity parameter and models both intra- and interspecific competition.