Two Dimensional Lotka-Volterra Models and Superprocesses

Neuhauser and Pacala (99) introduced a particle system to model the dynamics of two competing types on an integer lattice. We study the model near the parameter values for which there is a cross-over from preference of one's own type to preference of the other type. The rescaled system converges to a super-Brownian motion with non-trivial branching and growth rates. As a corollary, local information about parameter values for which rare types can survive and both types can co-exist is obtained. These results had been obtained earlier for dimensions greater than 2. We extend many of these results to two dimensions where the mathematics is more delicate, and perhaps the biology more interesting. This is joint work with Ted Cox (Syracuse University).