Coexistence in a two-dimensional Lotka-Volterra model
J Theodore Cox, Syracuse University
Mathieu Merle, Université Paris VII (Diderot)
Edwin A Perkins, The University of British Columbia
Abstract
We study the stochastic spatial model for competing species introduced by
Neuhauser and Pacala in two spatial dimensions. In particular we confirm a con-
jecture of theirs by showing that there is coexistence of types when the competition
parameters between types are equal and less than, and close to, the within types
parameter. In fact coexistence is established on a thorn-shaped region in parameter
space including the above piece of the diagonal. The result is delicate since coex-
istence fails for the two-dimensional voter model which corresponds to the tip of
the thorn. The proof uses a convergence theorem showing that a rescaled process
converges to super-Brownian motion even when the parameters converge to those
of the voter model at a very slow rate.
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