A new model for evolution in a spatial continuum

Abstract

We investigate a new model for populations evolving in a spatial continuum. This model can be thought of as a spatial version of the Λ-Fleming-Viot process. It explicitly incorporates both small scale reproduction events and large scale extinction-recolonisation events. The lineages ancestral to a sample from a population evolving according to this model can be described in terms of a spatial version of the Λ-coalescent. Using a technique of Evans (1997), we prove existence and uniqueness in law for the model. We then investigate the asymptotic behaviour of the genealogy of a finite number of individuals sampled uniformly at random (or more generally ‘far enough apart’) from a two-dimensional torus of sidelength L as L 𡥒 ∞. Under appropriate conditions (and on a suitable timescale) we can obtain as limiting genealogical processes a Kingman coalescent, a more general Λ-coalescent or a system of coalescing Brownian motions (with a non-local coalescence mechanism). © 2010 Applied Probability Trust.