A new model for evolution in a spatial continuum

TL;DR

This paper introduces a spatial continuum model for evolving populations that combines small and large scale events, analyzing its genealogical properties and asymptotic behaviors as the population size grows.

Contribution

It develops a novel spatial Lambda-Fleming-Viot process, proving existence and uniqueness, and characterizes its genealogical limits under various conditions.

Findings

Genealogies converge to Kingman or Lambda-coalescents under certain conditions.

The model captures both local and large-scale extinction-recolonisation dynamics.

Asymptotic behaviors depend on spatial scale and sampling distance.

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 Lambda-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 Lambda-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 side L as L tends to infinity. Under appropriate conditions (and on a suitable timescale), we can obtain as limiting genealogical processes a Kingman coalescent, a more general Lambda-coalescent or a system of coalescing Brownian motions (with a non-local coalescence mechanism).