On spatial coalescents with multiple mergers in two dimensions

TL;DR

This paper studies the genealogy of spatially structured populations with large offspring variance, showing that spatial Lambda-coalescents approximate the genealogy and converge to Kingman coalescent in large two-dimensional tori, with implications for biological detectability.

Contribution

It introduces spatial Lambda-coalescents as models for genealogies in large spatial populations and proves their convergence to Kingman coalescent in two dimensions as the system size grows.

Findings

Spatial Lambda-coalescents approximate genealogies with large offspring variance.

Convergence to Kingman coalescent occurs as the spatial domain size increases.

Detectability of spatial structure diminishes in large systems, but remains observable in moderate sizes.

Abstract

We consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with multiple mergers, so called spatial Lambda-coalescents, for which ancestral lines migrate in space and coalesce according to some Lambda-coalescent mechanism, are shown to be appropriate approximations to the genealogy of a sample of individuals. We then consider as the graph G the two dimensional torus with side length 2L+1 and show that as L tends to infinity, and time is rescaled appropriately, the partition structure of spatial Lambda-coalescents of individuals sampled far enough apart converges to the partition structure of a non-spatial Kingman coalescent. From a biological point of view this means that in certain circumstances both the spatial structure as well as larger variances of the underlying offspring distribution are harder to detect from the sample. However, supplemental simulations show that for moderately large L the different structure is still evident.