On the connections between the spatial Lambda–Fleming–Viot model and other processes for analysing geo-referenced genetic data

Abstract

The introduction of the spatial Lambda-Fleming–Viot model ($\Lambda$V) in population genetics was mainly driven by the pioneering work of Alison Etheridge, in collaboration with Nick Barton and Amandine Véber about ten years ago (Barton et al., 2010; Barton et al., 2013). The $\Lambda$V model provides a sound mathematical framework for describing the evolution of a population of related individuals along a spatial continuum. It alleviates the “pain in the torus” issue with Wright and Malécot’s isolation by distance model and is sampling consistent, making it a tool of choice for statistical inference. Yet, little is known about the potential connections between the $\Lambda$V and other stochastic processes generating trees and the spatial coordinates along the corresponding lineages. This work focuses on a version of the $\Lambda$V whereby lineages move rapidly over small distances. Using simulations, we show that the induced $\Lambda$V tree-generating process is well approximated by a birth–death model. Our results also indicate that Brownian motions modelling the movements of lines of descent along birth–death trees do not generally provide a good approximation of the $\Lambda$V due to habitat boundaries effects that play an increasingly important role in the long run. Accounting for habitat boundaries through reflected Brownian motions considerably increases the similarity to the $\Lambda$V model however. Finally, we describe efficient algorithms for fast simulation of the backward and forward in time versions of the $\Lambda$V model.