Pair coalescence times of ancestral lineages of two-dimensional logistic branching random walks

TL;DR

This paper investigates the coalescence times of ancestral lineages in a two-dimensional logistic branching random walk system, showing their asymptotic behavior aligns with classical models and justifies population size approximations.

Contribution

It introduces a joint regeneration construction method to analyze coalescence times, connecting complex spatial models with classical population genetics approximations.

Findings

Coalescence times match two-dimensional stepping stone model results.

Probability of identity by descent aligns with Malécot's approximation.

Justifies replacing fluctuating populations with fixed effective sizes.

Abstract

Consider two ancestral lineages sampled from a system of two-dimensional branching random walks with logistic regulation in the stationary regime. We study the asymptotics of their coalescence time for large initial separation and find that it agrees with well known results for a suitably scaled two-dimensional stepping stone model and also with Mal\'ecot's continuous-space approximation for the probability of identity by descent as a function of sampling distance. This can be viewed as a justification for the replacement of locally fluctuating population sizes by fixed effective sizes. Our main tool is a joint regeneration construction for the spatial embeddings of the two ancestral lineages.