One-dimensional stepping stone models, sardine genetics and Brownian local time

TL;DR

This paper analyzes the genealogy of samples in one-dimensional stepping stone and voter models, showing convergence to a pair of Brownian motions with coalescence governed by local time and an exponential variable.

Contribution

It introduces a novel limit theorem describing genealogical convergence to Brownian motions with coalescence driven by local time and exponential waiting times.

Findings

Genealogy converges to Brownian motions with coalescence after local time exceeds an exponential variable.

Derived a differential equation with unique boundary conditions for coalescence time distribution.

Identified conditions for convergence based on model parameters M, ν, L, and K.

Abstract

Consider a one-dimensional stepping stone model with colonies of size $M$ and per-generation migration probability $\nu$, or a voter model on $\mathbb{Z}$ in which interactions occur over a distance of order $K$. Sample one individual at the origin and one at $L$. We show that if $M\nu/L$ and $L/K^2$ converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.