Coalescent approximation for structured populations in a stationary random environment

TL;DR

This paper proves that the genealogy of a structured population with randomly changing migration probabilities converges to the Kingman coalescent, introducing a new formula for the effective population size in a random environment.

Contribution

It establishes convergence to the Kingman coalescent in a structured population model with stochastic migration, and introduces a novel quenched effective population size formula.

Findings

Convergence to Kingman coalescent under random migration.

Introduction of a quenched effective population size formula.

Comparison between quenched and annealed EPS.

Abstract

We establish convergence to the Kingman coalescent for the genealogy of a geographically - or otherwise - structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model - random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration probabilities over possible environments.