Mutation in Populations Governed by a Galton-Watson Branching Process

TL;DR

This paper models population genetics using a multitype branching process, deriving equations for mutation effects, stationary distributions, and phase transitions, with validation through numerical simulations.

Contribution

It introduces a multitype branching process framework for population genetics, deriving new diffusion equations and analyzing phase transitions between drift and mutation dominance.

Findings

Derived the diffusion limit forward Kolmogorov equation for neutral mutations.

Obtained the asymptotic stationary distribution partitioning populations by mutation rates.

Confirmed the approximate solutions with numerical simulations.

Abstract

A population genetics model based on a multitype branching process, or equivalently a Galton-Watson branching process for multiple alleles, is pre- sented. The diffusion limit forward Kolmogorov equation is derived for the case of neutral mutations. The asymptotic stationary solution is obtained and has the property that the extant population partitions into subpopulations whose relative sizes are determined by mutation rates. An approximate time-dependent solution is obtained in the limit of low mutation rates. This solution has the property that the system undergoes a rapid transition from a drift-dominated phase to a mutation-dominated phase in which the distribution collapses onto the asymptotic stationary distribution. The changeover point of the transition is determined by the per-generation growth factor and mutation rate. The approximate solution is confirmed using numerical simulations.