Mutation in Populations Governed by a Galton-Watson Branching Process

2017/07/17 by Conrad J. Burden, Yi Wei, Burden, Conrad J. +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Social Sciences · #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Biological sciences #Populations and Evolution (q-bio.PE) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1707.04980

openalex publication_date 2017/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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