Genetic drift in populations governed by a Galton–Watson branching process

Abstract

Most population genetics studies have their origins in a Wright–Fisher or some closely related fixed-population model in which each individual randomly chooses its ancestor. Populations which vary in size with time are typically modelled via a coalescent derived from Wright–Fisher, but use a nonlinear time-scaling driven by a deterministically imposed population growth. An alternate, arguably more realistic approach, and one which we take here, is to allow the population size to vary stochastically via a Galton–Watson branching process.We study genetic drift in a population consisting of a number of distinct allele types in which each allele type evolves as an independent Galton–Watson branching process. We find the dynamics of the population is determined by a single parameter κ0=(2m0/σ2)logλ, where m0 is the initial population size, λ is the mean number of offspring per individual; and σ2 is the variance of the number of offspring. For 0≲κ0≪1, the dynamics are close to those of Wright–Fisher, with the added property that the population is prone to extinction. For κ0≫1 allele frequencies and ancestral lineages are stable and individual alleles do not fix throughout the population. The existence of a rapid changeover regime at κ0≈1 enables estimates to be made, together with confidence intervals, of the time and population size of the era of mitochondrial Eve.