Dynamics of Fst for the island model

Fst is a measure of genetic differentiation in a subdivided population. Sewall Wright observed that in a haploid diallelic infinite island model, where N is the effective population size of each deme and m is the migration rate. In demonstrating this result, Wright relied on the infinite size of the population. Natural populations are not infinite and therefore they change over time due to genetic drift. In a finite population, Fst becomes a random variable that evolves over time. In this work we ask, given an initial population state, what are the dynamics of the mean and variance of Fst under the finite island model? In application both of these quantities are critical in the evaluation of Fst data. We show that after a time of order N generations the mean of Fst is slightly biased below . Further we show that the variance of Fst is of order where d is the number of demes in the population. We introduce several new mathematical techniques to analyze coalescent genealogies in a dynamic setting.