Dynamics of FstFst for the island model

FstFst is a measure of genetic differentiation in a subdivided population. Sewall Wright observed that Fst=11+2Nm in a haploid diallelic infinite island model, where NN is the effective population size of each deme and mm 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, FstFst 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 FstFst under the finite island model? In application both of these quantities are critical in the evaluation of FstFst data.We show that after a time of order NN generations the mean of FstFst is slightly biased below 11+2Nm. Further we show that the variance of FstFst is of order 1d where dd is the number of demes in the population. We introduce several new mathematical techniques to analyze coalescent genealogies in a dynamic setting.