Mean and variance of F_ST in a finite number of incompletely isolated populations

In the presence of migration F_ST in a finite number of incompletely isolated populations first increases, but after reaching a certain maximum value, it starts to decline and eventually becomes 0. The mean and variance of F_ST in this process are studied by using the recurrence formulas for the moments of gene frequencies in the island model of finite size as well as by using Monte Carlo simulation. The mean and variance in the early generations can be predicted by the approximate formulas developed. On the other hand, if we exclude the cases of an allele being fixed in all subpopulations, the mean of F_ST eventually reaches a steady-state value. This value is given by 1 - 2N_T(1 - λ) approximately, where N_T is the total population size and λ is the rate of decay of heterozygosity at steady state. It is shown that the mean and variance of F_ST depend on the initial gene frequency and when this is close to 0 or 1, Lewontin and Krakauer's test of the neutrality of polymorphic genes is not valid.