Identity by descent in island-mainland populations

A new model is presented for the genetic structure among a collection of island populations, with fluctuating population sizes and continuous overlapping generations, using a stochastic birth, death and immigration (BDI) process. Immigrants enter each island from a large mainland population, with constant gene frequencies, according to a Poisson process. The average probability of identity by descent (IBD) for two haploid individuals randomly selected from an island population is f0 = (phi f1 + lambda)/(phi + lambda), where f1 is the probability of IBD for two randomly selected immigrants, lambda is the birth-rate for each individual, and phi is the arrival rate of immigrants into each island. The value of f0 is independent of the death process, time and N. The expected level of genetic differentiation among island populations is FST = (1 - 1/n)lambda/(phi + lambda), where n is the total number of islands receiving immigrants. Because f0 and FST are independent of the death process, for a BDI model, the population genetic structure for several general demographic situations may be examined using our equations. These include stochastic exponential, or logistic (regulated by death rate) growth within islands, or a ""source-sink" population structure. Because the expected values of both f0 and FST are independent of time, these are achieved immediately, for a BDI model, with no need to assume the island populations are at genetic equilibrium.