First passage time to allopatric speciation.

Allopatric speciation is a mechanism to evolve reproductive isolation; it is caused by the accumulation of genetic differences between populations while they are geographically isolated. Here, we studied a simple stochastic model for the time until speciation caused by geographical isolation in fragmented populations that experience recurrent but infrequent migration between subpopulations. We assumed that mating incompatibility is controlled by a number of loci that behave as neutral characters in the accumulation of novel mutations within each population. Genetic distance between populations was defined as the number of incompatibility-controlling loci that differ between them. Genetic distance increases through the separate accumulation of mutations in different populations, but decreases after a successful migration event followed by genetic mixing between migrants and residents. We calculated the time to allopatric speciation, which occurs when the genetic distance exceeds a specified threshold. If the number of invasive individuals relative to the resident population is not very large, diffusion approximation provides an accurate prediction. There is an intermediate optimal rate of migration that maximizes the rate of species creation by recurrent invasion and diversification. We also examined cases that involved more than two populations.