Critical migration rate in the evolutionary dynamics of subdivided populations

We investigate the evolution model which consists of subdivided populations of individual agents confined to three islands. The individuals are haploid individuals with three types. In subdivided populations, migration acts in parallel with natural selection to determine the evolutionary dynamics. We call this model the (3, 3) parallel migration-selection (ParaMiSe) model. The individuals on each island reproduce with the probability proportional to their fitness values and die at random, and migrate between two islands. The evolutionary dynamics of the (3, 3) ParaMiSe model is formulated as a master equation and can be approximated as the Fokker-Planck equation and the stochastic differential equations (SDEs). By taking the infinite population limit, we obtain the deterministic parts of the SDEs and observe that there exists a second-order continuous phase transition in the population density from the monomorphic state to the polymorphic state at the critical migration rate. We find the critical migration rate as a function of the median fitness value on the islands and predict the critical migration rate for the general (D, D) ParaMiSe model with D > 3. We also carry out Monte Carlo simulations using the Gillespie-type algorithm and find that the results of Monte Carlo simulations are in good agreement with the analytic predictions.