Demographic-noise-induced fixation in subdivided populations with migration

We investigate the stochastic dynamics of a model which consists of subdivided populations of individuals confined to a set of islands. In subdivided populations, migration acts with selection and genetic drift to determine the evolutionary dynamics. The individuals are assumed to be haploid with two types. They reproduce according to their fitness values, die at random, and migrate between the islands. The evolutionary dynamics of an individual-based model is formulated in terms of a master equation and is approximated as the multidimensional Fokker-Planck equation (FPE) and the coupled non-linear stochastic differential equations (SDEs) with multiplicative noise. We first analyze the deterministic part of the SDEs, to obtain the fixed points and determine the stability of each fixed point. We find that there is a continuous phase transition in the population distribution when the migration rate is equal to the selection strength in the antisymmetric selection scheme. Next, we take demographic stochasticity into account, and analyze the FPE by eliminating the fast variable to reduce the coupled two-variable FPE to the single-variable FPE. We derive a quasi-stationary distribution of the reduced FPE, and predict the fixation probabilities and the mean fixation times to absorbing states. We also carry out numerical simulations, in the form of the Gillespie algorithm, and find that the results of simulations are consistent with the analytic predictions.