Evolutionary dynamics with migration by using the fast-mode elimination method

Abstract

We investigate the evolutionary dynamics of a Moran 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 only one stable fixed point to which all populations started from any initial distribution other than the unstable absorbing points will evolve in both the symmetric and antisymmetric selection schemes. 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.