Evolutionary game dynamics in finite populations with migration

We investigate the impact of migration on the evolutionary game dynamics of finite populations with community structures in the framework of evolutionary game theory. Rather than deterministic dynamics, our model incorporates stochastic factors induced by the finite population size. Depending on analysis of the stationary distribution of the evolutionary process in the limit of rare mutations, we prove that migration has no effect on the imitation process of strategies but has a significant influence on the competition process of communities. The population spends most time in the homogeneous state where all individuals belong to the community with the lower migration rate. Furthermore, we find that reducing the difference between migration rates of two communities can increase the first hitting time to the homogeneous absorbing state, promoting the conservation of biodiversity indirectly.