Exact Solution of Population Redistributions in a Migration Model

We study a migration model, in which individuals migrate from one community to another. The choices of the source community i and the destination one j are proportional to some power of the population of i (k^α_i) and j (k^β_j), respectively. Both analytical calculation and numerical simulation show that the population distribution of communities in stationary states is determined by the parameters α and β. The distribution is widely homogeneous with a characteristic size if α > β. Whereas, for α < β, the distribution is highly heterogeneous with the emergence of condensing phenomenon. Between the two regimes, α = β, the distribution gradually shifts from the nonmonotonous (α < 0) to scale-free (α > 0).