Analytical solution of metapopulation dynamics in a stochastic environment.

We study a discrete stochastic linear metapopulation model to understand the effect of risk spreading by dispersion. We calculate analytically the stable distribution of populations in different habitats. The simultaneous distribution of populations in habitats has a complicated self-similar structure, but the population in each habitat follows a log-normal distribution. A class of discrete stochastic matrix models was mostly dealt with numerically. Our analytical predictions are robust in the wide range of parameters. Qualitative predictions of the current results should hold in the case of multiple habitats. We thus conclude that environmental stochasticity always promotes dispersal.