Stochastic growth rates for populations in random environments with rare migration

Abstract: The growth of a population divided among spatial sites, with migration between the sites, is sometimes modelled by a product of random matrices, with each diagonal elements representing the growth rate in a given time period, and off-diagonal elements the migration rate. The randomness of the matrices then represents stochasticity of environmental conditions. We consider the case where the off-diagonal elements are small, representing a situation where migration has been introduced into an otherwise sessile meta-population. We examine the asymptotic behaviour of the long-term growth rate. When there is a single site with the highest growth rate, under the assumption of Gaussian log growth rates at the individual sites (or having Gaussian-like tails) we show that the behavior near zero is like a power of $\epsilon$, and derive upper and lower bounds for the power in terms of the difference in the growth rates and the distance between the sites. In particular, when the difference in mean log growth rate between two sites is sufficiently small, or the variance of the difference between the sites sufficiently large, migration will always be favored by natural selection, in the sense that introducing a small amount of migration will increase the growth rate of the population relative to the zero-migration case.