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The growth of a population divided among spatial sites, with migrationbetween the sites, is sometimes modelled by a product of random matrices, witheach diagonal elements representing the growth rate in a given time period, andoff-diagonal elements the migration rate. The randomness of the matrices thenrepresents stochasticity of environmental conditions. We consider the casewhere the off-diagonal elements are small, representing a situation wheremigration has been introduced into an otherwise sessile meta-population. Weexamine the asymptotic behaviour of the long-term growth rate. When there is asingle site with the highest growth rate, under the assumption of Gaussian loggrowth rates at the individual sites (or having Gaussian-like tails) we showthat the behavior near zero is like a power of $\epsilon$, and derive upper andlower bounds for the power in terms of the difference in the growth rates andthe distance between the sites. In particular, when the difference in mean loggrowth rate between two sites is sufficiently small, or the variance of thedifference between the sites sufficiently large, migration will always befavored by natural selection, in the sense that introducing a small amount ofmigration will increase the growth rate of the population relative to thezero-migration case.

Stochastic growth rates for populations in random environments with rare migration