  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.
