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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. If the sites are reinterpreted as ageclasses, the same model may apply to a single population with age-dependentmortality and reproduction.  We consider the case where the off-diagonal elements are small, representinga situation where there is little migration or, alternatively, where adeterministic life-history has been slightly disrupted, for example byintroducing a rare delay in development. We examine the asymptotic behaviour ofthe long-term growth rate. We show that when the highest growth rate isattained at two different sites in the absence of migration (which is alwaysthe case when modelling a single age-structured population) the increase instochastic growth rate due to a migration rate $\epsilon$ is like $(\log\epsilon^{-1})^{-1}$ as $\epsilon\downarrow 0$, under fairly genericconditions. When there is a single site with the highest growth rate thebehavior is more delicate, depending on the tails of the growth rates. For thecase when the log growth rates have Gaussian-like tails we show that thebehavior near zero is like a power of $\epsilon$, and derive upper and lowerbounds for the power in terms of the difference in the growth rates and thedistance between the sites.

Stochastic growth rates for life histories with rare migration or diapause