Stochastic growth rates for life histories with rare migration or diapause

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. If the sites are reinterpreted as age classes, the same model may apply to a single population with age-dependent mortality and reproduction. We consider the case where the off-diagonal elements are small, representing a situation where there is little migration or, alternatively, where a deterministic life-history has been slightly disrupted, for example by introducing a rare delay in development. We examine the asymptotic behaviour of the long-term growth rate. We show that when the highest growth rate is attained at two different sites in the absence of migration (which is always the case when modelling a single age-structured population) the increase in stochastic growth rate due to a migration rate $\epsilon$ is like $(\log \epsilon^{-1})^{-1}$ as $\epsilon\downarrow 0$, under fairly generic conditions. When there is a single site with the highest growth rate the behavior is more delicate, depending on the tails of the growth rates. For the case when the log growth rates have 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.