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We follow up on a companion work that considered growth rates of populationsgrowing at different sites, with different randomly varying growth rates ateach site, in the limit as migration between sites goes to 0. We extend thiswork here to the special case where the maximum average log growth rate isachieved at two different sites. The primary motivation is to cover the casewhere `sites' are understood as age classes for the same individuals. Thetheory then calculates the effect on growth rate of introducing a rare delay indevelopment, a diapause, into an otherwise fixed-length semelparous lifehistory.  Whereas the increase in stochastic growth rate due to rare migrations wasfound to grow as a power of the migration rate, we show that under quitegeneral conditions that in the diapause model --- or in the migration modelwith two or more sites having equal individual stochastic growth rates --- theincrease in stochastic growth rate due to diapause at rate $\epsilon$ behaveslike $(\log \epsilon^{-1})^{-1}$ as $\epsilon\downarrow 0$. In particular, thisimplies that a small random disruption to the deterministic life history willalways be favored by natural selection, in the sense that it will increase thestochastic growth rate relative to the zero-delay deterministic life history.

Stability of fixed life histories to perturbation by rare diapause