Stability of fixed life histories to perturbation by rare diapause

We analyze the behavior of an age-structured population subject to stochastically varying linear survival and reproduction at age-dependent rates, in the special case where births occur only when organisms attain a fixed maximum age $d$, so that generations have a constant length $d$. We show that perturbing this fixed-length life history by a small diapause — a delay in development, corresponding to adding diagonal terms of size $ϵ$ to the matrix that updates the population vector from one time period to the next — increases the asymptotic stochastic growth rate by an increment of order $(\log {ϵ}^{-1}{)}^{-1}$, and at least $\frac{{\sigma }_{\ast }^2}{\pi d\log {ϵ}^{-1}}$, where ${\sigma }_{\ast }^2$ is a sum of variances of log ratios of survival and birth rates one age class apart. The growth rate is thus continuous but not differentiable at $ϵ=0$, which is why the question has resisted the standard perturbative methods. As this effect dominates any linear cost suffered by individuals who are subject to diapause, it follows that a small random disruption to the deterministic life history would be favored by natural selection, in the sense that it would increase the stochastic growth rate relative to the zero-delay deterministic life history. We prove this in the wider setting of matrix migration models in which two or more sites share the maximum mean growth rate — a degeneracy that the fixed life history forces, and that is excluded in models with a single optimal site, where the growth rate instead increases like a power of $ϵ$.