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Abstract: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 $\epsilon$ 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
$(\log\epsilon^{-1})^{-1}$, and at least $\frac{\sigma_*^2}{\pi d\log\epsilon^{-1}}$, where
$\sigma_*^2$ 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 $\epsilon=0$, 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 $\epsilon$.

Submission history

From: David Steinsaltz [view email]
[v1] Tue, 11 Sep 2018 16:49:39 UTC (26 KB)
[v2] Sun, 13 Sep 2026 16:04:05 UTC (36 KB)