固定生活史对稀有滞育扰动的稳定性
Stability of fixed life histories to perturbation by rare diapause
AI总结：
本研究证明，在固定最大年龄的种群模型中，引入小的随机滞育（发育延迟）可使渐近随机增长率增加至少 $\frac{\sigma_*^2}{\pi d\log\epsilon^{-1}}$，且该效应优于线性成本，表明自然选择会偏好此类扰动。
摘要：
我们分析了一个年龄结构种群在年龄依赖的随机变化的线性存活和繁殖率下的行为，特别关注于出生仅发生在生物体达到固定最大年龄 $d$ 时的情况，因此世代具有恒定长度 $d$。我们表明，通过一个小的滞育——即发育延迟，相当于在从一个时间段更新到下一个时间段的种群向量矩阵中添加大小为 $\epsilon$ 的对角项——来扰动这种固定长度的生活史，会使渐近随机增长率增加一个数量级为 $(\log\epsilon^{-1})^{-1}$ 的增量，并且至少为 $\frac{\sigma_*^2}{\pi d\log\epsilon^{-1}}$，其中 $\sigma_*^2$ 是相隔一个年龄等级的存活率和出生率的对数比率方差的求和。因此，增长率在 $\epsilon=0$ 处是连续的但不可微，这就是为什么该问题一直抵制标准微扰方法的原因。由于这种效应主导了滞育个体所承受的任何线性成本，因此，对确定性生活史的小随机扰动将受到自然选择的青睐，从某种意义上说，它将相对于零延迟的确定性生活史增加随机增长率。我们在更广泛的矩阵迁移模型背景下证明了这一点，其中两个或多个位点共享最大平均增长率——这是固定生活史所强制的退化，而在具有单个最优位点的模型中被排除，因为在那里增长率反而像 $\epsilon$ 的幂次一样增加。
英文摘要：
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 $(\logε^{-1})^{-1}$, and at least $\frac{σ_*^2}{πd\logε^{-1}}$, where $σ_*^2$ 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 $ε$.
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