Selection Against Demographic Stochasticity in Age-Structured Populations

It has been shown that differences in fecundity variance can influence the probability of invasion of a genotype in a population, i.e. a genotype with lower variance in offspring number can be favored in finite populations even if it has a somewhat lower mean fitness than a competitor. In this paper, Gillespie’s results are extended to population genetic systems with explicit age structure, where the demographic variance (variance in growth rate) calculated in the work of Engen and colleagues is used as a generalization of “variance in offspring number” to predict the interaction between deterministic and random forces driving change in allele frequency. By calculating the variance from the life history parameters, it is shown that selection against variance in the growth rate will favor a genotypes with lower stochasticity in age specific survival and fertility rates. A diffusion approximation for selection and drift in a population with two genotypes with different life history matrices (and therefore, different growth rates and demographic variances) is derived and shown to be consistent with individual based simulations. It is also argued that for finite populations, perturbation analyses of both the growth rate and demographic variances may be necessary to determine the sensitivity of “fitness” (broadly defined) to changes in the life history parameters.

💡 Research Summary

The paper extends Gillespie’s “selection against variance” concept to age‑structured populations by employing the demographic variance defined in the work of Engen et al. (2005). In a classic unstructured model, variance in offspring number can outweigh a modest reduction in mean fitness, allowing a genotype with lower reproductive variance to invade. However, natural populations are rarely age‑independent; survival and fertility differ across life stages, and thus a more general measure of stochasticity is required.

The authors model each genotype by a Leslie matrix (L_A and L_B) that encapsulates age‑specific survival (s_x) and fecundity (m_x). The dominant eigenvalue λ of each matrix gives the long‑term deterministic growth rate, while the associated right eigenvector w and left eigenvector v describe the stable age distribution and reproductive value, respectively. When the matrix elements are treated as random variables with known covariances, the variance of the stochastic growth rate (the “demographic variance” σ²) can be approximated by

 σ² ≈ (wᵀ Cov(L) w) / (wᵀ v)² .

Thus, both the mean growth rate r = log λ and the demographic variance σ² are functions of the underlying life‑history parameters.

Using diffusion theory, the change in allele frequency p of genotype A in a finite population of size N is expressed as

 dp = p(1‑p)