Selection Against Demographic Stochasticity in Age-Structured Populations

Abstract

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 article, 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 agespecific 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 mean 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 mean and the variance in growth rate may be necessary to determine the sensitivity of fitness to changes in the life-history parameters.

A Stochastic Model for Annual Reproductive Success

“…This "demographic variance" (Engen et al 1998) has a large influence on the risk of extinction due to demographic stochasticity: all else being equal, a higher variance leads to higher extinction risk (e.g., Bartlett 1955). Furthermore, in small populations, natural selection will tend to reduce the demographic variance if this can be done without reducing the mean fitness (Gillespie 1975(Gillespie , 1977; although initially developed for small, unstructured populations, this model has recently been expanded to spatially structured and agestructured populations [Lehmann and Balloux 2007;Shpak 2007]). In both applications it is important to have a model that does not unnecessarily constrain the relationship between the variance and the mean of reproductive success.…”

“…This stochasticity often reduces the long-term population growth rate and increases extinction risk (Bartlett 1955) and may be selected against (Gillespie 1975(Gillespie , 1977Lehmann and Balloux 2007;Shpak 2007). There are exceptions to this general pattern (e.g., Doak et al 2005), and stochasticity can, under appropriate circumstances, promote the coexistence of competing species (e.g., Chesson 2000).…”

The probability of evolutionary rescue: towards a quantitative comparison between theory and evolution experiments

“…We use a Feller diffusion process [23], also called 'continuous state branching process' (see the electronic supplementary material, appendix S1), to approximate the demographic process. This approximation reduces all the complexity of the life cycle into two key parameters: the mean reproductive output r (growth rate or Malthusian fitness), and the variance of this output s (for use of similar diffusion approximation in age-structured populations, see Shpak [24]). This diffusion approximation and its range of validity are more detailed in the electronic supplementary material, appendix S1.…”

Reproductive Value and Fluctuating Selection in an Age-Structured Population

“…In stochastic models of genetic drift in a finite population, when using the Ito calculus to compute the infinitesimal mean and variance of a diffusion approximation, it is often assumed that the environment influences each genotype identically (Engen et al 2005a;Shpak 2007). However, in models of fluctuating selection (Lande 2007(Lande , 2008 stochastic environments exert distinct influences on the demography of different genotypes, producing a positive environmental variance contributing to the infinitesimal variance b(p) driving stochastic changes in allele frequencies even in populations sufficiently large to neglect genetic drift.…”

“…Genetic drift in a small age-structured population can be represented by an additional term in the infinitesimal variance of the diffusion approximation (Engen et al 2005b;Shpak 2007). The model can then be used to analyze the probability of fixation and the time to fixation as the boundaries now become accessible to the diffusion process.…”