The Contributions of Maternal Age Heterogeneity to Variance in Lifetime Reproductive Output.

AbstractVariance among individuals in fitness components reflects both genuine heterogeneity between individuals and stochasticity in events experienced along the life cycle. Maternal age represents a form of heterogeneity that affects both the mean and the variance of lifetime reproductive output (LRO). Here, we quantify the relative contribution of maternal age heterogeneity to the variance in LRO using individual-level laboratory data on the rotifer Brachionus manjavacas to parameterize a multistate age × maternal age matrix model. In B. manjavacas, advanced maternal age has large negative effects on offspring survival and fertility. We used multistate Markov chains with rewards to quantify the contributions to variance in LRO of heterogeneity and of the stochasticity inherent in the outcomes of probabilistic transitions and reproductive events. Under laboratory conditions, maternal age heterogeneity contributes 26% of the variance in LRO. The contribution changes when mortality and fertility are reduced to mimic more ecologically relevant environments. Over the parameter space where populations are near stationarity, maternal age heterogeneity contributes an average of 3% of the variance. Thus, the contributions of maternal age heterogeneity and individual stochasticity can be expected to depend strongly on environmental conditions; over most of the parameter space, the variance in LRO is dominated by stochasticity.

A demographic and evolutionary analysis of maternal effect senescence.

Maternal effect senescence-a decline in offspring survival or fertility with maternal age-has been demonstrated in many taxa, including humans. Despite decades of phenotypic studies, questions remain about how maternal effect senescence impacts evolutionary fitness. To understand the influence of maternal effect senescence on population dynamics, fitness, and selection, we developed matrix population models in which individuals are jointly classified by age and maternal age. We fit these models to data from individual-based culture experiments on the aquatic invertebrate, Brachionus manjavacas (Rotifera). By comparing models with and without maternal effects, we found that maternal effect senescence significantly reduces fitness for B. manjavacas and that this decrease arises primarily through reduced fertility, particularly at maternal ages corresponding to peak reproductive output. We also used the models to estimate selection gradients, which measure the strength of selection, in both high growth rate (laboratory) and two simulated low growth rate environments. In all environments, selection gradients on survival and fertility decrease with increasing age. They also decrease with increasing maternal age for late maternal ages, implying that maternal effect senescence can evolve through the same process as in Hamilton's theory of the evolution of age-related senescence. The models we developed are widely applicable to evaluate the fitness consequences of maternal effect senescence across species with diverse aging and fertility schedule phenotypes.

Age × stage-classified demographic analysis: a comprehensive approach.

This paper presents a comprehensive theory for the demographic analysis of populations in which individuals are classified by both age and stage. The earliest demographic models were age classified. Ecologists adopted methods developed by human demographers and used life tables to quantify survivorship and fertility of cohorts and the growth rates and structures of populations. Later, motivated by studies of plants and insects, matrix population models structured by size or stage were developed. The theory of these models has been extended to cover all the aspects of age-classified demography and more. It is a natural development to consider populations classified by both age and stage. A steady trickle of results has appeared since the 1960s, analyzing one or another aspect of age × stage-classified populations, in both ecology and human demography. Here, we use the vec-permutation formulation of multistate matrix population models to incorporate age- and stage-specific vital rates into demographic analysis. We present cohort results for the life table functions (survivorship, mortality, and fertility), the dynamics of intra-cohort selection, the statistics of longevity, the joint distribution of age and stage at death, and the statistics of life disparity. Combining transitions and fertility yields a complete set of population dynamic results, including population growth rates and structures, net reproductive rate, the statistics of lifetime reproduction, and measures of generation time. We present a complete analysis of a hypothetical model species, inspired by poecilogonous marine invertebrates that produce two kinds of larval offspring. Given the joint effects of age and stage, many familiar demographic results become multidimensional, so calculations of marginal and mixture distributions are an important tool. From an age-classified point of view, stage structure is a form of unobserved heterogeneity. From a stage-classified point of view, age structure is unobserved heterogeneity. In an age × stage-classified model, variance in demographic outcomes can be partitioned into contributions from both sources. Because these models are formulated as matrices, they are amenable to a complete sensitivity analysis. As more detailed and longer longitudinal studies are developed, age × stage-classified demography will become more common and more important.

Lifetime reproductive output: individual stochasticity, variance, and sensitivity analysis.

Lifetime reproductive output (LRO) determines per-generation growth rates, establishes criteria for population growth or decline, and is an important component of fitness. Empirical measurements of LRO reveal high variance among individuals. This variance may result from genuine heterogeneity in individual properties, or from individual stochasticity, the outcome of probabilistic demographic events during the life cycle. To evaluate the extent of individual stochasticity requires the calculation of the statistics of LRO from a demographic model. Mean LRO is routinely calculated (as the net reproductive rate), but the calculation of variances has only recently received attention. Here, we present a complete, exact, analytical, closed-form solution for all the moments of LRO, for age- and stage-classified populations. Previous studies have relied on simulation, iterative solutions, or closed-form analytical solutions that capture only part of the sources of variance. We also present the sensitivity and elasticity of all of the statistics of LRO to parameters defining survival, stage transitions, and (st)age-specific fertility. Selection can operate on variance in LRO only if the variance results from genetic heterogeneity. The potential opportunity for selection is quantified by Crow's index I , the ratio of the variance to the square of the mean. But variance due to individual stochasticity is only an apparent opportunity for selection. In a comparison of a range of age-classified models for human populations, we find that proportional increases in mortality have very small effects on the mean and variance of LRO, but large positive effects on I . Proportional increases in fertility increase both the mean and variance of LRO, but reduce I . For a size-classified tree population, the elasticity of both mean and variance of LRO to stage-specific mortality are negative; the elasticities to stage-specific fertility are positive.