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

Abstract

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)agespecific 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 sizeclassified 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.

Variance in animal longevity: contributions of heterogeneity and stochasticity

“…Partially because of its evolutionary implications, much of the interest in unobserved heterogeneity in fitness components focuses on accounting for variance (e.g., Caswell 2011Caswell , 2014Steiner and Tuljapurkar 2012;Vindenes and Langangen 2015;Cam et al 2016;Hartemink et al 2017;van Daalen and Caswell 2017;Jenouvrier et al 2018). In this case, we found that heterogeneity could typically account for less than half of the variance in longevity (35%, with interquartile range 23-44%).…”

Age × stage‐classified demographic analysis: a comprehensive approach

“…But in general, lifetime offspring production in age‐stage models is a more diverse and nuanced concept than R 0 . Note that much more information about lifetime reproductive output, including variances, higher moments, and sensitivity analysis can be obtained using Markov chains with rewards (Caswell , van Daalen and Caswell , ). The application of these methods to age × stage‐classified models will be explored elsewhere.…”

General conclusion to the special issue Moving forward on individual heterogeneity

“…Nonetheless, recent developments in stochastic models (so-called Markov chains with rewards) provide a powerful and general approach to analyzing lifetime reproductive success for age-classified, stage-classified, and multistate models, for any kind of reproductive output distributions, and including a general sensitivity analysis (van Daalen and Caswell 2017). These analyses have shown that the stochasticity within the individual life cycle produces much more variation in lifetime reproductive output than might be expected (Caswell 2011, van Daalen andCaswell 2017), with consequences to be explored at the population level (Caswell and Vindenes 2018).…”

“…These analyses have shown that the stochasticity within the individual life cycle produces much more variation in lifetime reproductive output than might be expected (Caswell 2011, van Daalen andCaswell 2017), with consequences to be explored at the population level (Caswell and Vindenes 2018).…”