The natural history of luck: A synthesis study of structured population models.

Chance pervades life. In turn, life histories are described by probabilities (e.g. survival and breeding) and averages across individuals (e.g. mean growth rate and age at maturity). In this study, we explored patterns of luck in lifetime outcomes by analysing structured population models for a wide array of plant and animal species. We calculated four response variables: variance and skewness in both lifespan and lifetime reproductive output (LRO), and partitioned them into contributions from different forms of luck. We examined relationships among response variables and a variety of life history traits. We found that variance in lifespan and variance in LRO were positively correlated across taxa, but that variance and skewness were negatively correlated for both lifespan and LRO. The most important life history trait was longevity, which shaped variance and skew in LRO through its effects on variance in lifespan. We found that luck in survival, growth, and fecundity all contributed to variance in LRO, but skew in LRO was overwhelmingly due to survival luck. Rapidly growing populations have larger variances in LRO and lifespan than shrinking populations. Our results indicate that luck-induced genetic drift may be most severe in recovering populations of species with long mature lifespan and high iteroparity.

To Prosper, Live Long: Understanding the Sources of Reproductive Skew and Extreme Reproductive Success in Structured Populations.

AbstractIn many species, a few individuals produce most of the next generation. How much of this reproductive skew is driven by variation among individuals in fixed traits, how much by external factors, and how much by random chance? And what does it take to have truly exceptional lifetime reproductive output (LRO)? In the past, we and others have partitioned the variance of LRO as a proxy for reproductive skew. Here we explain how to partition LRO skewness itself into contributions from fixed trait variation, four forms of "demographic luck" (birth state, fecundity luck, survival trajectory luck, and growth trajectory luck), and two kinds of "environmental luck" (birth environment and environment trajectory). Each of these is further partitioned into contributions at different ages. We also determine what we can infer about individuals with exceptional LRO. We find that reproductive skew is largely driven by random variation in lifespan, and exceptional LRO generally results from exceptional lifespan. Other kinds of luck frequently bring skewness down rather than increasing it. In populations where fecundity varies greatly with environmental conditions, getting a good year at the right time can be an alternate route to exceptional LRO, so that LRO is less predictive of lifespan.

Hurricanes affect diversification among individual life courses of a primate population.

Extreme climatic events may influence individual-level variability in phenotypes, survival and reproduction, and thereby drive the pace of evolution. Climate models predict increases in the frequency of intense hurricanes, but no study has measured their impact on individual life courses within animal populations. We used 45 years of demographic data of rhesus macaques to quantify the influence of major hurricanes on reproductive life courses using multiple metrics of dynamic heterogeneity accounting for life course variability and life-history trait variances. To reduce intraspecific competition, individuals may explore new reproductive stages during years of major hurricanes, resulting in higher temporal variation in reproductive trajectories. Alternatively, individuals may opt for a single optimal life-history strategy due to trade-offs between survival and reproduction. Our results show that heterogeneity in reproductive life courses increased by 4% during years of major hurricanes, despite a 2% reduction in the asymptotic growth rate due to an average decrease in mean fertility and survival by that is, shortened life courses and reduced reproductive output. In agreement with this, the population is expected to achieve stable population dynamics faster after being perturbed by a hurricane ( ρ = 1.512 ; 95% CI: 1.488, 1.538), relative to ordinary years ρ = 1.482 ; 1.475 , 1.490 . Our work suggests that natural disasters force individuals into new demographic roles to potentially reduce competition during unfavourable environments where mean reproduction and survival are compromised. Variance in lifetime reproductive success and longevity are differently affected by hurricanes, and such variability is mostly driven by survival.

When the going gets tough, the tough get going: Effect of extreme climate on an Antarctic seabird's life history.

Individuals differ in many ways. Most produce few offspring; a handful produce many. Some die early; others live to old age. It is tempting to attribute these differences in outcomes to differences in individual traits, and thus in the demographic rates experienced. However, there is more to individual variation than meets the eye of the biologist. Even among individuals sharing identical traits, life history outcomes (life expectancy and lifetime reproduction) will vary due to individual stochasticity, that is to chance. Quantifying the contributions of heterogeneity and chance is essential to understand natural variability. Interindividual differences vary across environmental conditions, hence heterogeneity and stochasticity depend on environmental conditions. We show that favourable conditions increase the contributions of individual stochasticity, and reduce the contributions of heterogeneity, to variance in demographic outcomes in a seabird population. The opposite is true under poor conditions. This result has important consequence for understanding the ecology and evolution of life history strategies.

Snared in an Evil Time: How Age-Dependent Environmental and Demographic Variability Contribute to Variance in Lifetime Outcomes.

AbstractTo what degree is lifetime success determined by innate individual quality versus external events and random chance, whether success is measured by lifetime reproductive output, life span, years that a tree spends in the canopy, or some other measure? And how do external events and chance interact with development (survival and growth) to drive success? To answer these questions, we extend our earlier age partitioning of luck in lifetime outcomes in two ways: we incorporate effects of external environmental variation, and we subdivide demographic luck into contributions from survival and growth. Applying our methods to four case studies, we find that luck in survival, in growth, or in environmental variation can all be the dominant driver of success, depending on life history, but variation in individual quality remains a lesser driver. Luck in its various forms is most important at very early ages and again close to the time when individuals typically first begin to be successful (e.g., entering the canopy, reaching reproductive maturity), but different forms of luck peak at different times. For example, a favorable year can boost a tree into the canopy, while luck in survival is required to take full advantage of that fortunate event.

Demystifying individual heterogeneity.

Among-individual variation in vital rates, such as mortality and birth rates, exists in nearly all populations. Recent studies suggest that this individual heterogeneity produces substantial life-history and fitness differences among individuals, which in turn scale up to influence population dynamics. However, our ability to understand the consequences of individual heterogeneity is limited by inconsistencies across conceptual frameworks in the field. Studies of individual heterogeneity remain filled with contradicting and ambiguous terminology that introduces risks of misunderstandings, conflicting models and unreliable conclusions. Here, we synthesise the existing literature into a single and comparatively straightforward framework with explicit terminology and definitions. This work introduces a distinction between potential vital rates and realised vital rates to develop a coherent framework that maps directly onto mathematical models of individual heterogeneity. We suggest the terms "fixed condition" and "dynamic condition" be used to distinguish potential vital rates that are permanent from those that can change throughout an individual's life. To illustrate, we connect the framework to quantitative genetics models and to common classes of statistical models used to infer individual heterogeneity. We also develop a population projection matrix model that provides an example of how our definitions are translated into precise quantitative terms.

Time and Chance: Using Age Partitioning to Understand How Luck Drives Variation in Reproductive Success.

AbstractOver the course of individual lifetimes, luck usually explains a large fraction of the between-individual variation in life span or lifetime reproductive output (LRO) within a population, while variation in individual traits or "quality" explains much less. To understand how, where in the life cycle, and through which demographic processes luck trumps trait variation, we show how to partition by age the contributions of luck and trait variation to LRO variance and how to quantify three distinct components of luck. We apply these tools to several empirical case studies. We find that luck swamps effects of trait variation at all ages, primarily because of randomness in individual state dynamics ("state trajectory luck"). Luck early in life is most important. Very early state trajectory luck generally determines whether an individual ever breeds, likely by ensuring that they are not dead or doomed quickly. Less early luck drives variation in success among those breeding at least once. Consequently, the importance of luck often has a sharp peak early in life or it has two peaks. We suggest that ages or stages where the importance of luck peaks are potential targets for interventions to benefit a population of concern, different from those identified by eigenvalue elasticity analysis.

Variance as a life history outcome: Sensitivity analysis of the contributions of stochasticity and heterogeneity.

Variance in life history outcomes among individuals is a requirement for natural selection, and a determinant of the ecological dynamics of populations. Heterogeneity among individuals will cause such variance, but so will the inherently stochastic nature of their demography. The relative contributions of these variance components - stochasticity and heterogeneity - to life history outcomes are presented here in a general, demographic calculation. A general formulation of sensitivity analysis is provided for the relationship between the variance components and the demographic rates within the life cycle. We illustrate these novel methods with two examples; the variance in longevity within and between frailty groups in a laboratory population of fruit flies, and the variance in lifetime reproductive output within and between initial environment states in a perennial herb in a stochastic fire environment. In fruit flies, an increase in mortality would increase the variance due to stochasticity and reduce that due to heterogeneity. In the plant example, increasing mortality reduces, and increasing fertility increases both variance components. Sensitivity analyses such as these can provide a powerful tool in identifying patterns among life history stages and heterogeneity groups and their contributions to variance in life history outcomes.

Pluck or Luck: Does Trait Variation or Chance Drive Variation in Lifetime Reproductive Success?

While there has been extensive interest in how intraspecific trait variation affects ecological processes, outcomes are highly variable even when individuals are identical: some are lucky, while others are not. Trait variation is therefore important only if it adds substantially to the variability produced by luck. We ask when trait variation has a substantial effect on variability in lifetime reproductive success (LRS), using two approaches: (1) we partition the variation in LRS into contributions from luck and trait variation and (2) we ask what can be inferred about an individual's traits and with what certainty, given their observed LRS. In theoretical stage- and size-structured models and two empirical case studies, we find that luck usually dominates the variance of LRS. Even when individuals differ substantially in ways that affect expected LRS, unless the effects of luck are substantially reduced (e.g., low variability in reproductive life span or annual fecundity), most variance in lifetime outcomes is due to luck, implying that departures from "null" models omitting trait variation will be hard to detect. Luck also obscures the relationship between realized LRS and individual traits. While trait variation may influence the fate of populations, luck often governs the lives of individuals.

Unexpected Nongenetic Individual Heterogeneity and Trait Covariance in Daphnia and Its Consequences for Ecological and Evolutionary Dynamics.

Individual differences in genetics, age, or environment can cause tremendous differences in individual life-history traits. This individual heterogeneity generates demographic heterogeneity at the population level, which is predicted to have a strong impact on both ecological and evolutionary dynamics. However, we know surprisingly little about the sources of individual heterogeneity for particular taxa or how different sources scale up to impact ecological and evolutionary dynamics. Here we experimentally study the individual heterogeneity that emerges from both genetic and nongenetic sources in a species of freshwater zooplankton across a large gradient of food quality. Despite the tight control of environment, we still find that the variation from nongenetic sources is greater than that from genetic sources over a wide range of food quality and that this variation has strong positive covariance between growth and reproduction. We evaluate the general consequences of genetic and nongenetic covariance for ecological and evolutionary dynamics theoretically and find that increasing nongenetic variation slows evolution independent of the correlation in heritable life-history traits but that the impact on ecological dynamics depends on both nongenetic and genetic covariance. Our results demonstrate that variation in the relative magnitude of nongenetic versus genetic sources of variation impacts the predicted ecological and evolutionary dynamics.

Stochasticity, heterogeneity, and variance in longevity in human populations.

Inter-individual variance in longevity (or any other demographic outcome) may arise from heterogeneity or from individual stochasticity. Heterogeneity refers to differences among individuals in the demographic rates experienced at a given age or stage. Stochasticity refers to variation due to the random outcome of demographic rates applied to individuals with the same properties. The variance due to individual stochasticity can be calculated from a Markov chain description of the life cycle. The variance due to heterogeneity can be calculated from a multistate model that incorporates the heterogeneity. We show how to use this approach to decompose the variance in longevity into contributions from stochasticity and heterogeneous frailty for male and female cohorts from Sweden (1751-1899), France (1816-1903), and Italy (1872-1899), and also for a selection of period data for the same countries. Heterogeneity in mortality is described by the gamma-Gompertz-Makeham model, in which a gamma distributed "frailty" modifies a baseline Gompertz-Makeham mortality schedule. Model parameters were estimated by maximum likelihood for a range of starting ages. The estimates were used to construct an age×frailty-classified matrix model, from which we compute the variance of longevity and its components due to heterogeneous frailty and to individual stochasticity. The estimated fraction of the variance in longevity due to heterogeneous frailty (averaged over time) is less than 10% for all countries and for both sexes. These results suggest that most of the variance in human longevity arises from stochasticity, rather than from heterogeneous frailty.

We Happy Few: Using Structured Population Models to Identify the Decisive Events in the Lives of Exceptional Individuals.

In any population, some individuals make it big: they are among the few that produce many offspring, grow to large size, and so on. What distinguishes the lives of these happy few? We present three approaches for identifying what factors distinguish those "lucky" individuals who come to dominate reproduction in a population without fixed differences between individuals (genotype, site quality, etc.): comparing life-history trajectories for lucky and unlucky individuals and calculating the elasticity of the probability of becoming lucky to perturbations in demographic rates at a given size or a given age. As examples we consider published size-structured integral projection models for the tropical tree Dacrydium elatum and the semiarid shrub Artemisia ordosica and an age-size-structured matrix model for the tropical tree Cedrela ordosica. We find that good fortune (e.g., rapid growth) when small and young matters much more than good fortune when older and larger. Becoming lucky is primarily a matter of surviving while others die. For species with more variable growth (such as Cedrela and Ordosica), it is also a matter of growing fast. We focus on reproductive skew, but our methods are broadly applicable and can be used to investigate how individuals come to be exceptional in any aspect.

Variance in animal longevity: contributions of heterogeneity and stochasticity.

Variance in longevity among individuals may arise as an effect of heterogeneity (differences in mortality rates experienced at the same age or stage) or as an effect of individual stochasticity (the outcome of random demographic events during the life cycle). Decomposing the variance into components due to heterogeneity and stochasticity is crucial for evolutionary analyses.In this study, we analyze longevity from ten studies of invertebrates in the laboratory, and use the results to partition the variance in longevity into its components. To do so, we fit finite mixtures of Weibull survival functions to each data set by maximum likelihood, using the EM algorithm. We used the Bayesian Information Criterion to select the most well supported model. The results of the mixture analysis were used to construct an age × stage-classified matrix model, with heterogeneity groups as stages, from which we calculated the variance in longevity and its components. Almost all data sets revealed evidence of some degree of heterogeneity. The median contribution of unobserved heterogeneity to the total variance was 35%, with the remaining 65% due to stochasticity. The differences among groups in mean longevity were typically on the order of 30% of the overall life expectancy. There was considerable variation among data sets in both the magnitude of heterogeneity and the proportion of variance due to heterogeneity, but no clear patterns were apparent in relation to sex, taxon, or environmental conditions.

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.