Lifetime reproductive output: individual stochasticity, variance, and sensitivity analysis
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Abstract
Lifetime reproductive output (LRO) determines pergeneration 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, closedform solution for all the moments of LRO, for age and stageclassified populations. Previous studies have relied on simulation, iterative solutions, or closedform 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 \(\mathcal {I}\), 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 ageclassified 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 \(\mathcal {I}\). Proportional increases in fertility increase both the mean and variance of LRO, but reduce \(\mathcal {I}\). For a sizeclassified tree population, the elasticity of both mean and variance of LRO to stagespecific mortality are negative; the elasticities to stagespecific fertility are positive.
Keywords
Lifetime reproductive output Matrix population models Individual stochasticity Sensitivity analysis Markov chains with rewards Opportunity for selection Interindividual varianceIntroduction
Like all men in Babylon, I have been proconsul; like all, I have been a slave. I have known omnipotence, ignominy, imprisonment … I owe this almost atrocious variety to an institution which other republics know nothing about, and which operates among them imperfectly and in secret: the lottery. Jorge Luis Borges, The Lottery in Babylon
Lifetime reproductive output (LRO) is, as the name implies, the total production of offspring over the lifetime of an individual^{1} and is one of the most important characteristics of an individual life history. The expectation of LRO, calculated in terms of female offspring per female, is the net reproductive rate R _{0}. In ecology, the critical value R _{0} = 1 defines the boundary separating population growth and persistence from population decline and extinction. In epidemiology, R _{0} for a pathogen determines whether a disease will or will not cause an outbreak. In evolutionary biology, R _{0} is a critical component of fitness (sometimes considered to be fitness, although that is sometimes problematic).
Empirical studies of individual LRO routinely find large variance and usually a positive skew. Most individuals produce few, or no, offspring, while a few rare individuals produce many offspring (CluttonBrock 1988; Newton 1989). These differences in LRO have two possible sources. One is heterogeneity—differences in the properties of individuals—including genetic heterogeneity, physiological differences, phenotypic plasticity, and environmental heterogeneity.
However, the differences may also be due to individual stochasticity (Caswell 2009, 2011, 2014; van Daalen and Caswell 2015). Individual stochasticity refers to differences among individuals due to the accumulation of random outcomes of the stochastic processes of mortality, growth, development, breeding, etc. Individual stochasticity would lead to variance among individuals even if they were totally identical and experienced exactly the same demographic rates.^{2} Depending on the outcome under consideration, variance due to individual stochasticity can be as great as or even exceed that caused by unobserved heterogeneity (Tuljapurkar et al. 2009; Steiner et al. 2010; Caswell 2011, 2014; Hartemink et al. 2017). Thus, before invoking heterogeneity as the explanation for differences among individuals, it is important to calculate the variation due to stochasticity as a kind of “neutral model” for variation (Steiner and Tuljapurkar 2012).
Individual stochasticity itself has two components. Consider an individual at some stage in its life cycle (say, at birth). The growth, development, and eventual death of this individual define a path through the stages of the life cycle. The pathways of two or more identical individuals, subject to the same rates at every stage of the life cycle, will differ randomly among themselves, and this variation among pathways is one source of individual stochasticity. At each step along an individual’s path, it may or may not reproduce. If it reproduces, the number of its offspring will be drawn from some probability distribution. This withinpathway stochastic variation in (stagespecific) fertility is the second source of individual stochasticity.
Our goal is to calculate the variance (and other statistics) of LRO, due to individual stochasticity, from basic demographic information, just as R _{0} can be calculated from ageclassified or stageclassified demographic models (Rhodes 1940; Cushing and Zhou 1994; De CaminoBeck and Lewis 2007; Cushing and Ackleh 2012). To calculate the variance in LRO, we need to account for the generally infinite number of pathways through the life cycle, calculate the probabilities of each path, calculate the distribution of reproductive output at each stage on each path, and then integrate those probabilities to calculate the mean, variance, etc. of LRO.
The calculation of variance in LRO has been approached in several ways. Early studies used simulation to generate random trajectories through stages, including stages defined by reproductive output, to create a sample of lives from which variance in LRO could be calculated (e.g., Tuljapurkar et al. 2009; Steiner et al. 2010). An analytical solution for all the moments of LRO was provided by Caswell (2011) and provides the framework for our results here. That result took the form of an iterative calculation rather than a closed form expression. Steiner and Tuljapurkar (2012) presented a closed form analytical solution for all the moments of one component of LRO, using a moment generating function approach. They also reported simulations of the complete distribution of LRO and explored effects of heterogeneity. However, their solution included only part of the variance in LRO. They assumed the fertility of each age or stage to be a fixed deterministic quantity, neglecting the withinpathway component of variance. For example, the agespecific fertility of a 26yearold Japanese woman in 1950 was 0.25. The analysis of Steiner and Tuljapurkar (2012) assumes that every 26yearold woman, without exception, produces one fourth of a baby. In our framework, every 26yearold woman, without exception, produces one baby with a probability of 0.25 and zero babies with a probability of 0.75 (Caswell 2011). This component of variance is biologically realistic and quantitively important. For example, analysis of 40 developed countries during the second demographic transition found that, as life expectancy increased, the fraction of variance in LRO due to withintrajectory randomness increased from ∼ 50% to ∼ 99% (Van Daalen and Caswell 2015). We show further examples and present the methodology for decomposing variance into these components, below.
All the studies published so far agree in finding that the variance in LRO due to individual stochasticity can comprise a high proportion of the observed variance in LRO, and that to ignore it is to miss a major source of variation (e.g., Tuljapurkar et al. 2009; Steiner et al. 2010; Steiner and Tuljapurkar 2012; Caswell 2011; van Daalen and Caswell 2015).
In this paper, we present exact, closed form, analytical formulae for all the moments of LRO, extending and replacing the iterative formulae of Caswell (2011). We include both the betweentrajectory and withintrajectory components of variance and incorporate variance in stagespecific fertility either empirically or by a statistical model. We also present the sensitivity and elasticity analysis of means, variances, and all moments of LRO due to individual stochasticity. Our results can be applied to any age or stageclassified matrix population model and to constant, periodic, or stochastic environments.
Our results rely on a mathematical model called a Markov chain with rewards, which we describe in “Markov chains with rewards as a model for LRO.” The construction of a Markov chain with rewards requires demographic information on survival, stage transitions, and fertility. Section “The statistics of LRO” presents the calculation of all the moments and other descriptive statistics of LRO. Section “Sensitivity analysis of LRO” presents the sensitivity and elasticity analysis of LRO. Section “Examples” presents examples of ageclassified and stageclassified populations, and “Discussion” concludes with a discussion of results and possible extensions. Proofs and derivations appear in Appendix A.
Markov chains with rewards as a model for LRO
Notation
Matrices are denoted by uppercase boldface letters (e.g., P), and vectors by lowercase boldface letters (e.g., ρ). Vectors are column vectors by default; X ^{T} is the transpose of X. The vector 1 _{ n } is a n × 1 vector of ones, I _{ n } is the identity matrix of order n, and e _{ i } is the ith unit vector, with a 1 in the ith entry and zeros elsewhere. The matrix E is a matrix of ones, and E _{ i j } is a matrix with a 1 in the (i,j) entry and zeros elsewhere. The diagonal matrix with the vector x on the diagonal and zeros elsewhere is denoted \(\mathcal {D}(\textbf {x})\). The symbol ∘ denotes the Hadamard, or elementbyelement product, and ⊗ denotes the Kronecker product. The vec operator vec X stacks the columns of an m × n matrix X into an m n × 1 column vector. The vecpermutation matrix K _{ m,n } satisfies vec X ^{T} = K _{ m,n }vec X. In cases where it will help understanding, we indicate the dimension of displayed matrix expressions.
The life cycle as a Markov chain
Our approach uses a mathematical model called a Markov chain with rewards. These models have a long history in stochastic process theory (e.g., Howard 1960; Puterman 1994; Sheskin 2010) but have only recently been applied to study lifetime reproduction (Caswell 2011; Van Daalen and Caswell 2015).
Reproductive rewards
An individual experiences a sequence of transitions according to the probabilities in P. Associated with each transition (including the transition of remaining in a stage) is a “reward” representing, in our case, reproductive output. The reproductive reward is a random variable with a specified set of moments. Rewards accumulate over the lifetime of the individual; the total accumulation at the time of death is the LRO. We make the reasonable assumption that individuals in the absorbing state stop accumulating rewards.^{3}
 Empirical distribution.

The moments can be calculated empirically from data on individual reproduction. Such data are frequently obtained, but typically only mean reproductive output is reported. In the absence of this information, the reward matrices can be modelled by any probability distribution that is determined by its mean, including the following.
 Bernoulli distribution.
 When only a single offspring is produced, mean offspring production equals the probability of reproducing. The matrices of second and third moments satisfy$$ \textbf{R}_{3} = \textbf{R}_{2} = \textbf{R}_{1}. $$(5)
 Poisson distribution.
 When multiple offspring are produced, the Poisson distribution describes a situation where every individual has the same chances of producing those offspring. The moments satisfy$$\begin{array}{@{}rcl@{}} \textbf{R}_{2} &=& \textbf{R}_{1} + \left( \textbf{R}_{1} \circ \textbf{R}_{1} \right) \end{array} $$(6)$$\begin{array}{@{}rcl@{}} \textbf{R}_{3} &=& \textbf{R}_{1} + 3 \left( \textbf{R}_{1} \circ \textbf{R}_{1} \right) +\left( \textbf{R}_{1} \circ \textbf{R}_{1} \circ \textbf{R}_{1} \right). \end{array} $$(7)
 Fixed rewards.
 It is possible to eliminate variance in fertility by creating reward matrices where every individual in a given stage produces exactly the mean number of offspring, in which case$$\begin{array}{@{}rcl@{}} \textbf{R}_{2} &=& \textbf{R}_{1} \circ \textbf{R}_{1} \end{array} $$(8)$$\begin{array}{@{}rcl@{}} \textbf{R}_{3} &=& \textbf{R}_{1} \circ \textbf{R}_{1} \circ \textbf{R}_{1}. \end{array} $$(9)
The statistics of LRO
In this model, every individual is subject to the same rates at any given stage, so there is no heterogeneity. Even so, each individual may experience a different life course. The resulting variation among individuals is due to the individual stochasticity implied by the vital rates in P and the fertility process described by the R _{ i }. Our task is to derive the statistics of LRO from this information.
In terms of these quantities, the moments of LRO are given by the following theorem.
Theorem 1
The moment vectors \(\tilde {\boldsymbol {\rho }}\) of lifetime accumulated reproductive output are
and, in general,
where N =(I _{ τ } −U)^{−1} is the fundamental matrix of the Markov chain.
Proof
See Appendix A □
Partitioning variance: within and between pathways
Sensitivity analysis of LRO
The statistics of LRO depend on the life history parameters that determine the transition matrices U and M, and the moment matrices R _{1}, R _{2}, …. Sensitivity analysis reveals how these parameters affect LRO; our goal is to derive the sensitivity and elasticity of LRO to changes in any parameters. Although sensitivity analysis of the net reproductive rate R _{0} has been presented before (Matser et al. 2009; Caswell 2009), there has been no such analysis for the variance or other measures of variation in LRO.
Theorem 2
Let P, U, and M define the absorbing Markov chain in Eq. 3, with τ transient states, α absorbing states, and s = τ + α total states. Let R _{ i } contain the ith moments of the reproductive rewards corresponding to each transition. Let 𝜃 be a vector of parameters. The vector \(\tilde {\boldsymbol {\rho }}_{m}\) contains the m th moments of remaining LRO for each of the τ transient stages. The sensitivity of \(\tilde {\boldsymbol {\rho }}_{m}\) to 𝜃 is
Proof
Proof is given in Appendix A. □
Sensitivity of the statistics of LRO
The moments \(\tilde {\boldsymbol {\rho }}_{i}\) provide the statistics (21)–(24) describing the interindividual variability of LRO, including the variance, standard deviation, coefficient of variation, and the scaled variance (Crow’s index) (Caswell 2011; Van Daalen and Caswell 2015).
Elasticity
Some special perturbations
Here we consider some perturbations that are of special interest: the survival and transitions in stageclassified models and sensitivity to means and variances of stagespecific fertility.
Mortality and transitions in stageclassified models
Sensitivity to means and variances of fertility
Perturbations of fertility appear in Eqs. 27–29 as derivatives of the reproductive reward matrices R _{ i }. When the distributions of stagespecific fertilities are specified by a parametric distribution, the moments may be linked, so that changes in mean fertility also affect the variance (e.g., in the Poisson distribution, the variance is equal to the mean). Sometimes, however, it is of interest to treat the mean and variance of fertility as independent traits and calculate the sensitivity of LRO to the mean, holding the variance fixed, and to the variance, holding the mean fixed. This subtle but important distinction was emphasized by Tuljapurkar et al. (2003) and Haridas and Tuljapurkar (2005) in the context of the elasticity of the stochastic growth rate to the entries of a stochastically varying matrix. One might compute the effect of changing one of the matrix entries (perhaps by a change in energy allocation strategy), recognizing that this would change both the mean and the variance. Or, one might be interested in the effects of variance per se and manipulate the moments to calculate elasticities with respect to the mean and variance independently.
Sensitivity to mean fertility, variance fixed
Sensitivity to variance in fertility, mean fixed
A protocol for the analysis of lifetime reproductive output
The results presented to this point provide a protocol for analysis of lifetime reproductive output, applicable to any matrix population model. A stepwise version of this protocol is given in Table 1. In the next section, we present ageclassified and stageclassified examples of the analysis.
A stepbystep protocol for analysis of lifetime reproductive output and its sensitivity, from any stage or ageclassified matrix population model
1. Obtain a transition matrix U, perhaps from decomposing a population projection matrix as A = U + F. 
2. Locate reproductive transitions. 
(a) If fertility is transition specific, identify the transitions (e.g., to reproductive states). 
(b) If fertility is stagespecific, extract the vector f from F. 
3. Obtain statistical moments of fertility: 
(a) From empirical measurements of the moments of stagespecific fertility, or 
(b) From an assumption of Bernoulli [see equation (5)], or Poisson [ see Eqs. 6 and 7], or fixed [see Eqs. 8 and 9] reproduction. 
4. Construct reward matrices from Eq. 11. 
7. Sensitivity analysis 
(a) Specify parameter vector 𝜃 of interest 
(b) Calculate derivatives of U, and R _{ i } to 𝜃. Take advantage of Eqs. 35 or 90–92 to compute derivatives of P to 𝜃. 
(c) If the matrix is stageclassified, 
i. Decompose U = GΣ. 
ii. Use Eq. 46 to compute the derivative of U to mortality. 
iii. Use Eq. 49 to compute the derivative of G to 𝜃, including compensation to preserve column sums of G 
(d) Compute derivatives of the moment vectors \(\tilde {\boldsymbol {\rho }}_{i}\) for the moments of interest (i = 1, 2 suffice to analyze the variance, standard deviation, 
CV, and \(\mathcal {I}\)). 
ii. Compute derivatives of \(\tilde {\boldsymbol {\rho }}_{i}\) using Theorem 2. 
(e) Compute sensitivity of desired statistics of LRO using Eqs. 38–41. 
(f) If desired, compute elasticities of statistics of LRO using Eq. 42. 
Examples
This section presents examples of the calculation of the statistical properties of lifetime reproductive output and its subsequent sensitivity analysis, for both ageclassified and stageclassified population. In “Ageclassified human populations,” we analyze a set of ageclassified human populations that span a wide range of demographic conditions. In “A stageclassified tree population,” we analyze a sizeclassified model for Canadian hemlock (Tsuga canadensis), a coniferous tree.
Ageclassified human populations
The transition matrix U for an ageclassified model contains survival probabilities on the subdiagonal and zeros elsewhere. One absorbing state, death, is included, and the mortality matrix M is calculated according to Eq. 34.
We present results for nine populations: the Netherlands (1950 and 2009), Sweden (1891 and 2010), Japan (1947 and 2009), two huntergather populations (the Ache of subtropical Paraguay (Gurven and Kaplan 2007; Hill and Hurtado 1996) and the Hadza of the Tanzanian savanna Blurton Jones 2011), and the Hutterites of North America. The Netherlands, Sweden, and Japan are typical of developed countries progressing through the demographic transition. The huntergatherer populations have higher mortality, lower life expectancy, and higher fertility than the developed countries. The Hutterites, an Anabaptist religious community in the United States and Canada, are known for having the highest total fertility for any known human population (Eaton and Mayer 1953), but are assumed to have experienced mortality rates similar to that of the USA around 1946–1950.
Data for the Netherlands, Sweden, and Japan were obtained from the Human Mortality Database (2013) and the Human Fertility Database (2013). Data for the Ache were obtained from Gurven and Kaplan (2007) and Hill and Hurtado (1996), and for the Hadza from Blurton Jones (2011). The fertility and mortality schedules for the Hutterites were taken from Eaton and Mayer (1953).
Variance in LRO
The statistics of lifetime reproductive output for the Netherlands (NLD), Sweden (SWE), and Japan (JPN), with two points in time for each country, two huntergatherer populations, the Hadza and the Ache, and a population of highfertility Hutterites
Population  Mean  V  V _{between} (%)  V _{within} (%)  SD  CV  \(\mathcal {I}\)  Life exp. 

NLD 1950  2.96  2.91  12.5  87.5  1.71  0.58  0.33  73.1 
NLD 2009  1.78  1.61  1.4  98.6  1.27  0.72  0.51  83.1 
SWE 1891  3.00  5.60  55.6  44.4  2.37  0.79  0.62  53.0 
SWE 2010  1.97  1.79  1.4  98.6  1.34  0.68  0.46  84.0 
JPN 1947  3.50  6.10  54.8  45.2  2.47  0.71  0.50  54.2 
JPN 2009  1.35  1.26  0.9  99.1  1.12  0.83  0.69  86.9 
Hadza  3.13  11.30  78.9  21.1  3.36  1.07  1.15  34.6 
Ache  4.48  17.23  81.1  18.9  4.15  0.93  0.86  38.0 
Hutterites  7.53  8.58  41.3  58.7  2.93  0.39  0.15  70.0 
Not surprisingly, recent populations in developed countries have higher longevity and lower mean lifetime reproductive output. The huntergatherer and Hutterite populations show the highest mean LRO. In each of the developed countries (the Netherlands, Sweden, and Japan), the reductions in mean LRO and increases in life expectancy are accompanied by reductions in the variance in LRO. The Ache and Hadza show the highest variance in LRO, and the variance for the Hutterites is higher than any of the developed countries.
Most of the variance in the huntergatherer populations is due to variance among pathways, which in an ageclassified model is determined by survival from birth through the reproductive ages. The most recent years in developed countries show extremely low betweenpathway variance because almost all women survive through their reproductive years (Van Daalen and Caswell 2015). The Ache and Hadza, with the lowest life expectancy, show very high V _{between}. The Hutterites have a long life expectancy, but their high fertility amplifies the effect of differences in longevity, so V _{between} is similar to that of Japan in 1947 and Sweden in 1891.
The opportunity for selection \(\mathcal {I}\) varies less (about sevenfold) among these populations than does the variance in LRO (about 13fold). The Hutterites showed the lowest opportunity for selection of all the populations we included, yet the other highfertility populations, the Ache and the Hadza, show the highest values for the opportunity for selection. Taking into account the fact that the huntergatherer populations have the highest variances in LRO, mostly due to variation in the pathways individuals take through life, we posit that the high opportunity for selection reflects room for improvement in survival rates from birth to reproductive ages.
The variance in LRO documented in Table 2 is calculated on the assumption that every individual experiences the same vital rates at every age and is thus due to individual stochasticity. Crow’s \(\mathcal {I}\) is a measure of the potential relative increase in fitness per generation, but the variance here is stochastic, not genetic, so the opportunity for selection is only apparent, not real.
Sensitivity analysis
The variance in LRO increases with an increase in mortality rate for most populations; the effect is greatest for the Hutterites (Fig. 1). For the developed countries, the sensitivity is positive over most of the reproductive ages. For the Hadza and the Ache, variance in LRO decreases with higher agespecific mortality rates. Variance in LRO increases with increasing fertility rates for all countries. The Hutterites, Sweden, the Netherlands, and Japan show a reduced sensitivity of variance in LRO to fertility around the reproductive ages.
Crow’s opportunity for selection \(\mathcal {I}\) combines both the mean and the variance. Increased mortality during the reproductive period increases \(\mathcal {I}\) in all the populations. It is most sensitive to mortality in the Ache and Hadza populations and least sensitive in the Hutterites. An increase in fertility reduces \(\mathcal {I}\) in all the populations. Thus, the net result of environmental changes that affect both mortality and fertility cannot be predicted a priori.
A stageclassified tree population
Results
The statistics of lifetime reproductive output for T. canadensis
Mean  V  V _{between} (%)  V _{within} (%)  SD  CV  \(\mathcal {I}\)  Life exp. 

1.42  1.41 × 10^{3}  99.9  0.1  37.54  26.39  696.23  12.16 
Sensitivity analysis
The mean and variance, the contribution to the variance of different processes, and the sensitivity of these indices to parameters differ between these examples. This reflects the different life history strategies of trees and humans, the difference between between ageclassified populations with low fertility, and a sizeclassified population with strongly sizedependent fertility, and the difference between assumptions of Bernoulli or Poisson distributed rewards. Vastly different life histories can be incorporated into the Markov chain with reward framework, allowing for the investigation of life history in many species, and from different perspectives.
Discussion
Lifetime reproductive output is an outcome of the life cycle. Any demographic model implies a distribution for LRO, just as it implies more familiar measures such as R _{0} and life expectancy. Even in a deterministic environment, the LRO is a random variable; the stochasticity arises from two sources: the random pathway that the individual follows through its life and the random fertility it exhibits at each stage on that pathway.
Our results in “Markov chains with rewards as a model for LRO” provide the analytical machinery needed to calculate all the statistical properties of LRO that follow from a specified set of stages, a transition matrix, and the moments of stagespecific fertility. These statistical properties include measures of variability among individuals (the variance, CV, skewness, opportunity for selection, etc.). It is important to recognize that this variability does not reflect heterogeneity, genetic or otherwise. Every individual is subject to the same set of vital rates at any stage in the life cycle. Only the outcomes of applying those rates vary; the variation is thus due to individual stochasticity.
The variance, or standardized variance, calculated from demographic models for a variety of species, is large (see also Caswell 2011; van Daalen and Caswell 2015; Steiner et al. 2010). Individual stochasticity creates a large apparent opportunity for selection that is not, in fact, a true opportunity. In this sense, as emphasized by Steiner and Tuljapurkar (2012), the calculations of individual stochasticity provide a neutral model for LRO. A number of comparisons of calculated variance (due to stochasticity) and observed variance (due to some mix of stochasticity and heterogeneity) have shown that stochasticity may explain a significant amount, or all, of the observed variance (Caswell 2011; Van Daalen and Caswell 2015; Steiner et al. 2010).
It is important to remember what neutral model results do, and do not, imply (Caswell 1976). The results show that a certain amount of variance can be accounted for by stochasticity, and hence, that the mere observation of such variation is no evidence for heterogeneity. It does not prove that the observed variance is stochastic, or that there is no heterogeneity, as pointed out by Steiner and Tuljapurkar (2012). It calls for a comparison with models explicitly incorporating heterogeneity, either observed or unobserved, as suggested by Cam et al. (2016). For examples of this approach, see Caswell (2014); Hartemink et al. (2017); Jenouvrier et al. (2016).
The results of Theorem 1 provide an exact solution to the calculation of the statistics of LRO, including both components (within and between pathways; see “Partitioning variance: within and between pathways”). We also provide a complete sensitivity analysis for LRO. Theorem 2 makes it possible to calculate the sensitivity and elasticity of all the moments of LRO and all the statistics calculated from those moments, with respect to changes in mortality, transition probabilities, and the moments of stagespecific fertility. The results include the sensitivity to changes in mean fertility (holding variance constant) and variance in fertility (holding the mean constant).
The formulae for the moments in Theorem 1 and the sensitivities in Theorem 2 are complicated and opaque, because the relationships between LRO and the life cycle structure, the moments of reproduction, the probabilities of survival, and the infinite diversity of pathways through the life cycle, are complicated. Simplifications that permit qualitative generalities are always welcome, and more work on this will be valuable.
We applied sensitivity analysis to several populations of humans and a population of trees. The patterns of sensitivity and elasticity of LRO that we report for these populations have not been described before. Some suggestive patterns appear; they warrant further investigation.
In longlived ageclassified populations with low reproductive output, as diverse as the nineteenth century Swedes, midtwentieth century Hutterites, the twentyfirst century Dutch, and Hadza and Ache huntergatherers, the sensitivity of mean LRO to mortality is negative. Most populations show a positive sensitivity of variance in LRO to mortality for the first 40 years of life, only showing a small negative sensitivity between ages 20 and 40. The Ache and Hadza have more pronounced negative sensitivity around these ages, with the Hadza showing negative sensitivity across the first 40 years of life. The sensitivity of Crow’s \(\mathcal {I}\) to mortality in the first 40 years of life is positive, showing that an increase in mortality would increase the apparent opportunity for selection of lifetime reproduction. Patterns for elasticities are similar, though smaller in magnitude.
All populations show broadly similar patterns for the sensitivity of LRO to fertility. The sensitivities of mean LRO and variance in LRO to fertility are positive. The sensitivity of Crow’s \(\mathcal {I}\) to fertility is negative. The elasticity of mean and variance to fertility is positive, but the elasticity of Crow’s \(\mathcal {I}\) is negative. According to these results, populations in modern countries would reduce the apparent opportunity for selection of LRO with increasing fertilities, and highfertility populations such as Hutterites and huntergatherers would only slightly reduce the opportunity for selection should fertilities increase.
On average, the variance in LRO is 59% between pathways and 41% within pathways. However, the individual populations differ significantly in these contributions by sources of variance. In the twentyfirst century in Japan, Sweden, and the Netherlands, only 1.2% of variance in LRO is between pathways and 98.8% is within pathways. Variance in LRO of the highfertility populations, the Ache, Hadza, and Hutterites, is 71% between pathways and 29% within pathways.
In contrast, in T. canadensis, with sizedependent demography, high fertility, and strongly increasing fertility with size, we find that the elasticities of both mean and variance in LRO to mortality are negative across all size classes. The elasticities of these statistical properties to changes in fertility in the reproductive classes are positive. In case of the elasticity to growth transition rates, mean and variance in LRO once again show similar patterns. Elasticity to stasis, i.e., not growing, is negative for the first five size classes, and positive for the last size class.
Lifetime reproductive output interests population ecologists and epidemiologists (for whom R _{0} is a measure of population growth), evolutionary biologists (for whom variance in LRO is a measure of potential selection), and human demographers (for whom declines in LRO following the demographic transition pose serious social policy challenges). It is thus important that our analysis is not restricted to any one class of population models. It applies equally to agestructured, stagestructured, and multistate models, and to any reproductive strategy. It also applies to periodic and stochastic timevarying models, by applying Theorems 1 and 2 to the models in Caswell (2011). The method can also be applied to rewards other than reproductive output, including health status (Caswell and Zarulli 2015) and economic transfers (Caswell and Kluge 2015).
The transition matrix U is part of any population projection matrix; the Compadre and Comadre matrix databases provide many examples (SalgueroGómez et al. 2016; SalgueroGómez et al. 2015). The mean reproductive reward matrix R _{1} can be obtained from the projection matrix, but the higher moments cannot and require assumptions of a parametric distribution for fertility. We encourage researchers with the appropriate reproduction data to report not only mean fertility but also the higher moments, or even the entire distribution.
Individual stochasticity arises in both reproductive output and in survival or longevity. Our results here complement the analysis of variation in longevity using Markov chain methods, which are widely used in ecology (e.g., Cochran and Ellner 1992, Caswell 2001, 2006, 2009, Horvitz and Tuljapurkar 2008, Tuljapurkar and Horvitz 2006) and have welldeveloped sensitivity analyses. The Markov chain with reward model now permits a similarly deep analysis of lifetime reproduction and its sensitivity analysis.
Footnotes
 1.
This is a slightly more general concept than lifetime reproductive success (CluttonBrock 1988) because it can accommodate many different operational definitions of reproduction, as are often encountered in ecological and demographic analysis.
 2.
Tuljapurkar et al. (2009) independently and at the same time introduced the term “dynamic heterogeneity” to refer to the same random variation . We continue to use individual stochasticity because it more accurately describes the process creating the interindividual variance and allows us to distinguish heterogeneity that is static from that which changes over the life of an individual.
 3.
If the Markov chain is ergodic, or if it is absorbing but the absorbing states continue to collect rewards, then rewards will continue to accumulate forever. Lifetime accumulated rewards in such cases converge only if a discount factor is imposed, making delayed rewards less valuable than immediate rewards. We will present results for ergodic chains elsewhere.
Notes
Acknowledgments
This research was supported by ERC Advanced Grant 322989 and NWO Project ALWOP.2015.100. We thank two anonymous reviewers for helpful comments.
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