Neutral theory for life histories and individual variability in fitness components Ulrich Karl Steinera,b,1 and Shripad Tuljapurkara a Department of Biology, Stanford University, Stanford, CA 94305; and bInstitut National de la Santé et de la Recherche Médicale U1001, Université Paris Descartes, 75014 Paris, France
Edited* by Burton H. Singer, University of Florida, Gainesville, FL, and approved February 3, 2012 (received for review December 3, 2010)
Individuals within populations can differ substantially in their life span and their lifetime reproductive success but such realized individual variation in fitness components need not reflect underlying heritable fitness differences visible to natural selection. Even so, biologists commonly argue that large differences in fitness components are likely adaptive, resulting from and driving evolution by natural selection. To examine this argument we use unique formulas to compute exactly the variance in life span and in lifetime reproductive success among individuals with identical (genotypic) vital rates (assuming a common genotype for all individuals). Such individuals have identical fitness but vary substantially in their realized individual fitness components. We show by example that our computed variances and corresponding simulated distribution of fitness components match those observed in real populations. Of course, (genotypic) vital rates in real populations are expected to differ by small but evolutionarily important amounts among genotypes, but we show that such differences only modestly increase variances in fitness components. We conclude that observed differences in fitness components may likely be evolutionarily neutral, at least to the extent that they are indistinguishable from distributions generated by neutral processes. Important consequences of large neutral variation are the following: Heritabilities for fitness components are likely to be small (which is in fact the case), small selective differences in life histories will be hard to measure, and the effects of random drift will be amplified in natural populations by the large variances among individuals. demography dynamics
| matrix model | age-stage structure | Markov chain |
L
ongitudinal studies of humans and other species show that individuals within a population often differ greatly in age at death and lifetime reproduction (1–4). For example, the SD of human age at death was ∼40% of life expectancy (∼35 y) in the early 1800s and ∼20% in the late 1900s even with a much higher life expectancy (∼70 y) (3). Lifetime reproduction of human females in high-fertility societies had a SD-to-mean ratio [coefficient of variation (CV)] of ∼0.5–1, and there was much greater variability in male lifetime reproduction (5, 6). Such variation obviously leads us to ask, Do the longest-lived individuals (or the most prolific reproducers) differ from other individuals because of heritable genetic differences? Many researchers assume the answer is “yes” (7–10), because both age at death and lifetime reproduction should be related to fitness (although they are not fitness, which is a property of genotypes as we discuss below) (Box 1). However, the desired genetic variation need not exist. For example, populations of genetically identical nematodes in the laboratory display substantial variation in age at death with CV ∼ 0.2. This variation is clearly not due to genetic differences, but is generated by stochastic variation during development (11). In this and other examples, individual fates are shaped by a sequence of stochastic events over the life course that generate substantial variation in the age at death (12–14). Many other examples of the importance of stochastic events are discussed by Finch and Kirkwood (15). In general, then, should we expect large stochastic variation in age at death, lifetime reproduction, and other measures of individual performance? Here we show that the answer is 4684–4689 | PNAS | March 20, 2012 | vol. 109 | no. 12
yes and that this stochastic variation has significant implications for both ecological and evolutionary studies. We begin by showing precisely how much stochastic variation is expected in stage- and age-structured life histories if we assume that all individuals are genotypically and phenotypically identical (Box 1). We analyze population models in which individuals follow different trajectories of stages that (along with age) affect mortality and reproduction (Box 1). Stages are developmental, behavioral, or physiological, [e.g., body mass (in nonhuman mammals), body mass index (in humans), or size (in plants)]. Stage–age models provide a rich description of the life-history phenotype as a set of transition rates between stages and ages (as detailed later in this paper and in Box 1), have been successfully applied to many longitudinal studies of natural populations (16– 18), and are being applied to humans (19). For general models of this form, we present exact formulas for the moments of age at death and lifetime reproduction. We show in an empirical example that our formulas describe the variation observed in nature and, more strikingly, that simulations of our stochastic models match the observed distributions of age at death and lifetime reproduction. Our formulas reveal similarly large variation in purely age-structured models (often used for humans and other species) or stage-structured models (often used for plants). These results march with a growing literature emphasizing the importance of stochastic events in generating variation in many biological processes, from gene expression to organism (11, 12, 20, 21). Next, we present the broader implications of our results. In practice, a natural population is genetically heterogeneous, and genotypes will, in part, determine life-history phenotypes (the transition rates in our model) (Box 1). The dynamics of this genetically heterogeneous population determine fitness as a genotype-specific measure that determines gene frequency change (22). Empirical studies record individual values of measures such as age at death and lifetime reproduction; these are not fitness but are individual components of fitness (Box 1). In many studies on natural populations we lack information on individual genotypes, but many studies use the values of these individual components of fitness to infer underlying differences in genotypic fitness and thence to study evolutionary change. These studies have used diverse methods: statistical models of unobserved differences at birth (23–25); studies of the correlation among fitness components (26–28); and the use of pedigrees to determine heritability, the fraction of the variance in fitness components that is due to underlying additive genetic variation (1, 2, 29). Our results bear on all of these studies. We have shown that individuals that share a life-history phenotype (common transition rates in our models) (Box 1) will display large stochastic variation in individual fitness components. We conclude, first, that large stochastic variation will make it difficult to detect small fitness
Author contributions: U.K.S. and S.T. designed research, performed research, analyzed data, and wrote the paper. The authors declare no conflict of interest. *This Direct Submission article had a prearranged editor. 1
To whom correspondence should be addressed. E-mail:
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This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1018096109/-/DCSupplemental.
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Fitness: A property of a genotype predicting the rate of change of genotype frequencies in a genetically heterogeneous population. Fitness depends on the mapping from genotype to life-history phenotype (i.e., genotypic vital rates: rates of survival, stage transitions, and reproduction). For weak selection in a constant environment, fitness for a genotype equals the long-run growth rate r of the corresponding stage–age-structured phenotype (22). Individual fitness component: A measure of individual performance, such as the realized age at death or lifetime reproductive success of an individual. Such components are not fitness; fitness is a property of a genotype or group of individuals, not of an individual (55). Genotypic vital rates: Survival, stage-transition, and reproduction rates, specified as the transition rates in our model (44). Trajectory: A sequence of stages experienced during the life course of an individual (35). Life-history phenotype: A complete set of transition rates. The rates describe probabilities to move among stages that can be age, stage, or age–stage specific. Age-structured population: A population that is described only by age-specific survival and reproduction; such populations are frequently described by Leslie matrices (56). Stage-structured population: A population where information on quantitative characters of individuals is tracked; such information is left out of classical age-only demography. Stages can comprise behavior, physiology, development, morphology, location, and similar characteristics. Such populations are sometimes described by Lefkovitch matrices (57). Age–stage-structured population: A population that is most accurately described by stage transition rates that change with age.
differences without studying large populations, typically much larger than those in current longitudinal studies. This point is especially relevant given that genotype–phenotype associations show that genotypic differences often lead to small phenotypic differences (30). Second, our work on stage–age models shows that stochastic variation by itself will lead to correlations among fitness components such as age at death and lifetime reproduction, so that such correlations are not a reliable signature of genotypic differences or phenotypic differences or differences in transition rates (see Box 1 for definitions). Third, our results show that the total variance in fitness components (genetic, plus stochastic, plus environmental) must be large because the stochastic component is large, which implies that heritability (the fraction of variance due to additive genetic effects) will likely be small unless we are lucky enough to find genes with large additive effects (31, 32). Finally, and to us most importantly, our results provide a quantitative description of the amount of variation in fitness
components that we must expect under a neutral model, (i.e., if we assume that all genotypes produce the same life-history phenotype) (the same transition probabilities in our model) (Box 1). Such a neutral assumption provides an essential null model against which to test the assumption that genotypes produce distinct phenotypes; this assumption has played a critical role in population genetics (33). Neutral models play a similarly important role in community ecology (34) and, we claim, in population ecology and life-history evolution. Our results also connect with dynamic models of neutral evolution: Stochastic events, by generating large variation in fitness components, amplify the effects of random genetic drift. In other words, they lead to a much smaller effective population size in models of evolutionary change. This result has the effects of slowing down the rate of phenotypic evolution and shows that considerable phenotypic variation may be maintained in nature by neutral processes alone. Results and Discussion Computing Neutral Variation in Fitness Components. We begin by
defining precisely a life-history phenotype (following our earlier work) (4, 18, 35, 36). We measure age and time in discrete intervals and describe the state of an individual at a given age a and time t by a set Z of discrete-valued stage variables. These variables may include developmental state, physiological measures, behavior types, location, and so on; for continuous variables like size we use small intervals of value. With a total of, say, na age classes and ns stage classes, every individual at a given time occupies one of the na × ns possible stage–age classes and follows a sequence of these over time (a trajectory). A life-history phenotype is a set of transition rates between these stage–age classes. Reproduction (which may in some species include vegetative or clonal reproduction) is described by the rates at which a living individual in a given stage– age class produces offspring in one or more birth stages. For individuals already born, the transition rates include survival probability and the probabilities of moving from the current stage to the same or other stages as result of ontogenetic change, plastic variation, etc. A life-history phenotype, in this paper, is a complete set of these transition rates [extending the description of agestructured phenotypes in, (e.g., ref. 22)]. We first present unique exact formulas for the variances of age at death and lifetime reproduction for individuals with an identical life-history phenotype, illustrating our results with an example of seabirds. Next we illustrate our results for stagestructured populations with no age variation, using an example of a tropical herb, and finally our results for the familiar setting of age-structured demography, using an example of a human hunter–gatherer population. Stage- and age-structure and neutral variation in fitness components. We present our unique exact formulas for the moments of age at death and lifetime reproduction in Tables 1 and 2. Our results apply to complex life histories, including those with multiple types of offspring, (e.g., seedlings and suckers in plants, or different birth sizes in animals). We illustrate our analysis for an age–stage-structured life history on the kittiwake, a colonial seabird species (Tables 1 and 2) (37, 38). Stages here are defined as levels of ontogeny and
Table 1. Exact formulas for the moments of age at death Age at death Mean value (exact) in years Variance, exact Second moment (SD)/(Mean)
Formulas
Seabird Rissa tridactyla
Herb Calathea ovandensis
b i E½Ti ¼ e Ne Vi ¼ EðTi2 Þ − ðE½Ti Þ2 b − IÞNe b i EðTi2 Þ ¼ eT ð2N
8.24 19.32 87.218 0.53
1.9932 1.9797 5.9527 0.71
T
There are ns discrete stages and na discrete ages. In general, the matrices here are of size (nsna) × (nsna), organized by age and stage within age. Stage (and age) transitions are described by matrix U whose i, j element is the transition probability from stage j to stage i; b ¼ ðI − UÞ−1 and the identity matrix is I (43, 54). e denotes a column vector of 1s; and ei denotes a column vector of zeros the matrix N except for the value 1 in element i, and its transpose is eTi . Steiner and Tuljapurkar
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Box 1. Fitness and individual fitness components
Table 2. Exact formulas for the moments of lifetime reproduction Formulas
Seabird R. tridactyla fledglings
Herb C. ovandensis seeds
b i E½Ri ¼ e FNe Vi ¼ EðR2i Þ − ðE½Ri Þ2 b i Ne b − IÞG b i EðR2i Þ ¼ eT Fð2N
3.19 14.24 24.416 1.18
0.8681 360.8538 361.6075 21.88
Lifetime reproductive success Mean value (exact) given birth in stage i Variance (exact) given birth in stage i Second moment (SD)/(Mean)
T
There are ns discrete stages and na discrete ages. In general, the matrices here are of size (nsna) × (nsna), organized by age and stage within age. Stage (and b ¼ ðI − UÞ−1 and the identity matrix age) transitions are described by matrix U whose i, j element is the transition probability from stage j to stage i; the matrix N is I (43, 54). Reproduction is described by a matrix F whose i, j element is the number of stage i offspring produced by a stage j parent; all individuals are born b i is a diagonal matrix whose ith entry is P Fðk; iÞ; e denotes a column vector of 1s; and ei denotes a column vector of zeros except for the value 1 in at age 1; G k element i, and its transpose is eTi .
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gatherers in Africa, their vital rates should be roughly similar to those of early humans. Clearly individuals with the same phenotype (identical survival and fertility rates) can have vastly different individual performance (Tables 4 and 5), (i.e., different
A
250
Observed Neutral model
Female Kiwake
200
Heterogenous 0.15 Heterogenous 0.025
150
100
50
0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
>15
Lifeme reproducve success (fledglings)
B
180
Observed Neutral model
160
Heterogenous 0.15 140
Female Kiwake
reproductive success. Central to our approach is the stage dub that computes the expected time an individual ration matrix N spends (at each age) in each stage given its birth stage. To illustrate our argument about neutral variation we compare distributions of lifetime reproductive success (LRS) and life span generated by simulating neutral variation (assuming a common life-history phenotype) with observations on kittiwakes (Fig. 1, SI Materials and Methods, and Tables S1–S3). We have no analytical formula for the predicted distribution (although we can analytically compute higher moments as indicated in SI Materials and Methods). The distributions predicted by our neutral model and the observed distribution differ at most by 5% in means and 11% in variances (Fig. S2). The observed and neutral distributions do not differ statistically (Table 3), supporting an argument that all observed variation in fitness components can be explained by a simple neutral process (first-order Markov process). Stage-structure and neutral variation in fitness components. In many natural populations, it can be challenging to determine the age of an individual whereas it may be feasible to track individual stages, such as size, behavior, physiology, or location, using, for instance, capture mark–recapture methods (39). Stages can also be a more accurate descriptor of individuals than age per se, (e.g., when stages are tightly linked to fertilities), as size in plants or instars in insects, or when individuals remain in certain resting stages such as seed banks. In such cases, our formulas directly yield variances in realized fitness components for individuals sharing a life-history phenotype described purely by stage transitions. We illustrate in Tables 1 and 2 the amount of variation obtained, using the example of a perennial understory herb, Calathea ovandensis, that reproduces only by seeds. Here stages are categorized by ontogeny and size and include seeds, seedlings, juveniles, prereproductives, and four different size categories for reproductives (40). Most individuals die at the seedling stage and the coefficient of variance in life expectancy is close to that of the kittiwake example; however, the individuals that make it to a reproductive stage have high expected fertilities and therefore variance in reproductive success is large. The assumption that all individuals have equal vital rates (common genotype and phenotype) shows that some seeds that germinate and make it to large individuals are lucky rather than special in their heritable properties. Age-structure and neutral variation in fitness components. Finally, we show that even in the relatively simple case of an age-structured population, the neutral variation in fitness components can be large with coefficients of variation reaching ∼1; this age-structure–related variance provides a lower limit in the sense that adding stage structure must increase variation. Recall from basic demography the familiar mortality rate μðaÞ and fertility rate mðaÞ (expected female offspring per female parent) at age a. Females with these rates have life expectancy e0 , and average lifetime reproduction (of females) is given by the net reproductive rate R0 . For clarity, we present in Tables 4 and 5 formulas for the age-only case (see SI Materials and Methods for derivations). Our illustration uses vital rates for a human population, the Hadza of central Tanzania (41). As the last hunter–
Heterogenous 0.025
120 100 80 60 40 20 0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 >20
Age at death (years)
Fig. 1. Lifetime reproductive success (Upper) and life-span distribution (Lower) for the kittiwakes. The observed population comprises 637 female kittiwakes of a natural population in Brittany, France. The distribution generated by a neutral model is based on the same genotypic vital rates (assumes a common genotype). The distribution for the genetically heterogeneous models is based on two equal-sized groups that differ in their transition rates. For these (genetically) heterogeneous populations, survival differences are either large (high-heterogeneity population: perturbation from the neutral model ±0.15) or small (lowheterogeneity population: perturbation from the neutral model ±0.025) (Fig. S1, and Table S4). The neutral model and heterogeneous model distribution is generated by simulating 50 synthetic populations of size equal to the observed population (SI Materials and Methods). Not all individuals completed their lifetime by the end of the study period and therefore their lifetime reproductive success is biased low and the distribution is skewed. Individuals that died in their first years of life before ever being resighted in the colony are not considered, which also skews the distribution. Both effects have been taken into account in the simulations.
Steiner and Tuljapurkar
Table 3. Means, variance, and coefficient of variation for the different distributions shown in Fig. 1 Genetic heterogeneity within a population
LRS LRS LRS LRS K-S
mean SD variance CV
LS mean LS SD LS variance LS CV K-S
Observed
Neutral
High
Low
1.99 2.76 7.64 1.39
2.09 2.68 7.19 1.28 D = 0.167 P > 0.43 7.18 3.57 12.74 0.50 D = 0.15 P > 0.62
3.01 3.91 15.29 1.30 D = 0.37 P < 0.01 8.78 5.47 29.92 0.62 D = 0.37 P = 0.07
2.09 2.64 6.70 1.26 D = 0.23 P > 0.19 7.13 3.51 12.35 0.49 D = 0.10 P > 0.81
7.07 3.37 11.35 0.476
High reduced survival
High increased survival
Low reduced survival
Low increased survival
1.26 1.74 3.04 1.39
4.40 4.29 18.37 0.97
2.29 2.79 7.79 1.22
1.86 2.34 5.48 1.26
5.81 2.29 5.23 0.39
10.75 5.62 31.62 0.52
6.82 3.17 10.06 0.46
7.39 3.65 13.30 0.49
K-S, one-sided Kolmogorov–Smirnov testing for simulated data to be distributed more to the extremes (high and low LRS or LS) compared with the observed data; LRS, lifetime reproductive success; LS, life span. The neutral model is based on the same genotypic vital rates, whereas the distribution for the genetically heterogeneous models is based on two equal-sized groups that differ in their transition rates to different extents (high heterogeneity or low heterogeneity, respectively). Further details are provided in Fig. 1, the main text, SI Materials and Methods, and Table S4.
under the neutral model we expect a positive correlation between age at death and lifetime reproductive success. It is clearly not necessary that long-lived individuals must be of higher “quality” (have genotypes with higher fitness) than those that die sooner. Finally, supposing that individuals differ by other (nonheritable) variables, say height or weight, in addition to age, the variances in Tables 4 and 5 can only increase. [Imagine that there is some neutral variation in realized fertility (nonheritable variance in fitness component); any such variation can only increase the variances in Tables 4 and 5.] Selective Variation in Fitness Components. We emphasize that we are not suggesting that all life-history variation in any natural population is neutral. Such an assertion would be silly in the light of evolution. We know that natural populations are composed of multiple genotypes that differ in fitness and have to be characterized by different genotypic vital rates (transition rates, Box 1). We are saying that chance, (i.e., neutral processes), makes a substantial contribution to variation, as it does in other neutral theories (33, 34). Neutral theories yield null hypotheses that can be used to test for the presence and strength of other evolutionary and environmental processes (42). Selective variation. The currency for natural selection is genotypic fitness, determined by the mapping from genotype to life-history phenotype. For weak selection in a stable environment, fitness for a genotype is given by the asymptotic population growth rate r that is determined by the life-history phenotype (43). When distinct genotypes map into distinct life-history phenotypes, and
Table 4. Alternative way of computing neutral variation in fitness components for age-structured populations (age at death) Age at death Mean value, exact Variance, exact (SD)/(Mean) Variance, approximation
Formulas
Hadza hunter–gatherers
Ð∞ e0 ¼ 0 lðaÞda Ð∞ VT ¼ 2 0 alðaÞda − e20 VT ≈ ð1 − lðAÞÞe20 þ lðAÞVA , where A is the end of the high-mortality juvenile period, and VA is variance of age at death if death occurs after age A. These approximations apply as long as lðAÞ is not very small (say >0.1).
34.9 y 1,019.7 y2 0.92 810.2 y2
The formulas in Table 1 compute the same exact values. da, mortality at age a; lðaÞ, survivorship to age a.
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realized fitness components). It is not surprising that some individuals live twice as long as others or have twice as many offspring as the average. We can use our results to gain broad insights into variances in fitness components across a range of age-structured life histories. Many life histories display relatively high juvenile mortality up to some age of maturity A, beyond which mortality is much lower until senescence eventually raises mortality. In such cases it makes more sense to think about age at death conditional on surviving to age A. The variance in this conditional life span is called VA and depends on the rate of senescence; but the value of VA can be quite high unless deaths are all tightly clustered around one age. Because all females have the same fertility rate mðaÞ (additional stochastic variation in fertility is not included in standard demographic models or in our present analysis), by assumption, their reproductive success varies only if they die at different ages. We provide approximations for such life histories to estimate neutral variances in Tables 4 and 5. We can distinguish life histories qualitatively similar to humans, including primates and mammals, from those for many invertebrates, amphibians, fish, and plants in which the early juvenile period can be nearly lethal. In all cases, the coefficient of variation of age at death will typically be much higher than 0 and can reach or exceed 1.The variance in lifetime reproductive success (approximations in Tables 4 and 5) increases as the probability to reach maturity falls and can easily be very large. Consider the correlation between individual fitness components. In our neutral model, an individual who (purely by chance) lives a long time also has high lifetime reproductive success. Thus,
these phenotypes differ in values of r, it is the variation in r that determines gene frequency change. How much does selective heterogeneity affect variation in individual fitness components? Are variances in individual fitness components
substantially larger in heterogeneous populations where individuals differ in their stage-age specific genotypic vital rates? Returning to our kittiwake example, we show the answer is no, as illustrated in Fig. 1, where we analyze heterogeneous populations constructed as two equal-sized groups differing in their survival. The illustrations in Fig. 1 show a high-heterogeneity population and a low-heterogeneity population, with heterogeneity only in survival. In our example, individuals that fledge one chick (stage 4) have a survival rate of either 0.675 or 0.975 for the high-heterogeneity population and of either 0.80 or 0.85 for the low-heterogeneity population (the neutral model survival for stage 4 is 0.825; we perturb the neutral rates by 0.15 or 0.025, respectively). When heterogeneity is high, differences in fitness components can be large (Fig. 1) with means differing between 20% and 30% and variances differing between 50% and 60% (Table 3). However, low heterogeneity changes the means and variances of fitness components very modestly and cannot be distinguished from a neutral model or the observed distribution (Fig. 1 and Table 3). Our illustrations use simulations whose results are confirmed by exact calculations for the kittiwakes, showing that means and variances of fitness components increase only modestly with increased “modest” heterogeneity (Fig. S3, Tables S5–S8). This result holds for different types of heterogeneous populations, including heterogeneity in fecundity or trade-offs in fecundity and survival (SI Materials and Methods, Figs. S3–S5, and Tables S5–S8). Heterogeneity can influence the expected lifetime reproductive success ER, generation time T, and fitness r (Fig. S2). Detecting selection. Suppose we observe individual values of ages at death and lifetime reproduction in two populations of size N and seek to find a difference in r. We know that r is determined to a good approximation from the average lifetime reproduction and average generation time (22, 43, 44). When lifetime reproduction has variance V, an estimate ofpmean ffiffiffiffiffiffiffiffiffiffi lifetime reproduction from a sample of size N has SE, V =N . Our results tell us that stochastic variation alone produces a large value of V. For small values of r, to detect a difference in expected fitness of δr at a 5% confidence level would require a sample size larger than V =½TðERÞδr2 . Assuming T being the generation time and ER ≈ 1 (small r), to detect moderately strong selection of δr ≈ 0:01 (45) requires a sample size of at least 2,000 female kittiwakes. This is a large number in any practical context. Magnifying random drift. Neutral variation in realized life histories affects the progress of natural selection. In a finite population selection has to be balanced against random drift that is measured by 1/n, where n is the observed population size (46). When there are neutral differences in lifetime reproduction, the effective population size is reduced by a factor approximately equal to the annual variance in lifetime reproduction, V/T, where V is
variance in lifetime reproduction and T is generation time (22). Selection with coefficient s will dominate drift if s > V/nT (22, 46–48). For female kittiwakes (V = 14.24, T = 8.39) this condition is s > 2/n, magnifying drift. This effect may explain why in some natural populations we do not observe evolutionary change predicted by selection differentials in many traits. Heritabilities. The amount of neutral variation also influences the estimation of heritabilities. The total observed variation in any trait linked to fitness or in a fitness component comprises both neutral variation and selective variation. Heritabilities are estimated as the ratio of the additive genetic variance over the total phenotypic variance and therefore have to be small if neutral variation is large (49). We have shown that small differences in genotypic vital rates result only in small differences in the amount of variation in fitness components (our low-heterogeneity population shows rather large selective differences from a population genetic point of view). Hence in a heterogeneous population, these total variances will be dominated by the large amount of neutral variation, which implies that heritabilities will be small. Thus, we predict that heritabilities for life-history traits will be low, in agreement with many studies in natural populations (29). These studies argue that additive genetic variances are small (31, 50) or total variances are large (1, 2, 31, 51), both consistent with our analyses. Conclusion To understand evolutionary processes we must disentangle observed variation in fitness components into stochastic (demographic, environmental) and selective components; this is a challenging task, especially in natural populations. Our exact results on neutral variation provide precise insights that can be used to study and test the evolutionary significance of life-history variation. They can help empiricists to estimate sample sizes to detect selective variation, test for the possible selective significance of phenotypic variation against a neutral alternative, and help evaluate the factors underlying estimates of heritability. The relevance of our results to evolutionary dynamics rests upon assumptions of a stable environment and weak selection. Further analysis may be needed when selection in natural populations violates these assumptions, but will require new theoretical understanding of strong selection in stage–age-structured populations. We show that a great deal of variation in fitness components is necessarily expected in natural populations, much larger than empiricists might expect from simple intuition. We show that such large variances are indeed observed in nature. Several factors may act to increase such neutral variation, including lifehistory structuring by additional stages and spatial or temporal environmental variation. Other factors may act to reduce variation such as correlations between individual genotypic vital rates due to environmental or social structure. Our results are an essential step in exploring ecological and evolutionary forces in shaping life histories and population dynamics.
Table 5. Alternative way of computing neutral variation in fitness components for age-structured populations (lifetime reproductive success) Lifetime reproductive success Mean value, exact Variance, exact
(SD)/(Mean) Variance, approximation
Formulas
Hadza hunter–gatherers
R0 ¼ 0 lðaÞmðaÞda Ð∞ VR ¼ ð1=2Þ 0 mðaÞMðaÞlðaÞda − R20 , where VR is variance of lifetime reproduction, and M(a) is cumulative reproduction to age a.
2.097 3.7
Ð∞
VR ≈ ð1 − lðηÞÞR20 =lðηÞ, where η is mean age of reproduction, and lðηÞ is the probability of reaching age η.
0.92 4.5
The formulas in Table 2 compute the same exact values. da, mortality at age a; lðaÞ, survivorship to age a; mðaÞ, fertility at age a.
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Steiner and Tuljapurkar
Our neutral theory for life histories uses results about variation generated by stochastic processes that are determined by (i) age–stage-specific survival and reproductive rates (vital rates) for the kittiwake example (ref. 35 and SI Materials and Methods, Tables S1–S3), (ii) stage transition rates (stage-specific vial rates) for the herb example (see ref. 40 for vital rate estimation), or (iii) age-specific transition rates for the example of the Hadza (see ref. 41 for vital rate estimation). The stochastic process described in both stage-structured examples follows a first-order Markov process. For our detailed illustration of the kittiwake example, we analyzed mark recapture data on known-age individually marked birds nesting at the nature reserve of Goulien Cap-Sizun (Brittany, France) (38). Observed life histories were classified in discrete reproductive stages as immature (I), nonbreeders (NB), failed breeders (FB), successful breeders fledging one chick (F1), or successful breeders fledging two or (very few that fledged) three chicks (F2) (35, 37, 38, 52). Observed stage sequences 1. Merilä J, Sheldon BC (2000) Lifetime reproductive success and heritability in nature. Am Nat 155:301–310. 2. Kruuk LEB, et al. (2000) Heritability of fitness in a wild mammal population. Proc Natl Acad Sci USA 97:698–703. 3. Edwards R, Tuljapurkar S (2005) Inequality in life spans and a new perspective on mortality convergence across industrialized countries. Popul Dev Rev 31:645–674. 4. Tuljapurkar S, Steiner UK, Orzack SH (2009) Dynamic heterogeneity in life histories. Ecol Lett 12:93–106. 5. Eaton JW, Mayer AJ (1953) The social biology of very high fertility among the Hutterites; the demography of a unique population. Hum Biol 25:206–264. 6. 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PNAS | March 20, 2012 | vol. 109 | no. 12 | 4689
POPULATION BIOLOGY
Materials and Methods
