## Abstract

Population dynamics and evolutionary change are linked by the fundamental biological processes of birth and death. This means that population growth may correlate with the strength of selection, whereas evolutionary change can leave an ecological signature. We decompose population growth in an age-structured population into contributions from variation in a quantitative trait. We report that the distribution of body sizes within a population of Soay sheep can markedly influence population dynamics, accounting for up to one-fifth of observed population growth. Our results suggest that there is substantial opportunity for evolutionary dynamics to leave an ecological signature and visa versa.

Ecological and evolutionary processes have traditionally been considered to operate at such different time scales that ecologists could ignore evolutionary dynamics, while evolutionary biologists could overlook ecological processes (*1*). Recently, however, there has been growing interest in the effects that ecological and evolutionary processes have on each other (*2*–*4*). For example, genetic variation at one allozyme locus influences population dynamics in a butterfly metapopulation (*5*), and evolutionary change in body and beak size has contributed more than ecological processes to population growth in a Darwin's finch (*4*). In parallel, evolutionary biologists have shown that selection can fluctuate with ecological processes and that this can generate evolutionary change. In the same population of Darwin's finch that was the focus of (*4*), varying ecological conditions in different decades impacted on the strength, direction, and outcome of selection (*6*). Given that ecological and evolutionary processes are intertwined, it is necessary to develop methods to capture their relation. A first step in doing this is to characterize the association between the phenotypic variation on which selection operates and population growth (*7*). Here we ask how quantitative trait variation impacts population growth in a population of Soay sheep (*8*) and how selection varies with population growth.

The population of Soay sheep on Hirta, St. Kilda, has been studied in detail since 1985 (*8*). The structure and size of the population is known for each year (fig. S1). Birth weight (kg) is collected each spring, and adult body weight (kg) and hind leg length (mm)—a measure of skeletal size—are collected annually from individuals caught in the summer catch (on average ∼50% of the population) (*9*). The sheep year runs from 1 August to 31 July, and recruitment is calculated as the number of lambs an individual produced in April that are still alive in August. Paternity is assigned using genetic markers to ∼60% of lambs with >80% confidence (*10*). Significant age- or environment-specific additive genetic variance exists for birth weight, August body weight, and hind leg length (*8*, *11*, *12*). We estimate heritabilities, *h ^{2}* (

*13*), using these values and the phenotypic variance estimated from our data set [supporting online material (SOM)].

To link variation in a quantitative trait to population growth requires an understanding of how variation in the trait influences survival and recruitment and how survival and recruitment influence population growth (*14*). One way of doing this is to calculate the proportion of variation in individual contributions to population growth, *p*_{t(i)}, accounted for by a quantitative trait. *p*_{t(i)} is calculated as the difference between observed population growth and population growth calculated with the contribution of a focal individual removed (*15*). This quantity describes how each individual contributed directly to observed population growth over a time step and is calculated as where *s _{t(i)}* and

*f*represent survival and recruitment of individual

_{t(i)}*i*at time

*t, s̄*and

_{t}*f̄*represent population means, and

_{t}*N*

_{t}represents population size. The quantity

*p*can be decomposed into contribution via survival,

_{t(i)}*S*

_{t(i)}, and recruitment,

*F*where and . In diploid systems,

_{t(i)}*f*and

_{t(i)}*f̄*are calculated as the number of offspring multiplied by ½. The proportion of variation (σ

_{t}_{p}) in

*p*accounted for by a quantitative trait is the contribution of variation in the trait to population growth. For quantitative traits, it is straightforward to calculate the contribution of additive genetic variation in a trait to population growth by multiplying σ

_{t(i)}_{p}by the

*h*of the trait.

^{2}*S*_{t(i)}, *F*_{t(i)}, and *p*_{t(i)} and body weight and hind leg length vary with ontogeny and sex in Soay sheep. Failing to correct for this variation would inflate estimates of the contribution of variation in these traits to population growth. To assess the contribution of quantitative traits to population growth, we conducted separate analyses within each age and sex class before combining results across classes. We considered males and females separately and divided each sex into four age-classes—lambs, yearlings, prime-aged adults (2 to 6 years), and senescent individuals (>6 years) (*16*). We examined the statistical relation between each trait and *p*_{t(i)}, *S*_{t(i)}, and *F*_{t(i)} using generalized additive models (GAMs) (*17*) in R (*18*). We estimated σ_{p} as the proportion of deviance explained by these GAMs. To calculate the contribution of variation in each trait to population growth across classes, we multiplied the σ_{p} value for each class by the proportion of the population in each class before summing these products across classes.

In Fig. 1, we show associations between body weight and individual contributions to population growth for data pooled across years. Body weight accounted for 4.7% of population growth, hind leg length 3.19%, and birth weight 1.69%. About two-thirds of the contribution of hind leg and body weight was via lambs and yearlings, while three-fourths of the contribution of birth weight was via lambs and yearlings (Fig. 2, A, B, and C). The relatively small contribution of variation in trait values to population growth occurs because none of the morphological traits account for much variation in *p*_{t(i)} in two of the more numerous demographic classes—prime aged and senescent females. In contrast, the small contribution of variation in trait values to population growth via adult males occurs because adult males constitute a small proportion of the population rather than because of a lack of an association between body weight and *p*_{t(i)} (Fig. 1).

Additive genetic variation for body weight contributed 0.88% of population growth, with values of 1.43% and 0.19% for hind leg length and birth weight, respectively. Because population growth is mean fitness (*19*), our results can also be interpreted as the heritability of fitness via the focal traits.

On average, trait variation contributed relatively little to population growth. However, trait variation interacts with environmental variation to influence survival and fecundity in the Soay sheep (*20*) and other ungulates (*21*). Consequently, we next looked for temporal variation in the contribution of variation in quantitative traits to population growth. There were insufficient data in some years to fit GAMs, so we first looked to see whether contributions varied between years when mean survival was low and when it was high (SOM). We now focus on body weight, as results for hind leg length are similar (fig. S2), given the high correlation between it and body weight (*r* = 0.84). In low-survival years, variation in body weight accounted for 9.23% of variation in population growth rate, with approximately equal contributions from all demographic classes (Fig. 2D). In contrast, body weight was much less influential in high-survival years, accounting for only 3.87%, with the contribution being primarily from lambs (Fig. 2E). These results suggest an interaction between the distribution of trait values and the environment in influencing population growth.

Although we were unable to fit GAMs through data for each year, we did examine how variation in traits influenced population growth in each year using linear regressions (Fig. 3). We used the *r ^{2}* of the associations between trait values and

*p*

_{t(i)},

*S*

_{t(i)}, and

*F*

_{t(i)}in each class to assess the proportion of variation accounted for. The contribution of body weight to variation in population growth varied substantially between years, contributing up to nearly one-fifth (18%) of observed population growth (Fig. 3A). Since 1995, the contribution of variation in body size to population growth was less pronounced than in earlier years. In the latter period, winter weather on St. Kilda was relatively good for sheep as the mean of the North Atlantic Oscillation (NAO) was lower than in the earlier period (0.039 versus 2.348). The NAO was significantly correlated with the contribution of variation in body weight to population growth (

*r*

^{2}= 0.23,

*t*= 2.13,

*P*= 0.049,

*n*= 16 years). This provides further support that environmental variation influences how much variation a quantitative trait contributes to population growth, which raises the possibility that the opportunity for evolution is greatest in harsh environments.

The associations we present (Fig. 1) also represent the strength of selection (*22*). Most evolutionary theory assumes that selection is linear (*13*), although nonlinear disruptive or stabilizing selection does dramatically alter evolutionary outcomes (*23*, *24*). We found evidence of nonlinear selection in each age and sex class and that selection operates via different demographic rates in males and females. For example, in prime aged adult females, the strength of selection on body size decreased with increasing weight (Fig. 4); with only individuals less than ∼20 kg experiencing significant selection; selection also operated almost entirely via survival in this demographic class (Fig. 1). In contrast, in prime aged males, selection operated primarily via recruitment and increased exponentially with weight (Figs. 1 and 4). The form of the selection functions and the normal distributions of individuals in each class (Fig. 4) suggest that nonlinear selection will not maintain additive genetic variation within this population.

The shape and form of selection varied with time in a pattern similar to the temporal fluctuations observed in the contribution of trait variation to population growth (Fig. 4C). Selection was stronger, and most nonlinear, in years when survival rates were low (SOM). For example, the selection on 2- to 6-year-old adult females seen at low weights (Figs. 1 and 4) was only apparent in crash years. Because selection did not vary in sign with environmental fluctuations, fluctuating selection cannot be the mechanism maintaining the genetic diversity observed in body weight. Because 20 years is a relatively short period, we suspect that the maintenance of additive genetic variance is due to patterns of selection not yet observed and antagonistic effects not yet detected, perhaps linked to traits not yet studied.

In this paper, we have done two things. First, we have shown how to calculate the contribution of variation in a quantitative trait to population growth. We found that trait variation can make a substantial contribution—up to nearly 20% in some years. Contributions via different demographic classes and rates varied with time; they were generally largest in years when many individuals died. A decreasing temporal trend in the contribution of variation in body weight may be due to environmental variation. Additive genetic variation made comparatively small contributions to population growth, but selection on these traits can generate nontrivial ecological effects. Second, our findings describe the modus operandi of selection: In each demographic class, there is evidence of nonlinear selection; selection operates predominantly via survival in adult females and via recruitment in adult males; selection on large adult females is weak but increases with body size in adult males; and the strength and pathways by which selection operates varies with time.

To extend these results to predict evolutionary change in size-related traits, we need to track both heritabilities and individual trajectories through trait space. However, given the nonlinearities we observed, making accurate predictions will be challenging. It would also be informative to examine how nonadditive genetic variance and genetic by environment interactions underpinning quantitative traits contribute to population growth. We found that, as the strength of selection increases, so too does the contribution of trait variation to population growth. This link between selection and the ecological consequences of evolutionary change has been apparent since Lande's pioneering work (*14*). Our results are a useful step toward extending stochastic evolutionary demography toward a theoretical framework that describes population dynamics in terms of the quantitative traits on which selection operates. Such a theory would describe the feedback between ecological and evolutionary processes in a stochastic environment and would illuminate the mechanisms shaping additive genetic variation and phenotypic variation.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/315/5818/1571/DC1

Materials and Methods

Figs. S1 to S4

Tables S1 and S2

References