The danger of applying the breeder's equation in observational studies of natural populations

The breeder's equation, which predicts evolutionary change when a phenotypic covariance exists between a heritable trait and fitness, has provided a key conceptual framework for studies of adaptive microevolution in nature. However, its application requires strong assumptions to be made about the causation of fitness variation. In its univariate form, the breeder's equation assumes that the trait of interest is not correlated with other traits having causal effects on fitness. In its multivariate form, the validity of predicted change rests on the assumption that all such correlated traits have been measured and incorporated into the analysis. Here, we (i) highlight why these assumptions are likely to be seriously violated in studies of natural, rather than artificial, selection and (ii) advocate wider use of the Robertson–Price identity as a more robust, and less assumption‐laden, alternative to the breeder's equation for applications in evolutionary ecology.     

Testing for biases in selection on avian reproductive traits and partitioning direct and indirect selection using quantitative genetic models

Key life history traits such as breeding time and clutch size are frequently both heritable and under directional selection, yet many studies fail to document microevolutionary responses. One general explanation is that selection estimates are biased by the omission of correlated traits that have causal effects on fitness, but few valid tests of this exist. Here, we show, using a quantitative genetic framework and six decades of life‐history data on two free‐living populations of great tits Parus major, that selection estimates for egg‐laying date and clutch size are relatively unbiased. Predicted responses to selection based on the Robertson–Price Identity were similar to those based on the multivariate breeder's equation (MVBE), indicating that unmeasured covarying traits were not missing from the analysis. Changing patterns of phenotypic selection on these traits (for laying date, linked to climate change) therefore reflect changing selection on breeding values, and genetic constraints appear not to limit their independent evolution. Quantitative genetic analysis of correlational data from pedigreed populations can be a valuable complement to experimental approaches to help identify whether apparent associations between traits and fitness are biased by missing traits, and to parse the roles of direct versus indirect selection across a range of environments.     

Selection on skewed characters and the paradox of stasis

Observed phenotypic responses to selection in the wild often differ from predictions based on measurements of selection and genetic variance. An overlooked hypothesis to explain this paradox of stasis is that a skewed phenotypic distribution affects natural selection and evolution. We show through mathematical modeling that, when a trait selected for an optimum phenotype has a skewed distribution, directional selection is detected even at evolutionary equilibrium, where it causes no change in the mean phenotype. When environmental effects are skewed, Lande and Arnold's (1983) directional gradient is in the direction opposite to the skew. In contrast, skewed breeding values can displace the mean phenotype from the optimum, causing directional selection in the direction of the skew. These effects can be partitioned out using alternative selection estimates based on average derivatives of individual relative fitness, or additive genetic covariances between relative fitness and trait (Robertson–Price identity). We assess the validity of these predictions using simulations of selection estimation under moderate sample sizes. Ecologically relevant traits may commonly have skewed distributions, as we here exemplify with avian laying date — repeatedly described as more evolutionarily stable than expected — so this skewness should be accounted for when investigating evolutionary dynamics in the wild.     

Evolutionary potential of morphological traits across different life‐history stages

Despite accumulating examples of selection acting on heritable traits in the wild, predicted evolutionary responses are often different from observed phenotypic trends. Various explanations have been suggested for these mismatches. These include within‐individual changes across lifespan that can create important variation in genetic architecture of traits and selection acting on them, but also potential problems with the methodological approach used to predict evolutionary responses of traits. Here, we used an 8‐year data set on tree swallow (Tachycineta bicolor) to first assess the effects of differences among three nestling life‐history stages on the genetic (co)variances of two morphological traits (body mass and primary feather length) and the selection acting on them over three generations. We then estimated the evolutionary potential of these traits by predicting their evolutionary responses using the breeder's equation and the secondary theorem of selection approaches. Our results showed variation in strength and direction of selection and slight changes in trait variance across ages. Predicted evolutionary responses differed importantly between both approaches for half of the trait–age combinations we studied, suggesting the presence of environmentally induced correlations between focal traits and fitness possibly biasing breeder's equation predictions. Our results emphasize that predictions of evolutionary potential for morphological traits are likely to be highly variable, both in strength and direction, depending on the life stage and method used, thus mitigating our capacity to predict adaptation and persistence of wild populations.     

Evolution of quantitative traits in the wild: mind the ecology

Recent advances in the quantitative genetics of traits in wild animal populations have created new interest in whether natural selection, and genetic response to it, can be detected within long-term ecological studies. However, such studies have re-emphasized the fact that ecological heterogeneity can confound our ability to infer selection on genetic variation and detect a population''s response to selection by conventional quantitative genetics approaches. Here, I highlight three manifestations of this issue: counter gradient variation, environmentally induced covariance between traits and the correlated effects of a fluctuating environment. These effects are symptomatic of the oversimplifications and strong assumptions of the breeder''s equation when it is applied to natural populations. In addition, methods to assay genetic change in quantitative traits have overestimated the precision with which change can be measured. In the future, a more conservative approach to inferring quantitative genetic response to selection, or genomic approaches allowing the estimation of selection intensity and responses to selection at known quantitative trait loci, will provide a more precise view of evolution in ecological time.     

THE CONTRIBUTION OF MATERNAL EFFECTS TO SELECTION RESPONSE: AN EMPIRICAL TEST OF COMPETING MODELS

Maternal effects can dramatically influence the evolutionary process, in some cases facilitating and in others hindering adaptive evolution. Maternal effects have been incorporated into quantitative genetic models using two theoretical frameworks: the variance‐components approach, which partitions variance into direct and maternal components, and the trait‐based approach, which assumes that maternal effects are mediated by specific maternal traits. Here, we demonstrate parallels between these models and test their ability to predict evolutionary change. First, we show that the two approaches predict equivalent responses to selection in the absence of maternal effects mediated by traits that are themselves maternally influenced. We also introduce a correction factor that may be applied when such cascading maternal effects are present. Second, we use several maternal effect models, as well as the standard breeder's equation, to predict evolution in response to artificial selection on flowering time in American bellflower, Campanulastrum americanum. Models that included maternal effects made much more accurate predictions of selection response than the breeder's equation. Maternal effect models differed somewhat in their fit, with a version of the trait‐based model providing the best fit. We recommend fitting such trait‐based models when possible and appropriate to make the most accurate evolutionary predictions.     

Using time series analysis to characterize evolutionary and plastic responses to environmental change: a case study of a shift toward earlier migration date in sockeye salmon

Environmental change can shift the phenotype of an organism through either evolutionary or nongenetic processes. Despite abundant evidence of phenotypic change in response to recent climate change, we typically lack sufficient genetic data to identify the role of evolution. We present a method of using phenotypic data to characterize the hypothesized role of natural selection and environmentally driven phenotypic shifts (plasticity). We modeled historical selection and environmental predictors of interannual variation in mean population phenotype using a multivariate state-space model framework. Through model comparisons, we assessed the extent to which an estimated selection differential explained observed variation better than environmental factors alone. We applied the method to a 60-year trend toward earlier migration in Columbia River sockeye salmon Oncorhynchus nerka, producing estimates of annual selection differentials, average realized heritability, and relative cumulative effects of selection and plasticity. We found that an evolutionary response to thermal selection was capable of explaining up to two-thirds of the phenotypic trend. Adaptive plastic responses to June river flow explain most of the remainder. This method is applicable to other populations with time series data if selection differentials are available or can be reconstructed. This method thus augments our toolbox for predicting responses to environmental change.     

EVOLUTION IN FLUCTUATING ENVIRONMENTS: DECOMPOSING SELECTION INTO ADDITIVE COMPONENTS OF THE ROBERTSON–PRICE EQUATION

We analyze the stochastic components of the Robertson–Price equation for the evolution of quantitative characters that enables decomposition of the selection differential into components due to demographic and environmental stochasticity. We show how these two types of stochasticity affect the evolution of multivariate quantitative characters by defining demographic and environmental variances as components of individual fitness. The exact covariance formula for selection is decomposed into three components, the deterministic mean value, as well as stochastic demographic and environmental components. We show that demographic and environmental stochasticity generate random genetic drift and fluctuating selection, respectively. This provides a common theoretical framework for linking ecological and evolutionary processes. Demographic stochasticity can cause random variation in selection differentials independent of fluctuating selection caused by environmental variation. We use this model of selection to illustrate that the effect on the expected selection differential of random variation in individual fitness is dependent on population size, and that the strength of fluctuating selection is affected by how environmental variation affects the covariance in Malthusian fitness between individuals with different phenotypes. Thus, our approach enables us to partition out the effects of fluctuating selection from the effects of selection due to random variation in individual fitness caused by demographic stochasticity.     

Variation,selection and evolution of function-valued traits

We describe an emerging framework for understanding variation, selection and evolution of phenotypic traits that are mathematical functions. We use one specific empirical example – thermal performance curves (TPCs) for growth rates of caterpillars – to demonstrate how models for function-valued traits are natural extensions of more familiar, multivariate models for correlated, quantitative traits. We emphasize three main points. First, because function-valued traits are continuous functions, there are important constraints on their patterns of variation that are not captured by multivariate models. Phenotypic and genetic variation in function-valued traits can be quantified in terms of variance-covariance functions and their associated eigenfunctions: we illustrate how these are estimated as well as their biological interpretations for TPCs. Second, selection on a function-valued trait is itself a function, defined in terms of selection gradient functions. For TPCs, the selection gradient describes how the relationship between an organism's performance and its fitness varies as a function of its temperature. We show how the form of the selection gradient function for TPCs relates to the frequency distribution of environmental states (caterpillar temperatures) during selection. Third, we can predict evolutionary responses of function-valued traits in terms of the genetic variance-covariance and the selection gradient functions. We illustrate how non-linear evolutionary responses of TPCs may occur even when the mean phenotype and the selection gradient are themselves linear functions of temperature. Finally, we discuss some of the methodological and empirical challenges for future studies of the evolution of function-valued traits.     

Cultural transmission and the evolution of human behaviour: a general approach based on the Price equation

Transmitted culture can be viewed as an inheritance system somewhat independent of genes that is subject to processes of descent with modification in its own right. Although many authors have conceptualized cultural change as a Darwinian process, there is no generally agreed formal framework for defining key concepts such as natural selection, fitness, relatedness and altruism for the cultural case. Here, we present and explore such a framework using the Price equation. Assuming an isolated, independently measurable culturally transmitted trait, we show that cultural natural selection maximizes cultural fitness, a distinct quantity from genetic fitness, and also that cultural relatedness and cultural altruism are not reducible to or necessarily related to their genetic counterparts. We show that antagonistic coevolution will occur between genes and culture whenever cultural fitness is not perfectly aligned with genetic fitness, as genetic selection will shape psychological mechanisms to avoid susceptibility to cultural traits that bear a genetic fitness cost. We discuss the difficulties with conceptualizing cultural change using the framework of evolutionary theory, the degree to which cultural evolution is autonomous from genetic evolution, and the extent to which cultural change should be seen as a Darwinian process. We argue that the nonselection components of evolutionary change are much more important for culture than for genes, and that this and other important differences from the genetic case mean that different approaches and emphases are needed for cultural than genetic processes.     

COMPARING STRENGTHS OF DIRECTIONAL SELECTION: HOW STRONG IS STRONG?

The fundamental equation in evolutionary quantitative genetics, the Lande equation, describes the response to directional selection as a product of the additive genetic variance and the selection gradient of trait value on relative fitness. Comparisons of both genetic variances and selection gradients across traits or populations require standardization, as both are scale dependent. The Lande equation can be standardized in two ways. Standardizing by the variance of the selected trait yields the response in units of standard deviation as the product of the heritability and the variance-standardized selection gradient. This standardization conflates selection and variation because the phenotypic variance is a function of the genetic variance. Alternatively, one can standardize the Lande equation using the trait mean, yielding the proportional response to selection as the product of the squared coefficient of additive genetic variance and the mean-standardized selection gradient. Mean-standardized selection gradients are particularly useful for summarizing the strength of selection because the mean-standardized gradient for fitness itself is one, a convenient benchmark for strong selection. We review published estimates of directional selection in natural populations using mean-standardized selection gradients. Only 38 published studies provided all the necessary information for calculation of mean-standardized gradients. The median absolute value of multivariate mean-standardized gradients shows that selection is on average 54% as strong as selection on fitness. Correcting for the upward bias introduced by taking absolute values lowers the median to 31%, still very strong selection. Such large estimates clearly cannot be representative of selection on all traits. Some possible sources of overestimation of the strength of selection include confounding environmental and genotypic effects on fitness, the use of fitness components as proxies for fitness, and biases in publication or choice of traits to study.     

Evolutionary potential of a widespread clonal grass under changing climate

Adaptive responses are probably the most effective long‐term responses of populations to climate change, but they require sufficient evolutionary potential upon which selection can act. This requires high genetic variance for the traits under selection and low antagonizing genetic covariances between the different traits. Evolutionary potential estimates are still scarce for long‐lived, clonal plants, although these species are predicted to dominate the landscape with climate change. We studied the evolutionary potential of a perennial grass, Festuca rubra, in western Norway, in two controlled environments corresponding to extreme environments in natural populations: cold–dry and warm–wet, the latter being consistent with the climatic predictions for the country. We estimated genetic variances, covariances, selection gradients and response to selection for a wide range of growth, resource acquisition and physiological traits, and compared their estimates between the environments. We showed that the evolutionary potential of F. rubra is high in both environments, and genetic covariances define one main direction along which selection can act with relatively few constraints to selection. The observed response to selection at present is not sufficient to produce genotypes adapted to the predicted climate change under a simple, space for time substitution model. However, the current populations contain genotypes which are pre‐adapted to the new climate, especially for growth and resource acquisition traits. Overall, these results suggest that the present populations of the long‐lived clonal plant may have sufficient evolutionary potential to withstand long‐term climate changes through adaptive responses.     

Evolutionary rates for multivariate traits: the role of selection and genetic variation

A fundamental question in evolutionary biology is the relative importance of selection and genetic architecture in determining evolutionary rates. Adaptive evolution can be described by the multivariate breeders'' equation (), which predicts evolutionary change for a suite of phenotypic traits () as a product of directional selection acting on them (β) and the genetic variance–covariance matrix for those traits (G). Despite being empirically challenging to estimate, there are enough published estimates of G and β to allow for synthesis of general patterns across species. We use published estimates to test the hypotheses that there are systematic differences in the rate of evolution among trait types, and that these differences are, in part, due to genetic architecture. We find some evidence that sexually selected traits exhibit faster rates of evolution compared with life-history or morphological traits. This difference does not appear to be related to stronger selection on sexually selected traits. Using numerous proposed approaches to quantifying the shape, size and structure of G, we examine how these parameters relate to one another, and how they vary among taxonomic and trait groupings. Despite considerable variation, they do not explain the observed differences in evolutionary rates.     

Evolutionary dynamics of a sexual ornament in the house sparrow (Passer domesticus): the role of indirect selection within and between sexes

The relative contribution of sexual and natural selection to evolution of sexual ornaments has rarely been quantified under natural conditions. In this study we used a long-term dataset of house sparrows in which parents and offspring were matched genetically to estimate the within- and across-sex genetic basis for variation and covariation among morphological traits. By applying two-sex multivariate "animal models" to estimate genetic parameters, we estimated evolutionary changes in a male sexual ornament, badge size, from the contribution of direct and indirect selection on correlated traits within males and females, after accounting for overlapping generations and age-structure. Indirect natural selection on genetically correlated traits in males and females was the major force causing evolutionary change in the male ornament. Thus, natural selection on female morphology may cause indirect evolutionary changes in male ornaments. We observed however no directional phenotypic change in the ornament size of one-year-old males during the study period. On the other hand, changes were recorded in other morphological characters of both sexes. Our analyses of evolutionary dynamics in sexual characters require application of appropriate two-sex models to account for how selection on correlated traits in both sexes affects the evolutionary outcome of sexual selection.     

Climate change shifts natural selection and the adaptive potential of the perennial forb Boechera stricta in the Rocky Mountains

Heritable genetic variation is necessary for populations to evolve in response to anthropogenic climate change. However, antagonistic genetic correlations among traits may constrain the rate of adaptation, even if substantial genetic variation exists. We examine potential genetic responses to selection by comparing multivariate genetic variance–covariances of traits and fitness (multivariate Robertson–Price identities) across different environments in a reciprocal transplant experiment of the forb Boechera stricta in the Rocky Mountains. By transplanting populations into four common gardens arrayed along an elevational gradient, and exposing populations to control and snow removal treatments, we simulated future and current climates and snowmelt regimes. Genetic variation in flowering and germination phenology declined in plants moved downslope to warmer, drier sites, suggesting that these traits may have a limited ability to evolve under future climates. Simulated climate change via snow removal altered the strength of selection on flowering traits, but we found little evidence that genetic correlations among traits are likely to affect the rate of adaptation to climate change. Overall, our results suggest that climate change may alter the evolutionary potential of B. stricta, but reduced expression of genetic variation may be a larger impediment to adaptation than constraints imposed by antagonistic genetic correlations.     

Evolutionary stasis of a heritable morphological trait in a wild fish population despite apparent directional selection

Comparing observed versus theoretically expected evolutionary responses is important for our understanding of the evolutionary process, and for assessing how species may cope with anthropogenic change. Here, we document directional selection for larger female size in Atlantic salmon, using pedigree‐derived estimates of lifetime reproductive success as a fitness measure. We show the trait is heritable and, thus, capable of responding to selection. The Breeder's Equation, which predicts microevolution as the product of phenotypic selection and heritability, predicted evolution of larger size. This was at odds, however, with the observed lack of either phenotypic or genetic temporal trends in body size, a so‐called “paradox of stasis.” To investigate this paradox, we estimated the additive genetic covariance between trait and fitness, which provides a prediction of evolutionary change according to Robertson's secondary theorem of selection (STS) that is unbiased by missing variables. The STS prediction was consistent with the observed stasis. Decomposition of phenotypic selection gradients into genetic and environmental components revealed a potential upward bias, implying unmeasured factors that covary with trait and fitness. These results showcase the power of pedigreed, wild population studies—which have largely been limited to birds and mammals—to study evolutionary processes on contemporary timescales.     

Genetic analysis of life-history constraint and evolution in a wild ungulate population

Trade-offs among life-history traits are central to evolutionary theory. In quantitative genetic terms, trade-offs may be manifested as negative genetic covariances relative to the direction of selection on phenotypic traits. Although the expression and selection of ecologically important phenotypic variation are fundamentally multivariate phenomena, the in situ quantification of genetic covariances is challenging. Even for life-history traits, where well-developed theory exists with which to relate phenotypic variation to fitness variation, little evidence exists from in situ studies that negative genetic covariances are an important aspect of the genetic architecture of life-history traits. In fact, the majority of reported estimates of genetic covariances among life-history traits are positive. Here we apply theory of the genetics and selection of life histories in organisms with complex life cycles to provide a framework for quantifying the contribution of multivariate genetically based relationships among traits to evolutionary constraint. We use a Bayesian framework to link pedigree-based inference of the genetic basis of variation in life-history traits to evolutionary demography theory regarding how life histories are selected. Our results suggest that genetic covariances may be acting to constrain the evolution of female life-history traits in a wild population of red deer Cervus elaphus: genetic covariances are estimated to reduce the rate of adaptation by about 40%, relative to predicted evolutionary change in the absence of genetic covariances. Furthermore, multivariate phenotypic (rather than genetic) relationships among female life-history traits do not reveal this constraint.     

Multivariate selection and intersexual genetic constraints in a wild bird population

When selection differs between the sexes for traits that are genetically correlated between the sexes, there is potential for the effect of selection in one sex to be altered by indirect selection in the other sex, a situation commonly referred to as intralocus sexual conflict (ISC). While potentially common, ISC has rarely been studied in wild populations. Here, we studied ISC over a set of morphological traits (wing length, tarsus length, bill depth and bill length) in a wild population of great tits (Parus major) from Wytham Woods, UK. Specifically, we quantified the microevolutionary impacts of ISC by combining intra‐ and intersex additive genetic (co)variances and sex‐specific selection estimates in a multivariate framework. Large genetic correlations between homologous male and female traits combined with evidence for sex‐specific multivariate survival selection suggested that ISC could play an appreciable role in the evolution of this population. Together, multivariate sex‐specific selection and additive genetic (co)variance for the traits considered accounted for additive genetic variance in fitness that was uncorrelated between the sexes (cross‐sex genetic correlation = ?0.003, 95% CI = ?0.83, 0.83). Gender load, defined as the reduction in a population's rate of adaptation due to sex‐specific effects, was estimated at 50% (95% CI = 13%, 86%). This study provides novel insights into the evolution of sexual dimorphism in wild populations and illustrates how quantitative genetics and selection analyses can be combined in a multivariate framework to quantify the microevolutionary impacts of ISC.