Exploring variation in fitness surfaces over time or space

As the number of studies estimating selection on multiple traits has increased in recent years, fitness surfaces have become a fundamental tool for understanding multivariate selection and evolution. However, rigorous statistical comparisons of multivariate selection surfaces over time or space have been limited to parametric analyses of selection coefficients estimated using a quadratic regression model. Although parametric comparisons are useful when selection is approximately linear or quadratic in nature, they are limited when confronting the complex nature of rugged fitness surfaces. Here, I present a novel solution to comparing nonparametric fitness surfaces over time or space. Using a Tucker3 tensor decomposition, which is essentially a higher order principal components analysis, I show how major features of fitness surfaces can be compared statistically. Combined with a bootstrap algorithm, I develop three statistical tests that identify (1) differences in the shape of nonparametric fitness surfaces, (2) differences in the contribution of each surface to variation in fitness across time or space, and (3) specific areas of the surfaces (trait combinations) that vary significantly over time or space. I illustrate the tensor decomposition and statistical analyses using idealized fitness surfaces.     

INFERRING FITNESS LANDSCAPES

Since 1983, study of natural selection has relied heavily on multiple regression of fitness on the values for a set of traits via ordinary least squares (OLSs), as proposed by Lande and Arnold, to obtain an estimate of the quadratic relationship between fitness and the traits, the fitness surface. However, well‐known statistical problems with this approach can affect inferences about selection. One key concern is that measures of lifetime fitness do not conform to a normal or any other standard sampling distribution, as needed to justify the usual statistical tests. Another is that OLS may yield an estimate of the sign of the fitness function's curvature that is opposite to the truth. We here show that the recently developed aster modeling approach, which explicitly models the components of fitness as the basis for inferences about lifetime fitness, eliminates these problems. We illustrate selection analysis via aster using simulated datasets involving five fitness components expressed in each of four years. We demonstrate that aster analysis yields accurate estimates of the fitness function in cases in which OLS misleads, as well as accurate confidence regions for directional selection gradients. Further, to evaluate selection when many traits are under consideration, we recommend model selection by information criteria and frequentist model averaging.     

Power and potential bias in field studies of natural selection

The advent of multiple regression analyses of natural selection has facilitated estimates of both the direct and indirect effects of selection on many traits in numerous organisms. However, low power in selection studies has possibly led to a bias in our assessment of the levels of selection shaping natural populations. Using calculations and simulations based on the statistical properties of selection coefficients, we find that power to detect total selection (the selection differential) depends on sample size and the strength of selection relative to the opportunity of selection. The power of detecting direct selection (selection gradients) is more complicated and depends on the relationship between the correlation of each trait and fitness and the pattern of correlation among traits. In a review of 298 previously published selection differentials, we find that most studies have had insufficient power to detect reported levels of selection acting on traits and that, in general, the power of detecting weak levels of selection is low given current study designs. We also find that potential publication bias could explain the trend that reported levels of direct selection tend to decrease as study sizes increase, suggesting that current views of the strength of selection may be inaccurate and biased upward. We suggest that studies should be designed so that selection is analyzed on at least several hundred individuals, the total opportunity of selection be considered along with the pattern of selection on individual traits, and nonsignificant results be actively reported combined with an estimate of power.     

Comparing complex fitness surfaces: among-population variation in mutual sexual selection in Drosophila serrata

Despite a dramatic increase in empirical estimates of phenotypic selection over the past two decades, we remain remarkably ignorant about variation in the multivariate fitness surfaces that shape the adaptive landscape. We develop a novel approach for quantifying patterns of spatial and/or temporal variation in multivariate selection that directly compares vectors of linear selection gradients (beta) and matrices of nonlinear selection gradients (gamma) that describe the multivariate fitness surface in each population. We apply this approach to estimates of sexual selection on a suite of cuticular hydrocarbons (CHCs) in males and females from nine geographic populations of Drosophila serrata. In males, variation in linear sexual selection was associated with the presence of the related species Drosophila birchii, suggesting that female mate preferences for male CHCs differ between sympatry and allopatry. This is consistent with previous experimental results suggesting that reproductive character displacement of male CHCs has resulted from selection caused by the presence of D. birchii. No significant associations were found for nonlinear sexual selection in males. In females, large-scale variation in both linear and nonlinear sexual selection was negatively associated with assumed-neutral population genetic structure, suggesting a key role for chance events in male mate preference divergence.     

Nomograms Aid Interpretation of Complex Regression Models

ABSTRACT Ecologists often develop complex regression models that include multiple categorical and continuous variables, interactions among predictors, and nonlinear relationships between the response and predictor variables. Nomograms, which are graphical devices for presenting mathematical functions and calculating output values, can aid biologists in interpreting and presenting these complex models. To illustrate benefits of nomograms, we developed a logistic regression model of elk (Cervus elaphus) resource selection. With this model, we demonstrated how a nomogram helps scientists and managers interpret interactions among variables, compare the relative biological importance of variables, and examine predicted shapes of relationships (e.g., linear vs. nonlinear) between response and predictor variables. Although our example focused on logistic regression, nomograms are equally useful for other linear and nonlinear models. Regardless of the approach used for model development, nomograms and other graphical summaries can help scientists and managers develop, interpret, and apply statistical models.     

A tale of two matrices: multivariate approaches in evolutionary biology

Two symmetric matrices underlie our understanding of microevolutionary change. The first is the matrix of nonlinear selection gradients (gamma) which describes the individual fitness surface. The second is the genetic variance-covariance matrix (G) that influences the multivariate response to selection. A common approach to the empirical analysis of these matrices is the element-by-element testing of significance, and subsequent biological interpretation of pattern based on these univariate and bivariate parameters. Here, I show why this approach is likely to misrepresent the genetic basis of quantitative traits, and the selection acting on them in many cases. Diagonalization of square matrices is a fundamental aspect of many of the multivariate statistical techniques used by biologists. Applying this, and other related approaches, to the analysis of the structure of gamma and G matrices, gives greater insight into the form and strength of nonlinear selection, and the availability of genetic variance for multiple traits.     

Differences in the temporal dynamics of phenotypic selection among fitness components in the wild

The balance of selection acting through different fitness components (e.g. fecundity, mating success, survival) determines the potential tempo and trajectory of adaptive evolution. Yet the extent to which the temporal dynamics of phenotypic selection may vary among fitness components is poorly understood. Here, we compiled a database of 3978 linear selection coefficients from temporally replicated studies of selection in wild populations to address this question. Across studies, we find that multi-year selection through mating success and fecundity is stronger than selection through survival, but varies less in direction. We also report that selection through mating success varies more in long-term average strength than selection through either survival or fecundity. The consistency in direction and stronger long-term average strength of selection through mating success and fecundity suggests that selection through these fitness components should cause more persistent directional evolution relative to selection through survival. Similar patterns were apparent for the subset of studies that evaluated the temporal dynamics of selection on traits simultaneously using several different fitness components, but few such studies exist. Taken together, these results reveal key differences in the temporal dynamics of selection acting through different fitness components, but they also reveal important limitations in our understanding of how selection drives adaptive evolution.     

ESTIMATING AND VISUALIZING FITNESS SURFACES USING MARK–RECAPTURE DATA

Understanding how selection operates on a set of phenotypic traits is central to evolutionary biology. Often, it requires estimating survival (or other fitness‐related life‐history traits) which can be difficult to obtain for natural populations because individuals cannot be exhaustively followed. To cope with this issue of imperfect detection, we advocate the use of mark‐recapture data and we provide a general framework for both the estimation of linear and nonlinear selection gradients and the visualization of fitness surfaces. To quantify the strength of selection, the standard second‐order polynomial regression method is integrated in mark‐recapture models. To visualize the form of selection, we use splines to display selection acting on multivariate phenotypes in the most flexible way. We employ Markov chain Monte Carlo sampling in a Bayesian framework to estimate model parameters, assessing traits relevance and calculating the optimal amount of smoothing. We illustrate our approach using data from a wild population of Common blackbirds (Turdus merula) to investigate survival in relation to morphological traits, and provide evidence for correlational selection using the new methodology. Overall, the framework we propose will help in exploring the full potential of mark‐recapture data to study natural selection.     

Reducing environmental bias when measuring natural selection

Abstract.— Crucial to understanding the process of natural selection is characterizing phenotypic selection. Measures of phenotypic selection can be biased by environmental variation among individuals that causes a spurious correlation between a trait and fitness. One solution is analyzing genotypic data, rather than phenotypic data. Genotypic data, however, are difficult to gather, can be gathered from few species, and typically have low statistical power. Environmental correlations may act through traits other than through fitness itself. A path analytic framework, which includes measures of such traits, may reduce environmental bias in estimates of selection coefficients. We tested the efficacy of path analysis to reduce bias by re-analyzing three experiments where both phenotypic and genotypic data were available. All three consisted of plant species (Impatiens capensis, Arabidopsis thaliana , and Raphanus sativus) grown in experimental plots or the greenhouse. We found that selection coefficients estimated by path analysis using phenotypic data were highly correlated with those based on genotypic data with little systematic bias in estimating the strength of selection. Although not a panacea, using path analysis can substantially reduce environmental biases in estimates of selection coefficients. Such confidence in phenotypic selection estimates is critical for progress in the study of natural selection.     

How to quantify (the response to) sexual selection on traits

Natural selection operates via fitness components like mating success, fecundity, and longevity, which can be understood as intermediaries in the causal process linking traits to fitness. In particular, sexual selection occurs when traits influence mating or fertilization success, which, in turn, influences fitness. We show how to quantify both these steps in a single path analysis, leading to better estimates of the strength of sexual selection. Our model controls for confounding variables, such as body size or condition, when estimating the relationship between mating and reproductive success. Correspondingly, we define the Bateman gradient and the Jones index using partial rather than simple regressions, which better captures how they are commonly interpreted. The model can be applied both to purely phenotypic data and to quantitative genetic parameters estimated using information on relatedness. The phenotypic approach breaks down selection differentials into a sexually selected and a “remainder” component. The quantitative genetic approach decomposes the estimated evolutionary response to selection analogously. We apply our method to analyze sexual selection in male dusky pipefish, Syngnathus floridae, and in two simulated datasets. We highlight conceptual and statistical limitations of previous path‐based approaches, which can lead to substantial misestimation of sexual selection.     

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.     

Egg‐induced changes to sperm phenotypes shape patterns of multivariate selection on ejaculates

Sperm cells exhibit extraordinary phenotypic diversity and rapid rates of evolution, yet the adaptive value of most sperm traits remains equivocal. Recent findings suggest that to understand how selection targets ejaculates, we must recognize that female‐imposed physiological conditions often alter sperm phenotypes. These phenotypic changes may influence the relationships among sperm traits and their association with fitness. Here, we show that chemical substances released by eggs (known to modify sperm physiology and behaviour) alter patterns of selection on a suite of sperm traits in the mussel Mytilus galloprovincialis. We use multivariate selection analyses to characterize linear and nonlinear selection acting on sperm traits in (a) seawater alone and (b) seawater containing egg‐derived chemicals (egg water). Our analyses revealed that nonlinear selection on canonical axes of multiple traits (notably sperm velocity, sperm linearity and percentage of motile sperm) was the most important form of selection overall, but importantly these patterns were only evident when sperm phenotypes were measured in egg water. These findings reveal the subtle way that females can alter patterns of selection, with the implication that overlooking environmentally moderated changes to sperm, may result in erroneous interpretations of how selection targets phenotypic (co)variation in sperm traits.     

Synthetic analyses of phenotypic selection in natural populations: lessons, limitations and future directions

There are now thousands of estimates of phenotypic selection in natural populations, resulting in multiple synthetic reviews of these data. Here we consider several major lessons and limitations emerging from these syntheses, and how they may guide future studies of selection in the wild. First, we review past analyses of the patterns of directional selection. We present new meta-analyses that confirm differences in the direction and magnitude of selection for different types of traits and fitness components. Second, we describe patterns of temporal and spatial variation in directional selection, and their implications for cumulative selection and directional evolution. Meta-analyses suggest that sampling error contributes importantly to observed temporal variation in selection, and indicate that evidence for frequent temporal changes in the direction of selection in natural populations is limited. Third, we review the apparent lack of evidence for widespread stabilizing selection, and discuss biological and methodological explanations for this pattern. Finally, we describe how sampling error, statistical biases, choice of traits, fitness measures and selection metrics, environmental covariance and other factors may limit the inferences we can draw from analyses of selection coefficients. Current standardized selection metrics based on simple parametric statistical models may be inadequate for understanding patterns of non-linear selection and complex fitness surfaces. We highlight three promising areas for expanding our understanding of selection in the wild: (1) field studies of stabilizing selection, selection on physiological and behavioral traits, and the ecological causes of selection; (2) new statistical models and methods that connect phenotypic variation to population demography and selection; and (3) availability of the underlying individual-level data sets from past and future selection studies, which will allow comprehensive modeling of selection and fitness variation within and across systems, rather than meta-analyses of standardized selection metrics.     

Q ST MEETS THE G MATRIX: THE DIMENSIONALITY OF ADAPTIVE DIVERGENCE IN MULTIPLE CORRELATED QUANTITATIVE TRAITS

The Q _ST– F _ST comparison has become an increasingly common method for inferring adaptive quantitative trait divergence among populations. For cases in which there is divergence in multiple traits, most studies have applied the method by performing multiple univariate Q _ST– F _ST comparisons. However, because traits are often genetically correlated, such univariate analyses are likely to paint a simplified picture of adaptive divergence. Here we show how the multivariate analogue of Q _ST, F_STq, which accounts for genetic correlations among traits, can be used to supply a more detailed picture of multitrait divergence. We apply the method to naturally occurring genetic variation for a suite of sexually selected display traits in Drosophila serrata . The analyses suggest the operation of divergent multivariate selection that has influenced multiple independent axes of genetic variance in a sex-specific manner. Finally, we show how a comparison of the components of F_STq, the average within and among population genetic variance–covariance matrices, G_W and G_B, can be used as an additional test of the null expectation of neutral divergence, and allows for an investigation of whether natural populations have diverged along major or minor axes of genetic variance.     

A standardized approach to estimate life history tradeoffs in evolutionary ecology

As tradeoffs limit the maximum Darwinian fitness individuals can reach, measuring reliably the strength of tradeoffs using appropriate metrics is of prime importance to understand the evolution of traits under constraints. Tradeoffs involving phenotypic traits and fitness components, however, are difficult to quantify in free‐ranging populations because of confounding effects due to environmental variation and individual heterogeneity. Furthermore, although some methods have been used previously to quantify tradeoffs, these methods cannot be applied with respect to binary traits, which are common to describe life histories (e.g. probability of reproduction, nesting success, offspring survival). Here, we demonstrate how to measure reliably the strength of tradeoffs involving binary traits using (auto)correlation estimates obtained from generalized linear (mixed) models. We first propose a standardized approach that accounts for the variation in the nature of the tradeoffs being compared (e.g. continuous/binary traits, repeated/non‐repeated measures), and then apply this method to longitudinal data from two contrasting species of large herbivores. Empirical estimates of tradeoffs varied among traits, and between‐species comparisons suggested that reproductive tradeoffs between successive breeding attempts might only occur in capital breeders. The empirical results we obtained clearly demonstrate that the method we provide allows measuring reliably the strength of tradeoffs under most circumstances, including tradeoffs on binary traits. Our original approach therefore offers an important first step for comparing the strength and, hence, the relative importance of different tradeoffs, and opens the door to a better understanding of the evolution of life history traits in free‐ranging populations.     

Experimental evidence for multivariate stabilizing sexual selection

Stabilizing selection is a fundamental concept in evolutionary biology. In the presence of a single intermediate optimum phenotype (fitness peak) on the fitness surface, stabilizing selection should cause the population to evolve toward such a peak. This prediction has seldom been tested, particularly for suites of correlated traits. The lack of tests for an evolutionary match between population means and adaptive peaks may be due, at least in part, to problems associated with empirically detecting multivariate stabilizing selection and with testing whether population means are at the peak of multivariate fitness surfaces. Here we show how canonical analysis of the fitness surface, combined with the estimation of confidence regions for stationary points on quadratic response surfaces, may be used to define multivariate stabilizing selection on a suite of traits and to establish whether natural populations reside on the multivariate peak. We manufactured artificial advertisement calls of the male cricket Teleogryllus commodus and played them back to females in laboratory phonotaxis trials to estimate the linear and nonlinear sexual selection that female phonotactic choice imposes on male call structure. Significant nonlinear selection on the major axes of the fitness surface was convex in nature and displayed an intermediate optimum, indicating multivariate stabilizing selection. The mean phenotypes of four independent samples of males, from the same population as the females used in phonotaxis trials, were within the 95% confidence region for the fitness peak. These experiments indicate that stabilizing sexual selection may play an important role in the evolution of male call properties in natural populations of T. commodus.     

The quantitative genetics of floral trait variation in Lobelia: potential constraints on adaptive evolution

Abstract Although pollinator-mediated natural selection has been measured on many floral traits and in many species, the extent to which selection is constrained from producing optimal floral phenotypes is less frequently studied. In particular, negative correlations between flower size and flower number are hypothesized to be a major constraint on the evolution of floral displays, yet few empirical studies have documented such a trade-off. To determine the potential for genetic constraints on the adaptive evolution of floral displays, I estimated the quantitative genetic basis of floral trait variation in two populations of Lobelia siphilitica . Restricted maximum likelihood (REML) analyses of greenhouse-grown half-sib families were used to estimate genetic variances and covariances for flower number and six measures of flower size. There was significant genetic variation for all seven floral traits in both populations. Flower number was negatively genetically correlated with four measures of flower size in one population and three measures in the other. When the genetic variance-covariance matrices were combined with field estimates of phenotypic selection gradients, the predicted multivariate evolutionary response was less than or opposite in sign to the selection gradient for flower number and five of six measures of flower size, suggesting genetic constraints on the evolution of these traits. More generally, my results indicate that the adaptive evolution of floral displays can be constrained by tradeoffs between flower size and number, as has been assumed by many theoretical models of floral evolution.     

Estimating fluctuating selection in age-structured populations

In age-structured populations, viability and fecundity selection of varying strength may occur in different age classes. On the basis of an original idea by Fisher of weighting individuals by their reproductive value, we show that the combined effect of selection on traits at different ages acts through the individual reproductive value defined as the stochastic contribution of an individual to the total reproductive value of the population the following year. The selection differential is a weighted sum of age-specific differentials that are the covariances between the phenotype and the age-specific relative fitness defined by the individual reproductive value. This enables estimation of weak selection on a multivariate quantitative character in populations with no density regulation by combinations of age-specific linear regressions of individual reproductive values on the traits. Demographic stochasticity produces random variation in fitness components in finite samples of individuals and affects the statistical inference of the temporal average directional selection as well as the magnitude of fluctuating selection. Uncertainties in parameter estimates and test power depend strongly on the demographic stochasticity. Large demographic variance results in large uncertainties in yearly estimates of selection that complicates detection of significant fluctuating selection. The method is illustrated by an analysis of age-specific selection in house sparrows on a fitness-related two-dimensional morphological trait, tarsus length and body mass of fledglings.     

Natural selection, evolvability and bias due to environmental covariance in the field in an annual plant

Estimates of the form and magnitude of natural selection based on phenotypic relationships between traits and fitness measures can be biased when environmental factors influence both relative fitness and phenotypic trait values. I quantified genetic variances and covariances, and estimated linear and quadratic selection coefficients, for seven traits of an annual plant grown in the field. For replicates of 50 paternal half-sib families, coefficients of selection were calculated both for individual phenotypic values of the traits and for half-sib family mean values. The potential for evolutionary response was supported by significant heritability and phenotypic directional selection for several traits but contradicted by the absence of significant genetic variation for fitness estimates and evidence of bias in phenotypic selection coefficients due to environmental covariance for at least two of the traits analysed. Only studies of a much wider range of organisms and traits will reveal the frequency and extent of such bias.