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.     

EVOLUTION AND EXTINCTION IN A CHANGING ENVIRONMENT: A QUANTITATIVE-GENETIC ANALYSIS

Because of the ubiquity of genetic variation for quantitative traits, virtually all populations have some capacity to respond evolutionarily to selective challenges. However, natural selection imposes demographic costs on a population, and if these costs are sufficiently large, the likelihood of extinction will be high. We consider how the mean time to extinction depends on selective pressures (rate and stochasticity of environmental change, and strength of selection), population parameters (carrying capacity, and reproductive capacity), and genetics (rate of polygenic mutation). We assume that in a randomly mating, finite population subject to density-dependent population growth, individual fitness is determined by a single quantitative-genetic character under Gaussian stabilizing selection with the optimum phenotype exhibiting directional change, or random fluctuations, or both. The quantitative trait is determined by a finite number of freely recombining, mutationally equivalent, additive loci. The dynamics of evolution and extinction are investigated, assuming that the population is initially under mutation-selection-drift balance. Under this model, in a directionally changing environment, the mean phenotype lags behind the optimum, but on the average evolves parallel to it. The magnitude of the lag determines the vulnerability to extinction. In finite populations, stochastic variation in the genetic variance can be quite pronounced, and bottlenecks in the genetic variance temporarily can impair the population's adaptive capacity enough to cause extinction when it would otherwise be unlikely in an effectively infinite population. We find that maximum sustainable rates of evolution or, equivalently, critical rates of environmental change, may be considerably less than 10% of a phenotypic standard deviation per generation.     

Population differences in early life‐history traits in grayling

In this paper we test population differences in early life‐history traits in three grayling Thymallus thymallus populations. The grayling shared ancestors some 80–90 years ago. We performed common‐garden experiments at three temperatures (mimicking population‐specific summer temperatures), and measured survival and growth rates during early development. We found significant additive genetic variance in size (length and yolk‐sac volume) measured at hatching, swim‐up and termination of the experiment, and significantly different reaction norms for growth rate and survival during the period of first feeding. In general, each population did best at the temperature experienced in nature. These differences in early life‐history traits suggest that natural selection has resulted in local adaptation in a time period of 13–18 generations.     

The selective advantage of reaction norms for environmental tolerance

A tolerance curve defines the dependence of a genotype's fitness on the state of an environmental gradient. It can be characterized by a mode (the genotype's optimal environment) and a width (the breadth of adaptation). It seems possible that one or both of these characters can be modified in an adaptive manner, at least partially, during development. Thus, we extend the theory of environmental tolerance to include reaction norms for the mode and the width of the tolerance curve. We demonstrate that the selective value of such reaction norms increases with increasing spatial heterogeneity and between-generation temporal variation in the environment and with decreasing within-generation temporal variation. Assuming that the maintenance of a high breadth of adaptation is costly, reaction, norms are shown to induce correlated selection for a reduction in this character. Nevertheless, regardless of the magnitude of the reaction norm, there is a nearly one to one relationship between the optimal breadth of adaptation and the within-generation temporal variation perceived by the organism. This suggests that empirical estimates of the breadth of adaptation may provide a useful index of this type of environmental variation from the organism's point of view.     

A C++ Template Library for Efficient Forward-Time Population Genetic Simulation of Large Populations

fwdpp is a C++ library of routines intended to facilitate the development of forward-time simulations under arbitrary mutation and fitness models. The library design provides a combination of speed, low memory overhead, and modeling flexibility not currently available from other forward simulation tools. The library is particularly useful when the simulation of large populations is required, as programs implemented using the library are much more efficient than other available forward simulation programs.     

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.