Rapid evolution of quantitative traits: theoretical perspectives

Abstract

An increasing number of studies demonstrate phenotypic and genetic changes in natural populations that are subject to climate change, and there is hope that some of these changes will contribute to avoiding species extinctions ('evolutionary rescue'). Here, we review theoretical models of rapid evolution in quantitative traits that can shed light on the potential for adaptation to a changing climate. Our focus is on quantitative-genetic models with selection for a moving phenotypic optimum. We point out that there is no one-to-one relationship between the rate of adaptation and population survival, because the former depends on relative fitness and the latter on absolute fitness. Nevertheless, previous estimates that sustainable rates of genetically based change usually do not exceed 0.1 haldanes (i.e., phenotypic standard deviations per generation) are probably correct. Survival can be greatly facilitated by phenotypic plasticity, and heritable variation in plasticity can further speed up genetic evolution. Multivariate selection and genetic correlations are frequently assumed to constrain adaptation, but this is not necessarily the case and depends on the geometric relationship between the fitness landscape and the structure of genetic variation. Similar conclusions hold for adaptation to shifting spatial gradients. Recent models of adaptation in multispecies communities indicate that the potential for rapid evolution is strongly influenced by interspecific competition.

Keywords: Adaptation; climate change; habitat degradation; natural selection and contemporary evolution; phenotypic plasticity; population dynamics; population genetics; quantitative genetics.

Figure 1

Illustration of trait evolution in the one-dimensional moving-optimum model. (A) Solid and dotted gray curves represent the fitness landscape at different points in time (eq. 2), whose width is determined by ω. θ_t is the optimal phenotype, which moves at constant speed (θ_t = kt). The black curves represent the distribution of breeding values in the population (mean , variance ). The mean phenotype evolves according to eqn (1). At the dynamic equilibrium, it follows the optimum with a constant lag . (B) illustrates the relation between rate of evolution and extinction risk. The gray curves show the log mean fitness as a function of the mean phenotype for two different fitness functions with widths ω_s and ω_w, respectively. The rate of evolution, given by the horizontal arrows is determined by the fitness gradient β_t, indicated by the black lines. The vertical position of the population gives its mean (log) fitness. In the Figure, the optimum is assumed to move at rate k = 0.035, and the population placed at the narrow fitness curve follows at this pace while maintaining a positive growth rate (). With the wide fitness function, however, the same rate of evolution requires a larger distance from the optimum, such that the growth rate is negative and the population goes extinct.

Figure 2

Illustration of adaptation involving two genetically correlated traits. (A) Adaptation after a sudden environmental change; the new optimum θ_t is constant. Gray lines illustrate the fitness surface, defined by the matrix ω. The distribution of breeding values defined by the G-matrix is illustrated by the black ellipse, whose center is the mean phenotype and whose axis is the eigenvectors of G. The initial response to selection is biased toward the leading eigenvector, that is, the genetic line of least resistance (Schluter 1996). (B) Adaptation to a moving optimum. Gray circles show the fitness landscape at four different points in time. Black ellipses show the corresponding positions of the population (represented by the G-matrix). The insets at the top show the leading eigenvector of G, λ₁, the selection gradient β_t and the response to selection at time points 1, 2, and 3, respectively. Because the initial response is biased toward the leading eigenvector, the population ‘rises’ above the line of the moving optimum (i.e., the flying-kite effect; Jones et al. 2004). This rise comes to a halt as the tendency to follow the line of least resistance is balanced by the selection gradient, resulting in horizontal movement of the population.

Figure A1

The critical rate of phenotypic evolution, κ_crit = k_crit / σ_p (eqn A7) expressed in haldanes, for the one-dimensional moving-optimum model (A2) (after Bürger and Lynch 1995), as a function of the width of the fitness function (with ), for various values of the reproductive potential B. The top row shows results for three different values of heritability . In the bottom row, has been set to the value predicted by the stochastic house-of-cards (SHC) approximation under pure stabilizing selection for three values of the population size N. The SHC approximation is given by (Bürger and Lynch 1995), where V_m is the mutational variance, α² is the variance of the effect of new mutations, and N_e ≈ 2BN / (2B−1) (Bürger and Lynch 1995) is the effective population size. The figures are for V_m = 0.001 and α² = 0.05. The thin dotted line gives the heritability h² associated with (SHC).

Figure A2

Critical rates of environmental change k_crit (top row) and the corresponding rates of phenotypic evolution κ_crit (bottom row), under the premise that the population maintains a minimal size of N_crit individuals over t_crit generations. The case t_crit = ∞, N_crit = N₀ (where N₀ is the initial population size, which equals the carrying capacity; dashed line) corresponds to the case investigated by Bürger and Lynch (1995). The case t_crit = 50,N_crit = N₀ (gray line) is given by eqn A14. In the bottom row, solid and dashed gray lines are identical, because κ_crit (t_crit, N₀) = κ_crit (∞, N₀). Parameters are as in the bottom row of Fig. A1 with B = 2.

Figure A3

Observed generation-to-generation rates of phenotypic change κ in haldanes for the entire population (n = N, black line) or based on a sample of n = 100 individuals (gray line), for two simulation runs with carrying capacities (and initial population sizes) N₀ = 1000 and N₀ = 10000, respectively, and parameters as in Fig. A2. The inset shows the trajectories of the mean phenotype and the phenotypic optimum z_opt. The spike in κ around generation 40, which partially closes the large initial phenotypic lag (insert), is due to an increase in genetic variance (see main text). Fluctuations in the black line reflect genetic drift and environmental variance, whereas those in the gray line are largely due to sampling effects. In addition, rates measured in haldanes vary due to fluctuations in the phenotypic variance (for potential problems of scale, see Hereford et al. ; Hansen and Houle 2008).

Figure A4

Distribution of observed generation-to-generation rates of phenotypic change κ in haldanes, over 100 simulation runs similar to those in Fig. A3. In (A), rates based on the entire population (n = N) or on samples of size n = 100 are shown for various initial population sizes (and carrying capacities) N₀. In (B), N₀ = 10000 was kept constant and only sample size n was varied. Other parameters are as in Fig. A3.

Figure A5

Expected absolute rate of phenotypic change κ between generations due to genetic drift and environmental variance, which contribute and , respectively (see main text), as a function of heritability h² and population size N = N_e (inset). Environmental variance refers to the phenotypic variance caused by developmental instability and micro-environmental fluctuations.