Selection on skewed characters and the paradox of stasis

Abstract

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.

Keywords: Paradox of stasis; phenotypic skewness; response to selection; selection estimation; selection gradient.

Figure 1

Influence of phenotypic skewness on estimates of directional selection. The histograms show phenotype distributions under skewed environmental effects (a) (skewness α_e = 2, heritability h² = 0.1), or skewed breeding values (b) (skewness α_g = 2, heritability h² = 0.9). The dotted line represents the optimum phenotype θ for the stabilizing fitness function (solid curve, eq. (3)). The dashed line shows the mean phenotype. The solid straight line has a slope equal to the directional gradient estimated by OLS regression (as in Lande and Arnold (1983)).

Figure 2

Estimated directional selection gradients with skewed environmental effects. Standardized directional selection gradient (mean ± se) estimated by the OLS method (a) or the ADIRF method (b), and the contribution of phenotypic skewness to estimated selection with the OLS method (c), are represented against the skewness of environmental effects, for variable heritabilities (${\mathit{h}}^{\mathbf{2}}=\frac{\mathbf{1}}{\mathbf{2}}\mathbf{;}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{;}\frac{\mathbf{1}}{\mathbf{6}}\text{or}\mathbf{0}$ from black to light grey). Dotted lines show predictions from the model, eq. (13) for (a), 6-7 for (b), and (14) for (c).

Figure 3

Estimated directional selection gradients with skewed breeding values. Standardized directional selection gradient (mean ± se) estimated by the OLS method (a) or the ADIRF method (b), and the contribution of phenotypic skewness to estimated selection (c), are represented against the skewness of breeding values, for variable heritabilities (${\mathit{h}}^{\mathbf{2}}=\mathbf{1}\mathbf{;}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{;}\frac{\mathbf{1}}{\mathbf{3}}\text{or}\frac{\mathbf{1}}{\mathbf{6}}$ from black to light grey). Dotted lines show, eq. (13) for (a), 6-7 for (b), and (14) for (c).

Figure 4

Contribution of skewed breeding values to the selection response. The difference between the response to selection predicted by the Robertson-Price identity versus using the selection gradient estimated by the ADIRF method is shown for simulations (mean ± se), together with the theoretical prediction in eq. (15) in dashed lines, for the same simulations as used in Fig. 3.