The effects of stochastic and episodic movement of the optimum on the evolution of the G-matrix and the response of the trait mean to selection.

Theoretical and empirical results demonstrate that the G-matrix, which summarizes additive genetic variances and covariances of quantitative traits, changes over time. Such evolution and fluctuation of the G-matrix could potentially have wide-ranging effects on phenotypic evolution. Nevertheless, no studies have yet addressed G-matrix stability and evolution when movement of an intermediate optimum includes large, episodic jumps or stochasticity. Here, we investigate such scenarios by using simulation-based models of G-matrix evolution. These analyses yield four important insights regarding the evolution and stability of the G-matrix. (i) Regardless of the model of peak movement, a moving optimum causes the G-matrix to orient towards the direction of net peak movement, so that genetic variance is enhanced in that direction (the variance enhancement effect). (ii) Peak movement skews the distribution of breeding values in the direction of movement, which impedes the response to selection. (iii) The stability of the G-matrix is affected by the overall magnitude and direction of peak movement, but modes and rates of peak movement have surprisingly small effects (the invariance principle). (iv) Both episodic and stochastic peak movement increase the probability that a population will fall below its carrying capacity and go extinct. We also present novel equations for the response of the trait mean to multivariate selection, which take into account the higher moments of the distribution of breeding values.