The G-matrix of quantitative genetics plays a central role in theory for the evolution of phenotypic traits, especially those affected by many genes and environmental factors (Lande, 1979). The G-matrix provides a summary of additive genetic variances and covariances for a suite of phenotypic traits, so its importance stems from the central role of genetic variation in evolutionary processes. For example, the additive genetic variances recorded in the G-matrix provide an indication of the potential for trait means to be altered by evolutionary mechanisms such as natural selection or genetic drift (Arnold et al., 2001; Steppan et al., 2002). In addition, the genetic covariances summarized by the G-matrix provide a window into the extent to which evolutionary change in one trait is expected to affect the evolution of other measured traits, whose values may in part be determined by pleiotropic or linked loci (Lande, 1979; Arnold et al., 2001). Despite its central role, the evolution and stability of the G-matrix remain outstanding, incompletely resolved problems (Jones et al., 2003; Arnold et al., 2008). The lack of complete resolution reflects the difficulty of finding analytical characterizations of G as a function of underlying processes, as well as the limitations of empirical studies in which G-matrices are sampled from natural or experimental populations and compared (Turelli, 1988; Steppan et al., 2002; Arnold et al., 2008). Because of these difficulties and limitations, simulation studies of the G-matrix have emerged as a powerful and complementary alternative to analytical theory and comparative studies (Jones et al., 2003, 2004, 2007; Guillaume & Whitlock, 2007; Revell, 2007; Arnold et al., 2008; Yeaman & Guillaume, 2009). Although largely restricted to the case of additive genetic effects (the simplest genotype-to-phenotype map), simulation studies yield many insights into the evolution and stability of the G-matrix.
A significant limitation of the simulation studies conducted so far is that they have not fully explored conditions that may contribute to G-matrix instability. For example, consider the case of traits evolving on an adaptive landscape with a single adaptive peak or optimum. Turelli (1988) argued that G would be especially unstable if the optimum were prone to episodes of large displacements. Although other cases of moving optima have been studied – steady movement (Jones et al., 2004) or erratic movement about a fixed position (Revell, 2007) – the cases of episodic and stochastic movement have not been addressed, an unfortunate lapse because both might yield illuminating instances of instability. The neglect of the episodic case is particularly unfortunate because this model of peak movement appears to be more consistent with known evolutionary trends than any other proposed model (Estes & Arnold, 2007; Uyeda et al., 2011).
The goals of this article are to explore the evolution and stability of the G-matrix when an intermediate optimum moves episodically or stochastically, with the justification that these cases might be both common in nature and especially conducive to instability of G. We compare these results with those for an optimum that moves at the same rate every generation, thereby identifying the influence of the mode of peak movement on various descriptors of the size, shape and stability of the G-matrix. We provide new theory that demonstrates and quantifies the role of skewness of breeding values induced by multivariate selection. To achieve greater power to detect systematic effects, we employ larger population sizes, stronger stabilizing selection and longer simulation runs than in our past study of a steadily moving optimum (Jones et al., 2004). Finally, we evaluate the effect of G-matrix instability on data analyses that assume the G-matrix is constant. In particular, we determine how estimation of the net selection gradient (net-β; Lande, 1979) is affected by skewness of the breeding values and by the instability in G induced by different modes of peak movement.