The additive genetic variance–covariance matrix, or **G** matrix, has become an important tool in predicting the course of phenotypic evolution under natural selection and genetic drift (Lande and Arnold 1983; Arnold 1992; Arnold et al. 2001). **G** is a symmetrical matrix in which the diagonal elements are additive genetic variances for traits and the off-diagonal elements are additive genetic covariances (Lynch and Walsh 1998). Quantitative genetic theory comprises, at its base, a set of equations that use this matrix to describe the course and pattern of phenotypic evolution (Lande 1979; Lande and Arnold 1983; Arnold et al. 2001). Central among these equations is the multivariate version of the classic breeder's equation (Lande 1979), which provides the single-generation response to selection,

in which is a vector of single-generation changes in population means for traits and **β** is a vector of the partial regression coefficients of fitness on phenotype (i.e., the selection gradient; Lande 1979; Lande and Arnold 1983). As the response to selection depends both on the genetic variation for single traits and on the genetic correlations between them, the **G** matrix plays a central role in determining the course of adaptive evolution.

Lande (1979) also proposed that the net selection gradient (**β_{net}**) responsible for phenotypic change over many generations could be obtained by summing equation (1) from time 0 to time

*t*, such that

in which is a vector of trait means at time *t*. In addition, if the populations or species had phenotypically diverged under drift, then the covariance matrix for species mean trait values, the **D** matrix, has an expectation proportional to the time average of **G**,

in which *N _{e}* is the effective population size. An implicit assumption of equation (2) and a practical necessity of equation (3) is that

**G**is stable over the evolutionary time scale of interest. As such, successful retrospective selection analysis (eq. 2) and prospective prediction of diversification (eq. 3) depend on the evolutionary dynamics of

**G**(Turelli 1988; Arnold 1992; Arnold et al. 2001).

Theoretical considerations provide no concrete predictions about the long-term dynamics of **G** and consequently the stability of **G** is fundamentally an empirical question (Barton and Turelli 1987; Turelli 1988; Shaw et al. 1995; Roff 2000). Although there is accumulating evidence that **G** may be conserved, particularly at low taxonomic levels (Roff 1997; Roff and Mousseau 1999; Steppan et al. 2002; Bégin and Roff 2004; Revell et al. 2007), theoretical investigations are needed to explore the evolutionary dynamics of **G**. Because the dynamics of **G** are not particularly amenable to analytic exploration (see Jones et al. 2003), several authors have made successful use of individual-based, stochastic computer simulations to examine the shape and stability of **G** under various conditions of selection, mutation, and genetic drift (Bürger et al. 1989; Bürger and Lande 1994; Reeve 2000; Jones et al. 2003, 2004).

In the most thorough such study to date, Jones et al. (2003) showed that the stability of some aspects of the eigenstructure of **G** is increased by correlational natural selection and correlated effects of pleiotropic mutation. In particular, the stability of the orientation of the primary eigenvector of **G** is increased by correlational mutation or selection or both. They also note, however, that under several mutation–selection parameter combinations (henceforward “mutation–selection scenarios”), instability in the orientation of the primary eigenvector of **G** is associated with low matrix eccentricity and thus is probably inconsequential to evolution by natural selection (Jones et al. 2003; p. 1758). This is because evolution is not constrained by a matrix that is not eccentric and thus matrix orientation is irrelevant to evolution. They also show that stability in the magnitude of the elements of **G** is lower for small effective population size (Jones et al. 2003).

In this paper, I use eigenanalysis and the response to random selection vectors to measure stability and to explore the conditions under which **G** can be considered to be stable. The latter approach, which is adapted from the random skewers method of Cheverud et al. (1983; Cheverud 1996), measures the evolutionary dynamics of **G** in terms of the correlation between the response to random selection vectors. This measure of the dynamic of **G** differs from the eigenanalysis in which the stability of **G** is measured directly. As such, I will henceforward refer to the random skewers approach as measuring the evolutionary consistency of **G**. This is because random skewers measures the consistency of the expected response to selection given the same selection gradient imposed on two different **G** matrices. As this method uses the quantitative genetic equation (1), it may be useful in diagnosing differences between **G** matrices that are likely to influence the response to natural selection.

I also simulate previously unexplored conditions under which **G** may be more severely destabilized. In particular, I explore the degree to which the stability and evolutionary consistency of **G** are compromised by fluctuating coefficients of correlational selection and pleiotropic mutation, and by fluctuating effective population size. To my knowledge, no prior simulation study has investigated the consequences of fluctuating correlational selection and mutation on the evolution of the **G** matrix.

In summary, therefore, this study uses different methods to: (1) evaluate the evolutionary dynamics of **G** under various conditions when correlational selection, correlational pleiotropic mutation, and effective population size are constant over time; and (2) evaluate the evolutionary dynamics of **G** when correlational selection and mutation, and effective population size fluctuate over time.