Phenotypic diversification for quantitative traits during an adaptive radiation can be viewed as the result of the interplay between two multivariate processes, the multidimensional adaptive landscape and the **G**-matrix (Arnold et al. 2001, 2008). The adaptive landscape is a function relating the mean fitness of a population to the phenotypic trait means. The shape of the landscape represents underlying evolutionary features, such as correlational, stabilizing, or disruptive selection, and reveals where small fitness changes per unit of trait change exist, resulting in multivariate axes where peak movement is likely, termed the “selective lines of least resistance” (Arnold et al. 2001). It also clearly predicts how peak movement within the landscape induces directional selection and how the curvature and orientation of a moving peak generates nonlinear selection. In contrast, long-term stability of peak position promotes evolutionary stasis via stabilizing selection. Moreover, convergent phenotypic evolution may occur if a similar topology of the adaptive landscape exists in different geographic locations.

Phenotypic evolution over short timescales depends both on the position of a population within an adaptive landscape and the pattern of inheritance for multiple traits. This pattern of inheritance is usually summarized in the **G**-matrix, a square symmetric matrix composed of additive genetic variances for traits on the diagonal and covariances elsewhere (Lande 1979; Arnold 1992; Lynch and Walsh 1998). Covariances among characters can play an important role in shaping the course of evolution by natural selection (Schluter 1996; Arnold et al. 2001). The multivariate response to selection equation, , summarizes how genetic patterns of trait covariance translate natural selection into phenotypic evolution (Lande 1979; Lande and Arnold 1983; Björklund 1996). In this equation, is the vector of change in phenotypic mean trait values, **P** is a matrix containing the phenotypic variances and covariances, and **s** is the vector of selection differentials (Lande 1979).

Under most conditions, **G** will tend to bias the response to selection away from the direction maximizing the increase in mean fitness, summarized in the selection gradient (**β= P ^{−1}**s

**)**, and represented in the adaptive landscape as the steepest uphill direction relative to the position of a population. The phenotypic response to selection () will generally not be collinear with

**β**for most conditions of

**G**(Lande and Arnold 1983; Arnold 1992; Björklund 1996). Rather it will be biased toward the major eigenvector of

**G**, the direction of greatest genetic variation (

**g**), termed the genetic “line of least resistance” (Schluter 1996). This bias creates a curved evolutionary trajectory as the population approaches a local adaptive optimum and is expected to diminish over time (Lande 1980a; Björklund 1996; Schluter 1996, 2000; Arnold et al. 2001). Thus, evolutionary divergence of phenotypic characters represents a balance between the effects of natural selection, in the form of the adaptive landscape, and those of genetic constraint in the form of additive genetic variances and covariances (Lande 1979; Björklund 1996; Schluter 1996, 2000; Arnold et al. 2001; Blows et al. 2004).

_{max}Multivariate selection imposed by the adaptive landscape is also expected to modify elements of **G** (Lande 1980b, 1984; Cheverud 1984; Arnold 1992; Brodie 1992; Arnold et al. 2001, 2008; Jones et al. 2003; Blows et al. 2004; Revell 2007a). Although **G** is expected to be stable under some circumstances, it also may be quite unstable under others (Turelli 1988; Agrawal et al. 2001; Arnold et al. 2001, 2008; Phillips et al. 2001; Steppan et al. 2002; Jones et al. 2003; Björklund 2004; Revell 2007a). In particular, simulation studies show large effective population sizes enhance stability of the size, shape, and orientation of **G** (Jones et al. 2003). Furthermore, correlated pleiotropic mutation, strong correlational selection, and directional selection promote stability in the orientation of **G** (Jones et al. 2003, 2004), and empirical studies show correlational selection tends to cause genetic covariances to evolve in the direction of the sign of the correlation (Cheverud 1984; Tallis and Leppard 1988; Brodie 1992; Blows et al. 2004; McGlothlin et al. 2005). These expected patterns of response to selection lead to the prediction that species experiencing similar multivariate selection regimes will converge in genetic architecture, but this hypothesis has seldom been tested empirically (but see Marroig and Cheverud 2001; Roff 2002).

Comparative studies of phenotypic integration (**P)** can offer important insight into patterns of multivariate phenotypic evolution (e.g., Steppan 1997; Game and Caley 2006), and data from multiple species are needed to test patterns of divergence and convergence of **P** among species. Furthermore, if **P** and **G** are related, analysis of **P** can provide insight into the underlying genetic architecture. Comparative data for **G** are difficult to obtain because estimating genetic parameters requires large-scale breeding experiments or natural pedigrees, and even under ideal circumstances **G** is often inferred with large error. However, some evidence suggests that **P** can often be a reasonable approximation for **G** in evolutionary studies (Cheverud 1988, 1996; Roff 1995; Steppan et al. 2002). Because **P**=**G**+**E** (the matrix of environmental variances and covariances), **G** and **P** should often be correlated (Cheverud 1988; Roff 1995, 1997), particularly if heritabilities are high such as is found for many morphological traits like those in this study (Roff 1997, but see Hadfield et al. 2007). Furthermore, because the sample size for **P** is the number of individuals, whereas the effective sample size for **G** is a function of the number of families in the breeding experiment, **P** can usually be estimated with smaller error than can **G**. This has led to the argument that sometimes a precise measure of **P** might be closer to the “true”**G** than a genetic estimate made with large error (Cheverud 1988, 1996). Additionally, many “nonmodel” species are not amenable to breeding in the laboratory, so empirical estimates of **G** for multiple species can be difficult to obtain. For these reasons, we use **P** as a first approximation of **G** for some evolutionary inferences of this study, including testing for an effect of genetic architecture on phenotypic divergence. Objections to this approach have been raised on both empirical and theoretical grounds (see Willis et al. 1991); however, we argue that our substitution of **P** for **G** in species divergence comparisons is more likely to increase the type II rather than the type I error probability of those tests.