## Introduction

Sexual size dimorphism (SSD) is generally attributed to differences in the net selection acting on the two sexes, so that the sexes have different optimal sizes. When each sex is at its optimum, the population will be at an evolutionary equilibrium. If selection changes, the optimal value for one or both sexes may shift. As the genetic correlation between sexes is very high for most morphological traits (Lande, 1980b; Roff, 1997, Table 7.4), changes in the mean size of one sex are expected to be associated with similar changes in the other sex. Short-term laboratory experiments have demonstrated that artificial selection on the size of one sex produces a nearly parallel response in the other (Alicchio & Palenzona, 1971; Reeve & Fairbairn, 1996). Thus, it is possible that the SSD found in natural populations might often be maladaptive, if long periods of time are required to overcome genetic constraints. How long this maladaptive state persists is important in terms of understanding the adaptive significance of SSD. Quantitative genetics provides a useful framework for understanding and predicting the evolutionary trajectories of traits such as body size, and sex differences in these trajectories.

Lande has introduced a number of equations for predicting response to selection under a variety of settings. Most are variations of the ‘multivariate response equation’ (Lande, 1979)

where Δ**Z** is the change in trait means, **G** is the genetic variance–covariance matrix and **β** is the selection gradient (all symbols are also defined in Table 1). Equation 1 is the multivariate equivalent of the ‘breeder’s equation’

*n*= number of traits.

where *R* is the response in the trait mean, *h*^{2} is the heritability and *S* is the selection differential. Equations 1 and 2 are valid only for a single generation of selection. Extending the predictions over longer periods requires two assumptions, one controversial and the other generally unwarranted. First, the genetic parameters (**G** or *h*^{2}) must remain constant over time. How likely this is in natural populations has been the subject of considerable debate (e.g. Shaw *et al*., 1995). Second, selection (**β** or *S*) must remain constant, which will not generally be true. For example, under stabilizing selection, **β** will change as the trait means move relative to their optimal values.

To deal with changes in **β** under a model of multivariate Gaussian (stabilizing) selection, Lande (1980a) introduced a ‘peak-shift’ equation

where **W** is a matrix of stabilizing (diagonal) and correlational (off-diagonal) selection coefficients, **P** is the phenotypic variance–covariance matrix, **θ** is the vector of optimal trait values, and –1 indicates matrix inversion. This equation models evolution as being driven by changes in the optimal value(s) of one or more traits in a population. From this equation, it is clear that selection in any generation depends in part on how far the trait means are from their optimal values, and therefore will not be constant unless the population means are at an equilibrium (i.e. there is no further response).

Each of the above equations assumes that the sexes are identical. To model the evolution of SSD, Lande (1980b) used a variant of eqn 3 that requires an additional matrix of genetic variances and covariances between the sexes (**B**), separate input values for each sex (subscripts m and f), and a term for the strength of sexual selection (**c**):

where T denotes matrix transposition.

Lande considers the situation where both sexes are initially under stabilizing selection for their optimal trait values, and the population is at equilibrium. Sexual selection of a constant intensity is then added, favouring an increase in male size. Iterating eqn 4 over many generations results in male and female trajectories that Lande describes as occurring in two phases. In the first, rapid phase, the two sexes evolve nearly in parallel, and the mean size quickly reaches a point between the male and female final equilibrium values. Although the optimal size for females does not change, mean size in females temporarily increases due to the genetic correlation with males. In the second phase, which takes much longer than the first if the correlation between sexes is high, the two sexes slowly diverge. Females return to their original optimum, and males move to a new optimum determined by a balance between viability selection favouring their original size, and sexual selection favouring increased size.

Reeve (2000) used a stochastic genetic simulation model to test predictions from eqn 3 for a single-sex system of three genetically correlated traits. The accuracy of predictions depended critically on assumptions concerning the distribution of mutational effects in the simulation model. Predictions were very good under ‘Gaussian’ assumptions (high mutation rates, low mutational variance). However, it is likely that such conditions are unrealistic (Turelli, 1984). Using more realistic ‘house-of-cards’ conditions (lower mutation rates, higher mutational variance), predictions were less reliable. In particular, equilibrium values were reached much more quickly than predicted. This was caused by an increase in genetic variance in the early generations of directional selection, which is a result of the leptokurtic distribution of allelic effects expected under house-of-cards conditions (Turelli, 1984).

In this paper, a similar genetic simulation model, with house-of-cards assumptions, is used to test the accuracy of eqn 4 in predicting the evolution of SSD. Predictions are made based solely on information available at generation zero of sexual selection. Although the simulation is set up to match the basic conditions of Lande’s analytical model in terms of modes of selection and life-cycle stages, it differs in one important respect. As the individuals in the simulation have finite numbers of loci, changes in allele frequencies result in changes in phenotypic and genetic distributions. Therefore, several variables (**G**, **P**, **B** and **c**_{m}) which are assumed constant in eqn 4, are free to vary in the simulation. As all real populations violate the constancy assumptions to at least some extent, we are interested in testing the effect of such violations on the quality of predictions from the analytical model. We focus on two main questions: (1) does eqn 4 accurately predict the equilibrium SSD? and (2) does it accurately predict the rate of separation between the sexes in the initial generations of selection?

We compare the evolution of SSD produced through frequency-dependent (sexual) selection on increased size in males with that produced through frequency-independent (natural) selection for a higher optimal size in males, as might occur under the dimorphic niche model for the evolution of SSD (Slatkin, 1984; Shine, 1989). Equation 4 is used to make predictions for both types of selection, and the results of these predictions are compared with the simulated responses.