### Abstract

- Top of page
- Abstract
- Introduction
- The model
- Results
- Discussion
- Acknowledgments
- Bibliography

Lande’s (1980b, *Evolution***34**: 292–305) equations for predicting the evolution of sexual size dimorphism (SSD) through frequency-dependent sexual selection, and frequency-independent natural selection, were tested against results obtained from a stochastic genetic simulation model. The SSD evolved faster than predicted, due to temporary increases in the genetic variance brought about by directional selection. Predictions for the magnitude of SSD at equilibrium were very accurate for weak sexual selection. With stronger sexual selection the total response was greater than predicted. Large changes in SSD can occur without significant long-term change in the genetic correlation between the sexes. Our results suggest that genetic correlations constrain both the short-term and long-term evolution of SSD less than predicted by the Lande model.

### Introduction

- Top of page
- Abstract
- Introduction
- The model
- Results
- Discussion
- Acknowledgments
- Bibliography

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’

Table 1. Symbol definitions (with dimensionality). *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.

### The model

- Top of page
- Abstract
- Introduction
- The model
- Results
- Discussion
- Acknowledgments
- Bibliography

Although Lande’s SSD model can deal with multiple traits, our simulations focus on the evolution of a single trait, for simplicity of presentation. However, we retain the matrix notation of eqn 4 for referring to the parameters and variables, to be consistent with Lande’s equations. For the single trait case examined here, **G** and **P** are equivalent to the genetic and phenotypic variances, respectively, and **B** is equivalent to the genetic covariance between the sexes. The basic simulation model is described fully in Reeve (2000), but several details have been modified, in particular those required for implementing sexual selection. Each individual in the population is represented by 50 unlinked, autosomal, diploid loci. Allelic values are initially assigned by drawing numbers at random from a normal distribution with mean of 0 and a variance of 1.0. Genotypic trait values are purely additive, with no dominance or epistasis. For each sex, three of the 50 loci do not contribute to the genotypic value (i.e. are not expressed). These three loci are different in males and females, so that sex differences are caused by differences between the sum of the three ‘unique-to-female’ and the three ‘unique-to-male’ loci. This number of sex-limited loci was chosen so as to produce a genetic correlation between the sexes of approximately 0.9, as is typically found in empirical estimates. For each individual, the genotypic value of its single trait is calculated as the sum of the allelic values at its 47 contributing loci. Phenotypic values are created by adding to each genotypic value a random normal deviate, with a mean of zero, and a variance chosen to give an initial (pre-equilibrated) heritability of 0.5. Populations consist of 2000 individuals of each sex, and generations are nonoverlapping. All figures are based on average values of 10 replicates, unless otherwise stated. The simulations were initiated from base populations that were equilibrated for 20 000–50 000 generations of stabilizing selection (i.e. no sexual selection), where **θ** was set at 0.0 for both sexes. During equilibration, the heritabilities fall to equilibrium levels determined by the strength of stabilizing selection, the mutation rate, the variance of mutational effects, the number of loci, and the effective population size. In eqn 4, **Z** is the only variable, as all the other symbols are assumed to remain constant. In the simulation, **θ**, **W**, **c**_{f}, and k (a coefficient that determines the strength of the linear relationship between mating success and relative body size in males, and thus is largely responsible for the magnitude of **c**_{m}) are the only fixed parameters. The variables (**P**, **G**, **B**, and **c**_{m}) are free to change over time. The 10 replicates are sampled from the appropriate base population at 3000 generation intervals, starting at generation 20 000. Therefore, they are replicating the population parameters but the variables vary amongst replicates. The term ‘generation 0’ in the text and figures refers to the populations at the start of either sexual selection or natural selection for a new optimum.

#### Life-cycle

Each generation consists of three stages:

**1** *Frequency-independent viability selection*. Each individual is assigned an ‘expected’ fitness (*w*), according to

(Lande, 1980a), where **z** is the vector of the individual’s trait values. Equation 5 yields values between 0.0 and 1.0, and can be interpreted as the probability of survival. Fitness for this stage is at a maximum when the phenotype (**z**) is at the natural selection optimum.

**2** *Frequency-dependent sexual selection for increased size in males*. Expected fitness for this stage is based on a linear function of each male’s rank,

where the rank of the largest=1, and the smallest=*N*/2, *N* being the total population size. The slope of this ‘mating success’ function can be adjusted by changing the value of the coefficient *k*. Lifetime expected fitness in males is the product of the fitness in stages 1 and 2. Female lifetime expected fitness is based entirely on stage 1. There is no fecundity selection in females independent of that caused by differences in survivorship. In our purely natural selection simulations, life-cycle stage (2) is omitted.

**3** *Offspring production*. Parents are sampled (with replacement) with a probability proportional to their lifetime expected fitness, with each set of parents producing one offspring of each sex each time they are sampled. Hence, family size for each individual parent is approximately Poisson distributed. Offspring consist of a random haploid compliment of genes from each parent. Offspring phenotypes are assigned as above. This procedure is repeated until there are enough offspring to replace the parental population.

#### Parameter estimates

All parameter estimates for eqn 4 were made during generation zero. **P, G**, and **B** are based on the population before selection is applied (i.e. they are not fitness weighted statistics). The strength of sexual selection, **c**_{m}, is defined (in single-trait equations) as the mean size of males after sexual selection minus the mean size after natural selection (=the sexual selection differential, **S***), divided by the phenotypic variance (Lande, 1980b). We calculate this from fitness-weighted means. Although the weighting after natural selection is determined from eqn 5, that after sexual selection is based on actual lifetime fitness (the total number of offspring produced by each individual), and is therefore a probabilistic function of eqns 5 and 6. **c**_{f} is set to 0.0. All figures except Fig. 4B,C use values measured before stage one of the life-cycle (see above). As the base populations have a mean near 0.0 in each sex, we define SSD as the difference between male and female size, rather than as their ratio.

Genetic covariances and correlations between the sexes are estimated as follows. The covariance between the genotypic values of females, and the genotypic values of those same females if their genes were expressed in male bodies, is calculated. This is repeated for male genotypes and males expressed as if in females. The average of these two covariances is then used.

#### Mutations

We use a continuum-of-alleles ‘stepwise’ mutation method, whereby the mutations are normally distributed, have a mean value of zero, and are added to the allele’s premutational value. The mutation rate is set at 1 × 10^{–4} per haploid locus, as in several previous simulation studies (e.g. Burger *et al*., 1989; Wagner, 1989; Wagner *et al*., 1997; Reeve, 2000). The mutational variance is set so that the mutational heritability is approximately 0.001 times the environmental variance (*V*_{E}), a value consistent with empirical findings (Lynch, 1988; Houle *et al*., 1996).

### Results

- Top of page
- Abstract
- Introduction
- The model
- Results
- Discussion
- Acknowledgments
- Bibliography

Figure 1A-D shows predicted and observed responses in the means, as well as corresponding changes in several genetic values, under sexual selection for increased male size. The same results are also shown for simulations where the shift in optimal male size was achieved through purely natural selection (Fig. 1E–H). In the latter case, **θ**_{m} (eqn 4) was changed to equal the same number of phenotypic standard deviations (SDs) as the equilibrium deviation for the simulations in Fig. 1A, and **c**_{m} was set to 0.0. The same set of initial base populations was used for both types of simulation.

In the sexual selection simulations, the equilibrium male size and therefore equilibrium SSD are larger than predicted (Fig. 1A). From generations 500 to 5000, males and females are predicted to change size at approximately the same rate, but in opposite directions. However, in the simulated populations over the same time period, males increased in size by more than three times the amount by which females decreased. This occurs because equilibrium male size is determined in part by the intensity of sexual selection (**c**_{m}) which, in the simulations, increases as male mean size is driven further from the stabilizing selection optimum (see below for an explanation of this). The net result is that, contrary to expectations from eqn 4, males tend to be further from their final equilibrium size than females for most of their evolutionary trajectories, and the equilibrium values for males and SSD are higher than predicted.

It is clear that equilibrium values are reached much more quickly in simulations with pure natural selection (Fig. 1E). This is mainly due to the fact that with sexual selection, the net selection intensity for increased size is reduced due to the conflict with stabilizing selection. With pure stabilizing selection, there are no conflicts and the realized selection intensity is therefore stronger. The equilibrium sizes of both sexes are predicted correctly as they depend only on **θ**_{m} and **θ**_{f}, which are parameters for both the equation and the simulation.

Under both sexual and natural selection, the trait means respond much more rapidly than predicted over the first few hundred generations. This initial accelerated response is caused by a rapid increase in both the genetic variance within each sex (Fig. 1B,F), and the covariance between the sexes (Fig. 1C,G) in the early generations of directional selection. The larger increase in (co)variances in the natural selection simulations is due to the stronger net directional selection (**β**) under purely stabilizing selection. The rapid increase in (co)variances is due to the leptokurtic distribution of allelic effects at the initial equilibrium under house-of-cards conditions (Fig. 2).

A further difference between the two simulation types is that under natural selection, the genetic variances and covariance (Fig. 1F,G) equilibrate at approximately the same level found in the base populations. Under sexual selection (Fig. 1B,C) the (co)variances eventually equilibrate at levels considerably lower than that of the base populations.

The genetic correlation between the sexes is defined as the genetic covariance between sexes divided by the product of the male and female SDs. Large changes in dimorphism may be achieved without producing any permanent significant change in the genetic correlations (Fig. 1D,H), although short-term changes in correlations can be large and nonintuitive.

Figure 3 compares the trajectories of predicted and observed SSD evolution for two different strengths of stabilizing and sexual selection. We refer to the two strengths tested within each selection type as ‘strong’ and ‘weak’. Our two levels of sexual selection correspond to expected mating successes that differ between the largest and smallest members of the population by factors of 4.96 (*k*=0.8=‘strong’) and 1.66 (*k*=0.4=‘weak’) times. The strength of stabilizing selection (**W**) is set at either 19 (‘weak’) or 9 (‘strong’) times the environmental variance (i.e. the commonly used term *V*_{s} (=environmental variance + **W**) is set to 20 or 10). These values are within the range found in typical experimental estimates (Johnson, 1976; Turelli, 1984), and produced average initial (equilibrated base population) heritabilities of 0.34 and 0.18 for weak and strong stabilizing selection, respectively.

Under strong sexual selection (Fig. 3A,C), equilibrium male size is larger than predicted, so the equilibrium SSD is also larger. Under weak sexual selection (Fig. 3B,D), the predictions for equilibrium SSD are very accurate. For all of the simulations, the rate at which SSD evolves in the early generations is faster than predicted due to an initial increase in genetic variance.

Equilibrium size in males is determined by

As **θ**_{m} is equal to 0.0. Because **W**_{m} is constant, error in the predicted levels of SSD is caused by final values of the phenotypic variance (**P**_{m}) and/or the sexual selection differential (**S***_{m}) that are different from those estimated at generation zero. Figure 4 plots initial vs. final estimates of **P**_{m}, **S***_{m}, and **c**_{m}, and observed vs. predicted values for equilibrium male body size (**Z**_{m}). The final values are based on measurements taken every 10 generations from 9900 to 10 000, and are assumed to be estimates of the ‘true’ equilibrium value of the variables. The initial values are based on the single set of estimates at generation zero, because that is the information on which the predictions are based. Here, the increased predicted values for **Z**_{m} (and therefore equilibrium SSD) in the simulations with strong sexual selection (ws and ss in Fig. 4D) can be seen to be caused by an increase in **c**_{m} (Fig. 4C) compared with the generation zero estimate. This, in turn, is caused by the combination of an increase in **S*** (Fig. 4B) and a decrease in **P**_{m} (Fig. 4A). The increase in **S*** is due to the interaction between the two modes of selection. Frequency-dependent selection (eqn 6) is an increasing function of rank, so its shape is independent of mean size. Frequency-independent selection (eqn 5) changes from a concave downward to a near-linearly decreasing shape, as male mean size moves away from its optimum. The net result is that after the population mean has moved a small distance from its optimal value, for a constant **P**, **S*** increases as the population mean increases. This was also true for types of sexual selection other than the rank-selection used in our main simulations, e.g. truncation selection, and selection based on number of phenotypic SDs from the mean (results not shown). However, the phenotypic variance (**P**) at the sexually selected equilibrium tends to decrease with increasing strength of sexual selection. This is because the two modes of selection produce balancing selection for the new equilibrium, rather than pure stabilizing selection as is found at generation zero. The resulting fitness function is steeper than that in the base population before sexual selection. Figure 5 shows this change for one of the replicates from Fig. 3C. The strength of ‘stabilizing’ selection at the sexually selected equilibrium can be estimated through multiple regression methods (Lande & Arnold, 1983). In Fig. 5B, this estimate (–0.09) indicates stronger stabilizing selection at the sexually selected equilibrium than in the generation zero population of Fig. 5A (–0.05). (Note – these values are not equivalent to the coefficients of **W**, where larger magnitudes indicate weaker selection, and stabilizing selection has positive coefficients.)

In the trajectories shown in Fig. 1A,E, there is a large displacement in female mean size, caused by the sudden increase in the male optimal size. However, if the optimal size in males is incremented by a small amount each generation (Fig. 6), female size shows much less displacement (less than one-sixth the amount seen in Fig. 1E). The trajectories produced when the intensity of sexual selection is incremented slightly each generation are very similar (results not shown). Thus, the maladaptive evolution of female size away from its optimum may largely be an artifact of the assumption (from Lande’s, 1980b paper) that the shift in selective pressures in males is sudden and large. When selection on males is small and incremental, the correlated selection on females is almost immediately balanced by direct selection in the opposite direction for the female phenotypic optimum.

### Discussion

- Top of page
- Abstract
- Introduction
- The model
- Results
- Discussion
- Acknowledgments
- Bibliography

Equilibrium SSD levels were predicted very accurately for weak sexual selection, but under stronger sexual selection, the discrepancy between predicted and simulated SSD increased. In addition, the rate of separation between the sexes in the initial generations of selection was much faster than predicted. Our results suggest that genetic correlations constrain both the short-term and long-term evolution of SSD less than predicted by the Lande model, if population sizes are relatively large. The overall quality of the predictions, however, is surprisingly good, given the sometimes large changes that are occurring in the genetic and phenotypic variances.

The accelerated response in male size, and therefore SSD, in the early generations of selection is related to changes in the distributional properties of allelic effects. It is known from earlier studies (Barton & Turelli, 1987; Burger, 1993; Reeve, 2000) that under house-of-cards conditions, directional selection can lead to an initial increase in genetic variance. This occurs because, at mutation-stabilizing selection-drift balance (our populations at generation zero), the allelic effects at each locus will be normally distributed only if mutation rates are very high (‘Gaussian’ conditions). Most quantitative genetics equations for predicting response to selection rely on the assumption of normally distributed allelic effects (Barton & Turelli, 1987). However, under house-of-cards equilibrium conditions, these effects tend to be leptokurtically distributed. Most alleles have values near the mean value for that locus, but some rare alleles will have values quite distant from the mean. With directional selection, favourable rare alleles are selected, and as they increase in frequency, genetic variance can increase dramatically. When a new equilibrium is reached, those once rare alleles (or their descendents) become the norm, and allelic effects again become leptokurtically distributed about the loci’s new mean values. The initial rise in genetic variance causes an increase in phenotypic variance, and a response that is faster than expected from base population parameters. Such accelerated responses are seldom seen in selection experiments, but this could be due to a lack of rare alleles caused by small effective population sizes.

Fisher (1958) and Lande (1980b) suggest that the genetic correlation between the sexes for sexually selected traits should diminish greatly with time. No mathematical justification is presented by either author, and the suggestion appears to be based on an intuitive idea of how such correlations should evolve. However, a reduction in the between-sex genetic correlation is not required (or permitted) in Lande’s model, and it does not occur in our simulations. Changes in the correlation are brought about solely through changes in allele frequencies in our simulation model. In nature, changes in the between-sex genetic correlation could also occur through the ‘capture’ or ‘loss’ of genes by either sex. Wright (1993) has suggested that there may be an extra phase to SSD evolution in some organisms, in addition to the two discussed in Lande (1980b). In this third phase, selection for newly evolved sex-specific genes (e.g. for sex-hormones or their receptors in vertebrates) could increase the rate at which SSD evolves. As our simulation model does not allow for the evolution of the genetic architecture itself (i.e. which genes control which sex), we cannot address this suggestion. However, our simulations show that rapid rates of SSD evolution can occur even in a very simplistic model with no provisions for such genetic changes.

The consequences of nonconstant **G** and **P** matrices in quantitative genetics models such as eqn 4 have not been adequately explored. In our simulation, changes in the genetic and phenotypic variance not only cause the accelerated initial response, but also contribute to the attainment of an equilibrium male size, and hence SSD, that is greater than predicted. In populations undergoing strong sexual selection, the difference between the initial and equilibrium values of **c**_{m} was due not only to changes in the phenotypic variance, but also to changes in the shape of the net fitness function acting over the two episodes of selection (viability and sexual). Therefore, even if we assume that the functional relationship between body size and mating success (*k*) remains constant over time, it is highly likely that the actual intensity of sexual selection (**c**_{m}), increases as the mean sizes of the two sexes diverge.

Whether the changes in variances are permanent depends on the mode of selection. For equilibrium shifts caused solely by frequency-independent natural selection, there is no permanent change in the magnitude of the net selective forces acting on the population. Therefore, the genetic variance can return to the value present before directional selection. In contrast, frequency-dependent sexual selection causes a decrease in genetic variance that is a function of the relative strengths of the two modes of selection and the distance that equilibrium male size has shifted. Although it is only the male lifetime fitness function that is permanently altered by sexual selection, female genetic and phenotypic variances at equilibrium are also lowered. This is because all the genes from fathers are passed on to both sons and daughters. However, in our simulations females have three loci that never directly experience the effects of the increased selection intensity on males, because they do not contribute to the male’s phenotype. The larger equilibrium genetic variance in females (for instance, ~10% higher for Fig. 1B) is entirely due to the increased variance present at these female-specific loci.

In addition to the results shown in this paper, many simulations were run with different starting conditions. Decreasing the mutation rate to 10^{–5} (while maintaining the mutational heritability of 10^{–3}*V*_{E} by increasing the number of loci) had no qualitative effect on the results reported in this study. Changing the genetic correlation between the sexes by changing the number of sex-specific genes, or by adding antagonistically pleiotropic loci, did not produce qualitative differences in the results. Simulations that included up to six traits (genetically correlated within and between sexes), with sexual selection acting on one of the male traits also produced the same general pattern of results – a faster than predicted response, less time to equilibrium, and a larger SSD for the selected trait at equilibrium.

When population sizes of *N*=400 were used, patterns of response were quite different. There are two main characteristics of small populations which combine to produce these differences. First, small populations have many fewer rare alleles segregating at equilibrium. Therefore the increase in variance and the accelerated response of the means, declines as population size decreases. Second, as there is a smaller amount of standing variance in small populations, it is more quickly eroded, and response to continued directional selection quickly becomes dependent on input from new mutations. This is much slower than the response due to standing variance. The net result is that for small populations, response is predicted fairly accurately for a small number of generations, but then tends to lag behind the predicted response if sexual selection is strong. Equilibrium SSD in such cases takes much longer to reach than predicted by eqn 4. Even with large populations, this lag effect can be produced by increasing the intensity of sexual selection, although here the initial accelerated response is still present. For all population sizes tested, equilibrium SSD levels were underestimated by eqn 4 when there was strong selection.

Even under conditions where the accuracy of predictions was very good, it must be remembered that the results are based on an average of 10 replicates. It can be seen from Fig. 4D that there is much more variation in the predicted than the observed equilibrium values for male size. This is mainly caused by random fluctuations in the variances in the base populations, and the effect that these have on estimates of **c**_{m}. Even in a population of size *N*=4000, there is a great deal of variance in the variances, a problem that has been dealt with at length elsewhere (e.g. Burger *et al*., 1989; Keightley & Hill, 1989).

For the simulated replicates, equilibrium values are a function of the characteristics of the genetic architecture of the population, not the value of the variables that happen to be present at generation zero. This architecture is determined by a combination of the fixed parameters (e.g. **W**, *N*, and the mutation rate), and the ‘developmental rules’ that specify which genes control size in each sex. In multitrait systems, these rules also determine the pleiotropic relationships between traits. In our simulations, these rules are determined by the number of loci that contribute to both sexes or to just males or just females. The rationale for not allowing mutations that change this architecture is that such developmental rules are likely to be more strongly canalized than are the loci that contribute to normal quantitative variation (Wagner, 1989).

Equation 4 cannot and should not be expected to predict long-term evolution in real populations. There are numerous reasons why this is an unrealistic goal. Most of the variables cannot be estimated with reasonable accuracy. Even if they could be, environmental changes probably cause **W** and **θ** to fluctuate constantly over time. The fitness surface itself is unlikely to ever be unimodal Gaussian. In addition, there will always be large numbers of traits under selection, and most of these will not be included in the analysis. Therefore, the results from our simulation analysis should not be viewed as a test of how well the equation can predict evolution, but of how robust the equation’s predictions are to violations of its assumptions caused by nonconstant genetic and phenotypic variances. This does not mean that the equations are of no practical importance. They, like other predictive models in quantitative genetics, should be viewed as a starting point for trying to understand and possibly quantify the evolutionary forces that were responsible for current patterns of trait distributions. In our simulations, predictions were often fairly accurate despite large changes in genetic and phenotypic variances. This result suggests that the greatest impediment to predicting long-term response in real populations is our ignorance of patterns of temporal variation in the forces of selection, rather than the relatively minor error caused by changes in the **G** matrix.