## Introduction

There has been considerable speculation about the degree of variation among individuals in their developmental stability, their ability predictably to complete development to an optimum state ( Palmer, 1996; Møller & Swaddle, 1997). The principal hurdle to empirical studies of developmental stability is that, in most cases, we do not know what the optimum state of a trait is, so we cannot say how much of the variation is due to variation in the optimum and how much to a failure to develop to that optimum. When the same structure develops on either side of a symmetrical body however, we can assume that the optimum state is often one of perfect symmetry. Where this symmetry of the optimum can be assumed, asymmetry is referred to as fluctuating asymmetry (FA) ( Palmer & Strobeck, 1986; Palmer, 1994). A large body of work now assumes that FA is a good indicator of developmental stability ( Møller & Swaddle, 1997).

Surprisingly, given these attractions of FA as an indicator of developmental stability, the precise relationship between the two has never been addressed analytically. Previous studies have, with few exceptions, relied on the negative relationship between FA and developmental stability that exists by definition. There are several barriers to studying this relationship. First, developmental stability is currently a hypothetical entity that has no unique relationship with observable properties of organisms. For example, FA is known to be affected by the environment an individual develops in ( Parsons, 1990) as well as by variation in the developmental stability of individuals. It is also not clear that there is any single property of an individual that can be labelled developmental stability, as each trait of an individual may to some degree have different stability properties from every other trait ( Møller & Swaddle, 1997, pp. 53–55; Leung & Forbes, 1997). For example, traits that develop at different times, or in different parts of the body, may differ in their developmental stability. Taken together, these complications have so far precluded empirical characterization of the relationship between developmental stability and FA, despite a number of demonstrations that variation in developmental stability of some characters does exist ( Parsons, 1990; Whitlock & Fowler, 1997; Gangestad & Thornhill, 1999).

In the absence of detailed empirical data, we must depend on models to develop our intuition. The majority of workers have adopted the following standard model of fluctuating asymmetry ( Palmer & Strobeck, 1986; Palmer, 1996). Each individual is characterized by its developmental variance, the converse of developmental stability. Paired structures are assumed to develop independently towards the same expected size but to show some normally distributed variation around that expectation. The amount of variation around the expected size is determined by the developmental variance. In this model, the difference between paired structures is therefore also normally distributed, with variance twice that of the variance of each paired structure.

Several recent explorations of this standard model suggest that the expected relationship between FA and developmental stability is very weak. FA essentially measures developmental variance, the variance in size of body parts when they develop in the same environment, which is the converse of developmental stability. Variances are more difficult to estimate well than means, so we should expect that more sampling effort would be needed to study FA than a typical trait. Unfortunately, FA must usually be estimated from a single pair of measurements, yielding a poor estimate of the variance and therefore of the proportion of variation in FA that could be due to real variation in developmental stability ( Whitlock, 1996; Houle, 1997).

The likelihood that only a small amount of information about individual developmental stability is gained from a single measure of FA has led to two efforts to quantify how much of the variation in FA could be due to variation in developmental stability, both assuming the simple model of the development of paired structures outlined above. One such effort used the relationship between the mean and variance of FA ( Whitlock, 1996), whereas the other exploited the kurtosis in FA expected to arise from variation in developmental stability ( Gangestad & Thornhill, 1999).

Whitlock (1996) observed that there is a simple relationship between the mean FA and the variance in FA for individuals with the same developmental stability. Therefore, one can calculate by subtraction the proportion of the total variance in FA that can be due to differences in developmental stability. This quantity is familiar from quantitative genetics as the repeatability. Although this insight is correct, the formulas given by Whitlock (1996) were incorrect; corrected formulas have now been published ( Van Dongen, 1998b; Whitlock, 1998). The repeatability provides an intuitive measure of the reliability of individual measurements. More importantly, the repeatability sets an upper limit to the heritability, the proportion of the variance that can be due to genetic causes. It also sets an upper limit on the phenotypic correlation between the asymmetries of different pairs of traits on the same individual. Whitlock showed that the maximum repeatability of FA is 0.64 and that the coefficient of variation of FA is sometimes so large that the repeatability would be expected to approach this maximum value ( Whitlock, 1996). Paradoxically, because FA is such a poor measure of variance, even traits with low repeatability and small correlations of FA among traits may reflect a great deal of variation in developmental stability.

Using simulations, Gangestad & Thornhill (1999) derived an empirical relationship between repeatability and kurtosis in signed FA, the difference in size between paired structures. They come to conclusions similar to those of Whitlock, arguing that, despite the low repeatability of many estimates of FA, the heritability of developmental stability itself may be high.

In this paper, I extend Whitlock’s (1996, 1998) work to consider the relationship between the distribution of developmental stabilities and the variance of FA. Whitlock’s approach leads to an estimate of the proportion of the variation in FA that can be due to variation in developmental stabilities, but it does not consider variation in developmental stability explicitly. The results presented here go the next step and allow inferences about the amount of variation in developmental stability from the repeatability of FA, based on the standard model. Previous work that has explicitly included variation in developmental stability has considered mixtures of individuals with two or three different stabilities ( Houle, 1997; Van Dongen, 1998b), rather than more realistic continuous distributions. Other work has relied on simulations ( Leung & Forbes, 1997; Van Dongen, 1998b; Gangestad & Thornhill, 1999), which are difficult to generalize. The principal result of this model is that, in order for measures of FA to have the substantial repeatabilities implied by some data, mean-standardized variation in developmental stability would have to be higher than for most previously studied traits.

In the next section I present an intuitive introduction to the model. The *Mathematical results* section then derives the relationship between developmental stability and measures of asymmetry based on this model. From these relationships, I then obtain *Numerical results*. The reader who wishes to obtain the main results without mathematical details may skip the *Mathematical results* section.