## Introduction

Population density and the corresponding intensity of intraspecific resource competition may influence the growth performance, and ultimately fitness, of differently sized individuals to different extents. Such size-asymmetric competitive effects may either provide smaller or larger individuals with an advantage. Competitive ability may be positively correlated with size, such that large individuals are less affected by increased competition (Uchmanski 1985; Lomnicki 1988). Alternatively, smaller individuals may have a superior performance under increased competition because of their lower *per capita* resource demands (Latto 1992; Persson *et al.* 1998; Pfister & Peacor 2003). A final possibility is that the effect of competitive intensity is completely symmetric and independent of the individual’s size, such that an increase in competition gives a proportional decrease in the performance in terms of growth rate or other fitness related rates. Because individual variation in performance can have important implications for population dynamics (Kendall & Fox 2002; Vindenes, Engen & Sæther 2008; Caswell 2009), it is important to understand how competitive intensity can shape this variation.

One commonly used approach to study how different individuals are influenced by competitive intensity has been to use population-specific measures of individual variation in growth rate or resulting size distributions. These usually employ measures of variance which are supposed to control for effects of competition on mean values, such as coefficients of variation (CV), or standard deviations (SD) of log-transformed data (plants: reviewed by Weiner & Thomas 1986; Weiner *et al.* 2001; animals: Rubenstein 1981; Wall & Begon 1987; Ziemba & Collins 1999; Sogard & Olla 2000; Peacor & Pfister 2006; Huss, Persson & Byström 2007). For example, Weiner & Thomas (1986) reviewed this topic for plant populations and found that for the majority of studies, measures of variation in size (CV or the closely correlated Gini Coefficient) increased with increasing population density. This was suggested to show that competition effects were asymmetric with respect to size and that larger individuals were less affected by density than smaller ones.

Although there is an appealing simplicity to the approach described earlier, interpretations of such data have some apparently unrecognised underlying complexities. First, although relationships between competitive regimes (e.g. population density) and *body size* variation can be informative in their own right, it is important to realise that one should not infer directly from this how variation in *growth rate* changes with competition intensity. A number of studies have used measures of variation in final body size as indicating how the population density influences the variation in growth performance, which in turn is used to infer whether some individuals are more influenced by increased competition than others (Weiner & Thomas 1986; Ziemba & Collins 1999; Sogard & Olla 2000; Keeley 2001; Einum, Sundt-Hansen & Nislow 2006; Huss, Persson & Byström 2007; Imre, Grant & Cunjak 2010; Lobon-Cervia 2010; Kvingedal & Einum 2011). Yet, as we will show, effects of competitive intensity on CV of growth rate vs. CV of final body size may even be qualitatively different.

A second issue with this approach, which also applies to studies of growth rate, is that it is difficult to know what the appropriate null-hypothesis is regarding the relationship between the chosen measure of variance (in growth or body size) and competitive intensity. The reason for this is that competitive intensity also influences mean growth, and variance and mean values are commonly correlated even if all individuals are equally influenced by competition. It is this latter relation that constitutes the appropriate null-hypothesis (i.e. no effect of competition on relative individual performance) when testing for effects of differential effect of competition on individuals. Observations of biological data suggest that SD increases with increasing mean, and hence should decrease with increasing competitive intensity, even in the absence of differential effects of competition on individuals. If the SD increases *proportionally* with the mean, then the CV (or the SD of log-transformed data) will be independent of the mean. Testing for relationships between the CV and competitive intensity should then give insights into effects on individual variation in performance, which is why this metric is the most commonly used one in such studies. However, for growth rate, SD is unlikely to scale exactly proportionally with the mean for many organisms. This is because their individuals may have negative growth, such that variation is expected even if the mean is zero. In this case, CV will depend on the mean even if the differences in mean are not caused by different intensities of competition.

Here we use simulations and empirical data from salmonid fishes to illustrate these problems. Salmonid fishes are well suited for this particular issue because they have been used extensively to study the effects of population density on competition intensity and growth performance (e.g. Gardiner & Shackley 1991; Crisp 1993; Jenkins *et al.* 1999; Nordwall, Naslund & Degerman 2001). In general, such studies show consistent negative effects of increased population density on mean growth rate. Furthermore, increasing size variance with increasing density has been suggested to indicate that different individuals are unequally influenced by density (Keeley 2001; Einum, Sundt-Hansen & Nislow 2006; Lobon-Cervia 2010; Kvingedal & Einum 2011; but see Imre, Grant & Cunjak 2010). Given the apparent increasing interest in studying the performance variation in relation to density (and a tendency for reviewers to request such analyses, pers. obs.), it appears timely to evaluate the validity of drawing such conclusions based on growth or size distribution data. We do this by first using a growth model to simulate the changes in final body mass after a period of growth with different levels of mean growth rates, and with different relationships between mean and SD growth rate. We then use empirical data from two salmonid species (Atlantic salmon *Salmo salar* and Arctic charr *Salvelinus alpinus*) to evaluate how the variation in growth rate scales with mean growth rate under situations where the mean growth rate is controlled by factors other than intensity of competition. Finally, we predict relationships between mean growth and CV growth, SD body mass and CV body mass based on our species-specific empirical estimates of relations between SD and mean growth rate.