### Introduction

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

Intrapopulation variation of individual growth rate is a widespread property demonstrated by the large size variation observed in organisms from the same cohort reared together in controlled conditions (Wilbur & Collins 1973; reviewed in Uchmanski 1985; Lomnicki 1988). Even in nearly equal conditions, large variation in size can be observed, as illustrated for green frog tadpoles in Fig. 1 (see Kooijman 2000 for an example photograph of fish), owing solely to differential growth among individuals. The same factors leading to growth rate variation can be expected to be equally important in populations of unequal age, although they are not as obvious.

Secondly, developing frameworks to understand the origin and development of size variation could elucidate important ecological factors (e.g. spatial heterogeneity, phenotypic effects on growth and fitness). Ecologists have performed numerous experiments in which individual variation was measured, or could have been measured without much additional difficulty. Frameworks to understand the origin of size variation could equip ecologists with an additional lens into interpreting ecological processes (Lomnicki 1988). The elucidation of processes from the examination of variation in a property is very familiar to ecologists, as the variation in population patterns across environmental gradients is used commonly to enhance our understanding of mean population processes (Connell 1961; Wellborn, Skelly & Werner 1996). This approach is also used commonly in physics, in which the variation of scattered particles reveal information about the particles, the object off which they scatter, and the particle–object interactions. We may be missing valuable information in ecological data when we focus on the mean and ignore the variation in animal size (or other responses) (Hassell & May 1985; Lomnicki 1988).

However, we have surprisingly little understanding of the origin of individual size variation. One explanation for a commonly observed increase in size variation through time is that larger individuals have an advantage that confers faster growth, and therefore growth rate and size are positively correlated through time (denoted ‘growth depensation’ by Ricker 1958). Statistically, this results in a positive correlation in growth over time, as individuals experiencing higher growth at one time interval will experience higher growth at the next time interval. Because the factor underlying this change in size variation is due purely to size effects on growth, we denote it a ‘size-dependent’ factor (*sensu *Pfister & Stevens 2002).

The development of size variation may extend beyond size-dependent factors. Phenotypically based differences in foraging traits (Fuiman & Cowan 2003) that have a genetic basis (Arnold 1981; Conover & Munch 2002) or are learned (Palmer 1984; Dukas & Bernays 2000) can also generate differences in growth rate, and therefore size variation. For example, genetic differences that underlie varying levels of boldness in fish (Coleman & Wilson 1997) could translate into differential resource acquisition and hence affect size variation. Further, if an individual does well at acquiring food in one time period, this could both affect growth in that time period but also lead to a good condition that leads to better growth at the next time period (DeAngelis *et al*. 1993; Ludsin & DeVries 1997). Additionally, resources or other factors that affect growth may be heterogeneous, especially for sessile organisms, which will lead to unequal resource use among individuals. Thus, a number of non-size-based factors can underlie persistent growth rate differences and hence affect size variation. We denote this broad group of factors ‘size-independent’ factors to distinguish them from the size-dependent factors. Size-independent factors have been described quantitatively, with ‘growth autocorrelation’ (Pfister & Stevens 2002, 2003), ‘residual autocorrelation’ (Fujiwara, Kendall & Nisbet 2004) and ‘memory’ (DeAngelis *et al*. 1993; Imsland, Nelson & Folkvord 1998). Theoretical studies suggest that size-independent factors could play a substantial role in the development of size variation (DeAngelis *et al*. 1993; Imsland *et al*. 1998; Pfister & Stevens 2002; Fujiwara *et al*. 2004).

In this study the development of size-variation was examined in wood frog tadpoles, with the goal of determining the influence of size-dependent and size-independent factors. The effect of competition, which has been shown to affect size variation (Uchmanski 1985), was examined by varying resource level and tadpole density. We develop a model to interpret the strong observed effect of competition observed, and estimate the varying contribution of size-dependent and size-independent factors.

### Discussion

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

The experiment demonstrated that tadpole size variation changed dramatically as the strength of competition varied. This result was clear when competition was changed by manipulating either tadpole density or resource supply rate. When competition was least, the relative size variation (CV) decreased as a function of size throughout the experiment. In contrast, when competition was higher, there was a marked increase in size variation as a function of mean size. To gain intuition into how these changes in the magnitude in CV represent changes in the relative performance of individual tadpoles, consider two tadpoles that are plus and minus one standard deviation from the mean. By definition, CV is equal to the SD divided by the mean. Thus, these two tadpoles would differ in size by 74% at the end of the high competition treatments in which CV ∼0·37, but by only 41% in the low competition (base) treatment when CV ∼0·21 (Figs 2 and 3). Therefore, increased competition almost doubled the difference in growth performance of paired tadpoles that differed by 2 SD.

The model analysis indicated that size-dependent differences in individual growth are primarily responsible for the observed relationship between mean size and size variation at low competition (Fig. 4b), whereas size-independent factors were not (Fig. 4a). The estimated value of the scaling exponent derived from the relationship between mean mass and size variation at low competition, 0·832, is close to the average value, 0·75, observed (and predicted theoretically) for animals (Brown *et al*. 2004). Thus, if we use this value from the literature for the scaling of individual growth and size, the model predicts the observed development of wood frog tadpole size variation fairly well, deviating from the actual data with a moderately steeper decrease in CV as a function of mean mass (Fig. 4b). This moderately steeper decrease suggests that the actual scaling factor for wood frogs is higher than 0·75, the average value for animals, or that other factors affected size variation. In particular, this deviation could suggest that size-independent factors are also contributing to growth variation. Using the same methodology as in the previous section, if we assume that the size-scaling relationship is 0·75, then a CV in *a*_{1} of 0·026 increases the size variation to the required amount to reproduce the empirical data (*r*^{2} = 0·988). Thus size-dependent scaling factors alone with *b* ∼ 0·83, or a moderate combination of size-independent factors combined with size-dependent factors, determined the observed decrease in size variation at low competition. In either case, size-variation is strongly dictated by a size-scaling relationship between mean size and growth rate similar to that determined from physiological studies.

The parameters needed to reproduce the relationship between mean size and size variation in treatments with higher competition are in stark contrast to those in the base treatment with the lowest competition. The model showed that an effect of competition on both size-independent factors, causing CV in coefficient in *a*_{1} approximately equal to 0·15, or an increase in size-dependent factors, by increasing the scaling relationship between size and growth (*b*_{1}) to approximately 1·3, could underlie this pattern. In the latter case, the competition changed the size advantage from being less than proportional to size (i.e. *b* < 1) to greater than proportional to size (i.e*. b* > 1). It is important to note that slower growth due to increased competition does not necessarily imply that the relationship between mean mass and mass variation will be modified. It is plausible that all tadpoles would grow at equivalent lower rates and therefore size variation and mean size would change at reduced rates, but the relationship between mean mass and size variation would be the same.

One reason why trait (including size) differences could have a less pronounced effect on individual growth variation at lower competition is that any trait disadvantages do not limit resource acquisition at relatively high resource levels associated with lower competition. Therefore, all individuals would acquire (find and garner) and assimilate resources at nearly equivalent rates; but when resources are scarce at heightened competition, the effect of trait advantages in acquiring and assimilating resources will be more pronounced, leading to larger individual differences in growth (Uchmanski 1985; Lomnicki 1988). Further, work is required to disentangle which phenotypic differences, e.g. size- or other trait-based, are responsible. The possibility that size-independent factors may play a role is supported by laboratory experiments using marked tadpoles, in which we observed persistent individual differences, over the course of a week, in wood frog tadpole foraging behaviour (Peacor, unpublished data). We do not postulate here whether the origin of potential size-independent factors had a genetic (Arnold 1981; Conover & Munch 2002) or other basis, such as learned (Palmer 1984; Dukas & Bernays 2000), or positive correlation between growth and condition (DeAngelis *et al*. 1993; Ludsin & DeVries 1997).

One method to distinguish size-dependent and size-independent factors is to mark and follow the growth of individuals. Given these data, regression should reveal whether size has a large effect on growth rate, and positive correlation in the residuals of the size vs. growth relationship has been referred to as ‘growth autocorrelation’ and is indicative of size-independent processes (Pfister & Stevens 2003). Although a powerful approach, it is not always practical to collect data repeatedly on the same individual. Thus models that interpret size-variation in the absence of individual growth data, such as we present here, are needed.

An increase in size variation at higher competition levels, as reported here for wood frog tadpoles (see also Wilbur & Collins 1973), has been observed in a number of other animals (reviewed in Uchmanski 1985; Lomnicki 1988), including fish (Rubenstein 1981; Sogard & Olla 2000), and grasshoppers (Wall & Begon 1987). While little is known about the relative contribution of size-dependent and size-independent factors in these cases, there is evidence that in some fish species, size-independent behavioural differences between individuals (Fuiman & Cowan 2003) contribute to growth rate differences (Imsland *et al*. 1998; Wilson 1998). In addition, size can play a large role in social hierarchies of fish that lead to increased interference and larger inequities of resource distribution at low resource levels (Magnuson 1962; Rubenstein 1981). It is unlikely that interference played a significant role in the effect of competition on size variation in our study because the tadpoles foraged independently over relatively large areas. Rather, trait differences (size or other) must have affected foraging in other ways, such as by affecting the relative ability to locate, garner or assimilate resources.

The results in the high competition treatments have implications to modelling animal population dynamics. Increasingly, ecologists are using individually based models (IBM) to model ecological systems (DeAngelis & Mooij 2005; Grimm & Railsback 2005). One of a number of advantages (there are also disadvantages) of IBMs over other models, is that IBMs can represent and examine the implications of individual variation in phenotype. For example, a number of studies examining the population dynamics of fish have used IBMs. It is a standard protocol to use growth rate equations that are similar in form to eqn 1, such as bionenergetic models, to represent individual growth as a function of size for individual fish (Rose *et al*. 1999). However, our results indicate that under some conditions (e.g. high competition), this approach will fail to account for individual differences in growth that could affect model results. Indeed, higher competition enhancing the size variation of fish has been reported (Rubenstein 1981; Sogard & Olla 2000), and thus we expect similar modifications to eqn 1 would be needed to describe individual fish growth. For example, in models in which differential growth is important due to a fraction of individuals attaining higher reproductive success, modelling the individual differences accurately is especially important. Our results suggest that a size-dependent scaling of growth rate on size well above that found from physiological studies, or introducing individual variation in the size-independent coefficients, may be required.

Although our analysis has assumed that resources are homogeneous and the mobility of wood frogs probably resulted in homogeneous resources, the basic growth equation has application to systems with heterogeneous resources. Of course, resource heterogeneity can lead to individual variation in growth if movement by individuals is limited. In addition, even in systems in which resources are initially homogeneous, restricted foraging can lead to persistent resource patchiness that can drive size differences for individuals with equivalent foraging abilities (Pfister & Peacor 2003). If modelled using eqn 1, persistent differences in resource availability, due to resource heterogeneity, can be represented by modifying coefficient *a*_{1} (Sebens 1987), with higher values corresponding to individuals that have had access to higher resources. Thus, variation in coefficient *a*_{1} can represent intrinsic organism trait differences among organisms or extrinsic differences due to conditions imposed by the environment.

Individual size variation is a ubiquitous feature of animal populations, and is predicted to affect numerous ecological processes (see Introduction). Size variation may also serve as an indicator of genetic variation proposed to affect community structure (Wilson 1998; reviewed in Agrawal 2003). Further, much in the way the variation in population patterns across environmental gradients enhances our understanding of mean population processes (Connell 1961; Wellborn *et al*. 1996), patterns of size variation may serve as a lens into processes that affect mean growth and consequently fitness (Lomnicki 1988). Building frameworks to understand how factors affect size-variation could help in the interpretation of many empirical studies that would typically focus on mean size. This study illustrated how the relationship between size variation and mean size can vary greatly as a function of competition, due to a shifting role for size-dependent or size-independent factors as competition varied. Finally, our results have important implications for modelling natural populations. Size is easily included as a state variable in population models and there is thus a rich history of modelling size-dependent demographics (Lefkovitch 1965; Caswell 2001). In contrast, variance among individuals of non-size-based traits is described less easily in analytical formulations and requires more theoretical development.