Individual variation in response to intraspecific competition: problems with inference from growth variation measures

Authors


Correspondence author. E-mail: sigurd.einum@bio.ntnu.no

Summary

1. The intensity of intraspecific competition may influence the growth of differently sized individuals to different extents. Measures of among-individual variation in growth and body size are commonly applied to assess such effects. The measure chosen is then supposed to control for the effect of mean (also influenced by competition) on variance [e.g. coefficient of variation (CV)].

2. Although there is an appealing simplicity to this approach, interpretations of such data have some apparently unrecognised underlying complexities. Here we combine empirical data from salmonid fishes and simulations to illustrate this.

3. Simulations show that even if all individuals are equally affected by competition, CV of body mass will not be constant. In fact, the CV of body mass may either decrease or increase with increasing competition.

4. Using empirical data from growth experiments, where differences among replicates in mean growth was controlled by temperature rather than competitive intensity, we show that two salmonid fish species differ with respect to how SD growth scales with mean growth. Because this slope represents a possible null-hypothesis, and will rarely be known, observed relations between competitive intensity and SD or CV growth are unlikely to yield information regarding size-specific effects of competition.

5. Based on the empirical data, simulation results with respect to body mass variation showed that the relation between mean and CV final mass varied from positive to negative across species and scenarios.

6. Insights regarding size-specific competitive effects appear unlikely to be obtained from measures of variation in either growth or body size. If attempts to do so are made, a positive correlation between SD body size and measures of competitive intensity (e.g. population density) does suggest that smaller individuals are more influenced by increased competition. However, this will be an extremely conservative test, because the appropriate null-hypothesis is a negative correlation of unknown strength.

7. Future attempts to quantify relationships between body size and impacts of competitive intensity should employ more traditional analyses of variance on individual data (i.e. interactions between body size and competition on growth rates).

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.

Methods

Simulations

In the first exploratory simulations, we choose to use Atlantic salmon as a case study. We created populations of individuals who differed consistently in growth rates. Growth was modelled according to the Ostrovsky model (Ostrovsky 1995), which takes into account the allometric relationship between body mass and growth. The mass standardised growth rate Ω (% per day) is expressed as follows:

image(eqn 1)

where M0 and Mt is the body mass (g) at the beginning and at the end of the period t (in days), and b is the allometric mass exponent for the relation between specific growth rate and body mass, estimated as 0·31 for Atlantic salmon (Elliott & Hurley 1997). The individuals (N = 1000) in each population were given a starting body mass of 0·15 g (Einum 2003), and a growth rate that was drawn randomly from a normal distribution based on the mean growth rate inline image of the population and a certain standard deviation SDΩ. Different scenarios for how SDΩ scales with inline image were simulated. For all of these we let inline image vary between zero and six among populations (this covers the range observed in laboratory experiments, Forseth et al. 2001). We then let SDΩ increase linearly with inline image, but with different slopes for different scenarios. SDΩ for the population with the highest growth within a given scenario was always 0·10 inline image, but the intercept was allowed to vary from 0 to 0·10 inline image. Thus, the different scenarios ranged from SDΩ increasing proportionally with inline image (intercept = 0, and hence CVΩ constant and independent of inline image) to SDΩ being constant and independent of inline image (intercept = 0·10 inline image). For all populations, equation (1) was rearranged into

image(eqn 2)

to calculate iteratively the change in body mass over time for each individual. We let the fish grow for a period of 100 days before assessing the resulting distributions of body mass.

Growth experiments

Growth experiments were conducted at the NINA Research Station, Ims, south-western Norway, during the years 1996–1999. Details of experimental setup and population origin are previously described in detail by Forseth et al. (2001), Jonsson et al. (2001) and Larsson et al. (2005). All experiments used juvenile Atlantic salmon and Arctic charr originating from various populations (salmon N = 5, charr N = 2) that had been hatched and reared at the station (first generation in hatchery with wild parents). The experiments were conducted over a range of water temperatures (4–24 °C) in standard hatchery tanks (45 × 45 cm wide, 60 cm deep), with common water depths (30 cm) water flow (3 L min−1) and light regime (70 lux at surface, 18 h light/6 h dark). The range of temperatures used covers most of the range within which these species can obtain a positive growth rate, and hence produces large variation in growth rates at the same fish density. The fish were fed ad lib using automatic feeders. The duration of each experiment was 20–21 days. Each tank received 10 individuals from a single population, which were individually marked using alcian blue dye. Body mass (± 0·01 g) was measured at the beginning and at the end of each experiment, and feeding was stopped in sufficient time (24 h) prior to any measurement to allow for stomach evacuation. Species-specific allometric mass exponents for the relation between specific growth rate and body mass [0·31 for Atlantic salmon (Elliott & Hurley 1997) and 0·30 for Arctic charr (Larsson et al. 2005)] were used in calculation of size standardised growth rate. Mean initial body mass was 6·92 and 11·34 g for salmon and charr, respectively, whereas within-replicate (tank) SD of initial body mass was 1·51 and 2·32 g. A total of 383 tank replicates were conducted, 160 with salmon and 215 with charr. Of these, 31 and 16 were excluded for salmon and charr, respectively, because of mortality of one or more individuals.

Statistical analyses

All statistical analyses were conducted using the statistical software R, v. 2·11·1. (R Development Core Team 2009). Within-replicate SDΩ from the growth experiments was modelled as a function of inline image, SD initial body mass (SDW), and the mean body mass of individuals during the experiment (i.e. mean of mean initial and final body mass, inline image). The latter two variables were included to control for any effects of variation in initial body mass on relative growth performance, and total biomass on level of competition, respectively. Any effects of these covariates, as well as intercepts, may be best described as differing among populations within species, between the two species (i.e. common parameters for different populations within species), or being identical for the two species. Thus, a large number of alternative models with different degrees of complexity exist. The full model can be written as follows:

image(eqn 3)

where j indicates replicate and i indicates either population or species, and ai represents the intercepts, which may also differ among populations or species. In the absence of population or species-specific slopes, a single estimated value is produced for β1, β2 and β3, respectively. We fitted all possible alternative models, from the full model down to a model that only estimated a single intercept (a total of 70 models) using the gls function within the nlme package (Pinheiro et al. 2009). Residuals from the full model showed some heteroscedasticity relative to population. We therefore applied the varIdent function, which successfully stabilised the residuals (visually assessed). For all the alternative models, we calculated their AIC value, and observed the AIC differences between alternative models to infer which one was the best in describing the data (Burnham & Anderson 1998).

Simulations with empirical parameters

Based on the model parameters from the growth experiment, a second set of simulations was conducted to examine the relationships between mean growth and the two measures of variation in growth and final body mass. These simulations were conducted as described earlier, but using the species-specific parameters for determination of SDΩ obtained from the best experimental based model, species-specific mean body mass equal to the mean used in the experiments, and species-specific allometric mass exponents for the relation between specific growth rate and body mass [0·31 for Atlantic salmon (Elliott & Hurley 1997) and 0·30 for Arctic charr (Larsson et al. 2005)]. Two scenarios were conducted for each species, one where the SD of initial body mass was equal to zero, and one where this SD was equal to the mean used in the experiments.

Results and discussion

Our simulations show that even if the SD growth is proportional to the mean, and hence CV growth will be independent of the mean, this does not hold for CV final body mass (Fig. 1a–d). Rather, SD of final mass will increase as a power function to the mean growth, and CV will be positively correlated with the mean. Furthermore, even if SD of growth is independent of the mean growth, SD of final mass will still be positively correlated with mean final mass (Fig. 1q,s). These findings have important implications for studies attempting to infer body size-specific effects of competition. The effect of competitive intensity on mean growth will in isolation cause a negative relation between competition and SD final body size (Fig. 2). Thus, if a positive relation between population density and SD body size is observed, this strongly suggests that smaller individuals are more affected by competitive intensity than larger ones. However, this represents an extremely conservative test, because the appropriate null-hypothesis is some unknown positive correlation between SD and mean final size, and hence a corresponding negative correlation between SD of body size and density. Furthermore, depending on how SD of growth scales with mean growth, CV of body size may either increase or decrease with increasing mean growth (e.g. compare Fig. 1l,p; Fig. 2). Thus, this latter measure should in no cases be used for inferring size-asymmetric competition effects.

Figure 1.

 Simulated effects of growth rate distributions on variance in body size of Atlantic salmon juveniles. The left panels (a, e, i, m, q) show five different scenarios for how the standard deviation of growth rate within a population scales with the mean growth rate, ranging from proportional (top) to constant (bottom). For the five scenarios, corresponding results are given for the effect of mean growth rate on CV growth rate, SD body mass after 30 days of growth, and CV body mass.

Figure 2.

 Summary of qualitative effects of competition intensity on variation in growth and final body mass as observed in simulations when competition only affects variation indirectly through effects on mean growth (Fig. 1). Signs indicate positive, neutral or negative relationships. Increased competitive intensity will decrease SD final body mass within a population indirectly through the effect on mean growth rate. This holds even if SD growth is independent of mean growth (Fig. 1q–t). If the relationship between mean and SD growth is sufficiently steep, SD final body mass will decrease faster than mean final mass with increasing competition, and CV final body mass will decrease; if not CV final body mass will increase.

In our empirical analyses, the top 10 models (i.e. those giving the lowest AIC values) all included an effect of mean growth rate on SD of growth (Table 1), and some variation in parameters between the two species. None of the models estimating population-specific parameters were among the top 10, suggesting little innate differences among populations within species in growth variation patterns. The best model was substantially better than the next best model, which received essentially no empirical support (delta AIC = 9·7, Burnham & Anderson 1998, p. 70). Furthermore, the species-specific slopes between mean growth and SD of growth were virtually identical among the different models that included these (Table 1). We therefore used the parameter estimates from the best model (Table 2) in the subsequent simulations. As expected from the observation that fish can have negative growth rates, substantial growth variation can be present even for a mean growth of zero (i.e. positive intercepts), and the slopes show a less than proportional increase in SD growth with increasing mean growth (Table 2).

Table 1.   AIC values for alternative models (best 10 models given) for the effects of mean growth (inline image), mean body mass (inline image) and SD initial body mass (SDW) on SD growth (log-transformed) within tanks of Atlantic salmon and Arctic charr
Modelinline imageinline imageSDWInterceptAICβ1, salmonβ1, charrβ1
  1. M, main effect (single common parameter for both species); S, species-specific effect; –, no effect. Parameters for the effect of mean growth (i.e. the slopes β1, see eqn 2) on SD growth are given. For models where these are estimated separately for the two species, species-specific slopes ± SE are given, otherwise a common slope is given

 1SSS417·50·24 ± 0·020·06 ± 0·03 
 2SMSS427·20·24 ± 0·020·05 ± 0·03 
 3SMS428·80·23 ± 0·030·09 ± 0·03 
 4SS432·80·22 ± 0·030·02 ± 0·02 
 5MSS433·3  0·16 ± 0·02
 6SSSS434·50·24 ± 0·020·06 ± 0·03 
 7MMS435·0  0·16 ± 0·05
 8SMS436·70·22 ± 0·030·08 ± 0·03 
 9SMMS438·50·23 ± 0·030·08 ± 0·03 
10SSS438·80·23 ± 0·030·05 ± 0·03 
Table 2.   Estimated effects of mean growth and SD initial body mass on SD growth (log-transformed) within tanks for Atlantic salmon (intercept) and Arctic charr from the best model (Model 1, Table 1)
 ParametertP
Intercept0·355 ± 0·0675·29<0·001
Mean growth0·238 ± 0·0259·62<0·001
SD initial body mass0·223 ± 0·0405·63<0·001
Charr0·373 ± 0·1292·890·004
Mean growth:charr−0·182 ± 0·037−4·88<0·001
SD initial body mass:charr−0·191 ± 0·043−4·41<0·001

The simulations based on the parameters from the experimental study give some baseline predictions regarding the relationship between mean and variance in growth and body mass in the absence of density effects for the two species. For growth rates, a common feature for both species was that the observed slope of the positive relationship between mean and SD growth caused negative correlations between mean and CV growth (Figs 3b,f and 4b,f). Thus, the appropriate null-hypothesis for body size-specific effect of competitive intensity is a certain increase in CV growth with increasing density. Because the slope of this null-hypothesis will rarely be known, observed relations between density and CV growth are unlikely to yield useful information regarding direct effects of competitive intensity on variation in individual performance. For body mass, the simulations predicted positive correlations between mean growth and SD final body mass (Figs 3c,g and 4c,g). Finally, whereas a strong positive effect of mean growth rate on CV final body mass can be predicted for salmon (Figs 3d and 4d), it will be much weaker for charr (Fig. 4h) and may even become negative under variable initial body mass (Fig. 3h). These simulation results with respect to body mass variation exemplifies that appropriate null-hypotheses may not only be unknown, but may even differ qualitatively among closely related species. Furthermore, it may also be argued that any such empirically derived null-hypothesis will be arbitrary. In the present study, we chose temperature as the environmental factor in shaping mean growth and found a corresponding null-hypothesis for the relationship between mean and variance. However, a different relationship may have been found if other environmental factors such as osmotic stress, parasite abundance or water velocity were manipulated. Thus, in reality, even an empirically developed null-hypothesis gives no solution to the problem.

Figure 3.

 Simulated effects of mean growth rate on variation in growth and body mass of Atlantic salmon (a–d) and Arctic charr (e–h) juveniles based on estimated effect of mean growth rate on SD growth from experiments. All simulations use a common mean and SD of initial body mass identical to that used in the growth experiments.

Figure 4.

 Simulation results as in Fig. 3, but with no variation in initial body mass.

Temperature was used as an environmental factor to provide estimates of the direct effect of mean growth on SD growth. It is possible that the relatively low increase in SD growth with increasing mean growth was caused by lower variation in competitive abilities at temperatures providing higher growth. However, this seems unlikely, as previous observations suggest salmonid aggressive interactions to be most pronounced at temperatures closer to those providing maximum growth rates (e.g. Vehanen et al. 2000; Vehanen & Huusko 2002). Thus, if anything, competitive abilities should be more variable at temperatures providing good growth because of more direct interactions among individuals. Furthermore, even if competitive abilities were less variable at temperatures providing higher growth, this does not alter our conclusion that environmental factors other than density or food availability that influence mean growth will indirectly influence measures of variation of growth or final body size.

In conclusion, it seems unlikely that insights regarding size-specific competitive effects can be obtained from measures of variation in either growth or body size. If any attempts to do so are made, they should test for a positive correlation between SD body size and measures of competitive intensity (e.g. population density). However, this will be an extremely conservative test, because the appropriate null-hypothesis is a certain negative correlation, and a negative result (i.e. lack of a positive relationship) will therefore be of little value. Thus, future attempts to quantify relationships between body size and impacts of competitive intensity should employ more traditional analyses of variance on individual data (i.e. interactions between body size and competition on growth rates).

Acknowledgements

We thank the staff at the NINA Research Station Ims for assistance with the experiments. Financial support to the experimental studies was provided by the Norwegian Research Council, Norwegian Institute for Nature Research and the European Commission under the FAIR-programme (CT95-0009).

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