Group size versus individual group size frequency ...



Group size versus individual group size frequency distributions: a nontrivial distinction

Roger Jovani a, *, Roddy Mavor b,1

a Estación Biológica de Doñana, CSIC

b Seabird Monitoring Programme, JNCC

Keywords: colony size crowding group living group size

individual group size seabird

Understanding group size variation is a major challenge in animal ecology. However, we argue that understanding group sizes from an individual point of view (i.e. individual group sizes) and the rela- tionship with population group sizes may be even more important. This may seem redundant, but in the present study we show that it is not. We analysed colony sizes of 20 seabird species breeding in Britain and Ireland from the Seabird 2000 project (19 978 colonies; 3 779 919 nests) comparing group (¼colony) size frequency distributions (GSFDs) with their individual group size frequency distribution (IndGSFD) counterparts. We did so for the first time for a number of species with semilogarithmic plots, and correlated eight statistics from each GSFDeIndGSFD pair. Shape-related variables (e.g. skewness) of GSFDe IndGSFD pairs were highly unrelated with only 1e15% of redundancy. In fact, species with similar GSFDs had individuals concentrating in either the largest or the medium-sized groups. There was a trend towards those species with higher group size variation having individuals living in a narrower range of group sizes. Some group size-related measures (e.g. mean group size) showed a tight linear correlation in logelog scatterplots between GSFDs and IndGSFDs. However, this correlation disappeared in linear scatterplots for two of the four measures. Moreover, group size-related measures were always a poor surrogate of corresponding individual group size measures. We discuss how animal grouping research could benefit from similar comparisons between GSFDs and IndGSFDs and how this can be carried out in a meaningful way.

Most animals live in groups either temporarily or permanently. Group size shapes the cost/benefit payoff of group living, with some group sizes often conferring higher fitness than others (Krause & Ruxton 2002). However, empirical and modelling approaches have shown that even when there is a clear peak in the fitness function of group sizes (i.e. there is an ‘optimal’ group size), a huge variation in group sizes still tends to exist. After decades of study, understanding this variation remains an unsolved challenge in animal ecology research (Giraldeau & Caraco 2000; Gerard et al.

2002; Krause & Ruxton 2002; Safran et al. 2007; Sumpter 2010).

A major driver of this research agenda has been the description of group size frequency distributions (hereafter GSFDs; e.g. Götmark

1982; Wirtz & Lörscher 1983; Brown et al. 1990; Stacey & Koenig

1990; Avilés & Tufiño 1998; Krause & Ruxton 2002; Jovani & Tella

2007; Serrano & Tella 2007; Jovani et al. 2008a,b). These studies

* Correspondence: R. Jovani, Department of Evolutionary Ecology, Estación

Biológica de Doñana, CSIC, Américo Vespuccio s/n, E-41092 Sevilla, Spain.

E-mail address: jovani@ebd.csic.es (R. Jovani).

1 R. Mavor is at the Seabird Monitoring Programme, JNCC, Inverdee House, Baxter

Street, Aberdeen AB11 9QA, U.K.

examined group sizes from a population point of view. However, group sizes can be viewed from an individual point of view as well. Describing individual group size selection, the reasons behind these choices and its constraints has proved to be a powerful mechanistic approach to explaining population group size patterns (Brown & Brown 2000; Safran 2004; Safran et al. 2007; Serrano & Tella 2007; Jovani et al. 2008b). However, surprisingly few studies have ana- lysed, per se, individual group size frequency distribution patterns (IndGSFD; but see Jarman 1974; Wirtz & Lörscher 1983; Weso1owski et al. 1985; Reiczigel et al. 2005, 2008). An illustrative example of this uneven attention to GSFDs versus IndGSFDs is the book on coopera- tive breeding in birds edited by Stacey & Koenig (1990) in which 14 of

18 chapters (each covering a study species) show a histogram of the GSFD of the population, but only one chapter (Emlen 1990) shows both the GSFD and the IndGSFD. This previous lack of attention paid to IndGSFDs could be because the properties (e.g. mean) of GSFDs and their IndGSFD counterparts are biologically redundant, thus pre- senting only a mathematical subtlety without biological relevance. In fact, some evidence would suggest that this might be the case.

First, a given GSFD has a unique IndGSFD counterpart. For instance, in a hypothetical population of 16 individuals distributed among five

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Group size

Figure 1. Semilogarithmic group (colony) size frequency distributions (in black; left Y axis) and corresponding individual frequency distributions (in grey; right Y axis) for 20 seabird species breeding in Britain and Ireland. Logarithmic bins of the form [Xn,Xnþ1 — 1] with n ¼ 0,1,2,3. are used; for instance, for X ¼ 2, bins are [1e1], [2e3], [4e7], [8e15]. The X axis shows the logarithmic midpoint of the bin (i.e. 10(log(minimum group size of the bin)þlog(maximum group size of the bin))/2), and the linear Y axis shows the number of groups (or individuals

groups of sizes 2, 2, 3, 4 and 5, specific individuals will be present in (i.e. experience) groups of sizes 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5,5 and5 (individual group sizes). Thus, GSFDeIndGSFD pairs are completely interlocked, and thus potentially redundant. Second, although the mean of an IndGSFD is always larger than that of its GSFD counterpart (Preston 1948, 1962; Lloyd 1967), the distinction between the two may be biologically meaningless. For instance, in the above example, mean group size is (2 þ 2 þ 3 þ 4 þ 5)/5 ¼ 3.2 and the mean individual group size is (2 þ 2 þ 2 þ 2 þ 3 þ 3 þ 3 þ 4 þ 4 þ 4 þ

4 þ5 þ 5 þ 5þ 5þ 5)/16 ¼ 3.625; surely not a large difference in

biological terms. Finally, Lloyd (1967) showed that the mean of an IndGSFD is larger than the mean of its GSFD as much as the variance/ mean of its GSFD, thus showing that one is the trivial predictable outcome of the other [e.g. in the above example mean IndGSFD ¼ 3.2 þ (1.36/3.2) ¼ 3.625]. Moreover, Iwao (1968) and recently Reiczigel et al. (2005, 2008) have shown a very tight linear correlation between log(mean GSFD) and log(mean IndGSFD) across different taxa, suggesting that mean GSFDs and mean IndGSFDs hold essentially the same biological information.

However, we show here that GSFD measures should not be used as surrogates of corresponding IndGSFD measures, and that the direct study of IndGSFDs combined with GSFDs can reveal interesting nonredundant information about group living. First, understanding IndGSFDs may be biologically even more important than under- standing group size variation. This is because most of the processes shaping the ecology and evolution of species (natural selection/ demography) have the individual rather than the group as the unit. For instance, if breeding success is lowered at large group sizes (negative density dependence), an important measure of the impact of these processes upon population demography will not be the proportion of large/small group sizes in the population, but rather the proportion of individuals breeding within such group sizes.

Second, contrary to the evidence stated above, IndGSFDs may not yield redundant information about their GSFD counterparts. This is because natural GSFD patterns do not follow the same and ideal theoretical distributions, but are considerably more complex. For instance, in a previous study we showed that similarly shaped GSFDs from 20 seabird species when plotted in standard histo- grams hide contrasting patterns that are unravelled when the same data are plotted with logarithmic bins (Jovani et al. 2008a). Thus, we predicted that GSFDs with different combinations of skewness, variability or maximum group sizes could have nontrivial impacts on their IndGSFDs. We reanalysed this seabird data set by comparing GSFDs and their IndGSFDs counterparts.

METHODS

We built on a previous study by Jovani et al. (2008a) in which we analysed the colony sizes (here also called group sizes) of seabird species breeding in Britain and Ireland. This is a data set from Seabird 2000, a collaboration between the Joint Nature Conserva- tion Committee, U.K., and the Royal Society for the Protection of Birds, U.K. The project involved over 1000 surveyors following detailed instructions for the census of each seabird species. No less important was the meticulous checking during the process of data entry, both by routine quality control by the Recorder 2000 soft- ware, and later by data entry personnel. The result is the highest-quality data on a snapshot (mainly 1998e2002) of bird colony sizes for a large area, and possibly the largest data set on animal group sizes considering different species in a large area.

Overall, it covers 20 seabird species, 19 978 colonies and 3 779 919 nests. For further details of Seabird 2000 see Mitchell et al. (2004), and of the data set analysed here see Jovani et al. (2008a).

Plotting Frequency Distributions

The data set of individual group sizes for each species was created from their GSFDs as explained in the hypothetical example in the Introduction, that is, with one value (the colony size in which the breeding pair was nesting) for each breeding pair of the species. Our unit of measure is typical in bird coloniality studies, that is, the breeding pair (the nest), and thus we used the nest as the ‘indi- vidual’ of IndGSFDs to lend comparability to other studies on group living. IndGSFDs were plotted in semilogarithmic plots (Fig. 1) following the same procedures as for GSFDs detailed in Jovani et al. (2008a), where we used Preston’s (1962) methods with slight modifications (see Fig. 1 and Pueyo & Jovani 2006 for details).

GSFD and IndGSFD Statistics

Seabird GSFDs are clearly not Gaussian distributions, but show distributions closer to log-normal and power laws (Jovani et al.

2008a). Thus, parametric measures such as the mean (and even log- normal measures such as the geometric mean) are not the most appropriate. The only parametric measure used was the mean to compare our results with those of Iwao (1968) and Reiczigel et al. (2005, 2008). Overall, we calculated eight statistics from each GSFD and IndGSFD. Our aim was to achieve a general description of the characteristics of GSFDs and IndGSFDs to be able to compare these two ways of looking at group size frequency distributions. For the size of the groups (or of individual group sizes) of each species we calculated the 5th percentile, the median, the mean and the 95th percentile. Minimum and maximum group sizes and individual group sizes were not measured because they are, by definition, the same for GSFDeIndGSFD pairs. To characterize the shape of the distributions we calculated the skewness (a measure of asymmetry), fit of GSFDs and IndGSFDs to a log-normal distribution as measured by the Kol- mogoroveSmirnov statistic, kurtosis (a measure of ‘peakedness’ around the mean), and population variability, a nonparametric counterpart of the coefficient of variation (CV) which quantifies the mean deviation of all group size pairs within populations (see Heath

2006 for details). Statistics were calculated with standard MatLab (MathWorks, Natick, MA, U.S.A.) functions applied to nontransformed group (and individual group) sizes. The fit to a log-normal distribution was calculated with log-transformed group sizes and individual group sizes. Population variability was calculated by modifying the code in version 1.1 of the variability calculator for MatLab by Heath (2006). All measures along with the data necessary to plot the frequency distributions were retrieved from the MatLab algorithm freely available from the Supplementary Material.

We used the Pearson productemoment correlation coefficient to calculate the linear correlation of each statistic between each GSFDeIndGSFD pair across the 20 analysed species. Although it is impossible to determine accurately whether our data (20 values for each statistic) follow a normal distribution, we used Pearson instead of rank correlation coefficients (e.g. Spearman correlation) because the latter do not test for the tightness of the correlation to a linear one (which is what we wanted to test) but rather for the level of correlation in the increase in x relative to y. From the scattering of data in Figs 2 and 3, we thought Pearson correlations were better

for IndGSFDs) for each bin. Note that all black bars must have their corresponding grey bar below, but because of the highly right-skewed distributions there are some grey bars that are too narrow to be visualized, e.g. in (g). Note the log scale only in the X axis. (a) Uria aalge; (b) Rissa tridactyla; (c) Fulmarus glacialis; (d) Alca torda; (e) Sterna paradisaea; (f) Hydrobates pelagicus; (g) Fratercula arctica; (h) Phalacrocorax aristotelis; (i) Chroicocephalus (¼Larus) ridibundus; (j) Phalacrocorax carbo; (k) Larus argentatus; (l) Cepphus grylle; (m) Larus canus; (n) Sterna albifrons; (o) Sterna hirundo; (p) Larus fuscus; (q) Puffinus puffinus; (r) Larus marinus; (s) Stercorarius skua; (t) Stercorarius parasiticus.

suited for Fig. 3, and thus we interpreted Pearson correlations from Fig. 2 with caution. Pearson correlation r ¼ 1 (or —1) would indicate that all data fall along the linear trend line fitted to the data, and values closer to 0 would indicate a complete scatter of values around

the fitted trend line. The coefficient of determination, R2 (r squared),

was calculated as a measure of the variance in each IndGSFD statistic (e.g. median individual group size) explained as a linear function of the corresponding GSFD counterpart (e.g. median group size of the population), that is, a measure of the redundancy, r ¼ 1, meaning that GSFDeIndGSFD pairs provide essentially the same information about the grouping patterns of the species.

RESULTS

GSFDs versus IndGSFDs Histograms

Figure 1 (black bars) shows the same GSFDs as those previously reported in Jovani et al. (2008a). These are semilogarithmic plots in which a distribution with a Gaussian shape thus corresponds to a log- normal distribution. Seabirds in Britain and Ireland show contrasting GSFDs, from clear log-normal distributions (e.g. Fig. 1a, b; Jovani et al.

2008a) to very skewed log-normal distributions (Fig. 1ret; following

power laws as detailed in Jovani et al. 2008a). However, species show different combinations of kurtosis and skewness (Figs 1, 2) so that many of the distributions depart from neat log-normals (e.g. Fig. 1g, k). This is

important because if all species followed the same distribution, IndGSFDs would be easy to predict from its GSFD (e.g. compare Fig. 1a and b). However, Fig. 1 shows that this is not so trivial for real animal grouping patterns. For instance, GSFDs in Fig. 1d and g are similar, but their IndGSFDs (grey bars in Fig. 1) are very different, while Fig. 1g and p show contrasting GSFDs but similar IndGSFDs. Note that these patterns remain hidden when we plot the same data in linear (standard) histograms (Appendix Fig. A1). This apparent lack of a general rule linking GSFDeIndGSFD pairs leads us to ask whether GSFDeIndGSFD pairs provide redundant or complementary information.

GSFD versus IndGSFD Statistics

Kurtosis and population variability showed a negative correla- tion between IndGSFDs and GSFDs (ca. —0.4), explaining ca. 15% of variance (Fig. 2, Table 1). This did not reach statistical significance, possibly because of low sample size, but also note the potential effect of outliers. In any case, the scattering of data around the trend line was considerable. In general, the shape of the IndGSFDs was not redundant with their GSFDs: only 1e15% of the variance in IndGSFD characteristics was explained by corresponding GSFD characteristics (Fig. 2, Table 1).

As expected, all IndGSFDs were more left skewed (with lower skewness values) than their GSFDs counterparts (Fig. 2). This is because the mean IndGSFD is constrained to being larger than the

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Group size frequency distribution

Figure 2. Correlation of shape-related statistics describing group size frequency distributions and their corresponding individual group size frequency distributions for the 20 seabird species studied. (a) Skewness, (b) KolmogoroveSmirnov, (c) kurtosis and (d) population variability. Species codes are the same as in Fig. 1.

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Group size frequency distribution

Figure 3. Correlation of log(size-related statistics) describing group size frequency distributions and their corresponding individual group size frequency distributions. See Table 1 for correlation statistics. Species codes are the same as in Fig. 1. (a) 5th percentile, (b) median, (c) mean and (d) 95th percentile. Grey vertical lines show the potential range of log(individual group size) values for a species according to its group size frequency distribution (see Discussion for more details). For instance, since the mean group size of species t was 3.3 and its maximum group size was 107, this species could only have a mean individual group size between 3.3 and 107, and shows an intermediate empirical value of 22.

However, species q could have values between 6269.3 and 120 000 and shows a value close to its potential maximum (82 431.7).

mean GSFD (see above), and thus the left tail of IndGSFDs extends further than in the corresponding GSFDs (Fig. 1). However, skewness of IndGSFDs and GSFDs was uncorrelated across species (Table 1, Fig. 2).

The other three shape-related characteristics showed indis- tinctly higher or lower values in GSFDs than IndGSFDs (Fig. 2). The fit to a log-normal distribution was uncorrelated between GSFDs and IndGSFDs. In other words, any combination was possible. This is easy to visualize comparing Fig. 1a, g and r. In Fig. 1a, a neat log-normal GSFD leads to a slightly left-skewed log-normal IndGSFD, but in

Table 1

Pearson correlation coefficients (r), coefficient of determination (i.e. variance explained by the linear model, R2), and P values for each graph in Figs 2, 3 and A2

Raw data Log (data)

r R2 P r R2

Fig. 1g a slight departure from a log-normal shape of the GSFD

produces a highly skewed IndGSFD. The opposite occurs in Fig. 1r.

Group size-related measures were also analysed for their corre- lation between GSFDs and corresponding IndGSFDs. This was done for raw variables and also for their logarithms (to compare with Reiczigel et al. 2008), because these are two approaches that give complementary information. Mean and 95th percentile group sizes were highly correlated in both linear and logelog plots (Fig. 3, Table 2, Appendix Fig. A2), with 75e95% of redundancy (Table 1). However, despite this strong linear correlation, mean and 95th percentile group sizes were very different between GSFDs and cor- responding IndGSFDs (Table 2). Median and 5th percentile group sizes were significantly correlated in logelog plots but not when raw

DISCUSSION

Our results show the first comparison of GSFD versus IndGSFD in semilogarithmic histograms. They have revealed a nontrivial relationship between the group sizes of a population and the group sizes in which individuals live, something difficult to appreciate in standard histograms (compare Fig. 1 and Fig. A1). This challenges

Table 2

Group size-related measures for group size frequency distributions (GSFDs) and corresponding individual group size frequency distributions (IndGSFDs)

5th percentile Median Mean 95th percentile

SCGSFDIndGSFDSCGSFDIndGSFDSCGSFDIndGSFDSCGSFDIndGSFDd11t18t322t1298l12r241r9142s302293g12s2195s13994r37983k14o6157l1550l50208m14m6300n1867m8011 219p16p63309h23217h841720r17l731m323767n85220i19n850o33310o1221033h114h868j51186e1664000j114k10295e52740k19010 129t117e14200k521565j200558s135i162500p1178005p23419 487n146j25125d1662728d61311 384o165d27961c1813055c71012 276e1579g2940 000i2104424i80014 575c250c31950b6133894b275911 077f2309f596800f84512 191g310459 471q23286q61101 800g122633 250f486627 297a5517b1542361a153414 856a734475 493b6165a2058679q626982 432q41 697120 000SC: species codes are the same as in Fig. 1.

previous evidence suggesting the redundancy of this double approach to animal group sizes (see Introduction).

Mean Group Size

We confirm the unavoidable mathematical fact that individuals live in larger groups than the average group size in their population, and the strong linear correlation between log(mean group sizes) and log(individual mean group sizes) previously reported by Iwao (1968) and Reiczigel et al. (2005, 2008). Reiczigel et al. (2008, page 719) argued that ‘Since mean group size tends to predict mean crowding (Fig. 3), this approach may also be useful as a rough approximation’. However, our results contradict this interpretation for the following reasons.

First, note that the apparent good fit shown in Fig. 3 (Table 1) and the similar Figure 3 in Reiczigel et al. (2008) is not so surprising when considering the potential individual group size values that a species can have with a given GSFD. This is what we have attempted to illustrate in Fig. 3 with the vertical grey lines. Given that a statistic (e.g. mean) of individual group sizes is always larger than its group size counterpart (Fig. 3; Preston 1962; Lloyd 1967), and that individual group sizes can never be larger than the maximum group size of the population (i.e. individuals cannot live in larger groups than the largest group of the population), these grey lines show the range of individual group size values that a given population can have. This shows that a tight linear fit to log(data) is simply a mathematical constraint imposed by the interlocked nature of GSFDeIndGSFD pairs: any random distribution of dots within the grey lines would create a tight linear correlation.

Second, even within the narrow range of values that a species could exhibit in Fig. 3, species differ considerably in their relative position within their corresponding grey line. This is very biologically relevant because of the logarithmic scale (compare Fig. 3 and Fig. A2), even hiding paradoxical situations (also acknowledged by Reiczigel et al.

2005, 2008): species with clearly larger group sizes can have individ- uals living in clearly smaller groups. For instance, Rissa tridactyla has a median group size of 154 nests, clearly larger than the 29 nests for Fratercula arctica (species b and g in Table 2, respectively). However, the median individual of species b lives in groups of 2361 nests and that of species g in groups of 40 000 nests. This could imply a huge difference in the ecology of the population (e.g. for the strength of negative density-dependent processes) and in the evolution of the species (e.g. behavioural adaptations to living in a given social scenario).

Third, even for variables showing a tight linear correlation between GSFDs and IndGSFDs on logelog (Fig. 3) and linear axes (Fig. A2), GSFD measures (e.g. mean group size) were a poor approximation of corre- sponding IndGSFD measures; to be at least a rough approximation they would need to be close to the x ¼ y line in Fig. A2 (see also raw data in Table 2). Note that if one is interested in the mean group size experi- enced by individuals in a given species, the mean group size is a poor predictor (Table 2). For instance, suppose Chroicocephalus ridibundus (species i) suffers a strong negative density dependence on breeding success when nesting in colonies larger than 2500 pairs. In that case, the median colony size would be highly misleading (i.e. 16 nests) in the evaluation of the demographic consequences of this density depen- dence, because it would suggest a negligible effect. However, in fact,

50% of the population breeds in colonies larger than 2500 nests (i.e. median individual colony size ¼ 2500; Table 2), thus having a probable effect on individual fitness and population demography.

Overall, this shows that while it is true that in a comparative (interspecific) study on seabirds, one can infer the log(mean indi- vidual group size) from the log(mean group size) of the species, it is also true that for a given species, one can only predict that mean individual group size will be larger than mean group size and lower than the maximum group size in the population, thus losing relevant information on the group sizes experienced by individuals. In any case, GSFD measures are a poor approximation of corresponding IndGSFD measures (Table 2).

Other Group Size Statistics

We have analysed not only the mean but also several other statistics of GSFDeIndGSFD pairs and we have found interesting new information potentially linking individual behaviour and population patterns. For instance, in half of the studied species, individuals live in a definite range of intermediate to large group sizes (e.g. Fig. 1a, h, l, r), avoiding the lower half of their GSFDs. In almost another half of the species, individuals cluster in the largest group sizes (e.g. Fig. 1f, g, p, q). In others, there does not seem to be a clear preference (e.g. Fig. 1t). In fact, Fig. 2 shows either a lack of correlation or a negative correlation between group size variation and individual group size variation. This challenges our view about the link between individual behaviour and group size population patterns and poses a paradox: species with larger group size variation have a larger proportion of individuals concentrated in particular group sizes.

Group size variation could come from two sources: from individual behaviour (e.g. owing to a larger underlying genetic predisposition for contrasting group sizes, Brown & Brown 2000; Serrano & Tella 2007), or because of formationedestruction dynamics (e.g. all large colonies started with a few nests). What these analyses tell us is that in species with larger colony size variation, individuals live in more specific colony sizes. The paradox is potentially solved by colony size dynamics: intraspecifically, variability in individual behaviour (e.g. owing to underlying genetics) could promote colony size variation. However, because of the imperative colony size dynamics, when a species shows a preference for breeding in large colonies, all smaller colonies also exist in the population (i.e. very small colony sizes are pervasive even in bird species with huge colonies; Brown et al. 1990; see Table 1 in Jovani et al. 2008a), thus leading to higher colony size variation even when most individuals prefer to live in some particularly large colonies.

The 5th percentile showed the weakest correlation for group size-related variables between GSFDs and IndGSFDs (Fig. 3, Table 1, Fig. A2). In fact, all species showed a lowest group size of fewer than

10 nests, but even species with a minimum group size of one nest showed contrasting 5th percentile individual group sizes from one to

579 nests. Since minimum group sizes of species often show very low values, often close to one nest, that is, solitary breeders (e.g. Brown et al. 1990; Krause & Ruxton 2002; this study), minimum colony sizes could scarcely be seen as a species-specific trait. However, our analyses show that seabirds in Britain and Ireland differ substantially in the smaller (5th percentile) group sizes in which individuals live, thus presenting the possibility that this could be a species-specific trait. This necessitates a study comparing populations of the same species in different parts of the world.

The Relevance of Logarithmic Binning

Animal GSFDs often do not follow normal distributions but show

contribution that opens the analysis to any statistic of IndGSFDs instead of only focusing on the mean group size experienced by individuals (i.e. the ‘typical group size’ of Jarman 1974). However, we prefer ‘individual group size’ because it is the logical individual counterpart of ‘group size’ without any connotation about the consequences of group size. ‘Crowding’ suggests that larger group sizes imply greater density. This is true when space is finite as occurs when parasite intensity increases in a host, or similarly sized hosts show different parasite intensities (Poulin 2007). However, this need not be the case in other situations. For instance, nest spacing in seabird colonies is often constant despite colony sizes ranging from tens to thousands of nests (Nelson 1980, page 125).

Conservation Implications

It was not necessary to plot IndGSFD for seabirds in Britain and Ireland to know that there are some species such as Puffinus puf- finus in which a few colonies harbour a large proportion of the total population, and, thus, are colonies of special conservation concern (Mitchell et al. 2004). However, we believe that plotting IndGSFDs of all species together as in Fig. 1 gives a more informed point of view on the degree in which this occurs in the different species. This is especially important because by only knowing the sizes of colonies (black lines in Fig. 1) it is difficult, without plotting them, to predict how concentrated, in a few large colonies, the population is (e.g. compare black and grey bars in Fig. 1d versus g or in r versus s). For instance, knowing the maximum colony sizes of a species is not enough to know the proportion of the total breeding population that will be lost if, for instance, the largest five colonies are destroyed (and birds do not move to other colonies; Fig. 4). Obvi- ously, species with the largest colonies are more sensitive to losing one of their five largest colonies. However, the correlation

2

highly right-skewed frequency histograms, with many small groups

(r ¼ 0.498, P ¼ 0.026) only explained R

¼ 0.248 of variation and

and very few large ones (Götmark 1982; Wirtz & Lörscher 1983; Brown et al. 1990; Stacey & Koenig 1990; Avilés & Tufiño 1998; Krause & Ruxton 2002; Jovani & Tella 2007; Serrano & Tella 2007; Jovani et al.

2008a,b). However, this apparent uniformity among species in their GSFDs is not real, but the result of a weak plotting history in animal grouping research. For populations/species with small ranges of group sizes (ca. 1e50), using linear bins (e.g. [1,5], [6,10], [11,15].) clearly highlights the underlying distribution even for highly skewed distri- butions (see several good examples in Stacey & Koenig 1990). Often, however, group sizes range from a few individuals to several hundred or even hundreds of thousands. This makes linear bins a poor choice for detecting differences in GSFD properties across time, space or taxa, because very large groups inevitably confine most of the groups in the

smallest one/few bins. Here, we have used logarithmic bins (see

20e80% of a population can be contained in the five largest colonies of a large-colony species (e.g. a, g, q), clearly a significant range for the purposes of conservation.

1

q

0.75

f

Methods). This approach has been key to unravelling the surprising nontrivial relationship between GSFDs and IndGSFDs. This is easy to appreciate in Fig. A1, where we have plotted in standard histograms the same data as in Fig. 1, and where the difficulty of visualizing the contrasting patterns within and between GSFDs and their IndGSFDs found in Fig. 1 is apparent. Also, semilogarithmic plots are a direct way of assessing how individuals are distributed across group sizes. This is

0.5

0.25

s p

m g

i n

o

e d

t j k a

a powerful way of identifying either possible preferences of individ-

uals for particular group sizes (something difficult to appreciate from

GSFD alone) or the relevance that particular processes (e.g. high

negative density dependence in survival in large group sizes) could 0

have upon a population.

r h bc l

Individual Group Sizes versus Crowding

We have not used the term ‘crowding’ recently coined by

Reiczigel et al. (2005). ‘Crowding’ has been an interesting

Maximum group size

Figure 4. Correlation between the maximum group (colony) size of each species and the proportion of all the breeding pairs of the species found in their largest five colonies.

Modelling Implications

Theoretical approaches are aimed at understanding group size variation, but not individual group size variation. An important (and apparently trivial and obvious) starting point of animal grouping models is that, ideally, mean group size in a population should be the group size conferring the highest fitness to the individuals (reviewed in Clark & Mangel 1986). Posterior modelling approaches, however, have questioned the validity of this assumption showing, for instance, that the difference between optimal and realized mean group sizes depends on whether group members have control over the entrance of newcomers to the group (Giraldeau & Caraco 2000). However, the initial assumption (i.e. that mean group sizes should be close to optimal group sizes) has been not questioned. This could be misleading because in genetically unrelated animals (e.g. a huge seabird colony) what should be expected is not that groups should be of an optimal size, but that most of the individuals of the population should live in such optimal group sizes, that is, show an adaptive behaviour. If group size variation is low, mean population group size and mean individual group size may be essentially the same (see hypothetical example in the Introduction, Lloyd 1967), and thus approaches modelling these kinds of GSFDs remain essentially equally valid. However, our results show that mean individual group sizes could be many times larger than population mean group sizes (e.g. Fig. 1f, g, i, p, q; Table 2). Thus, the empirical finding that group size is often larger than the optimal group size (Giraldeau & Caraco

2000; Krause & Ruxton 2002; Sumpter 2010) is even more intriguing when examined from the individual point of view.

More generally, since GSFDs and IndGSFDs have been shown to yield different information, models could be tested (and their design aided, Grimm & Railsback 2005) by how well they reproduce not only mean group sizes of GSFDs but also several of their properties (e.g. skewness), as well as for their IndGSFDs. These new approaches will surely benefit from current advances in the statistical treatment of IndGSFDs (Reiczigel et al. 2008; Neuhäuser

2009; Neuhäuser et al. 2010).

Finally, fitting theoretical models (e.g. power laws or truncated power laws) to empirical data has been shown to unravel interesting factors shaping population grouping patterns (e.g. Bonabeau et al.

1999; Sjöberg et al. 2000; Lusseau et al. 2004; Jovani et al. 2008b). Our study clearly shows that contrasting results can be found if indi- vidual group sizes are studied instead of population group sizes. Therefore, where the aim of the study demands it, it would be inter- esting to make this double approach to group sizes either to comple- ment group size analyses or to gain a new perspective on the causes and consequences of group living.

Acknowledgments

This and previous work on the Seabird 2000 data set would not have been possible without the collaboration between the Joint Nature Conservation Committee and the Royal Society for the Protection of Birds, and the over 1000 volunteers that have gath- ered the data. We also thank José L. Tella, Daniel Oro, José A. Donázar, Olga Ceballos, David Serrano, Jaime Potti and Ainara Cortés-Avizanda for discussion, and David Lusseau, Steve Oswald and an anonymous referee for interesting contributions. R.J. is supported by a Ramón y Cajal research contract (RYC-2009-03967) from the Ministerio de Ciencia e Innovación.

Supplementary Material

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.anbehav.2011.07.037.

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Figure A1. Group (colony) size frequency distribution (in black; left Y axis) and corresponding individual group size frequency distribution linear histograms (in grey; right Y axis) for 20 seabird species breeding in Britain and Ireland. Grey bars are always shown in their full length, but the first group size bin (the left-hand black bar of each graph) has been cut and its real value depicted by the left-hand value inside each graph. Each graph shows from left to right, 20 linear bins ranging from the smallest to the largest group size in the species. Bins are of different length between graphs, but constant within graphs. Bins are calculated as (maximum group size/20). For instance, in (e) the maximum group size (the number above the largest bin) is 4000. Thus, in (e), bin length is 4000/20 ¼ 200; thus, the first bin is (0e200], the second bin (200e400] and the last bin (3800e4000] (‘(‘ means that the number is not included in the bin and ‘]’ that it is included). See Fig. 1 for label details.

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Figure A2. Correlation of size-related statistics describing group size frequency distributions and their corresponding individual group size frequency distributions. (a) 5th percentile, (b) median, (c) mean and (d) 95th percentile. Species codes are the same as in Fig. 1. See Table 1 for correlation statistics. Dashed lines depict the x ¼ y line.

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| | |(b) |

| | | |

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|mb| | |

|er| | |

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| |100 000 | |

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| |data were analysed (compare Fig. 3 and Fig. A2, Table 1), and raw |

|P |data were highly different between GSFD and IndGSFDs (Table 2). |

|Shape-related | | | | |

|statistics |0.112 |0.013 |0.639 | |

|Skewness | | | | |

|KolmogoroveSmirnov |0.219 |0.048 |0.354 | |

|Kurtosis |—0.388 |0.151 |0.091 | |

|Population variability |—0.392 |0.154 |0.087 | |

|Size-related statistics| | | | |

|5th percentile |0.201 |0.041 |0.395 |0.610 |0.372 | ................
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