## Introduction

Metabolic rate is a fundamental property that dictates daily requirements for individuals and therefore has consequences for biomass and nutrient flow through communities and the structure and functioning of whole ecosystems. Metabolic rate has long been recognized to vary with body mass, M (Kleiber 1932; Peters 1983; Nagy, Girard & Brown 1999), typically expressed as

The tendency represented in (eqn 1) is enormously important at population (Ernest *et al*. 2003; Savage *et al*. 2004a), community (Cyr & Pace 1993; Brose *et al*. 2006, Reuman *et al*. 2008, 2009) and ecosystem (Brown *et al*. 2004) levels.

The large majority of past work that has empirically examined the metabolic rate vs. body mass relationship has used basal or resting metabolic rates (BMR or RMR) and/or has used species-averaged estimates of metabolic rate and body mass instead of individual measurements. However, field metabolic rates (FMR) and individual mass and rate phenotypes are more directly ecologically relevant and are probably more directly subject to selection than resting rates and species-average phenotypes, respectively. BMR measures organism metabolism in a calorimeter, but organisms live and interact in the field. Species-average quantities mask variation on which evolution can act, whereas individual analyses capture this variation. Researchers who use the scaling of metabolic rate as a component of their models ultimately seek to understand the behaviour of communities and ecosystems in the field. Individual-level FMR therefore appears to be a more ecologically and evolutionarily relevant measurement to use in the development of ideas about metabolism and its scaling with body size. We therefore compiled the first comprehensive database of measurements of FMR and body mass for individual birds and mammals. We here publish the data and use it to illuminate a series of questions that have long been important topics of debate for BMR/RMR, but that have not been systematically addressed for individual-level field metabolic rates.

For many years, great controversy focussed on whether the value of *b* is closer to 2/3 or 3/4 (reviewed by White & Seymour 2005). Scaling of 2/3 is predicted from the ‘surface law’ of metabolism (White & Seymour 2005; White 2011). The surface law is based on the ratio of volume to surface area, which affects the rates at which heat is produced and lost to the environment. This theory was called into question by empirical data from mammals suggesting that *b* is close to 3/4 (Kleiber 1932), leading to the adoption of ‘Kleiber's law’ of *b* = 3/4, a value that was more recently explained by a theory based on the scaling of circulatory systems and other biological networks (West, Brown & Enquist 1997). Heusner's (1982) analysis of 173 individuals of seven mammal species allowed each species to have a different value of *a*; he found that a value of *b* = 2/3 was appropriate for each of his seven species and argued that the value *b* = 3/4 is a statistical artefact of fitting a model that allows a single value of *a*. Feldman & McMahon (1983) analysed the same data using a different formulation of the same statistical analysis and found the same values but provided a different interpretation of the results, arguing that *b* = 3/4 and *b* = 2/3 are the appropriate inter- and intraspecific values, respectively, and concluding that *b* = 3/4 is a genuine trend, not an artefact. Further empirical studies have supported *b* = 2/3 (Heusner 1982; White & Seymour 2003, 2005), while others have supported *b* = 3/4 (Feldman & McMahon 1983; Savage *et al*. 2004b; Farrell-Gray & Gotelli 2005).

More recent studies focussed on whether a single value of *b* is even appropriate for all clades, and how *b* varies by clade. Such studies often account for nonindependence in the data resulting from shared evolutionary history. White, Phillips & Seymour (2006) examined basal rates of fish, amphibians, reptiles, birds and mammals and found significant heterogeneity in *b* among these groups. Capellini, Venditti & Barton (2010) investigated mammalian BMR and FMR and found wide variation in *b* among clades, with some having 3/4, some 2/3 and some significantly different from both values. Isaac & Carbone (2010) quantified the magnitude of variation in *b* for BMR at different taxonomic levels for a range of animals, finding a mean value of *b* close to 3/4 but large variation at the order level, with 5% of orders lying outside the range 0·54−0·95 and only small amounts of variation at the family and class levels. Analyses of mammalian and avian BMR (McNab 2008, 2009) have shown that phylogeny and various ecological factors can lead to variation in *b* between clades and found, once these factors had been accounted for, values of *b* = 0·694 for mammals (McNab 2008) and b = 0·689 for birds (McNab 2009). Glazier's (2005) meta-analysis of metabolic scaling within species, which was based on individual BMR/RMR data, revealed that ontogenetic scaling relationships are variable, often approaching isometry (*b* = 1) and sometimes appearing nonlinear (see also Killen *et al*. 2007; Moran & Wells 2007; Streicher, Cox & Birchard 2012). Individual-level analyses examining both the intra- and interspecific relationships in insects (Riveros & Enquist 2011) and terrestrial invertebrates (Ehnes, Rall & Brose 2011) have revealed large variation in *b*. Analysis of maximum metabolic rate data from mammals revealed *b*≈7/8 (White & Seymour 2005; Gillooly & Allen 2007; White *et al*. 2008), a value potentially explained by at least two recent competing theories (Glazier, 2005, 2008, 2010; Gillooly & Allen 2007). These studies illustrate the volume of research that has examined taxonomic heterogeneity of scaling coefficients, *b*, for data that has been on basal or resting rates or has been for species averages.

Of the much smaller collection of empirical studies that have investigated body mass dependence of FMR, all but one have used species-averaged data. These studies have found that *b* is close to 2/3 for birds, close to 3/4 for mammals and close to 8/9 for reptiles (Nagy, Girard & Brown 1999; Savage *et al*.'s 2004b; Anderson & Jetz 2005; Nagy 2005). Nagy (2005) reported that FMR scaling was steeper than BMR scaling for both birds and mammals, although the differences were small and not statistically significant. Anderson & Jetz (2005) argued that FMR has an upper limit determined by physiology and a minimum requirement driven by environmental factors. Capellini, Venditti & Barton (2010) phylogenetically informed investigation into mammalian FMR found that *b* was not statistically different from 2/3 for their data when considered as a whole but that different orders had confidence intervals that include both, one or none of the values 2/3 and 3/4. Speakman & Król (2010) performed both conventional and phylogenetic analyses of species-average FMR of endotherms and found values of *b* not significantly different from *b* = 0·63, the value predicted by their heat dissipation limit theory. The studies surveyed here serve to illustrate the prior work that has examined mass dependence of FMR, albeit for species-averaged data. Riek's (2008) is the only study we are aware of to analyse individual-level FMR. This study argued for the importance of including a random effect of study in statistical models, showing that a linear regression model and a mixed-effects model can give different estimates of *b* = 3/4 and *b* = 2/3 respectively (Riek 2008).

A gap in the existing literature is a comprehensive analysis of individual-level FMR data. Within-species scaling of FMR is of interest in its own right, but incorporating this variation into scaling models across species is also likely to be more robust than if it were simply treated as error variance, as in conventional analyses. We compiled the first comprehensive database of measurements of FMR and body mass for individual birds and mammals. We here publish our data and use it to answer four questions. First, what is the magnitude of variation in the exponent *b* among taxa, and at what taxonomic level does variation primarily occur when intraspecific variation is considered alongside variation among species and higher taxa? Second, after accounting for such variation, what are the mean scaling exponents for birds and mammals? Are the mean exponents for each class different from each other and are they closer to 2/3 or 3/4? Third, how does the extent of taxonomic variation in *b* compare to the magnitude of the difference between 2/3 and 3/4, and between the mean exponents for birds and mammals? Finally, what are the implications of our data for existing theory on metabolic rate scaling? These questions have been important in debates centred on species-averaged BMR data, but have not been systematically addressed for individual-level FMR data.

Based on earlier work using species-averaged FMR (Nagy, Girard & Brown 1999; Anderson & Jetz 2005; Nagy 2005; Capellini, Venditti & Barton 2010; Speakman & Król 2010), we posit the null hypothesis that taxonomic variance in *b* will be statistically meaningful and substantial relative to 3/4−2/3 = 1/12 and relative to the difference between bird and mammal mean slopes. As found by Isaac & Carbone (2010) for RMR, we hypothesize that variation will be more important at the order level of taxonomy than the family level. Based on earlier work using individual RMR (Glazier 2005), we posit the null hypothesis that species-level variation will also be important and comparable to 1/12. In testing the hypotheses that mean *b* is 2/3 or 3/4, we provide tests of the surface law of metabolism (White & Seymour 2005) and of modern theories predicting central tendency values of *b* ≈ 2/3 (Speakman & Król 2010) and *b* ≈ 3/4 (West, Brown & Enquist 1997; Banavar *et al*. 2002; Darveau *et al*. 2002; Ginzburg & Damuth 2008). In examining taxonomic heterogeneity in *b*, we provide tests of modern theories making predictions about variation (Kozłstrokowski, Konarzewski & Gawelczyk 2003; Glazier 2005, 2008; Savage, Deeds & Fontana 2008; Glazier 2010; Kolokotrones *et al*. 2010; Agutter & Tuszynski 2011). More broadly than testing some of the existing theories, this study provides the first comprehensive data set and systematic description of the individual-level FMR-vs.-body mass relationship for birds and mammals.