The relationship between body mass and field metabolic rate among individual birds and mammals

Summary The power-law dependence of metabolic rate on body mass has major implications at every level of ecological organization. However, the overwhelming majority of studies examining this relationship have used basal or resting metabolic rates, and/or have used data consisting of species-averaged masses and metabolic rates. Field metabolic rates are more ecologically relevant and are probably more directly subject to natural selection than basal rates. Individual rates might be more important than species-average rates in determining the outcome of ecological interactions, and hence selection. We here provide the first comprehensive database of published field metabolic rates and body masses of individual birds and mammals, containing measurements of 1498 animals of 133 species in 28 orders. We used linear mixed-effects models to answer questions about the body mass scaling of metabolic rate and its taxonomic universality/heterogeneity that have become classic areas of controversy. Our statistical approach allows mean scaling exponents and taxonomic heterogeneity in scaling to be analysed in a unified way while simultaneously accounting for nonindependence in the data due to shared evolutionary history of related species. The mean power-law scaling exponents of metabolic rate vs. body mass relationships were 0·71 [95% confidence intervals (CI) 0·625–0·795] for birds and 0·64 (95% CI 0·564–0·716) for mammals. However, these central tendencies obscured meaningful taxonomic heterogeneity in scaling exponents. The primary taxonomic level at which heterogeneity occurred was the order level. Substantial heterogeneity also occurred at the species level, a fact that cannot be revealed by species-averaged data sets used in prior work. Variability in scaling exponents at both order and species levels was comparable to or exceeded the differences 3/4−2/3 = 1/12 and 0·71−0·64. Results are interpreted in the light of a variety of existing theories. In particular, results are consistent with the heat dissipation theory of Speakman & Król (2010) and provided some support for the metabolic levels boundary hypothesis of Glazier (2010). Our analysis provides the first comprehensive empirical analysis of the scaling relationship between field metabolic rate and body mass in individual birds and mammals. Our data set is a valuable contribution to those interested in theories of the allometry of metabolic rates. The authors provide the first comprehensive empirical analysis of the scaling relationship between field metabolic rate and body mass in individual birds and mammals. The analysis reveals the importance of heterogeneity in the scaling exponent, with consequences for biomass and nutrient flow through communities, and the structure and functioning of whole ecosystems.


body mass, M, measured;
2. field metabolic rate, FMR, measured in the field or in field-like captivity, using the doubly labelled water method; 3. M and FMR presented for individual animals.
We omitted data when body mass was estimated rather than measured (e.g. some data from Costa & Prince 1987). Data presented graphically was digitised using DigitizeIt 1.5 (Digital River 2011).
The doubly labelled water method estimates volume CO 2 exhaled, which can be converted to energy by multiplying by an energy equivalent, the value of which is diet dependent (Butler et al. 2004). Bryant (1997) studied Turdus merula, Muscicapa striata and Parus major and presented just volume CO 2 exhaled. We used an energy equivalent value of 26.8 J/cm 3 CO 2 , taken from Ricklefs & Williams (1984), a study of Sturnus vulgaris, which has a similar diet to the three species investigated by Bryant (1997). Utter & LeFebvre (1973) used two plausible energy equivalent values for their study of Progne subis, resulting in two different estimates of FMR. We took the mean of these two estimates.
The same measurements of three individuals of Calidris minuta are presented by both Piersma et al. (2003) and Tulp et al. (2009). We included these data only once. Williams (1987) measured FMR of Passerculus sandwichensis in 1981and 1982. The presentation of the data is not completely clear with regard to whether the same individuals were measured in both years. In addition to this, Williams (1987) classified individuals as either being territorial, incubating eggs or rearing young and it is possible that individuals were measured performing more than one activity in the same year.
1 To avoid including pseudo-replicated data, we took data only for those individuals that were rearing young in 1982. Similarly, Williams (1988) studied Tachycineta bicolor engaged in a range of activities in 1981 and 1982; again, we took data only for individuals rearing young in 1982.
We took taxonomy for mammals from Wilson & Reeder (2005) and for birds from Dickinson (2003).
We ignored the sub-species level of taxonomy. Pontzer et al. (2010)

Appendix S2. Main models
The 27 mixed-effects models (discussed in 'The main set of models', in the main text) had fixed effects of taxonomic class on both slope and intercept. The most complex, 'global' model had random effects of order, family and species on both slope and intercept: FMR ijkl and M ijkl are the FMR and M of the l th individual in the k th species, j th family and i th order.
As described in the main text, the log 10 (M ijkl ) have been centred by subtraction of the grand mean.
c, o, f and b are taxonomic levels: class, order, family and species. t is the study from which the data point was taken. a c and b c are fixed effects of class on intercept and slope respectively. Fixed effects are estimated for each class in the data so the model has four fixed-effects parameters: a c = a Aves , a c = a Mammalia , b c = b Aves and b c = b Mammalia . α co , α cof and α cof b are random effects on intercept of order, family and species. α t is the random effect on intercept of the study from which the data point was taken. Likewise, β co , β cof , β cof b and β t are random effects on slope for the same taxonomic levels and for the study.
Random effects that act on just intercept are characterised by a normal distribution with mean of zero and a variance that is estimated during model fitting. Random effects that act on both slope and intercept, as do all in this global model, are characterised by bivariate normal distributions; the correlation between slope and intercept is also estimated. Σ co , Σ cof , Σ cof b and Σ t are symmetric, 2x2 covariance matrices and each contains three parameters. Therefore when fitting the global model in equation (S1), 17 parameters are estimated:

Appendix S3. Within-group-centred models
To test for systematic variation in slope at different taxonomic levels ('Supporting analyses' in the main text), we formulated 27 mixed-effects models that employed the within-group-centring method described by van de Pol & Wright (2009). This test required a model formulation that used several predictors. For brevity, we define where i, j, k, l are order, family, species and individual. From these values, we computed the mean log 10 masses at each taxonomic level. Each set of means was computed as an unweighted mean of the set of means at the level of the next higher taxonomic resolution. For instance, x ijk was defined as the mean log 10 mass of the k th species in the j th family, in the i th order, computed as the unweighted mean of x ijkl as the index l varies and the other indices are fixed. From the species-level means, we computed x ij , the mean log 10 mass of each of the families. Lastly, we computed x i , the mean masses of each of the orders, computed as the unweighted mean of the family means.
We formulated a model that uses the hierarchically computed means as predictors, allowing a 3 separate mean slope at each taxonomic level: This model has four fixed effects of class on slope: We compared the global model of equation (S4) to our main global model (equation (S1) in ) by fitting 27 models for each of these two equations, with random effects structures as described in the main text. As equations (S1) and (S4) have different fixed-effects structures, we fitted all these models using maximum likelihood. Results are in Table S1. Results demonstrate that the data do not display systematic variation of slope at lower taxonomic levels and that our main set of models derived from equation (S1) are appropriate for the data.
We here show that the fixed-effect structure of (S1) is a simplification of that of equation (S4).
The fixed-effects structure of equation (S4) is Assuming that all fixed effects of class are equal Cancelling, which is the fixed effects structure of equation (S1).     .92 for the two-intercept model. For the two-intercept models, intercepts for aquatic animals were higher. There was a fairly clear-cut distinction between aquatic and non-aquatic mammals in the data set, but not so for birds. Therefore, only penguins were considered aquatic birds for the plots and analyses shown here, but we tried a variety of other ways of defining aquatic versus non-aquatic for birds, spanning a range of permissiveness in the definition of "aquatic". Results were substantially the same in all cases. Our data Dunning, 2008Smith et al, 2003Nagy et al, 1999 Fig. S9: Distributions of body masses. Species-averaged body masses in our data together with the avian body mass data of Dunning (2008) with additions of Meiri, Raia & Phillimore (2011) and the mammalian body mass data of Smith et al. (2003), the best currently available databases of body masses of birds and mammals respectively. For comparison, we include the data used by Nagy, Girard & Brown (1999). Fig. S10: Sampling locations. Each circle represents a population that was measured. The area of the circles is proportional to the number of individuals sampled. The legend shows the area for 10 individuals.