Accounting for body mass effects in the estimation of field metabolic rates from body acceleration

Estimating the metabolic rates of animals in the laboratory is a well-established practice using either direct or indirect calorimetry; however, estimating field metabolic rate (FMR) is more challenging. Commonly used methods to estimate FMR include measuring CO₂ production via doubly labelled water (DLW; Speakman, 1997), and measuring heart rate (f_H) or dynamic body acceleration (DBA) as a proxy for oxygen consumption rate (Ṁ_O2). Of these, the DBA technique has come to the fore because of its wide taxonomic applicability, high temporal resolution (in contrast to DLW; Butler et al., 2004) and the logistical simplicity of attaching acceleration data-loggers, as opposed to the invasive surgery required for implantation of heart-rate loggers (Green et al., 2009). Of course, the DBA technique does have its own limitations, such as difficulties measuring the effects of digestion on energy use (reviewed in Wilson et al., 2020; but see Gleiss et al., 2011). Nonetheless, before using DBA to predict FMR, laboratory calibrations are necessary to establish predictive models that relate DBA to Ṁ_O2 (Gleiss et al., 2011; Wilson et al., 2006). Such calibrations have been conducted for numerous vertebrate and invertebrate species and show that a linear relationship between Ṁ_O2 and DBA holds across all taxa, where the intercept constitutes the baseline (i.e. basal or standard) metabolic rate (BMR) and the slope indicates how Ṁ_O2 changes with activity (Wilson et al., 2020). After appropriate calibrations, predictive models can then be applied to field-measured DBA data to predict the FMR of free-ranging animals.

To discern the utility of any given FMR estimation method, it is necessary to test its sensitivity to factors that are expected to influence Ṁ_O2 and to incorporate appropriate corrections into estimations. For instance, temperature and body size are considered the most important factors that influence metabolism, among other factors (e.g. age, sexual maturity, specific dynamic action), with both displaying positive exponential effects on Ṁ_O2 in ectotherms (Clarke and Johnston, 1999; Gillooly et al., 2001). As temperature is easily manipulated under laboratory conditions, the temperature dependence of the Ṁ_O2–DBA relationship has been tested by numerous studies. As such, it is known that the intercept of the Ṁ_O2–DBA relationship changes with temperature according to a van't Hoff or Arrhenius relationship, while the slope is unaffected (e.g. Lear et al., 2017; Lyons et al., 2013). Accordingly, temperature variation can be accounted for by incorporating a temperature-dependent intercept term into predictive Ṁ_O2 models. Although the level of temperature dependence varies between species, this type of correction has proven to be effective across ectothermic taxa, including fish (Lear et al., 2017, 2020; Wright et al., 2014), reptiles (Enstipp et al., 2011), crustaceans (Lyons et al., 2013) and molluscs (Robson et al., 2016).

While there is a reasonably well-defined hypothesis for the allometry of BMR, the kinematics that define animal movement and acceleration, and therefore govern the slope of the Ṁ_O2–DBA relationship, adhere to a different set of allometric laws. For example, physiological forces that govern variations in BMR could relate to factors such as fractal oxygen-transport networks or body surface area to volume ratios, which control the supply of metabolic substrates, and scale at allometric rates near 0.75 (West et al., 1997) and 0.67 (Kozłowski et al., 2003; West and Brown, 2005), respectively. In contrast, DBA is a product of body kinematics, such as stride frequency and amplitude, which scale at allometric rates of approximately −0.33 (Bale et al., 2014) and 0.33 (Videler, 1993), respectively. Furthermore, the fact that active metabolism (i.e. routine and maximum metabolic rate) has been found to scale at a different rate than BMR in all taxa (Bishop, 1999; Killen et al., 2007) suggests that the slope of the Ṁ_O2–DBA relationship, which is representative of energy expenditure due to activity, follows a fundamentally different allometric relationship than BMR. Nonetheless, current methods account for the effects of body mass by applying either isometric (Bouyoucos et al., 2017; O₂ kg⁻¹, e.g. Payne et al., 2011) or allometric (O₂ kg^−b, e.g. Enstipp et al., 2011; Lear et al., 2020) mass corrections to Ṁ_O2 estimates, where b is a species-specific BMR scaling exponent. Alternatively, some studies have accounted for effects of mass by simply including body size as a continuous covariate (with no interactions) when modelling the influence of DBA on Ṁ_O2 (e.g. Brownscombe et al., 2017). The former approach assumes an identical Ṁ_O2–DBA slope and intercept mass scaling, whereas the latter approach assumes no effect of mass on the Ṁ_O2–DBA slope. However, if the Ṁ_O2–DBA slope was to scale allometrically at a different rate than BMR, it is likely that these methods introduce significant bias into Ṁ_O2 estimates.

The present study set out to test the effect of mass on both the slope and intercept of the intraspecific Ṁ_O2–DBA relationship and to assess the accuracy of different approaches for incorporating mass effects into Ṁ_O2 predictive models for a species. To do this, we conducted respirometry experiments on individuals spanning one order of magnitude in body mass (1.74–17.15 kg), using lemon sharks (Negaprion brevirostris) as a model species. We then used a two-stage modelling process to test for independent mass scaling of the intercept and slope of the Ṁ_O2–DBA relationship. We incorporated the established mass-scaling effects into the coefficients of Ṁ_O2 predictive models and compared this model against other commonly used modelling approaches through a series of simulations that allowed assessment of the biases and error associated with each model.

Respirometry experiments were conducted on two groups of lemon sharks [Negaprion brevirostris (Poey 1868)]. The first group consisted of individuals <3 kg in mass [n=16, 69.5–86.0 cm total length (TL)], captured off Florida, USA, and housed at Mote Marine Laboratory (MML) in Sarasota, FL, USA, for the duration of experiments. Further capture and maintenance details for MML individuals are provided in Lear et al. (2017). The second group consisted of individuals >3 kg in mass (n=5, 107.0–154.0 cm TL), captured and housed in semi-captive pens off South Bimini, Bahamas, near Bimini Biological Field Station (BBFS), where they were subjected to natural environmental conditions (e.g. temperature, light and salinity). All sharks were fasted for at least 48 h prior to experiments to ensure a post-absorptive state and allow recovery from capture stress.

Research was conducted under permits from the Bahamas Department of Marine Resources (MA&MR/FIS/178), Murdoch University Animal Ethics (RW3119/19) and Mote Marine Laboratory Institutional Animal Care and Use Committee (09-09-NW1).

For MML sharks, respirometry was conducted between 2015 and 2016 as part of a separate study (Lear et al., 2017). A closed, annular, static respirometry system was constructed from a modified 2.45 m diameter fibreglass holding tank, as described in Whitney et al. (2016). Sharks were acclimated to the system for 12 h prior to trials. Trials began near 100% air saturation and were run until dissolved oxygen (DO) levels reached 80% air saturation. DO (% air saturation and mg l⁻¹) and water temperature (°C) were measured by a handheld multiparameter meter (Pro Plus, Yellow Springs Instruments, Yellow Springs, OH, USA) and recorded by researchers every 5 min throughout trials. To assess background respiration, a blank respirometer was measured for 4 h following each set of trials. For full details of the MML trial protocol, see Lear et al. (2017).

BBFS trials were conducted using a field-based respirometry system, similar to Byrnes et al. (2020). Two closed, annular, static respirometry systems were constructed from modified polyvinyl-lined metal-frame pools (Bestway Corp., London, UK), which were erected on a levelled section of beach on South Bimini, Bahamas. A 2.44 m diameter pool was used for sharks <110 cm TL, whereas a 3.66 m diameter pool was used for sharks >110 cm TL. Sharks were hand-netted from pens, measured and weighed, and transferred into the system 8 h prior to trials to allow them to acclimate. Trials began with air saturation between 80 and 110% and were run until saturation decreased by 20 percentage points or reached 70% saturation, whichever occurred first. Throughout trials, DO (% air saturation and mg l⁻¹) and water temperature (°C) were measured and recorded by a multiparameter meter (HI98196, Hanna Instruments, RI, USA) every 30 s. To assess background respiration, a blank respirometer was measured for at least 90 min immediately following the final trial for each shark.

Throughout all MML and BBFS trials, sharks were equipped with a Cefas G6a+ acceleration data logger (Cefas Inc., Lowestoft, UK), mounted through the base of the first dorsal fin as per Lear et al. (2017), which continuously recorded triaxial acceleration at 25 Hz (range: 2 g, resolution: 12-bit).

Vectorial dynamic body acceleration (VeDBA), a measurement of physical activity, was calculated from acceleration data recorded during respirometry. VeDBA was chosen as the acceleration metric for modelling instead of the commonly used overall dynamic body acceleration to allow predictive models to be applied to data collected by acceleration transmitters (see www.innovasea.com/fish-tacking/ for various applications), which process acceleration onboard as VeDBA. Additionally, VeDBA produces more robust estimations of activity when tag orientation may differ between applications, such as external mounting of loggers versus internal implantation of transmitters (Qasem et al., 2012).

where A_Xt, A_Yt and A_Zt are the raw acceleration values and G_Xt, G_Yt and G_Zt are the gravitational acceleration values observed for each axis at time t. Acceleration data were processed using Igor Pro (Version 7.08; Wavemetrics, Lake Oswego, OR, USA) and the Ethographer extension (Sakamoto et al., 2009).

Prior to building predictive models, VeDBA estimates were corrected for measurement error (i.e. acceleration irrespective of body movement), to ensure that the model intercept represented SMR of sharks. As previously mentioned, the intercept of the Ṁ_O2–DBA relationship represents zero movement (i.e. maintenance metabolic rate; Wilson et al., 2020); however, owing to factors causing measurement error (e.g. sensor noise, water movement; Gleiss et al., 2010; Whitney et al., 2010), Ṁ_O2 observations during rest tend to be greater than 0g. To account for this error, many studies have interpolated to zero movement to estimate SMR (e.g. Lear et al., 2017; Wright et al., 2014); however, such an interpolation likely underestimates SMR. Therefore, to account for measurement error, we rescaled all observations by subtracting the overall mean VeDBA of inactive intervals. Note that while baselining VeDBA in this manner may result in some minorly negative values, this correction was necessary to ensure the intercept was representative of SMR of sharks.

Owing to sharks frequently switching between inactive and active behaviours in MML trials, some sharks only provided one to two Ṁ_O2whole observations, precluding the ability to produce a reliable linear fit for each individual body mass (Table 1). To achieve a necessary number of Ṁ_O2whole estimations for regressions, MML sharks were grouped into 0.4 kg mass classes (1.60–1.9\dot{9}, 2.00–2.3\dot{9}, 2.40–2.7\dot{9} and 2.80–3.1\dot{9} kg). The midrange of a class was used as the mass for all sharks within the class during modelling. The BBFS sharks, in contrast, showed a greater range of activity within individual trials and had a larger range in body size, and were therefore kept as individuals rather than grouped into mass classes. We note that this grouping of individuals into mass classes precluded inclusion of ID as a random effect in models, which may have resulted in some degree of pseudoreplication in models, where there are multiple data points per shark.

As this equation incorporates mass effects into both the intercept and slope regression coefficients, it will hereafter be referred to as the ‘coefficient-corrected approach’.

To compare the accuracy of the coefficient-corrected approach, with other approaches used throughout the literature, we fit additional predictive equations based on methods commonly used throughout literature (see ‘Response-corrected approach’ and ‘Intercept-corrected approach’ subsections below). As previously mentioned, two other modelling approaches have been used to incorporate mass effects into Ṁ_O2 predictive equations. The more commonly applied approach uses mass-specific Ṁ_O2 (mg O₂ kg^−b h⁻¹) when modelling the Ṁ_O2–DBA relationship, rather than modelling Ṁ_O2whole (mg O₂ h⁻¹) (e.g. Enstipp et al., 2011; Lear et al., 2017). By using mass-specific Ṁ_O2, these methods incorporate mass effects into the response variable of predictive equations, as opposed to directly incorporating mass effects into the coefficients of predictive models. Accordingly, this approach will henceforth be referred to as ‘response-corrected approach’. The second modelling approach directly models Ṁ_O2whole as a function of body mass by including mass as a covariate within model formulations. However, unlike the coefficient-corrected approach here, previous applications have not considered interactive effects between mass and other covariates (e.g. Brownscombe et al., 2017), and therefore only incorporated an effect of mass on the intercept of the Ṁ_O2–DBA regression. Accordingly, this approach will henceforth be referred to as the ‘intercept-corrected approach’.

Two models were fitted using the response-corrected approach. For the first model, Ṁ_O2whole estimates were isometrically corrected by dividing Ṁ_O2whole by animal mass in kg (M), resulting in mass-specific Ṁ_O2 in mg O₂ kg⁻¹ h⁻¹. For the second model, Ṁ_O2whole estimates were allometrically corrected by dividing Ṁ_O2whole by animal mass in kg raised to an allometric BMR scaling exponent (M^b), resulting in mass-specific Ṁ_O2 in mg O₂ kg^−b h⁻¹. For this study, an allometric exponent of 0.86 was used, a value widely regarded as the universal elasmobranch SMR mass-scaling exponent (Sims, 2000). Models were fitted using the glm function in R, assuming a Gaussian error structure, with mass class included as a continuous covariate. Mass was not considered as a categorical variable for these models, as these methods are based on the most common modelling approaches used in calibrations of DBA data for predicting FMR of animals and were used for comparison with the coefficient-corrected approach. These two response-corrected models will hereafter be referred to as the ‘isometric response-corrected model’ and the ‘allometric response-corrected model’.

To allow comparison with models that estimate Ṁ_O2whole, mass-specific estimates from the isometric response-corrected model and allometric response-corrected model were corrected post hoc to Ṁ_O2whole by multiplying estimates by M and M^0.86, respectively.

A single model was fitted using the intercept-corrected approach. To do so, we used a linear mixed-effects model (lme4 package), which included VeDBA and mass class as continuous covariates to explain variation in Ṁ_O2whole. It is important to note that treating mass as a continuous covariate assumes mass has an isometric effect on the intercept of the modelling relationship, which may not accurately represent the species' specific BMR allometric scaling rate. Therefore, treating mass class as a categorical variable may produce a more accurate estimation of Ṁ_O2whole. However, treatment of mass as a categorical variable was not considered here because this approach was strictly based on the approaches of previous calibration studies and was used only for comparison with the coefficient-corrected approach.

To assess prediction error in more detail, we examined how prediction error changed as a function of body mass and activity level by repeatedly simulating a day in the life of 100 sharks per mass class, similar to Green et al. (2009). A sample size of 100 sharks per mass class was used to reduce likelihood of type II errors. In these simulations, bootstrapping was used to simulate an observation every 15 min over a 24-h period for each shark, resulting in 96 observations per shark. Respirometry data were split into two resampling pools, separating inactive (intervals where sharks rested) and active (i.e. intervals where sharks swam) data. Simulated observations were drawn at random with replacement (i.e. bootstrapped) from these resampling data pools. Inactive simulated observations were created by bootstrapping from the inactive data pool, whereas active simulated observations were created by bootstrapping from the active data pool. Ṁ_O2whole was predicted for all simulated observations based on the predictive equations from each of the modelling approaches, and the mean RMS (Eqn 11) and mean COV (Eqn 12) of each model were calculated individually for all body mass classes. This process was repeated for each shark, varying the amount of time sharks spent active between 0 and 100%, increasing by one percentage point for each successive iteration. The 12.5 and 14.65 kg mass classes were excluded from the simulation, as inactive and active data were not both available to bootstrap simulated datasets for these classes (Table 1).

All statistical analysis was carried out in R v3.6.3 (https://www.r-project.org/).

A total of 43 respirometry trials (mean±s.d.=1.95±0.88 per individual fish; mean duration=21.14±10.77 min; all means are presented ±s.d. unless otherwise indicated) were conducted on 21 individuals, ranging in body mass from 1.74 to 17.15 kg (mean=4.81±4.55 kg; Table 1). Once grouped by mass, a total of 10 individual mass classes were retained (Table 1). From these trials, 129 Ṁ_O2whole estimations (mean per class=12.90±10.23) were obtained for calibrations. However, five Ṁ_O2whole observations from Bimini sharks were deemed as biologically impossible because they were twice as high as other estimates with similar VeDBA. While it is unclear what caused such erroneous values, these may have been associated with short power outages, unnoticed by personnel, causing insufficient water flow over the DO probe. Nonetheless, these observations were removed from analysis, leaving 124 observations (mean per class=12.40±10.21), including 37 inactive (mean per class=3.70±5.85) and 87 active observations (mean per class=8.70±8.66) (Table 1). From these 124 observations, Ṁ_O2whole ranged from 286.57 to 6493.68 mg O₂ h⁻¹ (Table 1). After correction for acceleration sensor noise (0.018g; representing a mean 14.7% of active interval VeDBA), VeDBA ranged from −0.014 to 0.172g (Fig. 3). Water temperature during calibration intervals ranged from 27.32 to 31.42°C (mean=29.52±0.81°C) (Table 1).

The final predictive model explained 97.91% of the variance (marginal R²) in Ṁ_O2whole. Overall, the model tended to overestimate Ṁ_O2whole, with a mean error (COV) of 19.54% (RMS=329.99 mg O₂ h⁻¹). When only considering active data, the model had lower error (COV=17.07%, RMS=376.41 mg O₂ h⁻¹) than when only considering inactive data (COV=31.65%, RMS=78.34 mg O₂ h⁻¹).

where M is individual body mass in kg. Overall, the intercept-corrected model had a mean error of 22.47% (RMS=461.89 mg O₂ h⁻¹). Notably, the model tended to have substantially greater error for lower activity levels than higher activity levels across all body sizes (Fig. 3). When only considering active data, the model had lower error (COV=16.67%, RMS=461.11 mg O₂ h⁻¹) than when only considering inactive data (COV=85.12%, RMS=464.33 mg O₂ h⁻¹).

All models predicted similar Ṁ_O2whole for individuals near the midrange of masses; however, there was greater variation in model estimates for the larger and smaller masses (Fig. 4). For sharks smaller than 7.65 kg, the intercept-corrected model produced the lowest Ṁ_O2whole estimates at lower activity levels; however, this shifted as activity level increased, with the coefficient-corrected model producing the lowest estimates when animals were active more than 35% of the time. In contrast, for sharks larger than 7.65 kg, the intercept-corrected model produced the highest Ṁ_O2whole estimates across all activity levels and the allometric response-corrected model produced the lowest Ṁ_O2whole estimates (Fig. 4, Fig. S2).

Overall, the coefficient-corrected model consistently had the lowest error across all body sizes and activity levels. The models with the highest error varied by body mass, with the intercept-corrected model generally having the highest error for smaller body sizes and the allometric response-corrected model having the highest error for larger body sizes (Fig. 4). The simulation revealed systematic mass biases in the isometric response-corrected model, the allometric response-corrected model and the intercept-corrected model. Estimation error of the response-corrected models tended to increase with increased body mass. In contrast, estimation error of the intercept-corrected model tended to decrease with body mass (Fig. 4, Fig. S2). However, no mass-associated bias was present in the estimates of the coefficient-corrected model (Fig. 4, Fig. S2). Differences in COV between models were smallest for the middle mass classes but varied more widely between models as a function of activity at the lower and higher masses (Fig. 4). At lower masses, the difference between model COV increased with increased time spent active, whereas the difference between model COV at larger masses was relatively consistent (Fig. 4). Overall, model COV tended to decrease with increased time spent active, particularly in the coefficient-corrected model (Fig. 4).

Despite the prominent influence that body size exerts on many aspects of animal biomechanics and physiology, no study has assessed the allometric dependence of the relationship between Ṁ_O2 and DBA and its effect on the accuracy of Ṁ_O2 predictive equations. By individually assessing the scaling of the Ṁ_O2whole–VeDBA slope and intercept, we validated that DBA per unit metabolism is scale dependent, and this dependency varies from that of BMR. As such, we found that directly accounting for different body-mass scaling effects within model slope and intercept coefficients produces substantially more accurate estimates of oxygen consumption rates than other commonly applied modelling approaches. While all models performed similarly for individuals near the mean mass of calibrated animals, more commonly applied intercept-corrected and response-corrected approaches appeared to have substantial mass- and activity-associated biases that were not present within the coefficient-corrected approach. As a result, the coefficient-corrected approach consistently had the lowest prediction error across all body sizes (Fig. 4). Additionally, as activity levels of animals increased, the disparity in estimation error between the coefficient-corrected and response-corrected models tended to increase, whereas the disparity in estimation error between coefficient-corrected and intercept-corrected models tended to decrease. However, the covariate-corrected approach demonstrated the smallest activity bias, and again outperformed other modelling approaches across activity levels.

The differences in estimation error between modelling approaches could have large implications when calibrated predictive equations are applied to predict energy expenditure of animals in the wild. For example, for a 17.15 kg lemon shark that is active 50% of the time, the difference between the daily energy expenditure estimated by the isometric response-corrected model and coefficient-corrected model would be 17,114.83 mg O₂ day⁻¹. This difference converts to 232.591 kJ day⁻¹ (Jobling, 1995), approximately one yellow fin mojarra (Gerres cinereus), the primary prey of juvenile lemon sharks in Bimini (Pettitt-Wade et al., 2011), or a 25% increase in the daily energetic requirements. This difference would be proportionally even larger for smaller body sizes, where the disparity between model estimation error was greater. As a result, with such improvements over other modelling approaches for incorporating mass effects into Ṁ_O2 estimations, we strongly recommend the coefficient-corrected approach for establishing Ṁ_O2 predictive equations across different body sizes.

Although the coefficient-corrected approach provided more robust estimates of energy expenditure than currently used approaches, several opportunities for improvement remain. Foremost, although conducted pre hoc in this study, an ideal full model should incorporate a temperature correction factor in the intercept term. Additionally, more balanced sampling across activity levels and mass classes would allow for a mechanistic examination of error accumulation, which may help to identify and incorporate further correction factors into models. Of course, conducting respirometry experiments over a range of body sizes presents logistical and ethical complexities that may limit sample size, as was the case in this study. However, in respirometers with small system to animal volume ratios, which facilitate rapid respiration measurement response times (Clark et al., 2013), calibration interval durations could be decreased to facilitate larger sample sizes per trial. Additionally, when ethically viable, animals could be left in the respirometer for longer acclimation periods or more trials, potentially facilitating greater variability in behaviours as animals become more comfortable with the system. While more calibration intervals within trials may also increase variability of sampled activity levels, using forced activity protocols (e.g. swim tunnels or treadmills) may ensure that observations are obtained across all activity levels, although it is of note that such protocols may introduce other bias through alteration of natural gaits in many species (Lear et al., 2019; Whitney et al., 2018).

As the mechanics and external forces defining the Ṁ_O2–DBA relationship depend on numerous factors such as body form (Alexander, 2005; Bale et al., 2014), locomotory type (Alexander, 2005; Schmidt-Nielsen, 1972) and viscosity of the environmental medium (Schmidt-Nielsen, 1972; Wieser and Kaufmann, 1998), Ṁ_O2–DBA slope and intercept scaling exponents will inevitably vary between taxa. However, when corrected for body mass, the Ṁ_O2–DBA relationship may be similar across geometrically similar species, potentially allowing for common equations to be established for estimating energy expenditure from DBA (Gleiss et al., 2011). In fact, Halsey et al. (2009) and Miwa et al. (2015) both established common predictive equations for estimating the energy expenditure across a range of species, although the accuracy of estimations using these interspecifically derived equations was not assessed. Nonetheless, the establishment of common Ṁ_O2 predictive equations suggests that fundamental laws of biomechanics and physics may constrain the ratio of the Ṁ_O2–DBA slope to intercept scaling exponents to universal values. Further calibration experiments conducted with various taxa will determine the scatter in intraspecific scaling exponents and may help to identify such biomechanical and physical laws governing the Ṁ_O2–DBA relationship.

In circumstances in which respiration measurements cannot be acquired from a range of body masses (e.g. facility or animal availability limitations), and thus response-corrected models or intercept-corrected models must be extrapolated, the biases of such models must be carefully considered. Foremost, bias drastically increased with increasing difference in body mass between the median mass of individuals in the calibration experiments and individuals for which respiration rate was being estimated. Thus, to minimize estimation error, caution must be exercised when extrapolating equations to animals with substantially different body masses than those used in calibration experiments. Additionally, models demonstrated large activity bias, where the estimation error drastically changed as a function of increased activity, particularly for smaller body masses. However, the direction of the bias was opposite between models, where estimation error of the intercept-corrected model decreased with increased activity and estimation error of response-corrected models tended to increase with activity. This activity bias may partially be a product of the relatively larger spread of Ṁ_O2 and VeDBA measurements during active calibration intervals compared with inactive intervals. Using forced activity protocols to ensure more balanced sampling of different activity levels throughout calibration experiments would help to elucidate the source of this error. Nevertheless, this activity level associated bias indicates that as animals become more active, post hoc corrections based on isometric or allometric BMR scaling rates introduce increased error. In contrast, estimation error of the intercept-corrected model decreased as animals became more active, and estimates for larger body sizes had similar error as the coefficient-corrected approach at particularly high (>75%) activity levels. However, the rapid and nonlinear decrease in estimation error with increased activity could result in large and unpredictable variation in estimates with small changes in activity. Thus, it is imperative to independently account for separate scaling rates of the Ṁ_O2–DBA slope and intercept when estimating respiration of highly active species, especially for smaller body sizes, where a small change in activity may represent a relatively larger change in overall energy requirements.

We found that isometric mass-specific corrections produced more accurate estimates than using a universal allometric correction of M^0.86. However, the higher performance of the isometric correction was likely a product of the BMR scaling rate of lemon sharks in this study being closer to isometry. In species with lower BMR scaling exponents, such as mammals (White and Seymour, 2003), it is likely that a lower nonproportional allometric mass-correction would perform better. However, the intercept-corrected model, which included an isometric effect on the intercept only, outperformed the allometric response-corrected model at lower body sizes, indicating that additional factors led to the greater performance of the isometric response-corrected model. The higher performance of the isometric correction, alternatively, may also be due to the observed scaling rate of the Ṁ_O2–DBA slope falling closer to the isometric exponent than the universal BMR allometric exponent. Nevertheless, the coefficient-corrected approach circumnavigates such issues by separately identifying slope and intercept scaling rates and outperforms commonly applied response-corrected and intercept-corrected modelling approaches. Given that we confirmed that DBA per unit oxygen (i.e. regression slope) scales allometrically at a different rate to that of BMR, where possible it is essential to use the coefficient-corrected approach when establishing Ṁ_O2–DBA predictive models.

We thank Bimini Biological Field Station staff, including C. White, V. Heim, C. Mason, S. Hart, E. Richardson, H. Lintott, D. Warburton, W. Nambu, A. Warrior, K. Yang and J. Whicheloe, as well as L. Harlow and numerous other interns who were essential for capturing sharks and conducting field trials in Bimini. Mote Marine Laboratory staff, including J. Morris, A. Ontkos, A. Osowski, A. Andres and R. Hueter, and numerous interns were essential to captive maintenance of sharks and respirometry experiments at MML. Additionally, we thank R. Daly, S. H. Gruber and T. L. Guttridge for providing logistical support.