The significance of respiration timing in the energetics estimates of free-ranging killer whales (Orcinus orca)

Cetacean populations can have a significant influence on marine ecosystems (Jefferson et al., 1991; Estes et al., 1998; Plagányi and Butterworth, 2009), and estimates of their energetic requirements at sea are essential to assess their prey intake and role within food webs. Cetacean metabolic rates have also been studied to give insight on reproduction costs (New et al., 2013), migration costs (Rodríguez de la Gala-Hernández et al., 2008), activity costs (Goldbogen et al., 2011) and potential impacts of disturbance (Christiansen et al., 2014; Villegas-Amtmann et al., 2015). However, direct measurement of field metabolic rate (FMR) of free-ranging cetaceans remains challenging, if not unfeasible for most species.

Early respirometry studies on captive cetaceans and pinnipeds showed that respiration rate increased with increasing O₂ consumption rate (V̇_O2) (Scholander and Irving, 1941; Hampton et al., 1971; Hampton and Whittow, 1976). Respiration rate was therefore considered as a reliable metric for metabolic rate (MR; Yazdi et al., 1999; Sumich, 2001) and has been used widely (Irving et al., 1941; Scholander and Irving, 1941; Spencer et al., 1967; Olsen et al., 1969), including in key studies on free-ranging cetaceans (Sumich, 1983; Dolphin, 1987; Blix and Folkow, 1995; Rodríguez de la Gala-Hernández et al., 2008; Williams and Noren, 2009; Christiansen et al., 2014, Villegas-Amtmann et al., 2015).

where T_O2 is the amount of O₂ exchanged per respiration (l), and is the product of tidal lung volume (V_T, l respiration⁻¹) and the percentage of O₂ extracted from inhaled air (E_O2) (Wartzok, 2002). V̇_O2 can therefore be estimated by summing T_O2 across multiple respirations over a specified time period.

In studies that derive FMR from respiration rate, the observed respiration rate is typically multiplied by single average V_T and fixed E_O2 values, estimated from captive animals. These FMR estimates are then linked to behaviours such as feeding, travelling and resting (Dolphin, 1987; Blix and Folkow, 1995), or speed (Sumich, 1983; Blix and Folkow, 1995; Rodríguez de la Gala-Hernández et al., 2008; Williams and Noren, 2009; Christiansen et al., 2014). Correlations between respiration rate and activity level like travel speed tend to be very weak (Williams and Noren, 2009; Christiansen et al., 2014). A crucial assumption made in those studies is that every respiration has a constant level of V_T and E_O2 (Sumich, 1983; Dolphin, 1987; Blix and Folkow, 1995; Rodríguez de la Gala-Hernández et al., 2008; Williams and Noren, 2009; Christiansen et al., 2014). However, studies on cetaceans have shown that E_O2 can vary greatly between respirations (Olsen et al., 1969; Ridgway et al., 1969; Wahrenbrock et al., 1974; Sumich, 1994, 2001; Kriete, 1995; Fahlman et al., 2015, 2016). Also, V_T within an individual is not necessarily constant and has been shown to fluctuate (Spencer et al., 1967; Olsen et al., 1969; Wahrenbrock et al., 1974; Gallivan et al., 1986; Kriete, 1995; Fahlman et al., 2015, 2016). This oversight concerning variability in E_O2 and V_T decreases the potential accuracy of respiration rate as a metric for cetacean MR.

Over sufficiently long time scales, O₂ consumption by body metabolism must be balanced by O₂ uptake. However, on shorter time scales, significant deviations between V̇_O2 and FMR are expected to occur as breath-hold divers deplete stored O₂ to different amounts between respirations (Goldbogen et al., 2012). Net gain of O₂ decreases as the number of consecutive respirations increases, as O₂ stores [blood (haemoglobin), muscle (myoglobin) and respiratory tract] are replaced, and the partial pressure of O₂ (P_O2) in the blood approaches that of inhaled air (Boutilier et al., 2001; Fahlman et al., 2008, 2016). Apnoea duration influences P_O2 in the body and therefore the expected efficiency of E_O2 from inhaled air at the end of every dive. A prolonged apnoea, as occurs during longer dives, causes a larger O₂ storage depletion compared with that from shorter dives. The uptake efficiency increases swiftly for short dives and flattens for longer dives when the maximum possible uptake is approached (Sumich, 1994, 2001; Parkes et al., 2002; Wilson et al., 2003; Fahlman et al., 2008). A higher MR, dependent upon activity level, causes the O₂ store to deplete faster than a lower MR. Therefore, a higher MR causes a greater O₂ flux, and thus a greater expected E_O2, than after an equivalent apnoea with a lower MR.

In this study, we developed an alternative approach to estimate FMR from respiration events. Instead of assuming a fixed E_O2 per respiration, we applied a simple O₂-uptake function, in which E_O2 decreases as body O₂ stores become more saturated. For simplicity, we set V_T to be constant. The fact that over a sufficiently long time scale V̇_O2 and FMR should balance provides a tool to compare models that assume fixed versus variable T_O2 by correlating V̇_O2 with underwater activity over an appropriate time period. This study focused on the aerobic metabolism of free-ranging killer whales, Orcinus orca (Linnaeus 1758). Killer whales are ‘single-breathers’ that tend to continually dive between respirations, making it possible to detect respirations using depth recordings, in contrast to some deep-diving cetaceans like sperm whales (Physeter macrocephalus), which can remain passive at the surface during recovery periods. Though physiological data on killer whales are difficult (if not impossible) to collect at sea, some valuable physiological data have been collected on captive specimens (Table 1). Still, there exists a lack of solid values on energetic requirements of killer whales. The few studies that have been conducted on this species involving direct measurements of MR are based on a small number of animals and datasets (Kasting et al., 1989; Kriete, 1995; Worthy et al., 2014). Studies on the FMR of free-ranging animals have relied on respiration rates, assuming a constant E_O2. However, respiration rates correlated weakly with speed derived from killer whale surfacing positions (Kriete, 1995; Williams and Noren, 2009). Moreover, the relationship between cost of transport (COT) and speed is expected to be parabolic with a clear minimum at the optimal speed. This expected minimum in the killer whale COT curve has not been observed in previous studies (Williams and Noren, 2009). By including continuous data on respiration timing and underwater activity, both recorded by an animal-attached tag (Johnson and Tyack, 2003; Miller et al., 2010), a model of O₂ exchange and O₂ stores enabled a re-examination of predicted E_O2 dynamics. The objective was to investigate the potential influence of respiration timing, in addition to respiration rate, on killer whale FMR estimates using this model.

Animal experiments were carried out under permits issued by the Norwegian Animal Research Authority (permit no. 2004/20607 and S-2007/61201), in compliance with the ethical use of animals in experimentation. The research protocol was approved by the University of St Andrews Animal Welfare and Ethics Committee and the WHOI Institutional Animal Care and Use Committee.

For this study, we used 50.8 h of data recorded by digital acoustic recording tags (DTAGs; Johnson and Tyack, 2003) attached using suction cups to five adult male and five adult female free-ranging Norwegian herring-feeding killer whales between 2005 and 2009. Continuous acoustic and sensor recordings of the DTAG are synchronous (Johnson and Tyack, 2003), so the relative timing of sounds and motion can be determined precisely.

Characteristics information was collected for all tagged individuals, which were categorized into age–sex classes according to body size, as O₂ storage and use were expected to scale with body size in this sexually dimorphic species. For age–sex class, adult-sized animals with a tall dorsal fin were defined as adult males and those without a tall dorsal fin were classified as adult females. Though some of the latter could have been sub-adult males, their body size was smaller than that of adult males, which was the primary reason for sex categorization. Individual tag records ranged from 1.67 h to almost 12.5 h in length (Table S1). Some whales were tagged simultaneously and data records of four whales included time periods in which they were experimentally exposed to sonar (Table S1) (Miller et al., 2011), which probably affected the speed of these individuals during some periods of their tagging record (Miller et al., 2014).

DTAG data were processed and analysed using MATLAB^® (v7.5.0.342R2007b, MathWorks) and RStudio^® (v0.98.994, The R Foundation for Statistical Computing). Pressure, 3-axis magnetometer and 3-axis accelerometer measurements at 50 Hz sampling rate were converted to depth (m), magnetic field (μT) and acceleration (g), respectively, using calibration values, and condensed to a sampling rate of either 10 Hz for tags deployed in 2005 and 2006 or 5 Hz for tags deployed in 2009. Animal pitch, roll and heading data were derived from magnetometer and accelerometer values after correcting for tag orientation on the whale (Johnson and Tyack, 2003), which changed gradually and/or swiftly during seven suction-cup tag deployments.

DTAG depth data enable detection of surfacings, which are respiration events (Miller et al., 2010). Surfacings were automatically detected using depth criteria to define the start and end of each dive (see Miller et al., 2010, for details). The detected surfacings were manually checked by inspecting the dive profile, and acoustic recordings in rare cases of uncertainty.

Differences in respiration rate between sexes were investigated using generalized estimating equations (GEE). Respiration rates were modelled using a Gaussian distribution of errors and a one-step autoregressive correlations structure (AR1). The model structure was selected using the quasi-Akaike information criteria (QIC) (Pan, 2001).

Speed through water or swimming speed (U) was estimated by merging analyses of kinematic measurements and the sound pressure levels (SPLs) of the recorded low-frequency flow noise, as done previously by other authors (Goldbogen et al., 2006; Simon et al., 2009). The low-pass frequency filter, specified per tag record, ranged from 200 to 350 Hz. Kinematic speed was estimated for entire tag records by dividing the depth change rate over 1 s intervals by the sine of the instantaneous body pitch angle. A pitch angle threshold for these calculations was introduced per individual and per tag period, ranging from 45 to 65 deg, depending on data quantity and pitch angle occurrence and distribution. Flow noise SPLs and kinematic speed estimates from the sensor data were synchronized over 1 s intervals.

An exponential least-squares regression between kinematic speed and flow noise SPLs was fitted to data from depths >10 m, as surfacing events produced high noise levels unrelated to U. Separate regressions were performed for each tag record and periods between tag movements within tag records containing body pitch angles larger than the pitch angle threshold and a SPL range larger than 10 dB. For tag periods not meeting these requirements, the speed–SPL regression of the preceding period was applied. In the event of a gradual tag movement, average regression parameter values from the previous and following tag period without tag movements were used (Table S2). When either the previous or following period without tag movement, or both, did not contain large enough pitch angles or a SPL range larger than 10 dB, but the gradual movement period did, this latter period was handled as a period without tag movement (Table S2). The regression was then used to calculate U at 1 s intervals from the SPLs throughout each tag record. Again, to avoid overestimation errors in U calculation due to the elevated SPL caused by surfacings, U estimates during respiration events and the 3 s preceding them were replaced by the last estimate before that interval. U estimates for the 3 s after the respiration event were replaced with the following estimate.

Differences in U between sexes were investigated using GEEs with correlation structure ‘independence’. Model structure was selected using QIC.

Size-dependent FMR of the swimming whales was estimated by summing basal metabolic rate (BMR) and locomotion costs (MR_L) at a 1 s resolution. BMR was assumed to be equal to the standard metabolic rates quantified by Kriete (1995) (Table 1). A sex-specific relationship between U³ and energetic cost was fitted by combining equations used previously by Fish (1998) and Guinet et al. (2007).

where ρ is seawater density (1026 kg m⁻³) and S is the body surface area in m².

MR_L was modelled as k·U³, where k is a coefficient (l O₂ m⁻³ s²) determined using morphometrics according to sex (Table 1). MR_L was derived for U of 0–10 m s⁻¹. A linear function, with the intercept set at BMR, was calculated for both 3913 kg males (MR=0.006115×U³+0.1050) and 2800 kg females (MR=0.004922×U³+0.0731).

where TBO_t represents the total O₂ stored (l) in the lungs, blood and muscle together at time t, MR_t−1 is the total O₂ consumption (l) based upon BMR and MR_L during the preceding second, and T_O2,t is the O₂ (l) taken up if the whale respired at time t, or zero if no respiration occurred. T_O2,t was modelled in three different ways: model 1, a constant value from the literature; model 2, a constant fitted to the data to achieve a balanced O₂ budget over each animal's entire tag-recording period; and model 3, a broken-stick O₂-uptake function in which O₂ uptake depends upon the store at the time of the respiration. If at the time of respiration the O₂ store is lower than maximum TBO minus maximum T_O2, then T_O2 is the maximum possible uptake. Otherwise, T_O2 is maximum TBO minus the existing O₂ store (Fig. 1), which realistically caps the O₂ uptake according to the maximum TBO.

The model was initialized at the first second after the first breathing bout of at least six respirations within each tag record, at which point TBO was assumed to be saturated at the maximum TBO value (Table 1).

Using all models, V̇_O2 (estimated at a 1 s resolution) was calculated over successive 15 min time intervals to model against U³ over the same 15 min intervals. The 15 min time duration was considered to be sufficient to even out O₂ store fluctuations during typical dives, but short enough to capture temporal variation in underwater activity levels. To eliminate potential influence by biased start and stop points by filtering, both the first and last 15 min interval was excluded from analyses. Because O₂ consumption and uptake should be balanced over 15 min intervals, the average V̇_O2 and average activity level (U³) should be linearly related (Eqn 4). We therefore regressed O₂ uptake as a function of U³. Because BMR was fitted as a constant in the model, we regressed (V̇_O2−BMR) against U³ through the origin.

We used a GEE model with whale identity as random factor and a Gaussian family of distribution for the residuals. Correlation structure AR1 was included to account for auto-correlation among 15 min intervals. The model included sex as a factor.

Because of uncertainty in the parameter values, the sensitivity of the results of model 3 to the slope of MR_L versus U³ (k), BMR and the maximum T_O2 was tested by varying these parameters one at a time according to values derived from other studies where possible.

The slope of MR_L versus U³ was varied over a range of values (Table 1), with the highest value extracted from results by Williams et al. (1993), who found a relatively high MR for swimming captive bottlenose dolphins (Tursiops truncatus) compared with other studies on swimming cetacean MR. Because there are no published estimates below the values used, the lower limit of the sensitivity analyses was set at half the value (Table 1).

The values of BMR used were from Kriete (1995), who obtained measurements on just one adult male and one adult female killer whale. Upper limits for the sensitivity analysis of this parameter (Table 1) were derived from the equation by Kasting et al. (1989), which yielded relatively high BMR values compared with other studies on cetacean BMR. The lower limits were set according to recently found BMR values by Worthy et al. (2014), which are relatively low compared with other cetacean BMR study outcomes.

Both the upper and lower limits for the sensitivity analyses of T_O2 values for both sexes were derived from Kriete (1995), who measured a maximum and minimum V_T for both male and female killer whales which were multiplied by the percentage of O₂ in air (20.95%; Table 1).

Averages of the values initially used in model 3 and the upper and lower limit were also tested for all parameters to strengthen sensitivity analysis outcomes.

Estimated V̇_O2 over the 15 min intervals using the different O₂-uptake models were multiplied by the conversion factor of 20.1 kJ l⁻¹ O₂ to derive estimates in J. These values were divided by the speed measurements over the same 15 min interval to derive non-mass-specific COT (J m⁻¹). Finally, the non-mass-specific COT estimates were divided by the body mass per sex to calculate the mass-specific COT (J kg⁻¹ m⁻¹) for the different O₂-uptake models.

The mean±s.d. respiration rate was 1.54±0.22 respirations min⁻¹ and ranged from 1.08 to 2.18 respirations min⁻¹ (Table S3). The highest respiration rate was observed for one of the males, while the lowest rate was detected for one of the females (Table S3). Matching the previous finding of Miller et al. (2010), males exhibited a mean respiration rate (1.57±0.23 respirations min⁻¹) that did not differ significantly from that of the females (1.47±0.20 respirations min⁻¹; two-tailed Wald test: P=0.053).

The mean±s.d. U for all whales was 1.89±0.61 m s⁻¹, and ranged from 0.69 to 4.05 m s⁻¹, with both extremes being males (Table S3). The range of U estimates for females was somewhat smaller, ranging from 0.72 to 2.80 m s⁻¹ (Table S3). The mean±s.d. U for females (1.72±0.59 m s⁻¹) and males (1.96±0.61 m s⁻¹) was not significantly different (two-tailed Wald test: P=0.680).

Using the literature value for a fixed T_O2 (model 1), no correlation between V̇_O2 and U³ over the 15 min intervals was found for any of the individual whales or either sex when setting the intercept at BMR (Table 2). The fixed T_O2 resulted in an unrestrained and unrealistically high O₂ store for all individuals, well exceeding their maximum TBO, meaning there was no balance between O₂ use and uptake over the tag period (e.g. Fig. 2B). The fitted GEE model lay well below all data points for both sexes (Fig. 3A,D).

The estimated fixed values of T_O2 (model 2) for females and males ranged between 3.77 and 5.94 l O₂ respiration⁻¹, and 4.76 and 8.92 l O₂ respiration⁻¹, respectively (Table 2). All these estimated values were considerably lower than the values used by Williams and Noren (2009). The mean±s.d. estimated fixed T_O2 found for males (6.09±1.80 l O₂ respiration⁻¹) was somewhat higher (two-tailed t-test: P=0.137) than the mean±s.d. estimated value found for females (4.56±0.88 l O₂ respiration⁻¹; Table 2). Under model 2, there was a moderate correlation between estimated V̇_O2 and U³ for only one female and one male (Table 2). When grouping animals by sex, there was no relationship between V̇_O2 and U³ (Table 2, Fig. 3B,E). Although by definition the accumulated V̇_O2 over each time series was constrained to be equal to the accumulated MR, O₂ stores estimated under model 2 resulted in an unrealistically temporary excess of the estimated maximum TBO, and/or an unrealistically negative O₂ store for all individuals (e.g. Fig. 2C).

The model of O₂ uptake fluctuating per respiration according to O₂ store (model 3) led to a strong association between V̇_O2 and U³ for all individuals (Table 2). This association was weaker for all females grouped than for females individually, whereas for males the opposite was true (Table 2). Of the females, 05_322a had overall a higher V̇_O2. The data point representing the highest speed and V̇_O2, which shows the greatest deviation, belonged to this particular female (Fig. 3F).

MR and V̇_O2 were better balanced for all individuals when applying T_O2 as a function of O₂ store, without exceeding the estimated TBO capacity. Nor were negative O₂ store values obtained using this model. When the animals respired often in sequence after a longer dive, the first couple of respirations were sufficient to replenish O₂ stores. As the O₂ stores became more saturated, T_O2 decreased (e.g. Fig. 2D).

The V̇_O2–U³ correlation was sensitive to the diverse energetic parameter values derived from the literature (Table 3, Fig. 4A,D).

As expected, the occurrence of maximum T_O2 respirations increased when increasing the slope of MR_L versus U³ and vice versa. Average T_O2 decreased when decreasing the slope of MR_L versus U³ (e.g. Fig. S1).

Varying BMR according to BMRs derived from other studies caused only a minor change of the V̇_O2–U³ correlation (Table 3, Fig. 4B,E). Moreover, BMR values used originally in model 3 gained the highest r² values (Table 3).

When applying tested BMR values that were higher than the used values, the occurrence of maximum T_O2 respirations increased and vice versa. Average T_O2 decreased with decreasing BMR (e.g. Fig. S2).

The V̇_O2–U³ correlation became tighter when increasing the maximum T_O2, especially for females (Table 3, Fig. 4C,F).

The frequency of occurrence of maximum T_O2 was decreased by increasing this parameter value. Varying maximum T_O2 did not affect the frequency of occurrence of T_O2 values lower than the lowest tested value (e.g. Fig. S3). The low occurrence of T_O2 values between the lowest tested and other tested T_O2 values was equally distributed, with frequency decreasing with an increasing difference between the lowest tested and other tested T_O2 values (e.g. Fig. S3). Thus, though varying maximum T_O2 did not significantly change the overall results, O₂ uptake per respiration was limited by the set maximum T_O2, especially for the lower tested values (e.g. Fig. S3).

In summary, the sensitivity analyses demonstrated that, over the range of parameter values tested, the O₂ model with variable uptake (model 3) was especially sensitive to changes in the slope of MR_L versus U³. The results for females were somewhat more sensitive to parameter value changes than were results for males. Nonetheless, including the broken-stick O₂-uptake function in the O₂ model always yielded a better fit than using a fixed uptake per respiration (model 1).

For models 1 and 2, the relationship between COT and speed was best modelled by a power function with no minimum for both sexes (Table 4, Fig. 5B,D). In contrast, COT estimated by model 3 with fluctuating T_O2 was best represented by a second-order polynomial equation, with an underlying mechanism that revealed the expected U-shaped curve with a broad yet distinct minimum at a speed between 1.7 and 2.2 m s⁻¹ for females and between 1.9 and 2.4 m s⁻¹ for males (Fig. 5F). These estimated minimum COT speeds were the lowest for both sexes compared with results of model 1 and 2 (Table 4, Fig. 5F).

Minimum mass-specific COT estimated with fixed T_O2 (model 1) was the highest (approximately 0.8 J kg⁻¹ m⁻¹ for both sexes, closely matching the results of Williams and Noren, 2009), while minimum COT estimated by optimized fixed T_O2 (model 2) was the lowest (approximately 0.2 J kg⁻¹ m⁻¹ for both sexes; Fig. 5B,D,F). Estimated minimum COT according to model 3 was 0.44 and 0.42 J kg⁻¹ m⁻¹ for male and female killer whales, respectively. Minimum COT estimated using model 2 and 3 revealed no difference between sexes (Table 4, Fig. 5D,F).

Correlation between V̇_O2 and U over 15 min intervals, estimated with either model 1 or 2, was weak when fitting a simple linear regression as done by Williams and Noren (2009), especially for males (Table 4, Fig. 5A,C). V̇_O2 estimated using model 3 showed a strong non-linear relationship with increasing linear speed for both sexes as anticipated (Table 4, Fig. 5E).

In this study, we developed an alternative approach to the counting of respirations as a means to estimate O₂ uptake in free-ranging cetaceans. Rather than assuming that each respiration delivers an equivalent amount of O₂ (T_O2) to the body, we implemented a simple O₂-uptake function (broken-stick; Fig. 1), which more realistically limits O₂ uptake from respirations when body O₂ stores are predicted to be more fully saturated. O₂ uptake is predicted to increase linearly with U³, but as has been found in previous studies (Williams and Noren, 2009; Christiansen et al., 2014), relationships between activity levels (quantified as U³) and V̇_O2 were not apparent or were very weak when T_O2 was assumed to be constant. In contrast, we found strong associations between predicted V̇_O2 and underwater activity over 15 min intervals when uptake could vary across respirations using the broken-stick uptake function. The fixed T_O2 derived from Kriete (1995), as used by Williams and Noren (2009) (model 1), overestimated the predicted energetic requirements of both sexes, especially those of male killer whales. This could have been due to the parameter values used in the model rather than the fact that T_O2 was fixed. However, model 2, which balanced V̇_O2 and MR over the entire data recording period by specifying the value of T_O2, still caused unrealistic maximum TBO excesses and deficits in modelled O₂ stores and no association between estimated activity level (U³) and V̇_O2. This indicates that respiration rate alone is not a reliable indicator of energy requirements in killer whales. Allowing T_O2 to vary with O₂ store at the time of each respiration (model 3) led to more stable fluctuations in the modelled O₂ store. A much stronger association between estimated V̇_O2 and U³ was found for model 3, as expected, as that model includes a mechanism to constrain O₂ uptake by the MR predicted from BMR and estimated speed. Our results therefore demonstrate that the approach of applying an O₂-uptake function can overcome the weak relationships between activity levels and V̇_O2 found when T_O2 is fixed per respiration.

An important outcome of the present study is that by using the broken-stick O₂-uptake function we were able to describe a U-shaped COT curve with a clear minimum at an intermediate speed (Fig. 5F). In contrast, the COT curve based upon a fixed uptake per respiration was monotonically decreasing (Fig. 5B,D). Our model led to a much lower predicted FMR and COT than previous studies have predicted for killer whales (Kriete, 1995; Williams et al., 2004a; Guinet et al., 2007). However, because key parameter values such as true MR_L and BMR remain indeterminate for now, our focus was on the improved predictive power of a varying O₂-uptake model, rather than the specific MR or COT estimates. Our approach, however, could be useful to estimate FMR in the future, when more accurate estimates of parameters such as MR_L and BMR become available.

These results were for the resolution of 15 min intervals, which are relevant for MR variations at relatively fine time scales. This could include describing the relative FMR associated with different behaviours or the effects of disturbance events of relatively short duration. However, if one were interested in estimating the lifetime V̇_O2 of a whale, fitting a fixed T_O2 to 24 h or longer intervals would probably suffice. Further work could determine at what temporal scale the timing of respirations improves estimates of MR.

Speed derived from DTAG data showed the potential to function as an activity metric in the developed O₂ model. Speeds derived through flow noise SPL are approximate and the reliability of the measurements depends in part on tag placement and the proportion of diving behaviour including large body pitch angles. In addition, speed could not be derived during noisy surfacing events. Investigating the influence of ocean ambient noise, noise generated by oscillation of the flukes (dependent upon tag placement) and noise generated by the water surface recorded for logging whales on speed estimates would improve estimates of this relationship. Speed could also be measured more directly using fly-wheel speed sensor tags, rather than acoustic tags (Watanabe et al., 2011).

Speed through water (U) was then used to estimate locomotion costs (MR_L), which are predicted to increase as a function of U³, as hydrodynamic drag is predicted to increase as a function of U² (Vogel, 1994) and multiplication by speed converts drag force to power (work per unit time). The precise relationships between MR_L and U³ should be examined more thoroughly. Our derivation of expected MR_L followed the analysis of Guinet et al. (2007), but yielded total MR values that were substantially lower than other predictions (Williams et al., 2004a) of killer whale FMR. Fahlman et al. (2016) also found that estimates of MR using respiration rate and constant O₂ uptake were as much as 2–5 times higher than when using measured variation in O₂ uptake. We used the same BMR estimates as Williams et al. (2004a), so the difference in our estimates arises from the difference between BMR and FMR, which we assumed only included MR_L. Similar assumptions have been made for large marine mammals for which FMR measurements were possible (Williams et al., 2004b). We estimated MR_L to be quite small because of the very low C_d of this streamlined animal (Fish, 1998), but we neglected other metabolic requirements, e.g. overcoming active drag due to swimming motions and thermoregulation, which could cause MR to be greater than we estimated. It remains unclear whether extrapolations of FMR to killer whales from measurements of other marine mammals (Williams et al., 2004a) are accurate, which should be a focus of future research.

The results of this study were partially based upon the physiological parameters applied to the O₂ model for which exact values regarding killer whales remained uncertain. Sensitivity analyses showed that the O₂ model was most sensitive to variation in the slope of MR_L versus U³ for the range of parameter values tested. Fish (1998) found a theoretical C_d for swimming killer whales which was 3.8 times lower than the C_d calculated through Eqn 3. If this lower C_d from Fish (1998) were true, then using Eqn 3 would have yielded a V̇_O2 that was 3.8 times too large, and the overestimation of energetic requirements based on respiration rate would have been even more drastic. However, Guinet et al. (2007), using a similar theoretical approach to that of Fish (1998) to define free-ranging killer whale MR, estimated a relationship between MR and speed (see fig. 3 in Guinet et al., 2007). For a swim speed of 4.7 m s⁻¹ in females, this relationship predicted a V̇_O2 that was 1.6 times higher than the V̇_O2 estimated using model 3 of the present study for the same speed. This indicates once more the uncertainty in predicting actual MR in killer whales. During the sensitivity analyses, using extrapolated MR from data collected on other cetacean species (Otani et al., 2001; Shaffer et al., 1997; Williams et al., 1993, 2004b; O'corry-Crowe, 2009) in model 3 resulted in a higher MR than approximated using the estimated slope of MR_L versus U³. Other than for the two highest tested values of the slope of MR_L versus U³, the correlation between V̇_O2 and U³ was not greatly affected, which suggests that our conclusions are robust to uncertainty in the values of parameters we needed to employ in our study.

Differences in the correlation between V̇_O2 and U³ between sexes and between individuals of the same sex could be due to individual differences in physiology and behaviour. Foraging or hunting strategies will be decisive for respiration timing (Thompson et al., 2003). Species that are social and behave in groups, such as killer whales, are expected to be partly constrained by the decision making of conspecifics with diverse physiological abilities regarding body size. In all social groups to which the 10 tagged individuals belonged, at least one juvenile or calf was present. Miller et al. (2010) suggested that to maintain group cohesion, larger individuals could perform under their physiological limit (physiological compromise), or that hunting and foraging roles could be assigned to individuals according to physiological capacities (division of labour). The physiological compromise hypothesis was supported by our result that the O₂ stores of the 10 adult killer whales estimated through model 3 were never fully depleted. Female energetic requirements are higher during lactation (Fedak and Anderson, 1982; Costa et al., 1986; Noren et al., 2011) and when the calf is swimming in the echelon position because of the increase in drag (Noren, 2008), factors not included in our model.

Animals make behavioural decisions, driven by their physiology and ecological context, which shape their observable diving and respiration behaviours (Fedak and Thompson, 1993; Thompson et al., 1993). During high-level activity (porpoising at higher speeds), we found that respirations were relatively equally separated in time, and respiration intervals tended to be longer than those during surfacing periods between dives (Fig. 6E). Also, T_O2 was greater than during low activity (Fig. 6H,I). This observation could be explained by characteristics of the O₂-uptake model in combination with behavioural state. While porpoising, the animal is driven by the motivation to move fast (in this case as a response to sonar; Miller et al., 2014) and therefore respire in the most efficient way to reduce the number of surfacings, which lead to increased surface drag (Hertel, 1966; Costa and Williams, 1999). Thus, this male killer whale changed its respiration behaviour in a manner that appears to functionally increase T_O2 during a period of high-speed travel. High average T_O2 depends upon apnoea duration and MR. However, during lower activity periods, T_O2 was maximal for the first couple of respirations after a dive and decreased gradually according to O₂ store replenishment (e.g. Fig. 6F,H). A possible purpose of these later respirations could be delayed CO₂ offload rather than O₂ uptake after long dives (Boutilier et al., 2001; Wilson et al., 2003; Fahlman et al., 2008).

Noren et al. (2012) concluded that apnoea of 13.3 min in an adult male killer whale did not cause a rise in blood lactate levels. The maximum apnoea durations observed in our study were 6.1 and 5.3 min for males and females, respectively; it was assumed that lactate accumulation was not an issue. Nevertheless, it is hypothesized that anaerobic metabolism would be valuable to consider for future O₂ models concerning short-term anaerobic sprints or cetaceans that perform longer duration or high-activity dives that may exceed their diving lactate threshold.

COT has been derived from respiration rate in various studies on free-ranging cetaceans in which it was generally assumed that swimming effort was reflected in respiration rate (Sumich, 1983; Rodríguez de la Gala-Hernández et al., 2008; Williams and Noren, 2009; Christiansen et al., 2014). Williams and Noren (2009) and Christiansen et al. (2014) both found a weak linear correlation between respiration rate and speed, and argued that the absence of a minimum COT was caused by the lack of high-speed observations. In our study, COT estimated through the O₂ models with fixed T_O2 (Fig. 5B,D; model 1 and 2) also monotonically decreased with increasing speed. However, when COT was estimated by the O₂ model with the broken-stick O₂-uptake function, the expected curvilinear relationship between MR and speed was found, showing a clear minimum for speeds of approximately 1.7–2.2 m s⁻¹ for females and 1.9–2.4 m s⁻¹ for males. The speed of 1.7 and 2.0 m s⁻¹ estimated from flow noise in the present study for females and males, respectively, fell within these predicted optimal COT speed ranges. However, optimal COT speed depended somewhat on the slope between MR_L versus U³, which remained uncertain for this study.

Overall, sensitivity analyses demonstrated that the conclusions of this study were not dependent on any specific parameters in the implemented O₂ model. Only for the highest values of k, for which a wide range was tested during the sensitivity analysis, was there no relationship between V̇_O2 and U³. For other studies inferring a curvilinear correlation between V̇_O2, or respiration rate, and speed, a clear minimum COT was found as well (Sumich, 1983; Williams et al., 1993; Kriete, 1995; Yazdi et al., 1999; Otani et al., 2001). Consequently, it can be concluded that it was not just the lack of high-speed observations in previous studies of free-ranging killer whale FMR (Williams and Noren, 2009) that led to monotonically decreasing COT curves. The other key factor is that the assumption of fixed T_O2 within broad activity categories used by Williams and Noren (2009) fails to capture higher levels of O₂ uptake per respiration which were predicted to occur when whales travelled at higher speeds in our study.

During the present study, BMR referred to the basal metabolic rate. According to the equation established by Kasting et al. (1989), BMR would be 2.7 times higher than the values used for both sexes. Also, during other studies it was concluded that killer whale BMR was higher than predicted by the Kleiber (1975) regression (T. M. Williams, Mammalian Physiology Lab UCSC, personal communication). Guinet et al. (2007) estimated BMR using the equation provided by Motani (2002), taking into account an elevated BMR according to body mass, and obtained values similar to those used in the present study for both sexes. Extrapolated killer whale BMR measured by Worthy et al. (2014) yielded values considerably lower than those used here for both sexes. Nevertheless, the sensitivity analyses indicated that this uncertainty did not change the main conclusions of our study.

Despite uncertainties regarding E_O2 percentages in cetaceans, precise fixed E_O2 values (and V_T values) per respiration were used in free-ranging cetacean energetics studies using respirations. Assigned E_O2 values showed a large range between different baleen species from 11% for grey (Eschrichtius robustus) and humpback whales (Megaptera novaeangliae) (Sumich, 1983; Dolphin, 1987) to 45% for minke whales (Balaenoptera acutorostrata) (Blix and Folkow, 1995; Christiansen et al., 2014). Williams and Noren (2009) employed a fixed E_O2 of 41% and 38% for adult male and female killer whales, respectively. Kriete (1995) applied a fixed E_O2 per activity level, ranging from 35% to 47% and 33% to 44% for adult male and female killer whales, respectively. Also, large variations in E_O2 within respiratory cycles existed for captive specimens, ranging from 8% to over 80% (Irving et al., 1941; Olsen et al., 1969; Ridgway et al., 1969; Sumich, 1994; Kriete, 1995; Fahlman et al., 2016), depending on activity, respiration interval and respiration number in a sequence. Moreover, the proportion of O₂ that diffuses from the lungs into the blood during apnoea can vary (Olsen et al., 1969; Kriete, 1995), with excess O₂ remaining in the lungs being exhaled before inhalation. To account for this concern, mean instead of maximum E_O2 values were taken from Kriete (1995) and multiplied by V_T to estimate the maximum T_O2.

The broken-stick O₂-uptake function predicts the T_O2 depending upon the O₂ store at the time of each respiration. This represents a rather crude and unrealistic mechanism of how O₂ exchange actually occurs in the lungs. However, we sought to assess the potential benefit of using an uptake function which has the key features of maximum O₂ uptake per respiration (T_O2) when body stores are low, and very low T_O2 when body stores are more fully saturated. Derivation of a truly accurate uptake function was not the goal of our study, but we recommend that a more realistic uptake function could be predicted using a gas-exchange model that includes the different body tissues and lung collapse (Fahlman et al., 2006), which could further improve predictions of V̇_O2 in free-ranging cetaceans.

This study showed that an O₂ model including an O₂-uptake function has the potential to be a significant improvement over using fixed T_O2 (respiration rate) to derive consistent MR estimates from longitudinal observations of respiration times and underwater activity level in free-ranging killer whales. The assumption that respiration rate alone is an appropriate proxy for the level of MR, which is the foundation of an important part of fundamental research on free-ranging cetacean energetics, should be re-evaluated. However, it should be stressed that the presented O₂ model including a broken-stick O₂-uptake function can be substantially improved with additional data. One important constraint of the presented model is the paucity of concrete measurements on V̇_O2 in relation to speed. The model should be validated through quantifying kinematics, V̇_O2 and respiration-by-respiration E_O2 and V_T to the greatest extent possible with captive killer whales (Fahlman et al., 2016). Also, the inclusion of lactate accumulation and CO₂ in the gas-exchange model would be helpful for predicting alternative functions of respirations than O₂ uptake alone. Future versions of the proposed model could become useful tools to quantify the metabolism of free-ranging cetaceans and inform sustainable marine ecosystem management.

The data were collected as part of the 3S collaboration to study the effects of sonar on cetaceans. The authors wish to thank all the parties who helped to arrange and carry out the field work in which the data were collected. Thanks to Dave Thompson for helpful input on oxygen uptake functions and T. Williams and A. Fahlman for their useful comments on the manuscript.