Current estimates of marine mammal hydrodynamic forces tend to be made using camera-based kinematic data for a limited number of fluke strokes during a prescribed swimming task. In contrast, biologging tag data yield kinematic measurements from thousands of strokes, enabling new insights into swimming behavior and mechanics. However, there have been limited tag-based estimates of mechanical work and power. In this work, we investigated bottlenose dolphin (Tursiops truncatus) swimming behavior using tag-measured kinematics and a hydrodynamic model to estimate propulsive power, work and cost of transport. Movement data were collected from six animals during prescribed straight-line swimming trials to investigate swimming mechanics over a range of sustained speeds (1.9–6.1 m s−1). Propulsive power ranged from 66 W to 3.8 kW over 282 total trials. During the lap trials, the dolphins swam at depths that mitigated wave drag, reducing overall drag throughout these mid- to high-speed tasks. Data were also collected from four individuals during undirected daytime (08:30–18:00 h) swimming to examine how self-selected movement strategies are used to modulate energetic efficiency and effort. Overall, self-selected swimming speeds (individual means ranging from 1.0 to 1.96 m s−1) tended to minimize cost of transport, and were on the lower range of animal-preferred speeds reported in literature. The results indicate that these dolphins moderate propulsive effort and efficiency through a combination of speed and depth regulation. This work provides new insights into dolphin swimming behavior in both prescribed tasks and self-selected swimming, and presents a path forward for continuous estimates of mechanical work and power from wild animals.

There is a direct relationship between mechanical work performed by the musculoskeletal system and energetic cost. As such, observed movement patterns of animals are often explained using an analysis of locomotive energy economy (e.g. horses: Minetti et al., 1999; pumas: Williams et al., 2014; humans: Donelan et al., 2002; right whales: Nowacek et al., 2001; sperm whales: Miller et al., 2004). Experiments in humans and other terrestrial animals that compare the efficiency of different movement conditions are often conducted in controlled settings where the motion of the body and ground reaction forces can be measured directly (Full and Tu, 1991; Pontzer, 2007). Legged locomotion studies use these measurements, along with rigid body models of the system, to estimate the forces, moments, work and power at the individual joints and for the center of mass (Saibene and Minetti, 2003; Faraji et al., 2018; Voloshina et al., 2013).

In contrast, it has been challenging to create fully instrumented controlled experimental environments for large, swimming animals. External forces, such as thrust for propulsion or drag acting on the moving body, are particularly difficult to measure directly in the marine environment. To address this, biologging tags with sensors such as accelerometers, magnetometers, gyroscopes and pressure sensors have been used to measure animal kinematics (Johnson and Tyack, 2003; Goldbogen et al., 2008; Allen et al., 2016; Roos et al., 2016; Ware et al., 2016; Williams et al., 2017; Zhang et al., 2020). The measurements obtained are often used to infer energetic costs and long-term energy budgets using proxies such as overall dynamic body acceleration (ODBA; Wilson et al., 2006). However, ODBA-based estimates of energetic cost require empirical relationships that are constructed using controlled experiments where direct measurements of acceleration and metabolic cost are made simultaneously (John, 2020; Wilson et al., 2020). Proxies based on single-point accelerometer measurements have been developed and tested with terrestrial animals, free-diving sea lions and, recently, with bottlenose dolphins. In these studies, direct measurements of metabolic cost via respirometry are used to create data-driven maps between acceleration and energetic cost (Wilson et al., 2006; Fahlman et al., 2008; Halsey et al., 2009; Allen et al., 2022). Experimental validation of the relationships between acceleration, mechanical work and energetic cost remain limited for free-swimming cetaceans (Williams et al., 1993; Yazdi et al., 1999; van der Hoop et al., 2014; Allen et al., 2022). Further, comparisons between individuals or across types of propulsion-involved behaviors (e.g. varying gait modes, diving versus travel swimming, play behaviors) are unlikely to be accurate.

In addition to ODBA, kinematic parameters derived from tag-based measurements such as speed, fluking frequency and acceleration can be used with physics-based models of biological systems to estimate forces acting on the animal. For bottlenose dolphins, previous work has combined measured speed and kinematics of the fluke with hydrodynamic models to estimate external forces acting on the animals, such as drag (Fish, 1993; Fish and Rohr, 1999; Schultz and Webb, 2002). Particle image velocimetry has also been used to estimate thrust created by swimming bottlenose dolphins (Fish et al., 2014). More recently, per-stroke thrust and power have been studied with a Pacific white-sided dolphin (Lagenorhynchus obliquidens) during high-speed swimming through camera-based tracking and computational fluid dynamics (Tanaka et al., 2019). However, these approaches use video data to measure kinematics and are limited to when the animal is in the camera's field of view. As such, there currently exists a significant gap in our knowledge about the mechanical work performed by dolphins during daily life. Addressing this gap would aid researchers in understanding the energetic costs of individual behaviors and active responses to environmental stimuli (e.g. anthropogenic disturbances in the wild). Further, regular and distributed tag-based kinematics/energetics monitoring has the potential to provide population-scale estimates of physical fitness, which is key to wild animal conservation as it allows researchers to track the physical health of a population over time.

In this study, we investigated the energetics and swimming mechanics of bottlenose dolphins using tag-based estimates of locomotive work, power and cost of transport (COT) in an institutional setting. We expected that the animals would modulate energetic cost by swimming at depths that reduce surface drag during high-speed prescribed swimming, and select both speeds and depths that minimize COT during self-selected swimming. Prescribed straight-line travel was used to characterize the performance of six animals over a range of swimming speeds, enabling the investigation of a mid- to high-energy task that occurs infrequently during self-selected swimming. Four dolphins from the group were also tagged and monitored for an extended duration during daylight hours to quantify preferred swimming behavior throughout a typical day. The contributions of this work include: (1) the experimental investigation of mechanical work and power during consistent-speed (minimal variation in speed) swimming; and (2) the quantification of the undirected, long-duration swimming patterns of multiple animals in an institutional environment. Additionally, we demonstrate the viability of the approach for the analysis of swimming mechanics during daily life by estimating the energy budgets of the animals during unprescribed swimming, an important step for the use of this approach with dolphins in the wild.

Experimental setup

Experiments were conducted at Dolphin Quest Oahu (Oahu, HI, USA) with six bottlenose dolphins, Tursiops truncatus (Montagu 1821) (denoted T1–6; morphometric measurements are reported in Table 1). Data were collected during prescribed swimming (lap trials) and from four long-duration sessions where the animals swam freely (undirected sessions). For both cases, the dolphins were trained by the animal care specialists (ACSs) to wear biologging tags (movement tags, MTags) placed between the blowhole and dorsal fin, in the orientation indicated in Fig. 1. For the lap trials, the dolphins were asked to start at a floating dock (denoted ‘at station’), swim around an ACS located in the water typically 35 m from the dock, and then return to station to complete the lap, all underwater. Note: ‘asked’ is used in this paper to indicate that the dolphins were requested but not required to cooperate; positive reinforcement was presented if a dolphin complied, but it was free to decline an ACS request. During the lap trials, the animals completed up to 16 consecutive laps per session with 1–3 breaths taken between each lap, with session durations ranging from 4.4 to 13.8 min. An example lap trajectory is displayed in Fig. 2A, overlaid on a diagram of the lagoon, and a bathymetric map of the lagoon is shown in Fig. S1.

Fig. 1.

Diagram of forces on a swimming dolphin. (A) This research focused on the thrust and drag forces (Fthrust, Fdrag) that act in the animal's direction of travel, and assumed that the buoyancy and gravitational forces (Fbuoyancy, Fgravity) cancel. The approximate movement tag (MTag) placement on the animal was between the animal's blowhole and dorsal fin, with the fin of the tag parallel to the dorsal fin. Center of mass velocity (vCOM) and acceleration (aCOM), and body angle (θtorso) are indicated. (B) The location of the micro-turbine is indicated (forward speed, vtag), along with the x and z tag accelerometer axes. (C) Cross-sectional view of the tag placement on the animal, with animal-frame coordinate axes as they relate to tag-frame cylindrical axes .

Fig. 1.

Diagram of forces on a swimming dolphin. (A) This research focused on the thrust and drag forces (Fthrust, Fdrag) that act in the animal's direction of travel, and assumed that the buoyancy and gravitational forces (Fbuoyancy, Fgravity) cancel. The approximate movement tag (MTag) placement on the animal was between the animal's blowhole and dorsal fin, with the fin of the tag parallel to the dorsal fin. Center of mass velocity (vCOM) and acceleration (aCOM), and body angle (θtorso) are indicated. (B) The location of the micro-turbine is indicated (forward speed, vtag), along with the x and z tag accelerometer axes. (C) Cross-sectional view of the tag placement on the animal, with animal-frame coordinate axes as they relate to tag-frame cylindrical axes .

Fig. 2.

Example lap trial procedure and extracted kinematics data. (A) Dolphins were asked to perform laps in the Dolphin Quest Oahu Lagoon 2. A nominal lap trajectory is shown by the red loop. Laps began at the dock (beige), looped around an animal care specialist (ACS) in the water (hairpin turn), and ended at the same dock. (B) The depth (light gray), power (black), speed (red), forward acceleration (blue) and relative pitch (dark gray) of a sample lap (gray shaded area) are shown, with consistent-speed (CS) swimming regions highlighted in yellow.

Fig. 2.

Example lap trial procedure and extracted kinematics data. (A) Dolphins were asked to perform laps in the Dolphin Quest Oahu Lagoon 2. A nominal lap trajectory is shown by the red loop. Laps began at the dock (beige), looped around an animal care specialist (ACS) in the water (hairpin turn), and ended at the same dock. (B) The depth (light gray), power (black), speed (red), forward acceleration (blue) and relative pitch (dark gray) of a sample lap (gray shaded area) are shown, with consistent-speed (CS) swimming regions highlighted in yellow.

Table 1.

Lap trial metrics

Lap trial metrics
Lap trial metrics

For the undirected sessions, four dolphins (T2–5) were asked to wear MTags during daylight hours, from 08:30 h to 18:00 h. All animals were tagged between 08:00 h and 08:30 h, and ACS presence typically ceased after 18:00 h, starting the dolphins' transition from a non-fasted to fasted state. Only two dolphins were outfitted with MTags in this manner at one time, with T2 and T5 on one day, and T3 and T4 on another. These animals did not participate in lap trials on their respective days to observe the typical activities of a standard day: free-swimming (no instructions given by the ACSs to the dolphins), standard ACS interactions (behavior imprinting, husbandry, instructed play and feeding sessions) and public sessions (dolphins instructed through a set sequence of interactive and non-interactive behaviors with the public). Dolphin T2's session was further extended to cover a full 24 h day to provide a case study example of day-scale activity, metabolics and distance traveled for a bottlenose dolphin. Experimental data were collected between 13 and 28 May and 10 and 13 October in 2018, and 5 and 9 May in 2019. Animal mass was measured with an Altralite scale (Rice Lake Weighing Systems, Rice Lake, WI, USA) outfitted with a GSE 250SS indicator (GSE Scale Systems). All experimental work was approved by the University of Michigan Institutional Animal Care and Use Committee (IACUC #PRO00010632).

Experimental equipment

MTags are persistently monitoring biologging tags with internal electronics built on the OpenTag3 platform (Loggerhead Instruments, Sarasota, FL, USA). Kinematic sensors include: 3-axis accelerometer, 3-axis magnetometer, 3-axis gyroscope, temperature sensor and ambient pressure sensor. Forward speed was measured using a secondary circuit board with a 1-axis Hall-effect sensor, and a free-spinning uniaxial magnetic micro-turbine mounted in line with the tag fin (Fig. 1B). Rotations created by water moving past the turbine were recorded by the Hall-effect sensor. The accelerometer, magnetometer and gyroscope were sampled at 50 Hz, and all other sensors at 5 Hz, with the data samples archived in CSV format to an onboard microSD flash memory card. Forward speed [vtag (m s−1)] was obtained by converting the micro-turbine's spin rate to meters per second using the linear calibration function:
formula
(1)
where s is the spin rate in Hz. The calibration function used in this experiment was obtained from the MTag-specific results in Gabaldon et al. (2019, their table II, combined fit calibration). The procedure to obtain this calibration is also outlined in Gabaldon et al. (2019). The MTag electronics were powered by a 1100 mA h lithium-ion battery, enough to record continuously for ∼3 days. Four silicone suction cups were used to secure each tag to the animal (Fig. 1). Three MTag units were used for the 2018 experimental sessions, and three different units were used for the 2019 sessions. A detailed explanation of the methods used to prepare the MTag data is given in the Supplementary Materials and Methods (see ‘MTag Data Post-Processing’). Data processing was performed using MATLAB R2021a (version 9.10, MathWorks).

Power estimation

In this study, we assumed that the animal can control buoyancy to balance the gravitational force acting on the body, and combined the remaining forces into net thrust and net drag (Fig. 1), which are related to the animal's motion by:
formula
(2)
where m′ is the total effective mass and aCOM is the center of mass (COM) acceleration of the animal. Total effective mass is defined as the mass of the animal (m) plus the added mass of the fluid displaced by the animal during movement (madd): m′=m+madd. For a gliding animal, madd≈0.2ρV (Gero, 1952; Webb, 1975), while for a swimming animal, madd≈0.2βρV (Weihs, 2002), where β is the ratio of drag during active swimming versus gliding, ρ is the density of the fluid medium, and V is animal volume. Weihs (2002) notes that β ranges from 1 to 5 (Skrovan et al., 1999), but used the functional range of 1<β≤3. As explicit values of β are not known for these dolphins and swimming conditions, this paper operates with the assumption of β=2 as a conservative estimate:
formula
(3)
Here, ρ is the density of seawater (1030 kg m−3) and V was obtained from a 3D CAD model of an animal (dolphin T1 was used as the representative example). Power (P) is related to force (F) and velocity (v) through P=Fv, so after applying this definition to Eqn 3, the power generated by the animal during swimming locomotion is then:
formula
(4)
MTag-estimated speed (vtag) was used to approximate the COM velocity (vCOM), and aCOM was approximated by numerically differentiating vtag and smoothing the result using a 2 s moving average to reduce noise.
Drag acting on the animals was modeled according to Webb (1975):
formula
(5)
where As is the wetted surface area of the animal and CD is the drag coefficient. However, Eqn 5 does not account for the increased drag due to surface drag effects (Au and Weihs, 1980). A depth-dependent multiplier, γ, was applied to the drag coefficient to account for surface effects: CDd=CDγ. Applying the modified drag coefficient to Eqn 5 yields:
formula
(6)

Animal surface area was obtained using As=0.08m0.65, where m is animal mass (Fish, 1993). Non-depth-normalized drag coefficients were computed using CD=16.99Re−0.47 (Fish et al., 2014), where Re=Lvtag/ν, L is animal length and ν=1.044×10−6 m2 s−1 is the kinematic viscosity of seawater. Values for γ range from 1 to 5.05 for underwater swimming, and are dependent on the number of body diameters the animal is below the surface (Fig. 3B). The depth/body diameter to γ relationship was digitized from a graphical figure in Hertel (1966), and interpolated using a smooth spline (no explicit expression was available). Note that β is unnecessary in Eqn 6 as Fish et al. (2014) estimated CD during active swimming. As before, vtag is used as the best approximation of vCOM.

Fig. 3.

Illustration of animal drag multiplier γ and its relationship to depth. (A) Example dive profile for dolphin T2, with high-γ regions (γ≥1.5) in blue and low-γ regions (γ<1.5) in red. Depth is estimated for the location one animal-radius below the point of tag placement (star), so the depth will not read 0 m during a surfacing event (i.e. only the blowhole is at the surface with the rest of the body underwater). (B) Plot of the drag multiplier γ due to an animal's proximity to the surface, as a function of body diameter below the surface (high-γ in blue, low-γ in red). The multiplier maximizes at −0.5 depth/body diameter, at γ=5.05. Note: the true depth versus γ relationship presented here is specific to dolphin T2; only the depth/body diameter versus γ relationship applies in the general case. (C) Plot of the drag multiplier γ experienced by dolphin T2 during this dive profile, temporally aligned with A (color code as in A and B).

Fig. 3.

Illustration of animal drag multiplier γ and its relationship to depth. (A) Example dive profile for dolphin T2, with high-γ regions (γ≥1.5) in blue and low-γ regions (γ<1.5) in red. Depth is estimated for the location one animal-radius below the point of tag placement (star), so the depth will not read 0 m during a surfacing event (i.e. only the blowhole is at the surface with the rest of the body underwater). (B) Plot of the drag multiplier γ due to an animal's proximity to the surface, as a function of body diameter below the surface (high-γ in blue, low-γ in red). The multiplier maximizes at −0.5 depth/body diameter, at γ=5.05. Note: the true depth versus γ relationship presented here is specific to dolphin T2; only the depth/body diameter versus γ relationship applies in the general case. (C) Plot of the drag multiplier γ experienced by dolphin T2 during this dive profile, temporally aligned with A (color code as in A and B).

The final expression for thrust power is then:
formula
(7)
A non-dimensional form of thrust power was also computed to better compare animals of varying length and mass:
formula
(8)
The specific powers used in this non-dimensionalization were defined in order to apply a multidimensional normalization through both morphometrics [m (kg), L (m)] and a common physical constant (gravitational acceleration g=9.8 m s−2) to fully cancel the units of Pthrust (W). Expressions for both Pthrust and Pt,nd were generated in the form of weighted zero-intercept power fits in the format of , where b and c are fit coefficients. These are denoted and , respectively. In both cases, data samples were weighted according to the inverse of the number of samples per individual.

Work and COT estimation

Work estimates [Wthrust (J)] were obtained by numerically integrating sequences of Pthrust data using a trapezoidal sum. Only positive power components were included to ensure that periods where there was no active fluking did not artificially lower the work estimates (e.g. negative power/work during braking). Total distance traveled [TDT (m)] for discrete time intervals were calculated by numerically integrating vtag (m s−1), also using a trapezoidal sum. Instantaneous mechanical cost of transport [MCOTI (J kg−1 m−1)] was obtained with the expression:
formula
(9)
Time-interval mechanical COT [MCOTT (J kg−1 m−1)] was obtained using:
formula
(10)
To provide predictions for the animals' metabolic power, work and COT, additional factors must be considered. To account for the energy loss as chemical energy is converted into mechanical energy, the mammalian metabolic-to-muscle power efficiency (chemical) was taken to be ηms=0.25 (Massaad et al., 2007). Muscle-to-propulsion power efficiency (mechanical) was taken to be ηsp=0.85 for T. truncatus (Fish, 1998). Using these efficiencies, and the animals' resting metabolic rate (RMR), total metabolic power requirements for propulsion were predicted using:
formula
(11)
where PRMR is the RMR power (W), and the ‘hat’ notation serves to indicate the result is a prediction. Numerically integrating (again only using segments where Pthrust>0) yielded the metabolic work (J). Similar to MCOTI, the metabolic cost of transport [COTmet (J kg−1 m−1)] was predicted using:
formula
(12)

RMR

A previous study at Dolphin Quest Oahu assessed the mass-specific RMR of dolphins T2 and T3, along with two others (see table 1 of van der Hoop et al., 2014, non-fasted, units of ml O2 kg−1 min−1). For the dolphins with unreported RMR (T3 and T4), the group non-fasted mass-specific RMR (6.65 ml O2 kg−1 min−1) was used. RMR in ml O2 min−1 was computed using animal mass (Table 1). RMR was then converted to units of power using 20.1 kJ (or 4.8 kcal) per liter O2, to obtain each animal's respective PRMR,nf (W) (Schmidt-Nielsen, 1997, p. 170), reported in Table 2. A non-fasted/fasted ratio of 1.53 was also computed from the multi-animal RMR results presented in van der Hoop et al. (2014), which is necessary to evaluate T2's night-time metabolic expenditures (the dolphins are assumed to be non-fasted during the day and fasted otherwise). Dividing T2's non-fasted RMR by this ratio yielded a fasted mass-specific RMR of 4.13 ml O2 kg−1 min−1, or PRMR,f=289.0 W. T2 was fed throughout the daytime period, and was considered to be under the non-fasted RMR condition from 08:00 h to 18:00 h, transitioning from non-fasted to fasted from 18:00 h to 22:00 h (the 4 h following the daytime interval: food passage time through a bottlenose dolphin's digestive tract was estimated to be ∼4 h; Kastelein et al., 2003), and under the fasted condition otherwise. The transition from non-fasted to fasted RMR for T2 was then modeled as a linear trajectory from PRMR,nf (442.9 W) at 18:00 h to PRMR,f (289.0 W) at 22:00 h.

Table 2.

Parameters for metabolic cost of transport () curves for dolphins T2–5, including combined curve parameters

Parameters for metabolic cost of transport () curves for dolphins T2–5, including combined curve parameters
Parameters for metabolic cost of transport () curves for dolphins T2–5, including combined curve parameters

Consistent-speed interval segmentation

Periods of animal swimming with minimal speed fluctuations, defined here as consistent-speed (CS) intervals, were identified using a heuristically tuned (manually defined parameter) automated method. First, speed data (vtag) were smoothed using a 2 s Savitzky–Golay filter to produce . This filtering method was chosen for its ability to remove high-frequency noise while preserving a varying-frequency sinusoidal signal's shape. This preserves the low-frequency signal's original amplitude and zero-crossings (e.g. as opposed to a moving-average filter which ‘flattens’ this type of signal). A 2 s moving-window standard deviation was computed for , to produce as a continuous metric of non-noise animal speed variation. Regions with minimal variation in speed would then result in low . However, low alone was not sufficient to identify CS swimming, as small values also occurred when an animal was not moving. An additional check was therefore necessary to identify whether the animal was actively swimming (as opposed to gliding: in motion but minimal actuation). As fluking generates positive Pthrust, this secondary check was performed by eliminating low- segments that corresponded to low thrust power. Pthrust data were smoothed with a 2 s Savitzky–Golay filter to produce (for noise reduction), which was used for this analysis. A dolphin was then considered to be swimming in the CS condition when ≤0.045 m s−1 and ≥50 W, both thresholds heuristically determined. Conversely, an animal that was actively swimming but not in the CS condition was assumed to be in the variable-speed (VS) condition.

A note on CS versus steady-state swimming

Research on mammalian energetics involving second-scale measurements of metabolic expenditure (e.g. respirometry, heart rate) require that the measurements be taken when the subject has reached its aerobic steady-state activity level (Hoyt and Taylor, 1981; Williams et al., 1992). This ensures that the subject's metabolic rate is not underestimated: for example, for respirometry, O2 consumption is under-reported during anaerobic activity, and for heart rate monitoring, the beat frequency rises during anaerobic activity until steady-state aerobic activity is realized. In contrast, the approach presented in this paper does not rely on metabolic measurements, but attempts to predict them through a combination of fluid-dynamic estimates of mechanical power and a set of animal-specific assumptions (muscle and joint efficiency, animal RMR). Consequently, the results in this paper have no explicit relationship to or reliance on whether the animal is in the steady-state metabolic condition.

The CS condition is then postulated to represent a period of time where an animal was assumed to only engage in travel swimming, as VS swimming may include a mixture of travel and social behaviors. We propose the use of the CS condition as a proxy useful in predicting metabolic COT, similar to the results obtained by metabolic measurements during steady-state propulsion: in the circumstance where a subject's RMR is known, coupling it with an estimate of propulsive work while accounting for chemical and mechanical efficiencies can yield a total metabolic COT (Eqn 12 in this paper; complementary to the cost of locomotion computation in eqn 1 of Allen et al., 2022). This aligns with the established understanding of total COT being a combination of the propulsive cost (van der Hoop et al., 2014: ‘locomotor cost’; Allen et al., 2022: ‘cost of locomotion’) and an animal's RMR. In summary, while this approach relies on the assumption that the animal's chemical and mechanical efficiencies and RMR are accurately known, we promote this approach as a proxy for predicting an animal's total metabolic cost during propulsion for these two main reasons: (1) the results using this method are not heavily skewed during anaerobic activity because of the lack of reliance on direct metabolic measurements; (2) the combination of propulsive cost and RMR is an established approach for breaking down total metabolic COT into its constituent components. The limitations of this approach and its assumptions are further addressed in the Discussion.

Metabolic power and COT versus speed

In order to find direct expressions from animal speed to and for dolphins T2–4, the best-fit relationships between their undirected CS low-γ (γ<1.5) daytime (08:30–18:00 h) Pthrust and vtag data were computed per-animal and then applied to Eqns 11 and 12. A zero-intercept third-order polynomial was used for the initial fit (corresponding to the relationship), and fitted using the least squares method, yielding the expression:
formula
(13)
where q, r and s are fitting coefficients, and the ‘bar’ notation serves to indicate that the result is computed using a best-fit curve. These curve fits were applied to Eqn 11 to produce the metabolic power/speed relationship:
formula
(14)
was also applied to Eqn 12 to produce the general form of the metabolic COT/speed curve relationship:
formula
(15)
A separate thrust power fit was performed using the combined data of all four animals, with the following modifications: (1) speed was converted from m s−1 to body lengths/second (BL s−1) to normalize by animal size; (2) power values were normalized by mass prior to the fit; (3) data points used for the fit were weighted according to the inverse of the number of points per animal to equalize the dependency of the fit on each animal. The combined-data form of Eqn 15 is then:
formula
(16)
where lc is the mean length of dolphins T2–5, vbl=vtag/lc is the speed in BL s−1, and PRMR,c is the mean RMR of dolphins T2–5. Note the additional lc term in the denominator of the expression outside the parentheses: this is included to ensure remains in units of J kg−1 m−1.

Data aggregation and statistical comparisons

Swimming was characterized using animal speed, work/power, distance traveled, COT and achieved γ values. For the lap trials, swimming metrics were calculated for both the whole lap and identified periods of CS swimming for each dolphin. Metrics were also calculated for periods of low-γ (γ<1.5) CS swimming. The γ<1.5 threshold was heuristically defined, and was chosen to ensure less than 1/3 of the animal's effort during CS swimming was related to secondary drag modifiers (wave drag). Metrics were computed and then averaged with respect to each animal to produce the final results.

For the undirected swimming trials, parameters were calculated for periods of general, VS and CS swimming. General swimming parameters were computed from the entire dataset. VS intervals were first concatenated and then processed appropriately according to metric type (e.g. averaged, numerically integrated, etc.). CS metrics were computed in the same format as for the lap trials (first per interval, then averaged per animal) to facilitate comparison with results from the lap trials.

Statistical significance tests are used throughout the paper to lend support to findings that rely on differences (or lack thereof) in inter- or intra-animal propulsive behaviors, on comparisons of a particular result with a baseline value, or on statements of how an animal divided its time between differing swimming conditions. The two-sample t-test was used to evaluate the statistical significance between parameter means during different swimming conditions for one animal. When comparing one mean versus a target value (e.g. against γ=1.5), the one-sample t-test was used. The one-way analysis of variance (ANOVA) approach was used to compare parameter means between animals, and significance tests were performed using a multiple comparison of means. This approach was also employed to test the statistical differences between the lap trial speed versus power results and those found in existing literature. This was performed by computing the percent-error residuals of the speed versus power data samples from this research and from the literature (Fish, 1998; Fish et al., 2014), with respect to , and evaluating the differences in residual means through a one-way ANOVA. Inter- and intra-animal comparisons for the undirected sessions were evaluated using the same statistical tests as for the lap trials. Further comparisons were made between the achieved γ values for dolphins T2–5 during active swimming: full-lap and undirected general γ results were compared, and lap trial and undirected γ results while in CS swimming were compared. The statistical significance was again evaluated using the two-sample t-test. Built-in MATLAB functions ttest, ttest2, anova1 and multcompare were used for this analysis. Differences were evaluated using a significance level of α=0.01.

For both lap trials and undirected sessions, the statistical significance of metrics majorities (e.g. ‘T2 spent a significant majority of CS swimming in low-γ’) were evaluated using the binomial test. This was performed by evaluating the cumulative likelihood of each animal spending 0–50% of the swimming condition in question (e.g. CS) in the relevant sub-condition (e.g. low-γ), given the measured likelihood extracted from the data. The built-in MATLAB function binopdf was used for these computations, which were evaluated against a significance level of α=0.01.

Lap trials

General metrics for ‘full laps’, differentiated by animal, are reported in Table 1. Standard deviations are also reported for metrics computed as means. Each dolphin completed between 28 (T4) and 70 (T5) laps across all trials, for a total of 282 laps. Mean speed per lap varied between animals, from T4 at 2.6±0.4 m s−1 to T1 at 4.3±0.4 m s−1, with each dolphin exhibiting significant differences between at least three others (varying by individual) in the group. The mean power per lap ranged from T4 at 0.27±0.10 kW to T1 at 1.71±0.46 kW, with T1 and T4 significantly differing from all conspecifics and the other four not exhibiting significant differences from within that subgroup (T2, T3, T5, T6). The mean total distance traveled per lap was relatively consistent across dolphins T1–5, from 78.1±6.6 to 86.3±5.0 m, with T6 being the only animal with statistically significant differences from all conspecifics at 62.2±7.8 m. Providing a sense of animal travel efficiency, mean MCOT ranged from the lowest value of 0.61±0.25 J kg−1 m−1 for T4, up to 1.50±0.38 J kg−1 m−1 for T1, with T1 and T4 again significantly differing from all other conspecifics. In terms of animal wave drag mitigation tendencies, dolphins T2–6 maintained γ levels significantly under 1.5 (mean±s.d. 1.60±0.39).

CS segments were extracted from each lap, for a total of 469 segments. Individual animal metrics, including CS segment counts and percentage time (and distance) per lap spent in CS swimming, are reported in Table 1 (‘CS’ section). A sample lap in the lagoon is depicted in Fig. 2A, and close-up examples of CS swimming are shown in Fig. 2B (yellow highlight). The majority of the dolphins (T1–5) spent approximately one-third of their lap time (and distance traveled) in the CS condition, with T6 as the significant outlier at 7.5±13.0% (distance per lap: 7.4±12.7%), with T1–5 sustaining no significant differences for percentage of lap time or distance traveled during CS swimming. While mean CS γ values were under 1.5 for all animals, T1 was the only dolphin not significantly under this threshold (mean γ=1.42±0.59), and it had the highest percentage of CS segments in the γ≥1.5 condition at 25.6% (N=86). While T1's mean γ was not significantly lower than 1.5, T1 still spent a significant majority of CS swimming in the γ<1.5 condition, as did all other dolphins. Dolphins T2–6 maintained CS γ values close to 1, with T2–4 and T6 having all segments under 1.5, and T5 having one segment (0.8%, N=119) above this threshold. CS swimming typically demonstrated the dolphins' peak travel speeds under these conditions, and all animals produced higher average speeds than their full-lap values, with only dolphins T4 and T6 not increasing significantly. In contrast, there were no statistical differences for any dolphin in power or MCOT when comparing CS with full-lap swimming.

There were 446 low-γ CS intervals across all animals (Table 1, ‘CS for γ<1.5’ section). Fig. 3A illustrates an example dive for T2, with the low- and high-γ segments indicated by color. Inter-animal swimming metrics are more directly comparable using only low-γ versus all CS segments, as T1 had mixed high- and low-γ conditions, whereas T2–6 almost exclusively operated in low-γ. Mean low-γ CS interval speeds varied from 2.8±0.6 m s−1 for T4 to 5.2±0.5 m s−1 for T1, with T1 and T3 significantly differing from all others. Similarly, mean power ranged from 0.28±0.17 kW for T4 to 1.71±0.45 kW for T1, and mean MCOT from 0.61±0.25 J kg−1 m−1 for T4 to 1.32±0.25 J kg−1 m−1 for T1. For both metrics, T1 was significantly higher than the others, with T2–5 differing significantly from at least three other animals (varying by animal). Because of T6's smaller sample size, it differed insignificantly from all but T1 and T3 for speed, and from T1 alone for power and MCOT. Low-γ CS segment durations did not differ significantly, with a mean of 4.20±1.89 s for all segments across all animals.

The per-segment means of length-normalized dolphin speed (vbl=vtag/L) and non-dimensional thrust power (Pt,nd) data from the low-γ segments were correlated using a weighted zero-intercept power fit (Fig. 4A). The non-dimensionalized fit relationship was found to be (adjusted R2=0.93, N=446). The per-segment means of dolphin speed (vtag) and thrust power (Pthrust) data from the low-γ segments were correlated using a separate power fit for comparison with the literature (Fig. 4B: Fish, 1998; Fish et al., 2014). The fitted relationship was found to be (adjusted R2=0.93, N=446). These speed versus power results differed significantly from all but Chopra's hydrodynamic model (Fig. 4B filled gray circles; Fish, 1998).

Fig. 4.

Comparison of animal speed versus power for CS time segments. (A) Dimensionless animal swim trial data (open symbols) were fitted to a zero-intercept power curve (black line). Each data point corresponds to the average body length-normalized speed and dimensionless power of a single CS segment, and only segments with γ<1.5 were used (average duration 4.20 s). (B) The unmodified CS experimental data (open symbols) and power fit (black line) compared with results from existing literature (colored circles: Fish, 1998; Fish et al., 2014). Adjusted R2 values are reported.

Fig. 4.

Comparison of animal speed versus power for CS time segments. (A) Dimensionless animal swim trial data (open symbols) were fitted to a zero-intercept power curve (black line). Each data point corresponds to the average body length-normalized speed and dimensionless power of a single CS segment, and only segments with γ<1.5 were used (average duration 4.20 s). (B) The unmodified CS experimental data (open symbols) and power fit (black line) compared with results from existing literature (colored circles: Fish, 1998; Fish et al., 2014). Adjusted R2 values are reported.

Undirected extended sessions

Animal metrics for undirected extended sessions for dolphins T2–5 are reported in Table 3, for the following swimming modes: general, VS, CS and CS for γ<1.5 (as a subset of the CS dataset). Undirected swimming data used to generate Table 3 were collected from 08:30 h to 18:00 h for consistent comparisons between animals. To characterize active swimming profiles, only data where speed was non-zero were used. During the day, the total distance traveled by the dolphins ranged from 19.7 km for T4 to 45.23 km for T5, with total propulsive work estimates ranging from 1.1 MJ for T4 at 4.7 MJ for T2. For the general condition, mean γ values were above 1.5 for all but T5 at 1.48±0.97, with T3 the highest at 1.96±1.41. Percentage active time ranged from 59.4% for T4 to 86.2% for T5. General condition mean speed ranged from 1.0±0.57 m s−1 for T4 to 1.96±0.74 m s−1 for T2, propulsive power from 54±102 W for T4 to 212±227 W for T2, and estimates from 3.26±1.85 J kg−1 m−1 for T3 to 4.26±2.12 J kg−1 m−1 for T4. Estimates for mean power and speed were lower for all animals in the VS versus general condition, while γ and differences varied per individual. All inter-animal ANOVA tests, inter-condition two-sample t-tests, and threshold evaluation one-sample t-tests for these conditions were statistically significant, save for T4's general versus VS γ comparison.

Table 3.

Undirected monitoring session metrics

Undirected monitoring session metrics
Undirected monitoring session metrics

All four dolphins engaged in CS swimming for under 20% of their active time, from 2.7% (N=167) for T4 to 19.0% (N=1142) for T2. As a result, the work done and distance traveled while in the CS condition varied widely, from 0.1 MJ and 1.1 km for T4 to 1.0 MJ and 9.4 km for T2, respectively. Mean γ ranged from 1.59±1.04 for T5 to 2.52±1.55 for T3, with that for T3 being significantly larger than for the other dolphins, and all but T4 significantly over the γ=1.5 threshold. The number of segments ranged from 167 for T4 to 1142 for T2, for a total of 2695 segments. Mean CS condition speed and power for all four dolphins were significantly higher than their general and VS condition results. differences fluctuated per individual, with T2 and T4's CS results significantly lower than their general condition results, and T3 and T5 having insignificant changes. T3 and T5 had significantly higher γ while in the CS condition than in general or VS swimming conditions, while T2 and T4 did not have significant differences in either case. Time spent in CS under the γ<1.5 condition ranged from 43.0% (N=393) for T3 to 72.7% (N=993) for T5, with dolphins T2, T4 and T5 spending a significant majority of CS swimming in low-γ.

The low-γ condition was used for comparisons between the dolphins' CS metrics, as all but T3 operated in this condition for the majority of their CS time. The number of segments ranged from 121 for T4 to 784 for T2, for a total of 1796 segments. All four dolphins achieved their highest mean speeds of the undirected sessions during low-γ CS swimming, from 2.0±0.4 m s−1 for T3 to 2.5±0.5 m s−1 for T2, with values for T2 and T4 significantly higher than those for T3 and T5. Mean power ranged from 129±78 W for T5 to 242±137 W for T2, with values for T2 and T4 significantly higher than those for T3 and T5. Conversely, mean values ranged from 2.52±0.53 J kg−1 m−1 for T3 to 3.39±1.00 J kg−1 m−1 for T4, with all dolphins differing significantly from all others. Low-γ CS segment duration ranged from 3.25±1.35 s for T3 to 3.77±1.93 s for T2, with only T2 versus T3 demonstrating significant separation.

Per-hour mechanical work estimates between the hours of 09:00 h and 18:00 h are shown for dolphins T2–5 in Fig. 5, and are broken up into VS and CS components (these do not account for inefficiencies or RMRs). From 09:00 h to 18:00 h, dolphin T2 produced significantly higher mechanical work per hour than T3–5, regardless of whether VS, CS or total estimates were compared, while T3–5 did not differ significantly in either case. No clear temporal pattern of hourly work emerged, with each dolphin engaging in high and low levels of activity at differing times of day: T2 peaked from 10:00 h to 11:00 h, T3 from 14:00 h to 15:00 h, T4 from 10:00 h to 12:00 h and 14:00 h to 15:00 h, and T5 from 15:00 h to 16:00 h. As an example of undirected CS swimming, a sample of T2's session data is shown in Fig. 5E, with CS segments highlighted in yellow.

Fig. 5.

Comparison of per-hour work (propulsive thrust only) for dolphins T2–5 during extended-duration monitoring. (A) Total propulsive work performed by T2 for each hour interval for a 24 h monitoring session, with day (08:00–18:00 h) and night (18:00–08:00 h) time ranges indicated by the sun and moon symbols. Note: T2's session began at 08:55 h and ended at 09:18 h the next day; however, the final complete hour interval (08:00–09:00 h) was moved to the beginning of this plot for a simpler visualization. (B–D) Total propulsive work performed by dolphins T3–5, respectively, during the day time portions of their respective monitoring sessions. For T2–5, the work per interval bars are separated into variable-speed (VS) and CS components. (E) Extracted 2 min sample of T2's depth (gray), thrust power (black), forward speed (v; red), and forward acceleration (a; blue) to demonstrate T2's swimming profile during a period of ‘high’ activity. Surfacing events are indicated (starred dashed lines), with CS segments highlighted (yellow shading).

Fig. 5.

Comparison of per-hour work (propulsive thrust only) for dolphins T2–5 during extended-duration monitoring. (A) Total propulsive work performed by T2 for each hour interval for a 24 h monitoring session, with day (08:00–18:00 h) and night (18:00–08:00 h) time ranges indicated by the sun and moon symbols. Note: T2's session began at 08:55 h and ended at 09:18 h the next day; however, the final complete hour interval (08:00–09:00 h) was moved to the beginning of this plot for a simpler visualization. (B–D) Total propulsive work performed by dolphins T3–5, respectively, during the day time portions of their respective monitoring sessions. For T2–5, the work per interval bars are separated into variable-speed (VS) and CS components. (E) Extracted 2 min sample of T2's depth (gray), thrust power (black), forward speed (v; red), and forward acceleration (a; blue) to demonstrate T2's swimming profile during a period of ‘high’ activity. Surfacing events are indicated (starred dashed lines), with CS segments highlighted (yellow shading).

The per-hour mechanical work estimates from T2's complete full-day session are shown in Fig. 5A [with day (08:00–18:00 h) and night (18:00–08:00 h) indicated by the sun and moon, respectively]. During this 24 h period, dolphin T2 swam a total of 78.2 km (day: 47.9 km, night: 30.3 km), and had a total mechanical work of 6.64 MJ (day: 5.07 MJ, night: 1.57 MJ). Note: these day values do not align with T2's general estimate in Table 3, as the table used the time interval 08:30–18:00 h, while the full day interval is 08:00–18:00 h. T2's 08:00–09:00 h interval was sourced from the end of its 24 h session (immediately before tag removal in the morning). Accounting for efficiency losses and incorporating the varying RMR across the 24 h interval, T2 yielded a total metabolic work estimate of =62.9 MJ, or 15,030 kcal.

curves were generated for dolphins T2–5 by fitting to their low-γ CS condition speed versus power data in the form of Eqn 13 (for T2, only the daytime data were used for consistency). Applying their RMR estimates and the efficiency factors ηms and ηsp produced individual curves in the form of Eqn 15 (Fig. 6A–D). A combined-data curve was generated using a weighted fit combining data for T2–5 (Eqn 16), shown with the aggregate data in Fig. 6E, and with all individual curves in Fig. 6F. Probability density functions (PDFs) of each animal's low-γ CS condition speed data are included in their corresponding subplots of Fig. 6. The combined-data speed PDF was generated by averaging T2–5's individual PDFs, to account for the high sample count imbalance. All computed curves demonstrated good fits to their data, with a minimum adjusted R2 of 0.88 (T2, N=784). Each PDF's absolute peak is indicated with a vertical dashed line, and its intercept on the corresponding curve is indicated by an open circle. To compare these data with existing literature, the Yazdi et al. (1999)  curve is overlaid on each subplot of Fig. 6. The fit parameters, optimal range speed and results, maximum range speed intervals (MRSIs, defined as the speed interval on a curve within ±10% of the minimum) and speed PDF results for all fitted curves are reported in Table 2. Each dolphin's percentage of low-γ CS segments performed within their MRSI is reported in Table 2, ranging from 45.9% (N=784) for T2 to 84.0% (N=169) for T3. Despite dolphins T3–5 all spending over 50% of their low-γ CS condition in their MRSI, only T3 and T5's estimates represented significant majorities. The corresponding percentage for the combined-data case was 67.5% (N=1796). PDF peaks for all dolphins except T2 fell within their MRSIs; T2's PDF peak was 0.02 m s−1 over its upper bound. The combined-data PDF peak also fell within its MRSI bound. A sensitivity analysis was performed to evaluate the effects of potential errors in ηms and CD on the fit computation of (see Supplementary Materials and Methods, ‘Cost of Transport Sensitivity Analysis’, Fig. S2, Table S1).

Fig. 6.

Metabolic cost of transport curves for dolphins T2–5. (A–D) Individual animal predicted metabolic cost of transport () curves for animals T2–5, respectively, and the data points used to generate each curve. Each data point corresponds to a low-γ segment of CS swimming, performed between 08:00 h and 18:00 h (daytime). The maximum range speed intervals are denoted by the short vertical bars on each curve. Probability density functions (PDFs) of each animal's speed profile during these segments (low-γ daytime CS swimming) are shown at the bottom of each plot, and correspond to the right-hand y-axes. (E) Combined metabolic cost of transport () curve generated using aggregate data from dolphins T2–5, along with the data points for each animal. (F) Aggregate curve for dolphins T2–5, along with the individual curves for each animal. For E and F, the data and resultant curves have been normalized along the x-axis from units of m s−1 to BL s−1 to better compare animals of different sizes. The speed PDFs for E and F represent the combined distributions for the four dolphins. The x-location of the primary peak of each PDF is projected onto its corresponding curve (vertical dashed line and open circle), to indicate the animal's (or group's) general preferred speed as it relates to . Adjusted R2 values are reported for each generated curve in their respective plots. The COT curve from Yazdi et al. (1999) is included for reference on all plots (black), along with their reported maximum range speed interval (short vertical bars) and preferred animal speed (circle).

Fig. 6.

Metabolic cost of transport curves for dolphins T2–5. (A–D) Individual animal predicted metabolic cost of transport () curves for animals T2–5, respectively, and the data points used to generate each curve. Each data point corresponds to a low-γ segment of CS swimming, performed between 08:00 h and 18:00 h (daytime). The maximum range speed intervals are denoted by the short vertical bars on each curve. Probability density functions (PDFs) of each animal's speed profile during these segments (low-γ daytime CS swimming) are shown at the bottom of each plot, and correspond to the right-hand y-axes. (E) Combined metabolic cost of transport () curve generated using aggregate data from dolphins T2–5, along with the data points for each animal. (F) Aggregate curve for dolphins T2–5, along with the individual curves for each animal. For E and F, the data and resultant curves have been normalized along the x-axis from units of m s−1 to BL s−1 to better compare animals of different sizes. The speed PDFs for E and F represent the combined distributions for the four dolphins. The x-location of the primary peak of each PDF is projected onto its corresponding curve (vertical dashed line and open circle), to indicate the animal's (or group's) general preferred speed as it relates to . Adjusted R2 values are reported for each generated curve in their respective plots. The COT curve from Yazdi et al. (1999) is included for reference on all plots (black), along with their reported maximum range speed interval (short vertical bars) and preferred animal speed (circle).

To summarize, dolphin T2 had the second-shortest active time but had the second-longest distance traveled and performed the most propulsive work over the 9.5 h period. T2's average low-γ CS swimming speed (2.5±0.5 m s−1) fell just outside its MRSI (Fig. 6). This dolphin engaged in several bouts of extended swimming at these higher speeds, resulting in the greatest percentage time spent in the CS condition (∼19%) of the four animals. A significant majority of T2's CS swimming was at low-γ (N=1142), reducing excess drag during higher-speed swimming. This combination of high swimming speed and partially optimal depths yielded the second-highest overall (3.61±2.07 J kg−1 m−1).

In contrast, T4 spent a negligible amount of time swimming at in the CS condition (<3%), and swam at shallow depths with increased surface drag. T4 also had the lowest observed swimming speed during the undirected session. This resulted in the highest (4.26±2.12 J kg−1 m−1) of the four, where estimated basal metabolic expenditure dominated the overall cost. While T3 also spent a negligible amount of time in CS swimming (∼5%) and did not prioritize low-γ overall (with the highest general γ of the group), its preferred speed range yielded the lowest general (3.26±1.85 J kg−1 m−1) of the group. Finally, T5 presented itself as the endurance swimmer, with the second-highest general speed and highest active time while achieving a mean γ in the optimal range. During unprescribed swimming, T5 swam the longest distance (45 km) while maintaining the second-lowest overall (3.40±1.76 J kg−1 m−1). Further, T5 spent a non-negligible amount of time in the CS condition (12%), with a significant majority of CS swimming spent in low-γ (N=993), and a significant majority of that within its MRSI (N=722).

Comparison of lap trials versus undirected sessions

Dolphins T2–5 exhibited significantly higher speed, power and MCOT during the lap trials versus the undirected sessions. All dolphins swam in a state of higher γ when comparing the general undirected condition with the full lap condition, with all but T4's data differing significantly. Similarly, all dolphins had significantly shallower CS swimming depths (higher γ) during the undirected sessions. The percentage time spent in the CS condition was lower during the undirected sessions (2.7–19.0%) than in the lap trials (32.7–34.5%) for all animals. T2-5's lap trials all resulted in over 99% low-γ occupancy (N=373) while in the CS condition, while their undirected sessions showed between 43.0% (N=393) and 72.7% (N=993) of the CS swimming in this more optimal mode. When comparing the low-γ CS results, the lap trials showed significantly higher speed, power and MCOT than the undirected sessions, for all animals. For each dolphin, the percentage of low-γ CS segments within their MRSI can be used to evaluate how they utilized optimal travel speeds. For the undirected sessions, MRSI occupancy ranged from 45.9% (N=784) for T2 to 84.0% (N=169) for T3 (Table 2), while the lap trials showed as little as 0% (N=85) for T3 and as high as 6.3% (N=95) for T2.

This work presents the hardware and hydrodynamic analysis framework to enable the investigation of kinematics, kinetics and energetics of dolphins during prescribed and self-selected swimming. The drag-based hydrodynamic model used in this research included animal-specific morphometric measures and accounted for surface drag effects. When paired with continuous measurements of animal speed, acceleration and depth, this model enabled direction-of-travel work and power estimates during mid- to high-energy prescribed swimming tasks and lower-energy undirected swimming. Further, the near-instantaneous measurement capability of the biologging tags allows for tracking energetics below the 1 s scale, offering higher flexibility in both data analysis and experimental time frames when compared with traditional methods (e.g. doubly labeled water, respirometry). These data represent hundreds of fluke strokes, an order of magnitude greater than current results in the literature (Fish, 1998; Fish et al., 2014). This approach also enabled data collection for extended periods of time (9.5–24 h continuously) during daily life, providing an opportunity to investigate how energetic cost may influence movement. Our results indicate that the animals modulate their swimming depth to reduce surface drag during high-speed straight-line swimming, and tend to select speeds that minimize COT during the undirected condition. However, there was variability in the self-selected speeds that suggests energy optimization is not the only factor that drives animal behavior.

Efficiency prioritization in high-effort swimming

During the prescribed swimming trials, data from the 282 laps were collected and analyzed from 6 animals to investigate CS work and power during a straight-line swimming task. Average lap distance ranged from 36.7 to 98.6 m, resulting in 22.4 km of swimming data for analysis. Lap speed ranged from 1.9 to 6.1 m s−1, with the majority of these ACS-tasked speeds faster than self-selected travel speeds reported in the literature (Williams et al., 1993). While the aggregate thrust power results estimated from the lap trials' low-γ CS swimming bouts compared well to Chopra's hydrodynamic model, they occurred at significantly faster speeds than those reported for the computational and DPIV-based models (Fish, 1998; Fish et al., 2014; Fig. 4B).

Throughout the lap trials, all six dolphins swam at depths that reduced surface drag (γ<1.5) over a wide range of speeds, with dolphins T1–5 spending a consistent 33–37% of lap time in the CS condition. Additionally, dolphins T2–6 maintained hydrodynamically efficient depths (minimal deviation from γ=1) during high-effort CS swimming. Only T1 swam at depths with non-trivial penalties while in the CS condition (mean γ=1.42±0.59, 26% of laps γ≥1.5, N=86). Despite the ∼40% increase in drag at this average depth, T1 had the highest mean speed of 5.0±0.55 m s−1, along with the highest per-lap swimming power. The resulting average MCOT for T1 was more than twice as large as that of the most efficient swimmer, T4 (1.59 versus 0.61 J kg−1 m−1). No general group behavior was present in the low-γ CS swimming metrics. Animal speed, power and MCOT all differed significantly within the group to varying degrees, demonstrating an individualized approach to the same general task.

Task condition and its relationship to individual variability

Data collected during the day (08:30–18:00 h) from dolphins T2–5 were used to investigate swimming kinematics and energetics during undirected swimming (aligning with the Hoyt and Taylor, 1981, paradigm of allowing the animals to self-select gaits and speeds for theoretical optimality). Average CS swimming speed was under 2 m s−1 (<0.8 BL s−1), which is comparable to speeds reported in literature (Williams et al., 1993). However, all animals had bouts of deep (low-γ) CS swimming that were faster than expected if the main objective was to minimize cost, including bursts that exceeded 5 m s−1. estimates were evaluated in these inter-animal comparisons to assess the differences in how the dolphins balanced propulsive and basal metabolic expenditures.

While in the undirected sessions, the dolphins self-selected speeds that tended to minimize cost in general, but did not exclusively optimize speed and depth for maximal propulsive efficiency. T5 tended towards swimming parameters that improved its metabolic efficiency: optimal speeds and swimming depths that reduced surface drag (low γ). T3 reduced cost primarily through speed modulation but did not prioritize depths to reduce drag. While T2 swam at speeds at or above its estimated MRSI, it swam primarily at optimal depths at these faster speeds. In contrast, T4 did not appear to optimize speed or depth. However, while T4's overall energy expenditure was not efficient, it was the lowest of the group, potentially indicating that T4 prioritized energy expended per hour rather than per meter.

In contrast to the undirected sessions, during the lap trials the dolphins were given mid- to high-speed swimming tasks. The animals were free to choose the swimming depth for both condition types. The results demonstrated distinct behavioral differences between the two conditions that can be explained using an energetics approach. Average speeds were higher during the lap trials for all animals whether in CS or overall. This resulted in significant energetic penalties as the animals generated thrust to overcome drag forces that increased . To reduce costs at these faster speeds, all dolphins tended to select depths that reduced surface drag, which also had the potential to reduce the time spent accelerating the body mass through water. Notably, dolphins T2–5 swam at deeper depths in the lap trials than in the undirected sessions (overall and in CS swimming), and spent a greater percentage of time in CS swimming.

In contrast, all dolphins swam at reduced swimming speeds during undirected sessions, reducing costs, but swimming depth and time in the CS condition varied between animals. Three of the four dolphins had average CS swimming speeds that fell within their MRSIs, and the group average was close to the minimum of the predicted metabolic cost of transport curve . However, only T5 and T2 (during CS swimming) swam at depths that limited surface drag in any condition for a non-negligible amount of time. While energetics appeared to influence the speed and depth the animals selected during CS swimming, there were a non-trivial number of bouts outside each animal's MRSI during the undirected sessions. This presents a stark separation in their behavior concerning efficiency as it relates to differing conditions: when charged with swimming tasks that required high-cost propulsion, all four dolphins engaged in behavior that mitigated excess effort. When the high base cost requirement was eliminated, each animal settled into individualized swimming behaviors with differing approaches and priorities on overall energetic expenditure and metabolic COT.

Data from future undirected monitoring sessions will enable a further investigation of these initial results. Importantly, a more in-depth contextualization of behavior to animal condition will be key to understanding how environmental factors (e.g. enrichment, animal care specialists: Zhang et al., 2021) or interactions with conspecifics influence animal movement and behavior in addition to energetic cost. As absolute travel efficiency was not prioritized across the group, it will be important to evaluate the effects of social interactions on swimming behavior patterns on an individual basis.

Undirected 24 h session

Dolphin T2's 24 h session was presented as a case study example of a continuous measure of animal activity and estimated energy expenditure at the day scale. These data yielded an overall picture of T2's activity, with variable levels during the day and reduced activity at night. T2's activity peaked in the morning (10:00–11:00 h), decreased during the rest of the day, and dropped at night (Fig. 5A). This behavioral pattern has been reported in the literature for managed dolphins (Sekiguchi and Kohshima, 2003), where animal activity level is associated with the schedule put in place by ACSs. The 24 h session also provided an opportunity to collect baseline data about T2's movement in the lagoon environment. The 78 km that T2 traveled during this period was higher than expected, and represents the first persistently estimated 24 h range measurement for a bottlenose dolphin, as well as the first day-scale range estimate for a cetacean in a managed setting. Interestingly, T2 still swam ∼30 km at night when activity levels were significantly reduced. The longest distance traveled for wild dolphins reported in the literature is 47 km day−1 between two separate sightings (Defran et al., 1999). This was recorded as a minimum distance as the estimate only considered the regional travel distance and not local VS travel (foraging, diving, etc.). Bottlenose dolphins can have extended habitat ranges (Defran et al., 1999; Dinis et al., 2021), and the total distance traveled observed in this research indicates that current range observations for these animals may be severe underestimates. This is further supported by the fact that T5 swam over 45 km in only 9.5 h during its undirected session, only 4% below the per-day range observed by Defran et al. (1999).

T2's 24 h metabolic energy usage was estimated to be Wmet=15,030 kcal, which was compared with its recorded daily caloric intake. T2 was fed 13,337 kcal of a mixture of mackerel and squid each day for the month of May 2018. Assuming an assimilation efficiency of 87%, T2 had ∼11,600 kcal of available metabolic energy per day (Bejarano et al., 2017). By this measure, Wmet overestimated T2's metabolic energy use by 30% for that day. This estimate falls below all three allometric energy requirement estimates produced by the methods presented in Bejarano et al. (2017). When evaluated using T2's mass of 209 kg and accounting for assimilation efficiency, we obtain the following field metabolic rate (FMR) values: FMRmeasured=24.4 kcal day−1 (model based on direct measurements from bottlenose dolphins), FMRKleiber=18.0 kcal day−1 (adjusted general model based on Kleiber's scaling equation; Kleiber, 1975) and FMRbodymass=16.4 kcal day−1 (model based on percentage body mass consumed per day for delphinids in human care), which is closest to the estimate obtained in this study. As T2's 24 h prediction represents a single data point, it will be important to run additional 24 h monitoring sessions to produce generalized statistics.

comparisons with the literature

When comparing the best-fit curves for dolphins T2–5 individually and in the combined-data case (Table 2) with the curve from Yazdi et al. (1999), all models showed values that were higher over all speed ranges, penalized higher speeds more heavily, and had slower and smaller-range optimal speed regions (Fig. 6). T2 and T5's CS low-γ speed PDF peaks fell within the optimal speed region proposed by Yazdi et al. (1999), while those for T3 and T4 did not. The peak preferred speed for the dolphins in Yazdi et al. (1999) was inferred through their reported mean and their statement that speeds were normally distributed. T3–5's speed PDF peaks all fell within their MRSIs (with T2 0.02 m s−1 above), and all peaks fell within the combined-data MRSI. Only T3 (N=169) and T5 (N=722) spent a significant majority of their low-γ CS time within their MRSIs, while T3–5 (N=1012) spent a significant majority of this time within the combined-data MRSI.

It is important to note that Yazdi et al. (1999) measured metabolic expenditure through respirometry (exempting their data from efficiency conversions), and estimated swimming speed through direct observation (extrapolation from duration and distance traveled). They observed speeds in the range 0.8–2.6 m s−1, with speeds above this range (denoted ‘high speed’) assigned a general value of 3 m s−1 (there was stated uncertainty on the accuracy of the high-speed values as they could not be reliably measured). In contrast, the speeds used in our estimation ranged from 1.03 to 5.33 m s−1, all of which were directly measurable by the MTag's onboard sensor. A separate study performed by Williams et al. (1992) used electrocardiography to estimate bottlenose dolphin metabolic rates (using a data-driven relationship of heart rate to metabolic rate) as they kept pace with a boat traveling in open water. These speed and power data were used to produce COTmet estimates for their animals, and for higher speed travel (non wave-riding), Williams et al. (1992) found that their dolphins had a mean COTmet of 2.85 J kg−1 m−1 at 2.9 m s−1. At the same speed, the model in Yazdi et al. (1999)  predicted 1.21 J kg−1 m−1 (58% lower), and the model from our research predicts 3.48 J kg−1 m−1 (22% higher). As each study used widely different methods of estimating metabolic rate (respirometry, electrocardiography and fluid dynamics+RMR), a further investigation where all three data streams are used to estimate cost is warranted.

Limitations

The model presented in this paper uses a drag coefficient (CD) formula estimated from low-amplitude swimming (Fish et al., 2014). As effective drag coefficients vary dependent on the degree of fluke oscillation (Weihs, 2002; Fish et al., 2014), the model could be made more accurate by accounting for this variation, as opposed to using only small-amplitude CD values at present. Further, the added-mass coefficient was obtained from an estimate originally used for fish (Gero, 1952; Webb, 1975), so the inclusion of dolphin-specific coefficients would remove an additional source of error. Finally, there is evidence to indicate dolphin tails are capable of spring-loading during fluking (Pabst, 1996; Long et al., 1997), which may yield propulsive efficiency gains through mechanical optimization. Including this effect in the energetics model would require animal-specific soft-body mechanics simulations, which is outside the scope of this research.

In terms of the limitations concerning animal RMRs, during the 24 h trial, the assumed RMR accounted for a significant portion of the estimated daily metabolic energy requirements of the animal (50.3% for T2's 24 h session), and errors in this rate would impact the accuracy of the estimate. While this research uses estimates measured by van der Hoop et al. (2014), the tagging sessions occurred several years later, and updated measurements of the dolphins' RMRs would improve model accuracy. The RMR estimates were collected exclusively during the day, so additional samples must be collected at night to evaluate any RMR dependency on circadian rhythm.

Finally, it is necessary to discuss the limitations of predicting metabolic cost by coupling tag-estimated thrust, propulsive and chemical inefficiencies, and an animal's RMR. It has been established that swimming animal thermoregulation costs can be offset dependent on the waste heat produced by muscle activation (Hind and Gurney, 1997; Yazdi et al., 1999), and through intrabody heat flow attenuation during diving (Williams et al., 1999). As thermoregulation is a core component of mammalian RMR, accounting for these offsets may lower the RMR component of total energetic cost during activity. Further, circulatory system responses to varying activity levels, including differences between aerobic and anaerobic conditions, result in fluctuations in non-propulsive energetic costs (Williams et al., 1993), and digestive energetic costs vary significantly dependent on time since last feeding (Kastelein et al., 2003; van der Hoop et al., 2014). While the transition from fed to fasted was assumed to be linear for this research, an experimentally measured relationship is necessary to improve the RMR estimate during the day between feeding times and at night during the transition to the fully fasted state. In general, the missing or incomplete parameters discussed within this subsection were not addressed in this study as they have yet to be measured or estimated through simulation, and would be the focus of future research. However, it is important to note the limitations and assumptions of the proposed model, if only to provide an understanding of the current state of the research and a potential path forward.

Conclusions

Pairing measured kinematics with a hydrodynamic thrust model yielded insights into the propulsive behavior of managed bottlenose dolphins and how they modulate cost during swimming. When the dolphins were tasked with swim speeds outside their predicted maximum range speed interval, the animals reduced drag forces by swimming deeper and spending more time at consistent speeds. When the animals self-selected their swimming speeds, effort and efficiency were moderated through a blend of individualized approaches regulating speed and wave drag. The ability to measure the animals' kinematics using a biologging tag was a key enabler for the energetics framework used in this research. This approach resulted in data from hundreds of fluke strokes for analysis, and yielded new knowledge on the day-scale energy budgets and range capabilities of bottlenose dolphins in a managed environment. Estimated propulsive power combined with a resting metabolic rate from the existing literature were used to compute day-scale total metabolic expenditure and COT. Importantly, these tags and the proposed analysis framework can be used with the study animals' wild counterparts to create improved energy budget estimates. This analysis will be key to future investigations on animal energetics and swimming biomechanics, with the potential to facilitate an improved understanding of how environmental context may affect behavior and welfare (e.g. anthropogenic disturbances).

We would like to thank the team and the animals at Dolphin Quest Oahu for their help in facilitating this research. The animal care specialists were instrumental in acquiring the data that enabled this work. Particularly, Dolphin Quest Oahu provided in-kind support in the form of time working with the animals, and support from the animal care specialists in executing the experiments. A portion of the Results and Discussion in this paper are reproduced from the PhD thesis of J.T.G. (Gabaldon, 2021).

Author contributions

Conceptualization: J.T.G., J.R.-L., M.J.M., J.v.d.H., K.A.S.; Methodology: J.T.G., D.Z., J.R.-L., M.J.M., J.v.d.H., K.B., K.A.S.; Software: J.T.G., D.Z., K.A.S.; Validation: J.T.G., D.Z., K.B., K.A.S.; Formal analysis: J.T.G., D.Z., J.v.d.H., K.B., K.A.S.; Investigation: J.T.G., D.Z., J.R.-L., K.A.S.; Resources: J.T.G., D.Z., J.R.-L., K.A.S.; Data curation: J.T.G., J.R.-L.; Writing - original draft: J.T.G., D.Z., K.A.S.; Writing - review & editing: J.T.G., D.Z., J.R.-L., M.J.M., J.v.d.H., K.B., K.A.S.; Visualization: J.T.G., D.Z., J.v.d.H., K.B., K.A.S.; Supervision: J.R.-L., M.J.M., K.B., K.A.S.; Project administration: J.R.-L., M.J.M., K.B., K.A.S.; Funding acquisition: J.R.-L., M.J.M., K.B., K.A.S.

Funding

This research was funded by the Office of Naval Research (N00014-17-1-2747), The National Oceanographic Partnership Program (National Science Foundation via the Office of Naval Research, N00014-11-1-0113), a Contribution Agreement with the Department of Fisheries and Oceans Canada (DFO), and start-up funds from the University of Michigan.

Data availability

Lap-trial CS-section speed versus thrust power data are available from figshare: https://doi.org/10.6084/m9.figshare.21350496.v1

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Competing interests

The authors declare no competing or financial interests.

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