Unsteady flow affects swimming energetics in a labriform fish (Cymatogaster aggregata)

The energetic costs of locomotion comprise a large and variable component of the energy budgets of mobile organisms (Boisclair and Sirois, 1993). Environmental factors that influence locomotor costs can therefore have profound effects on individual fitness (Arnold, 1983; Irschick and Garland, 2001). In fishes, the energetic costs of swimming have traditionally been estimated by measuring oxygen consumption rates in steady (i.e. constant speed) water flows (e.g. Steffensen et al., 1984; Claireaux et al., 1995; Farrell et al., 2003; Clark et al., 2011). However, these measures may not reflect the true costs of swimming in nature, where the velocity of water flows can vary dramatically over short time scales (Liao, 2007; Webb et al., 2010). Increasingly, fish biologists and eco-physiologists are aware of the need to measure swimming costs in settings that approximate wild conditions both to improve our understanding of fish locomotion and for practical applications such as water and habitat management (Enders et al., 2003; Liao, 2007; Lacey et al., 2012).

In nature, water flow can be influenced by numerous physical variables including wind, gravity and obstructions below the surface, creating complex water flows (Liao, 2007; Webb et al., 2010). Terms used to describe flow hydrodynamics with regard to swimming are often not clearly defined, which makes generalizations about the effects of these complex flows on fishes difficult (see Liao 2007). Turbulence usually refers to the creation of vortices of variable strengths and sizes in flowing water, whereas unsteady flows can be near-laminar and are characterized by changes in fluid velocity over time at a given point in space (Liao 2007; Webb et al., 2010). Depending on the causal agent, turbulent flows may have an element of predictability that can be exploited by swimming fish (Liao et al., 2003a; Liao et al., 2003b; Liao, 2004; Beal et al., 2006; Cook and Coughlin, 2010; Taguchi and Liao, 2011). However, water flows with unpredictable and/or wide fluctuations in velocity are known to increase the costs of locomotion (Pavlov et al., 2000; Enders et al., 2003; Enders et al., 2005; Lupandin, 2005; Webb and Cotel, 2010). Whether unsteady flows represent advantages or disadvantages to swimming fishes remains an important area of research.

One major impediment to studying the effects of complex flows on fish locomotion is the challenge of creating describable and/or repeatable hydrodynamic perturbations in an experimental setting (Liao, 2007; Lacey et al., 2012). As a result, only a handful of studies have directly examined the metabolic costs of swimming in complex flows (Enders et al., 2003; Liao et al., 2003a; Cook and Coughlin, 2010; Tritico and Cotel, 2010; Taguchi and Liao, 2011). While extremely useful, these studies are restricted to fishes such as trout, salmon and minnows that use their body and caudal fin for propulsion (BCF swimmers). However, roughly 15–20% of all living fishes, including a large proportion of fishes in shallow marine (e.g. labrids, pomacentrids) and freshwater (e.g. cichlids, centrarchids) habitats, use their pectoral (i.e. paired) fins for swimming (MPF or labriform swimmers) (Westneat, 1996; Bellwood and Wainwright, 2001). Labriform swimmers are commonly found in habitats associated with complex flows (Fulton, 2010), especially inshore coastal habitats where wave-driven water motion varies considerably across local and regional gradients (Webb et al., 2010). For example, on the west coast of the United States, bays and sounds that are sheltered from large storm waves regularly experience wave-driven water flows ranging from 0 to 50 cm s⁻¹ (Finlayson, 2006; Gaylord et al., 2008). Similarly, shallow coral reef habitats in the tropics are

routinely subjected to wind-driven water motion (Denny and Gaylord, 2010) and can harbour over 60% of fish species that use labriform swimming as a primary means of locomotion (Fulton, 2010). Currently, we do not know how unsteady flows affect the swimming performance and energetics of labriform swimming fishes in coastal habitats.

We used intermittent-flow respirometry to compare the energetics, swimming performance and kinematics of a marine labriform swimmer, the shiner surfperch (Cymatogaster aggregata Gibbons), holding station in steady versus unsteady water flows. Our unsteady flow treatments mimicked a repeatable, unilateral wave surge scenario (sinusoidal variations in water flow velocity in a single direction, around a constant mean velocity) with low- and high-amplitude fluctuations in flow velocity. These two treatments are hereafter referred to as low and high unsteady flow. The vertical component of orbital waves was absent from the flow changes imposed, which therefore mimicked the horizontal component of waves. This movement (i.e. parallel to the seabed) is the dominant flow for travelling waves in shallow, coastal waters (Denny, 2006; Webb et al., 2010).

First, we tested whether fish swimming in unsteady flows incur greater swimming costs than fish in steady flow at the same mean velocity. Second, we estimated the values of oxygen consumption rate (Ṁ_O2) for fish swimming in the two unsteady flow treatments using simulations based on the non-linear relationship between swimming speed (U) and Ṁ_O2 obtained in steady flow. Third, we compared observed (Ṁ_O2) and estimated (Ṁ_O2E) oxygen consumption rates, where their difference may result from additional costs of accelerating and decelerating as well as maintaining stability in unsteady flow. Finally, we examined whether observed differences in Ṁ_O2 between the three flow treatments were related to pectoral fin kinematics measured on different time scales (time-averaged over 300 s versus instantaneous over the duration of one fin beat cycle) and body movements in the swim chamber to understand possible mechanisms underlying differences in oxygen consumption rates.

Calculations of Ṁ_O2E indicated that fish swimming in the low unsteady flow treatment should consume on average 2.75% (range 1.63–3.1%) more oxygen than fish in steady flow, whereas fish in high unsteady flow should consume on average 11.1% more (range 6.7–13.3%; Fig. 1). Observed Ṁ_O2 did not differ significantly from Ṁ_O2E for fish in low unsteady flow (LMM, t=−1.76, P=0.08) but was significantly higher than Ṁ_O2E for fish in high unsteady flow (LMM, t=2.05, P=0.04; Fig. 1).

Fish in the different flow treatments transitioned from a pectoral to a pectoral–caudal swimming gait (U_p–c) at different swimming speeds (ANOVA, F_2,17=7.18, P<0.01; Fig. 2). Fish in high unsteady flow reached U_p–c at lower swimming speeds than fish in steady flow (Tukey's HSD, P<0.05) and in low unsteady flow (Tukey's HSD, P<0.01; Fig. 2). There was no difference in U_p–c between fish in low unsteady and steady flow (Tukey's HSD, P=0.85). Critical swimming speed was also different among treatments (ANOVA, F_2,17=3.87, P<0.05). Fish in high unsteady flow reached U_crit at lower swimming speeds than fish in low unsteady flow (Tukey's HSD, P<0.05). The U_crit of fish in steady flow did not differ from that in either high (Tukey's HSD, P=0.13) or low (Tukey's HSD, P=0.88) unsteady flow.

Pectoral fin-beat frequency (f_p) increased with swimming speed (LMM slope, F_1,79=392.9, P<0.001) in a similar fashion for all three flow treatments (LMM interaction, F_1,79=0.78, P=0.46; Fig. 3A). In turn, f_p was a good predictor of oxygen consumption rates in all three treatments (LMM slope, F_1,79=315.6, P<0.001; LMM interaction, F_2,79=0.07, P>0.90; Fig. 3B), but fish in high unsteady flow consistently consumed more oxygen for a given f_p than fish in either steady or low unsteady flow (LMM intercept; F_2,16=6.55, P<0.01; Fig. 3B).

Pectoral fin-beat period (T) decreased with swimming speed (LMM slope, F_1,79=931.1, P<0.001) similarly for fish in all three flow treatments (LMM interaction, F_2,79=1.07, P=0.35). However, variation (CV) in T across swimming speeds differed among flow treatments and increased with the degree of flow unsteadiness (Fig. 4). Observed variations in T were consistently higher than the estimated variation (T_E) for fish in high unsteady flow, but not for fish in low unsteady flow (Fig. 4). There was a significant overall negative association between variation in T and Ṁ_O2 (LMM slope, F_1,14=66.0, P<0.001) for fish in high unsteady flow and the slope of this relationship did not differ among swimming speeds (LMM interaction, F_3,14=0.18, P=0.91; Fig. 5B). The 95% confidence interval of the estimates for both the within-speed slope (estimate=−0.261, 95% CI=−0.400 to −0.122) and between-speed slope (estimate=−0.556, 95% CI=−0.930 to −0.182) did not overlap zero, indicating that these slopes differed from zero. In contrast, associations between variation in T and Ṁ_O2 were generally non-significant and inconsistent across swimming speeds for fish in low unsteady flow (Fig. 5A). The 95% confidence interval for the estimate of the between-speed slope did not overlap zero (estimate=−0.997, 95% CI=−1.516 to −0.478); however, the within-speed slope estimate overlapped zero (estimate=−0.154, 95% CI=−0.340 to 0.032), indicating that variation in T had no effect on Ṁ_O2.

There was a significant overall negative association between fish displacement (horizontal and vertical body movement) and variation in T (LMM slope, F_1,14=7.16, P=0.02) for fish in high unsteady flow, and the slope of this relationship did not differ among swimming speeds (LMM interaction, F_3,14=0.20, P=0.89; Fig. 6).

Many labriform fishes live in marine environments and experience variable speed flows created by travelling waves in inshore habitats. Our experimental flow treatments mimicked unidirectional wave surge with either low- or high-amplitude velocity fluctuations and a period of 5 s, similar to waves in Puget Sound, Washington (Finlayson, 2006). We found that high unsteady flow tended to decrease fish swimming performance (Fig. 2) and increase swimming costs by an average of 25.3% compared with steady flow (Fig. 1). This increase in oxygen consumption rate also exceeded the 11.1% average increase expected based on calculations of Ṁ_O2E (i.e. Ṁ_O2 estimated using the non-linear relationship between swimming speed and Ṁ_O2 in steady flow) (Fig. 1). Together, these results suggest that estimates of fish swimming energetics based on steady flow conditions underestimate the costs of swimming in water flow with large velocity fluctuations (in the order of one body length). This discrepancy may be due to additional energy expenditure from fish accelerating and decelerating (see Kramer and McLaughlin, 2001; Minetti et al., 2001) as well as correcting for postural disturbances (see Webb, 2006; Webb et al., 2010) to maintain their position and stability during flow velocity changes. In contrast, swimming performance and oxygen consumption rates did not increase in low unsteady flow relative to steady flow. This result is consistent with the small estimated increase in oxygen consumption rate of only 2.75%, on average, based on Ṁ_O2E. Further experiments are necessary to establish the relative costs of maintaining stability and varying acceleration for fish swimming in unsteady flows.

Spatial and/or temporal fluctuations in water flow velocity can result in both energetic challenges and benefits for fishes (Lacey et al., 2012). Enders et al. (Enders et al., 2003) found that juvenile Atlantic salmon, Salmo salar, incurred higher energetic costs even at relatively low water velocity fluctuations in turbulent flow compared with laminar flow. In contrast, other studies have shown that BCF swimmers can exploit vortices that have an element of predictability: by bending their body around vortices, fish can generate forward thrust with less energy expenditure (e.g. Liao et al., 2003b; Taguchi and Liao, 2011). Our results support the general finding that disturbances from flow variations are important only if they are large relative to the size of the fish (Pavlov et al., 2000; Lupandin, 2005; Liao, 2007; Tritico and Cotel, 2010; Webb et al., 2010); irrespective of its mean swimming speed, C. aggregata appeared unaffected by relatively small water velocity fluctuations of 0.5 BL s⁻¹ while experiencing significantly higher energetic demands when subjected to larger fluctuations of 1 BL s⁻¹.

The relationship between mean swimming speed and pectoral fin-beat frequency (measured over 300 s) did not differ among flow treatments (Fig. 3A). Despite these similarities, for a given fin-beat frequency, fish in high unsteady flow consumed significantly more oxygen than fish in both low unsteady and steady flow (Fig. 3B). This suggests that a time-averaged measure of fin use is independent of observed differences in Ṁ_O2. Within a given 5 s period of water velocity fluctuations, we observed individuals in low and high unsteady flows altering their fin-beat movements: fish were beating their fins less frequently as the flow velocity decreased and, conversely, increased their fin-beat frequency as the flow velocity increased (see supplementary material Movie 1). Despite these adjustments in fin kinematics, the mean fin-beat frequency (f_p) was the same at any given mean swimming speed, regardless of the flow treatment. Because f_p was calculated as the average number of fin beats over a time scale of minutes, this time-averaged measure of fin use did not capture adjustments in the timing of fin beats made by fishes in unsteady flow. Therefore, measurements of fin kinematics on a shorter time scale are needed to explain differences in Ṁ_O2 among treatments (see below, Variation in instantaneous pectoral fin kinematics).

Pectoral fin-beat frequency is positively related to swimming speed and/or oxygen consumption in several species of labriform fishes (e.g. Mussi et al., 2002; Kendall et al., 2007; Tudorache et al., 2009; Johansen et al., 2010). Some authors have suggested that the relationships between fin-beat frequency, oxygen consumption rate and swimming speed in a given fish species may provide useful indicators of swimming energetics in the wild, which are extremely difficult to estimate in aquatic species (Steinhausen et al., 2005; Ohlberger et al., 2007; Tudorache et al., 2009; Layton, 2011). However, our results suggest that the use of time-averaged fin kinematics to predict oxygen consumption rates depends on the hydrodynamic context in which these estimates are made. As such, caution is warranted when inferring oxygen consumption rates from time-averaged measures of fin-beat frequency.

Although time-averaged f_p was similar across flow treatments, fin-beat period (T) clearly differed. The consistently timed beats of fish swimming in steady flow resulted in low variation in T, whereas the rapid speeding up and slowing down of fin beats in unsteady flow resulted in higher variations in T (Fig. 4). As mean swimming speed increases, fish must beat their fins faster in order to keep up with increasing flow speed, resulting in a shorter refractory or gliding period between fin beats. As a result, both the observed and predicted variation in T decreased with increasing swimming speed for fish in the unsteady flow treatments (Fig. 4). In contrast, variation in T remained constant in steady flow (Fig. 4), despite increases in f_p with higher swimming speeds (Fig. 3A).

In low unsteady flow, observed variations in T were greater than the estimated variation (T_E) only at low swimming speeds, where water velocity fluctuations would have had a minimal effect on fish (Fig. 4). In high unsteady flow, however, observed variations in T were consistently greater than estimated variations, across all swimming speeds (Fig. 4). These differences in fin-beat period variability between treatments are consistent with observed differences in Ṁ_O2 (Fig. 1).

Similarly, there was a clear relationship between Ṁ_O2 and variation in T in high unsteady flow (Fig. 5B) but not in low unsteady flow (Fig. 5A): fish with greater variation in T consumed less oxygen for a given mean swimming speed in high unsteady flow (Fig. 5A). While we did not quantify whether, within one 5 s wave period, accelerations in water velocity coincided with shorter fin-beat periods and vice versa, these relationships clearly indicate that larger variations in T were energetically advantageous in conditions of high flow variability. This also suggests that some fish are capable of adjusting the duration of their fin beats in response to changes in water flow velocity to maintain a relatively constant ground speed in the swim chamber (Fig. 6). Specifically, by increasing the refractory period of fin beats as the water decelerates, fish may be able to conserve energy by increasing their glide towards the end of the wave cycle. Fish with lower variation in T may not take advantage of decelerations in water flow velocity and may resort to more energetically costly behaviours to try and maintain their position in the swim chamber. Such behaviours were not assessed in this study but could include modifying the amplitude and/or the power output of fin strokes to accelerate and overcome resistance at various swimming speeds, as well as braking to control posture and position during deceleration.

In conclusion, our results suggest that swimming costs in unsteady flows depend on the magnitude of the water velocity fluctuations. When velocity fluctuations were relatively large, the energetic costs of swimming in unsteady flow exceeded the costs of swimming in steady flow at the same mean velocity. It is important to note that these costs apply to station-holding fishes, which swim to remain stationary relative to the substrate. In contrast, travelling fishes exposed to wave surge might be able to conserve energy by taking advantage of forward surges and varying their ground speed while maintaining a constant velocity relative to the water. Our results are also conservative because hydrodynamic perturbations in our experiments were unidirectional and designed to minimize turbulence. Coastal habitats are often characterized by turbulent, oscillatory wave-driven water motion, which may require fishes to expand more energy for postural control and stability. Swimming costs also depend on the ability of individual fish to adjust their fin kinematics to the flow environment and avoid displacement while station holding in the swim chamber. Individual variability in swimming performance has previously been observed in a number of species and has been shown to be both repeatable and biologically important (Kolok, 1999). It is possible that inter-individual differences in our study relate to differences in habitat use among fish; for example, individuals foraging high in the water column may experience greater water flows than individuals that remain closer to the substrate. Further studies should examine the learning potential of individual fish to modify their fin-beat kinematics via repeated exposure to variable water flows.

Adult Cymatogaster aggregata were collected using a beach seine net at Fourth of July and Jackson's Beach on San Juan Island, Washington, USA, in August 2011. Fish were held in flow-through aquaria at the University of Washington's Friday Harbor Laboratories at an ambient light regime. Tanks were continuously supplied with filtered seawater (salinity 34 ppt) at a mean temperature of 12°C (range 11 to 13°C). Given the proximity of the laboratory to the collection site, fish were not fed and were tested shortly after their capture, ensuring near-wild conditions during the experiments. Fish were fasted for a minimum of 24 h before the experimental trials to ensure that satiation was standardized across individuals (Niimi and Beamish, 1974; Johansen et al., 2010; Roche et al., 2013). The experimental protocol was approved by the University of Washington in accordance with Institutional Animal Care and Use Committee standards (IACUC permit no. 4238-04).

We measured oxygen consumption rates (Ṁ_O2; mg O₂ kg⁻¹ h⁻¹) for 20 fish (total length=14.84±0.49 cm; mass=46.3±6.3 g; means ± s.d.) swimming in an 8.31 litre clear Plexiglas Steffensen-type respirometer (Steffensen et al., 1984; Methling et al., 2011) with a working section of 9.0×26.0×10.0 cm (width×length×depth) (supplementary material Fig. S1). Oxygen levels in the respirometer were recorded using a fibre optic oxygen meter (PreSens Fibox 3, Regensburg, Germany) monitored with AutoResp V1 (Loligo Systems, Copenhagen, Denmark). We calibrated the flow in the working section of the respirometer from 0 to 80±0.5 cm s⁻¹ (mean ± s.e.m.) using a digital TAD W30 flow meter (Höntzsch, Waiblingen, Germany) at 1 cm intervals. The flow velocity profile varied less than 5% across the full cross-section of the working chamber and we did not observe fish favouring one corner or particular side of the working section during the swim trials. Changes in flow speed inside the swim chamber can lag behind changes in the rotational speed of the propeller; however, we measured flow velocity during oscillations in propeller speed and observed minimal dampening and attenuation of intended flow speed minima/maxima in the swim chamber. Solid blocking effects of the fish were corrected by the respirometry software (AutoResp V1); the mean fish cross-sectional area was 8.1% of the swim chamber cross-sectional area, corresponding to a 3.5–4% greater effective water velocity around the fish compared with the water velocity in the empty swim chamber (Webb, 1975). We used UV filtration to reduce bacterial growth in the system and regularly rinsed the respirometer in freshwater to ensure that bacterial respiration rates remained below 15% of the standard metabolic rate of fish. Three Ṁ_O2 determinations were run without fish before and after each trial to measure bacterial respiration in the test chamber. Background respiration rates were determined from the slope of the linear regression between initial and final measurements of background respiration rates and subtracted from each Ṁ_O2 determination.

We repeated the same step-wise procedure for the remaining 14 test subjects, but varied flow velocity by an amplitude of either 0.5 BL s⁻¹ (low unsteady flow; N=7) or 1.0 BL s⁻¹ (high unsteady flow; N=7) around the mean at each speed increment (e.g. 1.0±0.5 BL s⁻¹). These velocity fluctuations were adjusted for each individual fish based on body length, and approximated natural, wave-induced changes in flow speed on the coasts of Puget Sound [0–50 cm s⁻¹ (Finlayson, 2006)]. We used a computer-generated sine function with a period of 5 s (TracerDAQ Pro, Measurement Computing, Norton, MA, USA) to create repeatable variations in water flow speed by continuously controlling the voltage to the motor and the rotational speed of the propeller in the respirometer. Five seconds corresponds to moderate wave periods in the San Juan Islands (Finlayson, 2006). The computer was connected to a motor controller (Movitrac AC VFD, SEW Eurodrive, Lyman, SC, USA) via a USB-1208 ADDA converter (USB DAQ Data Acquisition, Measurement Computing). Flow straighteners were used to dissipate gross turbulence and produce flow with uniform micro-turbulence (Beamish, 1978). Eliminating micro-turbulence in flumes is practically impossible (Beamish, 1978); however, very small orbits with diameters much smaller than a fish's body length are unlikely to have notable effects on swimming performance (see Tritico and Cotel, 2010; Webb et al., 2010). We used flow visualization with neutrally buoyant particles (expandable polystyrene beads diameter <1 mm, Foamex Polystyrene, Revesby, Australia) in a similar respirometer to confirm the absence of large vortices. Given their small size, these particles allowed us to detect perturbations greater than 1 mm. Flow visualizations were filmed in high definition at 30 frames s⁻¹. Frame-by-frame measurements using the plugin MtrackJ in ImageJ (Meijering et al., 2012) revealed a relatively uniform flow profile and the absence of vortices with a diameter >0.5 cm (most were much smaller or almost nil), both in steady and unsteady flow conditions. Examples of flow characterizations in steady and high unsteady flow are provided in supplementary material Figs S2, S3, Table S1.

A computer-generated sine function that determined the rotational speed of the swim tunnel's propeller allowed us to precisely control four key parameters in the different flow conditions: the period, amplitude and wavelength of the water velocity fluctuations, and the mean flow velocity. Prior to the experiments, we calibrated the voltage from the computer-generated sine function with changes in water velocity in unsteady flow using a digital Höntzsch TAD W30 flow meter (Höntzsch, Waiblingen, Germany). We increased the mean water flow velocity from 1 to 4.5 BL s⁻¹ by increments of 0.5 BL s⁻¹, following a standard U_crit protocol. The period was 5 s for all treatments and the amplitude varied between 0, 0.5 and 1 BL s⁻¹ depending on the treatment. The wavelength varied between 5 BL and 22.5 BL because wavelength is a product of the period and the water flow velocity. The variance in water flow velocity was 0 for steady flow, 0.125 for low unsteady flow and 0.5 for high unsteady flow. The variance remained the same within each flow treatment as the amplitude of water velocity fluctuations was identical within treatment, across all mean swimming speeds. There was only small variation in the maximum and minimum flow velocities recorded around a given mean flow velocity, which indicated that the flow conditions were consistent (supplementary material Fig. S2, Table S1).

For each fish (N=20) and each speed increment, we examined the three 5 min video segments that corresponded to the Ṁ_O2 measurement cycles. Using ODlog (Macropod Software), we recorded the number of (1) pectoral fin beats, (2) combinations of pectoral and caudal fin beats and (3) caudle fin beats that resulted in burst-and-coast swimming. We calculated pectoral fin-beat frequency (f_p) in Hz as the number of pectoral fin beats performed divided by the time elapsed during the analysis period (300 s). Calculated in this way, fin-beat frequency represents a time-averaged measure of fin oscillations (Drucker and Jensen, 1996).

To examine finer-scale effects of water speed fluctuations on pectoral fin kinematics, we measured the period of individual pectoral fin beats (T) during the 5 s cycle of sinusoidal water velocity fluctuations. We used field-by-field video analysis to record values of T three times per fish (once per Ṁ_O2 determination) at each swimming speed. Fin-beat period is the duration of a pectoral fin-beat cycle, which begins and ends with consecutive onsets of pectoral fin abduction (Drucker and Jensen, 1996). It includes both the propulsive phase measured from the onset of abduction to the end of adduction, and a non-propulsive phase during which the fish glides until the onset of the next abduction (Drucker and Jensen, 1996). We calculated the mean and the coefficient of variation (CV; s.d./mean) of fin-beat period across the three Ṁ_O2 determinations for each fish and swimming speed. Because there was considerable individual level variation in oxygen consumption rates and fin-beat period among fish in high unsteady water flow (see Results), we tested whether fish that had a high variation in fin-beat period (i.e. longer and shorter fin-beat cycles as denoted by the CV of fin-beat period) in response to high fluctuations in water flow velocity consumed less oxygen than fish that had less variable fin-beat periods (see supplementary material Movie 1). We also tested whether these fish were better able to hold station in the swim chamber by maintaining their ground speed constant. We used field-by-field video analysis to measure horizontal and vertical body displacements during the 5 s cycle of sinusoidal water velocity fluctuation three times per fish (once per Ṁ_O2 determination) at each swimming speed.

In fishes, the response curve of oxygen consumption rate versus swimming speed is a positive, nonlinear function (e.g. Korsmeyer et al., 2002; Cannas et al., 2006); therefore, for the same mean swimming speed, fish experiencing cyclic changes in water flow velocity should consume more oxygen than fish swimming at a constant velocity (Fig. 7) (see Ruel and Ayres, 1999). To compare observed and estimated oxygen consumption rates, we calculated estimated changes in Ṁ_O2 values (Ṁ_O2E) as a function of swimming speed for the low and high unsteady flow treatments. This was achieved by integrating a sinusoidal function based on experimental variations in flow speed (period of 5 s, amplitude of 0.5 or 1 BL s⁻¹) into the equation for the Ṁ_O2–swimming speed relationship in steady flow. Comparisons of Ṁ_O2 and Ṁ_O2E values allowed us to determine whether potential differences could be attributed to factors other than the mathematical properties (i.e. non-linearity) of the swimming speed–Ṁ_O2 relationship. Such factors could include costs associated with stability and acceleration–deceleration. We used this same procedure to calculate the variation (CV) in estimated fin-beat period (T_E) as a function of swimming speed, based on the relationship between swimming speed and fin-beat frequency in steady flow. In steady flow, we considered variability in pectoral fin-beat period to be the natural amount of variation for individuals swimming in our flow chamber; therefore, we adjusted the CV of T_E for fish in the two unsteady flow treatments by adding the background variations in fin-beat period observed for fish in steady flow.

We used a general linear mixed effects model (LMM; lme function in R) to test for differences in the Ṁ_O2–swimming speed relationship across flow treatments. We specified the relationship between speed and Ṁ_O2 as a second-degree polynomial and individual fish as a random effect. Mixed models are useful as they control for temporal autocorrelation among data points in physiological response curves (Peek et al., 2002; Bolker et al., 2009; Nakagawa et al., 2013). This same model was used to test for differences between estimated (Ṁ_O2E) and observed Ṁ_O2 values in the two unsteady flow treatments. We tested for differences in swimming performance (U_p–c, U_crit) across flow treatments with two one-way ANOVAs and subsequent Tukey's honestly significant difference (HSD) tests. We used five distinct LMMs with fish as a random factor to examine: (1) differences in the relationship between swimming speed and pectoral fin-beat frequency across flow treatments, (2) the relationship between pectoral fin-beat frequency and Ṁ_O2 across flow treatments, (3) the relationship between swimming speed and pectoral fin-beat period, (4) whether individual variation in fin-beat period explained differences in Ṁ_O2 for fish swimming in the low and high unsteady flow treatments, and (5) the relationship between variation in fin-beat period and fish displacement (i.e. body movement) in high unsteady flow. We specified random intercepts and a first-order autoregressive covariance structure to account for equally spaced points in time. Where needed, we used log₁₀ and exponential transformations to linearize the data and meet the assumptions of normality and homoscedasticity. For mixed effects models, we determined R²_LMM(m), the proportion of variance explained by fixed factors, and R²_LMM(c), the proportion of variance explained by both fixed and random factors (Nakagawa et al., 2013). R²_LMM(m) values were almost identical to overall R² obtained for linear (non-mixed) models. We used within-group centring to compare estimates of within-group slope versus between-group slope and test relationships between variation in fin-beat period and Ṁ_O2 (van de Pol and Wright, 2009). All analyses were conducted in Rv2.11.1 (R Development Core Team, 2010). Data are deposited in figshare (public data repository, http://dx.doi.org/10.6084/m9.figshare.789064).

We thank the staff at the Friday Harbor Laboratories for logistical support, the 2011 Fish Swimming Class for fish collections and helpful discussion, and D. L. Kramer, M. D. Jennions, A. Roche, P. W. Webb and five anonymous reviewers for comments on earlier versions of the manuscript. L. Tremblay-Boyer and M. van de Pol provided helpful advice with statistical analyses.