Introduction

Newton’s laws of motion underpin animal locomotion, from fine-scale maneuvers to long-distance migrations (1). As speed increases or during migration involving sustained movement over long distances, animals often move in coordinated groups: e.g., migratory birds in V-formation (2), ducklings swimming in formation (3), cyclists in a peloton (4), elite marathon runners (5), and fish schools (6). To overcome gravitational or fluid dynamic resistance to forward motion, animals use both aerobic and glycolytic metabolism (oxidative and substrate-level phosphorylation at the cellular level) to generate energy and sustain movement in the air, on land, and in the water. In water, a fluid that is 50 times more viscous than air and contains much less O₂ per Kg than air, the need for aquatic animals to reduce fluid dynamic drag for energy conservation is even greater than for aerial or terrestrial locomotion. Hence, we use fish schooling behaviour as a model system to explore whether the fluid dynamics of collective movement (active and directional movement of animal groups along a common trajectory (7)) can enable energetic savings compared to locomotion by a solitary individual. Fish schooling behaviours are some of the most prominent social and group activities exhibited by vertebrates.

Fish may school for many reasons, including foraging, reproduction, and migration, and fish schools often exhibit high-speed movement during evasion from predators. Hence, natural selection can put pressure on the formation of schools and individual interactions within a school. Besides ‘safety in numbers’ where individuals in a large school have a lower probability of becoming prey, can there be more tangible energetic benefits of high-speed swimming as a school? Since fluid drag scales as velocity squared (8), movement at higher speeds places a premium on mechanisms to conserve energy. Fast and unsustainable swimming is fueled by aerobic metabolism with a major contribution from anaerobic glycolysis (9) (10). During high-speed swimming aerobic capacity is maximized (10), and energy use is more likely to compete with other activities than in low-speed swimming which only occupies a small portion of aerobic capacity. The direct consequences of unsustainably engaging anaerobic glycolysis include both fatigue and metabolic perturbation (such as changing blood acidity) that need time to recover after bouts of intense locomotion. Animals become vulnerable during recovery because of their hindered ability to repeat peak locomotor performance in the presence of predators.

Another factor of importance in the experimental evaluation of the energetic cost of collective locomotion in fishes is that endurance and efficiency are often key to achieving lifetime fitness, especially in fish species that undergo significant migrations. Fish groups usually undergo long-distance migrations at slow speeds ranging from 0.25–1.0 BL s^-1 as recorded by tags on migrating fish (Fig. S1). The common migratory speed of ∼1 BL s^-1 for marine and anadromous fish could be related to the minimum aerobic cost of transport (11). However, there is currently no direct measurement of the absolute cost of teleost fish locomotion demonstrating a minimum group metabolic rate (MO₂) at an optimal speed (U_opt). Fish schools in nature swim over a wide range of speeds (Fig. S1) but no study has directly characterized the total energy expenditure (TEE), and accounted for the potentially substantial anaerobic metabolic costs involved in higher speed locomotion. As a result, the swimming TEE performance curve of fish schools from the minimum swimming speed to the critical swimming speed (U_crit) has not yet been measured.

More broadly, despite the widespread notion that collective motion saves energy (6), very few studies have directly measured the energetic cost of collective movement compared to the cost of movement by solitary individuals. In the canonical case of V-formation flight in birds, energetic saving is exclusively inferred from indirect measurements, e.g., heart rate (2), and flapping frequency and phase (12) (Table 1). There are no direct measurements of metabolic energy consumption of collective movement in birds, and indeed this would be extremely challenging to accomplish. Although some energetic aspects of schooling fishes have been studied (13) (14) (15) (16) (17), these studies have analyzed a limited range of speeds and focused on aerobic metabolism (see Table 1). In fact, several field and laboratory kinematic studies suggest that both bird flocks and fish schools do not always conserve aerobic energy (17) (18) (19) and indicate that moving in a group can actually involve increased costs. Without directly measuring the energetic cost of movement, inferences solely based on kinematics do not include complex group interactions, and the interaction between collective behaviour and fluid dynamics. Consequently, we do not yet have a holistic and mechanistic view of where and how the collective movement by fish might conserve energy.

Table 1.

For swimming fish, fluid dynamic experiments have shown how collective movement can improve swimming efficiency due to interactions among deforming bodies and through interactions between moving animals and the local fluid environment (these energy-saving mechanisms are summarized in Fig. 1). Our current understanding of energy saving mechanisms during collective locomotion in fishes is largely based on computational fluid dynamic models with a few analyses using robotic systems (6) (20) (21) (22) (23) at low speed, but the link between such models and fish metabolic energy saving over a wide speed range is not yet established. We expect that the need to conserve energy in animals moving against air or water resistance should be greater at higher speeds. We hypothesize that fish in schools can reduce the total cost of high-speed locomotion relative to solitary movement, leading to reduced non-aerobic energetic costs and time of recovery for schooling fish swimming at speeds occupying the majority (> 50%) of their aerobic capacity. Our secondary objective is to determine the differences (if any) between the swimming performance curves (metabolism versus speed) of solitary fish and fish schools.

Fig. 1.

To evaluate these hypotheses, we directly measured both the aerobic and non-aerobic energy used by schooling fishes over a wide range of water velocities (0.3–8.0 body lengths s^-1; BL s^-1), and then compared the swimming performance curve to that of solitary fish. We equipped a high-resolution (see Methods for our approach to enhancing the signal-to-noise ratio in our respirometer) swim-tunnel respirometer with two orthogonal high-speed cameras to quantify three-dimensional (3-D) fish kinematics. This enabled us to simultaneously measure energetics, 3-D dynamics of fish schools (n=5 replicate schools), and the kinematics of solitary fish (n=5) of a model species, giant danio (Devario aequipinnatus), that exhibits an active directional group swimming from near still water to maximum sustained speeds (equivalent to a Reynolds number range of 6.4•10³ to 1.8•10⁵; Fig. S2), using a U_crit test.

Results

We discovered that both solitary fish and fish schools have a concave upward shaped aerobic metabolic rate (MO₂)-speed curve over the lower portion of their speed range (0.3–3 BL s^-1, Fig. 2C, D). The aerobic locomotor cost at 0.3–3 BL s^-1 showed no statistical difference between solitary fish and fish schools (F_1,80 ≤ 2.40, p ≥ 0.13). Fish schools swimming at ∼1.0 BL s^-1 consume less energy than at slower speeds (F_9,40 = 24.7, p ≤ 0.0007), while swimming at 3 BL s^-1 consumes a similar amount of energy to maintain position at a water velocity of 0.3 BL s^-1 (p = 0.85; Fig. 2D). Danio schools have a minimum aerobic cost (MO_2min = 212.9 mg O₂ kg^-1 h^-1) of 1.25 BL s^-1. MO_2min at this optimal speed (U_opt) was lower than both the MO₂ of aggregating behaviours (328.5 mg O₂ kg^-1 h^-1) exhibited before U_crit test (Table S1, Fig. S3) and for the group at rest in still water MO₂ (267.9 mg O₂ kg^-1 h^-1) (by 35% and 21% respectively; F_2,12 = 16.7, p ≤ 0.02, Fig. 2A, B). Swimming at U_opt reduces the energetic cost below that of swimming at either slower or higher speeds. Indeed, the Strouhal number of fish schools decreased from 2.1 at the lowest speed to 0.3 at the energetic minimum U_opt, and then increased to 0.4 at the higher U_crit.

Fig. 2.

We also characterized the kinematic features that result in elevated MO₂ at low water velocities. Danio maintaining body position in low-velocity aggregations had a higher tail beat frequency (f_TB) than solitary fish (F_1,80 ≥ 9.8, p ≤ 0.002). However, at U_opt and the highest speed of active directional schooling, fish in schools had a lower f_TB (F_1,80 ≥ 4.6, p ≤ 0.035) than solitary fish (Fig. 4B). The 3-D angular heading of schooling fish transitioned from omnidirectional to pointing against water flow when water velocity is above 0.75 BL s^-1, and individual fish orient into the flow starting at 0.3 BL s^-1 (Fig. 4C). Although the body angle and turning frequency of solitary fish and schools decreased with water velocity, schooling fish had a higher turning frequency (F_1,80 ≥ 15.6, p < 0.002) below 0.75 BL s^-1 than solitary fish (Fig. 4D,E).

Fig. 3.

Fig. 4.

We discovered that, across the entire 0.3–8 BL s^-1 range, MO₂-speed curves of fish schools are concave upward shaped (reached 1053.5 mg O₂ kg^-1 h^-1 at 8 BL s^-1) whereas solitary fish showed an upward concave curve that reaching a plateau of 760.2 mg O₂ kg^-1 h^-1 (∼10% CV in 6–8 BL s^-1, Fig. 2C), demonstrating that schooling dynamics results in a 44% higher maximum aerobic performance (F_1,80 = 30.0, p < 0.001). This increased maximum aerobic performance translated to a lower use of aerobic capacity compared to solitary fish over 0.3–8 BL s^-1 (see the discussion for how fish swimming in school can improve aerobic performance). Collectively, fish schools used a 36% lower proportion of their aerobic scope than solitary fish (Wilcoxon test: p = 0.0002, Fig. 2F). Fish schools used a 38% lower proportion of aerobic scope (34 vs. 55%) at 4 BL s^-1 (50% U_crit & > 50% aerobic scope onward), and consistently used ∼25% lower proportion of their aerobic scope at 6 and 7 BL s^-1 than solitary fish (F_1,103 ≥ 4.8, p ≤ 0.03, Fig. 2E).

Given that fish schools had a higher maximum aerobic performance, generally used a lower proportion of their aerobic scope, and individuals within the schools have a 14% lower f_TB than the solitary fish at 8 BL s^-1 (11.5 vs. 13.3 Hz, F_1,80 = 15.1, p < 0.001), we predicted that fish schools, compared with solitary fish, use less anaerobic energy to supplement aerobic energy for the high-speed movement approaching their aerobic limit. Hence, we measured post-exercise O₂ utilization, a majority of which is used to restore high-energy phosphate storage and glycolytically induced metabolic perturbations (EPOC: Excess Post-exercise O₂ Consumption). We discovered that fish schools have a 65% lower EPOC (0.69 vs. 1.95 mg O₂; t₈ = 4.5, p = 0.0021), and recover 43% faster than solitary fish (8 vs. 14 h; t₈ = 2.8, p = 0.025; Fig. 2G, H).

To estimate the relative proportions of aerobic and anaerobic energy contributions for locomotion at each swimming speed, we modeled EPOC in addition to MO₂, the aerobic cost (> 50% U_crit & aerobic scope, Fig. 3A) (Table S2). We also estimated the total O₂ cost during the entire swimming process for each school and individual and calculated total energy expenditure (TEE) (10). The TEE of fish schools was 38–53% lower than that of the solitary fish between 5– 8 BL s^-1 (F_1,103 ≥ 7.4, p ≤ 0.008; Fig. 3D). TEE of fish schools was only 42–143 % higher than the aerobic metabolic rate at 5–8 BL s^-1 (F_1,96 ≥ 3.5, p ≤ 0.001) and anaerobic metabolic energy only accounted for 29–58 % of the TEE depending on speed (Fig. 3B). In contrast, TEE of solitary fish was 131–465 % higher than the aerobic metabolic rate between 5–8 BL s^-1 (F_1,112 ≥ 10.7, p ≤ 0.001), where anaerobic energy accounted for 62–81% of the TEE depending on speed (Fig. 3C).

Schooling dynamics reduced the total cost (aerobic plus anaerobic) of transport (TCOT) by an average of 43% compared to swimming alone (F_1,103 = 6.9, p = 0.01), and most of this energy conservation happens at higher speeds when fish approach their aerobic limit. Schooling dynamics in danio enables an extremely shallow rate of increase in TCOT with speed compared to that of an individual (Fig. 3E). The TCOT of solitary fish increased by 490% (6.5 kJ km^-1 kg^- ₁) at 8 BL s^-1, whereas the schooling TCOT increased by only 200% (3.6 kJ km^-1 kg^-1) at 8 BL s^-₁. Therefore, individual fish form a more energy-efficient biological entity when they collectively move as a school.

To answer the question of how schooling dynamics reduces TEE, we combine video analysis of fish tail beat kinematics with simultaneous aerobic and anaerobic measurements to compute energy expended per tail beat (TEE•beat^-1), and compared values for fish in schools to those for solitary fish. Schooling fish reduced TEE•beat^-1 by 30–56% at higher speeds compared to solitary fish (F_1,81 ≥ 7.3, p ≤ 0.008), a substantial reduction in TEE•beat^-1 consumed both by the school, and by individual fish within a school (Fig. 4A). Notably, the energetic benefits during active directional schooling occur when fish schools become more streamlined, as the length of fish school increased with speed and plateaued beyond 2 BL s^-1 (Fig. 4F), while the 3-D distance among individuals stayed relatively constant at ∼1.2 BL (Fig 4G). These results directly demonstrate schooling dynamics benefits the swimming kinematics of individual fish within the school and results in a net outcome of up to 53% TEE reduction in fish schools compared to solitary fish.

Discussion

Simultaneous characterization of energetics and kinematics enables an integrated understanding of both the physiology and physics of fish schooling behaviour. Hydrodynamic models, kinematic measurements, and robotic analyses of fish schools indicate that the cost of swimming can be reduced when fish swim beside neighbouring fish (20), behind or in front of another fish (24), or behind leading individuals (6) (Fig. 1). We have demonstrated here that fish schools downshifted an overall concave upward swimming energetics performance curve at higher speeds by ∼43%. Energy use as swimming speed increases changes in a non-linear manner, with higher energy use at the lowest speeds (Fig. 2D), and the performance curve below 3.0 BL s^-1 is U-shaped. Energy saving occurs through a substantial reduction in the non-aerobic energy contribution when fish approach aerobic limits. One of the key ecological benefits of the reduced use of glycolysis is a faster recovery time from fatigue (Fig. 2H). This would enable fish schools in nature to repeat high-performance movements. Considering that high-speed maneuvers are extremely common in predatory evasion (25) food-searching group motion (26), and in fish schools in the ocean (Fig. S1), energy savings by fish schooling at high speed and the faster recovery that followed could have a considerable impact on lifetime fitness (27) (28). Moreover, considerable energy savings occur even though we observed that individual danio regularly change positions within the school and do not maintain stable inter-individual locations. We expect that this is because individuals within the school can use multiple hydrodynamic mechanisms to save energy depending on their location (Fig. 1). We propose that energetic savings by collective movement may not require either fixed positional arrangements among individuals or specific kinematic features (such as tail beat synchronization) compared to solitary locomotion. Based on these results, we regard fish schooling as a highly robust behaviour that provides considerable energetic benefits that are not necessarily sensitive to specific fish locations or movements. Future studies are necessary to decipher each of the specific hydrodynamic mechanisms for energy saving.

Schooling dynamics enhances aerobic performance and reduces non-aerobic energy use

We discovered that a significant amount of energy conservation for active directional swimming occurs at speeds above 3 BL s^-1. In nature, fish schools routinely exhibit active directional collective locomotion above ∼6 BL s^-1 (Fig. S1, Table S3) (29), a speed that engages anaerobic glycolysis. Yet, there are no previous measurements of the anaerobic cost of schooling in fish. Hence, important and previously unrecognized benefits of active directional collective locomotion are (1) increased aerobic performance, (2) a reduced use of aerobic capacity and anaerobic energy, and (3) a resultant faster recovery from the associated metabolic perturbations and costs of swimming at higher speeds. When animals are approaching their maximum metabolic rate, the highest attainable rate of O₂ uptake limits aerobic performance during the unsteady state of high-speed locomotion and anaerobic glycolysis is engaged to support the maximum metabolic demand.

Fish swimming at higher speeds (> 50% U_crit) routinely exhibit a burst-&-glide gait mode of locomotion (30, 31). Repeated bursting is substantially fueled by anaerobic glycolysis and the gliding phase enables peak MO₂ (32) to replenish the venous O₂ content (33). Fish within a school are known to increase gliding time and decrease burst time by 19% when they are trailing the leading fish (34), e.g, a lower f_TB of fish schools at 8 BL s^-1 is related to the extended gliding time (Fig. 4B). We reason that the higher proportion of time spent gliding likely enables more bouts of peak MO₂ and enhances the maximum aerobic performance (32) (33). This enhanced maximum aerobic performance increases the metabolic ceiling and enables fish to use a lower proportion of their aerobic scope for locomotion (Fig. 2E). Physiological studies of exercise metabolism and locomotion suggest that a lower proportional use of aerobic scope during movement relates to a reduced accumulation of anaerobic end-products (9). Collectively, our results showed that the increased aerobic performance (Fig. 2C) and the reduced use of the aerobic capacity (Fig. 2E) for swimming above 50% U_crit in fish schools likely plays a key role in reducing the use of glycolysis and the accumulation of lactate (e.g. EPOC, Fig. 2G) for high-speed swimming (10, 31).

We present direct evidence of substantial non-aerobic energy saving by demonstrating the 65% lower EPOC (Fig. 2G) used by fish that swam in schools. Anaerobic glycolysis is crucial in permitting continued movement when aerobic limits are reached at high swimming speeds (35). Often vertebrates at higher speeds use more fast twitch fibres that generate high-frequency contractile force in part through anaerobic glycolysis (36) (37). The higher EPOC for single fish is unlikely to have been confounded by any possible stress effect of swimming as a solitary fish. Stressed solitary fish would have elevated aerobic metabolic rates in low water velocity, which we show is not the case for our experiments (see Fig. 2D). Indeed, to mitigate the stress response, we acclimated fish to the solitary conditions and habituated fish to the respirometer for a quiescent day before the experiment (see more details in experimental animal and protocols). When vertebrates move at high speed, the typical ‘stress’ neurotransmitters (e.g. adrenaline, catecholamines & cortisol) usually increase (38) (39), regardless of whether vertebrates move as an individual or in a group. These endocrinological responses underpin the physiological mechanisms that enable higher aerobic performance (e.g. rising heart rate) (40) (41). Moreover, after the highest swimming speeds immediately followed by an active recovery process, fish likely prioritize the recovery of the essential functions (as the levels of the neurotransmitters decline fairly rapidly) for the potential of repeated locomotion performance (speedy recovery allows fish to restore their swimming performance) (31) (42) (43).

To present a complete energetic profile showing the total energy expenditure (TEE) of locomotion for fish schooling, we model the non-aerobic cost (EPOC) on top of the aerobic swimming performance curve. The model is rooted in the use of aerobic and glycolytic energy when fish swim above 50% U_crit (30) and a commonly observed inflection point of faster anaerobic end-product accumulation is at ∼50% aerobic scope (9) (see Methods for detailed physiological bases and criteria for EPOC modeling). As a result, we observed a TEE being ∼2.5 fold higher than the aerobic metabolic energy expenditure which agrees with the theoretical estimates based on individual yearling sockeye salmon (44). This model elucidates how aerobic and anaerobic metabolic energy constitute TEE, fuel muscles, generate locomotor thrust and overcome hydrodynamic drag during swimming.

We also demonstrated that the need for energy saving in fish schools at lower speeds (< 3 BL s^-1 or 38% U_crit Fig. 2E, 3D, E) is not as crucial as at higher speeds where we demonstrated substantial energy savings in the schools compared to solitary individuals. Fluid drag is exponentially less at lower speeds compared to higher speeds. Fish predominately use aerobic metabolism to support low-speed steady swimming for prolonged periods (10) which does not require lengthy recovery as swimming at higher speeds where glycolysis contributes to the energetic demands. The costs of low-speed swimming are often less than < 20% of aerobic capacity (Fig. 2E), which leaves the majority of aerobic capacity (>50%) for other activities. Therefore, the benefits of reducing the energetic cost of locomotion are likely not major factors underlying behaviours such as low-speed milling (45) and aggregation (46) in fishes, where other ecological drivers such as feeding, predator avoidance, and reproduction likely dominate the fitness landscape.

Schooling dynamics and energy conservation

Our 3D kinematic analyses shed light on the complex interactions between schooling and hydrodynamics that enable energy saving by fish collective movement. One of the key possible mechanisms is local interactions among individuals, where the kinematics of individual fish respond to the wakes shed by neighbours (47). Kinematic and simulation studies indicate that 2– 3 coordinated fish can save energy (48) (20) through local interactions. We directly demonstrate here that eight coordinated fish can save energy, and recent simulations show that schooling benefits can extend to at least 23 fish (21). Although these studies appear to suggest that the energetic benefits of schooling are scalable, we are still some ways from proving the exact mechanisms of how the large school size in nature is coordinated and whether or not local interactions are still one of the mechanisms for energy saving when scaling up school size. Future hydrodynamic and long-term 3-D tracking studies are needed to investigate the specific schooling formations and hydrodynamic mechanisms used by live fish for energy saving (as summarized in Fig.1) (49).

We show that danio in a school keep a relatively consistent 3-D distance from neighbours and have small variation in mean position even as individuals routinely change position within the school (Fig. 4G). Computational fluid dynamic analyses of fish schools show that a small 3D neighbouring distance (e.g. < 0.4 BL s^-1) increases drag and can increase swimming costs, whereas a large 3-D neighbouring distance (e.g. > 2 BL s^-1) reduces the hydrodynamic benefits (48). There may thus be an optimal mean distance among individuals within a school to maximize energetic savings. Whether inter-individual distance should change as swimming speed increases remains an open question and may be a function of school size. While we cannot completely preclude the possibility that elongation of danio fish school might be due to the weaker individuals falling behind, the fact that the school maintained a stable 3-D distance as speed increased and that fish continued to change position within the school suggests that the elongated school volume at higher speeds may reflect changing hydrodynamic interactions among individuals (Fig. 1).

To encapsulate the complex interaction among animal physiology, kinematics and fluid dynamics, more comprehensive quantification beyond simple kinematic metrics are necessary. For example, the wing flapping frequency of birds in a V-formation can be higher than for solitary flight despite strong indirect evidence of overall energy saving by V-formation flying (2) (12) (19), and flapping phase of adjacent birds may be informative for understanding the fluid dynamic advantages of V-formation flying (12). Likewise, phase matching of body motion between neighbouring fish can help individual fish within the school to boost thrust or reduce drag (20). Fish can also adjust their body stiffness and maintain f_TB while reducing the amount of muscle activity needed to generate movement (50) (51) (52). Thus, we did not observe consistent and substantial changes in the f_TB of fish within the schools compared to solitary swimmers despite demonstrating substantial energetic savings by the group at high speed. Although reduced f_TB has been used in past studies as an indirect indicator of energy saving in fishes (53) (54), we caution that f_TB is not necessarily an indicator of energy use in group dynamics where fish constantly and dynamically interact with complex vortices and flow generated by other individuals. By simultaneously measuring kinematics and energetics, we discovered a downshifted performance curve of TEE per tail beat in fish schools at higher speeds (Fig. 4A), even with limited alteration in f_TB at lower swimming speeds within the school. This suggests that the locomotor muscle fibres in the body musculature of fish within the school need to generate less force as indicated by the lower measured TEE. Fish can possibly fine-tune undulatory motion to harness kinetic energy from nearby vortices (20) (48), reducing biological energy contribution for thrust.

Aerobic metabolic rate–speed curve of fish schools

We discovered that the aerobic metabolic rate (MO₂)–speed curve at speeds less than 3 BL s^-1 is U-shaped, with swimming at 0.3 BL s^-1 utilizing the same aerobic energy as moving at 3 BL s^-1 (Fig. 2D). The entire aerobic energy-speed performance curve is concave upward in an extended J-shape (Fig. 2C), and non-linear, in contrast to several previous analyses of energy use with speed in fish which have shown linear relationships (55) (56) (57). By quantifying energy use over a wide speed range and by measuring aerobic energy use at 14 distinct speeds, we showed that locomotion at the lowest speeds involves increased energy use relative to swimming at speeds up to 3 BL s^-1, similar to previous results from skate locomotion (58). This suggests that analyses of fish energy use as speed increases would benefit from increasing the number of tested speeds, and that efforts to extend speed measurements to both the lowest and highest speeds would enable a greater level of precision in measuring fish locomotor performance.

One key finding in this regard is that fish schools show a minimal absolute energetic cost (MO₂) at a mean group swimming speed of ∼1.0 BL s^-1 that is higher than the minimum swimming speeds tested. Thus, fish schools can swim at a speed with the least amount of energy use and potentially extend the distance travelled with the same amount of metabolic substrate onboard. This swimming speed of minimum cost closely matches the migratory speeds of carangiform and subcarangiform migratory fishes (often as schools) which are in the range of 0.5 to 1.5 BL s^-1 as recorded by tags on migrating fish (59, 60) (Fig. S1; Tables S1, S3). Although D. aequipinnatus is not a migratory species, our results suggest that the migratory speed of fish schools likely occurs at the speed showing the minimum aerobic metabolic rate for long-distance locomotion. Indeed, the Strouhal number when fish schools swim at U_opt, showing a minimum aerobic energy cost of locomotion, is 0.3, a hallmark of efficient swimming for many fish species (61). Fish of different size swimming at 1 BL s^-1 will necessarily move at different Reynolds numbers, and hence the scaling of body size to swimming speed needs to be considered in future analyses of other species that differ in size.

The exact energetic mechanisms underpinning the chosen migratory speeds of fish schools would benefit from more in-depth studies. Since both fish schools and solitary individuals have U_opt of ∼1 BL s^-1, the MO₂–speed curve of a fish school might be the average of individual curves from fish within the school. However, pushing and pulling forces fromhydrodynamic interactions of neighbouring fish could help swimmers settle into stable arrangements, a phenomenon known as the Lighthill conjecture (62). Further studies will need to explore whether individuals within the school are more frequently located at points recommended by the Lighthill conjecture which could potentially result in minimum metabolic costs of locomotion at U_opt in fish schools (49). Research on fish swimming in large aquaculture circular tanks has discovered that holding a large school of salmonids at a water velocity of ∼1 BL s^-1 resulted in healthy growth and good conditions (63). This phenomenon seems to suggest that the minimum metabolic costs of locomotion yield the largest available aerobic capacity for growth and other functions.

Higher locomotor costs at the lowest speeds are likely caused by higher postural costs for active stability adjustments in the near-still fluid. The direct link between higher energetic costs caused by a higher 3-D body angle and higher turning frequency at the lowest speeds in solitary danio (Fig. 4D, E) supports previous results in skates (58) where energetic costs are high at the very low swimming speeds due to the increased energetic cost involved in maintaining body stability and generating lift in negatively buoyant fishes. We expect more species that move in the fluid to show a higher postural cost in near-still fluid but careful measurements of both body kinematics and position along with high-resolution respirometry and accurate low-speed flow control will be required to demonstrate this phenomenon in a diversity of fish species. Elevated costs should be indicated at the lowest speeds as a direct increase in MO₂ and not just as COT. When the denominator (i.e. speed) for deriving COT is less than 1, COT tends to skew towards the higher value at the lowest speeds. Mechanistically, fish at low fluid speeds have lower muscle and propeller efficiencies (44) with reduced fluid-assisted stability. Increased active stability adjustments such as fin movements (58) and tilting behaviours that enable the fish’s body to act as a hydrofoil (64) are used for stability control (see supplementary video). Moreover, danio do not show intermittent swimming gaits at low speed as observed in some labriform swimmers (65) (66), and hence gait-specific kinematics most likely do not play a role in the elevated locomotion cost that we observe at the lowest speeds. Also, the higher turning frequency at < 0.75 BL s^-1 in fish schools can relate to the need for coordination with neighbours. Nevertheless, as fish move more rapidly over a speed range, we note that the total cost of transport of ∼1.2 kJ km^-1 kg^-1 at 3 BL s^-1 is among the lowest recorded for aquatic organisms, and is less than the TCOT (1.5 kJ km^-1 kg^-1) for jellyfish (the most energetically efficient low-speed swimmers) (4) swimming at < 2 BL s^-1 (67).

In summary, our experiments on giant danio have demonstrated substantial energy conservation resulting from schooling dynamics across a wide range of speeds in fish. Direct measurement of both aerobic and non-aerobic energy use is critical for understanding the rapid collective movement of animals. Fish schooling in the high-drag viscous aquatic medium serves as a model for understanding how group movement by animals can be a more energy-efficient biological collective than movement by isolated individuals. By increasing maximum aerobic performance, fish schools save anaerobic energy and reduce the recovery time after peak swimming performance. Furthermore, by decreasing the proportion of metabolic capacity and recovery time devoted to locomotion, animals can apportion more energy to other fitness-related activities, such as digestion, growth and reproduction. More broadly, comprehending how the collective dynamics of animal movements in the water, land, and air can modify the energy use profiles of individuals provide a better understanding of the ecological and evolutionary implications of group locomotion.

Materials and Methods

Experimental animals

The experiments were performed on giant danio (Devario aequipinnatus) that were acquired from a local commercial supplier near Boston, Massachusetts USA (Table S4). Five schooling groups are randomly distributed and housed separately in five 37.9 l aquaria (n=8 per tank). The five solitary individuals are housed separately in five 9.5 l aquaria (n=1 per tank). The individual housing condition acclimated the single D. aequipinnatus to the solitary environment and helped to reduce any isolation stress that might elevate whole-organism metabolic rate. In fact, the aerobic locomotion cost of solitary individuals showed no statistical difference from (in fact, being numerically lower) that of fish schools at a very low testing speed. The flow speed is similar to some areas of the aerated home aquarium for each individual fish. This suggests that the stress of solitary fish likely does not meaningfully contribute to the higher locomotor costs (see experimental protocol for more details on mitigating the stress). The condition factor showed no difference between solitary fish and fish schools (0.81 vs. 0.99; t₈ = 2.14, p = 0.065). All aquaria have self-contained thermal control (28 °C), an aeration system (>95 % air saturation, % sat.) and a filtration system. Water changes (up to 50% exchange ratio) were carried out weekly. Fish were fed ad libitum daily (TetraMin, Germany). Animal holding and experimental procedures were approved by the Harvard Animal Care IACUC Committee (protocol number 20-03-3).

Integrated Biomechanics & Bioenergetic Assessment System (IBAS)

The core of our Integrated Biomechanics & Bioenergetic Assessment System (IBAS) (Fig. S4) is a 9.35-l (respirometry volume plus tubing) customized Loligo® swim-tunnel respirometer (Tjele, Denmark). The respirometer has an electric motor, and a sealed shaft attached to a propeller located inside the respirometer. Regulating the revolutions per minute (RPM) of the motor controls water velocity of the tunnel. The linear regression equation between RPM and water velocity (V) is established (V = 0.06169 • RPM – 5.128, R² = 0.9988, p < 0.0001) by velocity field measured by particle image velocimetry (PIV). Hence, the aerobic costs during locomotion are measured through the regulation of the water velocity.

The swim-tunnel respirometer is oval-shaped. The central hollow space of the respirometer increases the turning radius of the water current. As a result, the water velocity passing the cross-section of the swimming section (80 × 80 × 225 mm) is more homogenous (validated by direct particle image velocimetry (PIV) of flow in the working section of the respirometer following the procedures in (68)). Moreover, a honeycomb flow straightener (80 × 80 × 145 mm) is installed in the upstream section of the swimming section that was specifically designed to create relatively uniform flow across the working section (also confirmed by direct flow imaging). Based on our flow visualization analyses, the effective distance of slow-moving fluid due to boundary layer was <2.5mm at speeds above 2 BL s^-1. The boundary layer played an ever-diminishing role at higher speeds (> 4 BL s^-1) when energy saving of fish schools becomes more predominant. Danio (∼10 mm wide) cannot effectively hide in the narrow boundary layer created by our flow baffle system. In addition, the convex hull volume of the fish school did not change as speed increased, suggesting that the fish school was not flattening against the wall of the swim tunnel, a typical feature when fish schools are benefiting from wall effects. In nature, fish in the centre of the school effectively swim against a ‘wall’ of surrounding fish where they can benefit from hydrodynamic interactions with neighbours.

To standardize the experimental apparatus and avoid instrument variation, solitary fish and fish schools are measured in the same swim-tunnel respirometer, and the same overall experimental design is used as in previous studies (14) (13) (15). The ratios of respirometer:individual volume (r_RI) in our experiments was 2200 for individual fish (we used larger solitary D. aequipinnatus to increase the signal-to-noise ratio), and 693 for fish schools. The r_RI is essentially a balance between giving enough space for fish to exhibit natural locomotion and reducing wall effects and generating a reliable signal-to-noise ratio to measure the decline of dissolved O₂ (DO) in water (69). The increase in the signal-to-noise ratio can come from the better technology of the O₂ probe (we used an optical O₂ probe which has a higher measurement sensitivity than the Winkler method or Electrode & Galvanic probes used in the previous studies, see (70) for review) and modifications to the swim-tunnel respirometer. We used a water loop is installed 95 cm downstream of the propeller with water returned to the respirometer 240 cm upstream of the swimming section. Flow in the water loop moves (produced by an in-line circulation pump, Universal 600, EHEIM GmbH & Co KG, Deizisau, Germany) in the same direction as the water flow in the swimming tunnel. A high-resolution fibre optic O₂ probe (Robust oxygen probe OXROB2, PyroScience GmbH, Aachen, Germany) is sealed in the loop in the downstream of the circulation pump where there is better mixing to continuously measure the DO level in the water (recording frequency ∼1 Hz, response time < 15s). The designated water sampling loop together with the water mixing by the propeller and water pumps effectively reduces the noise-to-signal ratio. As a result, the respirometry system reaches a stable signal-to-noise ratio once the sampling window is longer than 1.67 mins (see Fig. S5), well within the duration of the velocity step to obtain a stable signal-to-noise ratio for calculating MO₂ (32).

The oxygen probe was calibrated to anoxic (0 % sat., a solution created by super-saturated sodium sulphite and bubbling nitrogen gas) and fully aerated water (100 % sat.). After fish removal, the background MO₂ in the swim-tunnel respirometer was measured for a 20-min sealed period before and after each trial to calculate the average background MO₂ (< 6% of fish MO₂), which was used to correct for the MO₂ of fish (69). The pre-filtered water (laboratory grade filtration system) is constantly disinfected by UV light (JUP-01, SunSun, China) located in an external water reservoir to suppress the growth of microbial elements. Water changes of 60% total volume occurred every other day and a complete disinfection by sodium hypochlorite is conducted weekly (Performance bleach, Clorox & 1000 ppm).

To simultaneously measure schooling dynamics and swimming kinematics, the customized oval-shaped swim-tunnel respirometer is located on a platform with an open window beneath the swimming section. The platform is elevated 243 mm above the base to allow a front surface mirror to be installed at a 45° angle. This mirror allows a high-speed camera (FASTCAM Mini AX50 type 170K-M-16GB, Photron Inc., United States, lens: Nikon 50mm F1.2, Japan) to record the ventral view. The second camera (FASTCAM Mini AX50 type 170K-M-16GB, Photron Inc., United States, lens: Nikon 50mm F1.2, Japan) is positioned 515 mm to the side of the swimming section to record a lateral view. Synchronized lateral and ventral video recordings were made at 125 fps, and each frame was 1024 by 1024 pixels. To avoid light refraction passing through the water and distorting the video recordings, the swim-tunnel respirometry is not submerged in a water bath. Temperature regulation of the respirometer is achieved by regulating room temperature, installing thermal insulation layers on the respirometer and replenishing the water inside the respirometer from a thermally regulated (28 °C, heater: ETH 300, Hydor, United States & chiller: AL-160, Baoshishan, China) water reservoir (insulated 37.9-l aquarium) located externally.

The aerated (100% sat., air pump: whisper AP 300, Tetra, China) reservoir water is flushed (pump: Universal 2400, EHEIM GmbH & Co KG, Deizisau, Germany) to the respirometer through an in-line computer-controlled motorized ball valve (U.S. Solid) installed at the in-flow tube. The other in-line one-way valve is installed at the out-flow tube. The out-flow tube is also equipped with a one-way valve. The valve is shut during the measurement period, a precautionary practice to eliminate the exchange of water between the respirometer and the external reservoir when the water moves at a high velocity inside the respirometer. This flushing was manually controlled to maintain DO above 80 % sat. Every time the respirometer was closed to measure MO₂, the water temperature fluctuates no more than 0.2 °C. The water temperature inside the respirometer is measured by a needle temperature probe (Shielded dipping probe, PyroScience GmbH, Aachen, Germany) sealed through a tight rubber port of the respirometer.

To allow fish to reach the undisturbed quiescent state during the trial (another stress mitigation practice), the entire IBAS is covered by laser blackout sheets (Nylon Fabric with Polyurethane Coating; Thorlabs Inc, New Jersey, United States). The room lights are shut off and foot traffic around IBAS is restrained to the absolute minimum. Fish are orientated by dual small anterior spots of white light (lowest light intensity, Model 1177, Cambridge Instruments Inc, New York, United States) for orientation (one to the top and the other to the side) of the swimming section. The test section is illuminated by infrared light arrays to allow high-speed video recording with minimal disturbance of fish behaviour.

Experimental Protocol

The energy use of vertebrates at lower speeds is primarily aerobic, while for high-speed movement anaerobic metabolic pathways are activated to supply the additional (largely shorter-term) energy needs (9). While whole-animal aerobic metabolism is measured by oxygen (O₂) uptake rate (MO₂), the non-aerobic O₂ cost (mostly through high-energy phosphate and anaerobic glycolysis) is measured as excess post-exercise O₂ consumption (EPOC)(10). Both metabolic energy sources contribute to the total energy expenditure (TEE) required for movement. For a given workload, the higher the maximum aerobic performance of the animal, the less the need for the anaerobic energy contribution (9, 35). The experimental protocol captures both metabolic energy contributions by measuring MO₂ during locomotion and EPOC afterwards.

We studied five replicate schools and five replicate individuals. Swimming performance test trials were conducted with Devario aequipinnatus fasted for 24 hours, a sufficient period for a small size species at 28 °C (i.e. high resting MO₂) to reach an absorptive state. In fact, we observed no specific dynamic action, the amount of O₂ consumed for digestion during the first diurnal cycle (Fig. S3). Prior to the swimming performance test, testing fish were gently weighed and placed in the swim-tunnel respirometer. The fish swam at 35% U_crit for 30 mins to help oxidize the inevitable but minor lactate accumulation during the prior handling and help fish become accustomed to the flow conditions in the swim-tunnel respirometer (43). After this time, the fish to be tested were habituated (>20 hours) to the respirometer environment under quiescent and undisturbed conditions. The long habituation period also helps to reduce the stress and further reduce the likelihood that the solitary fish might be more stressed than the fish schools and showed an elevated MO₂ (13). During this time, we used an automatic system to measure the resting MO₂ for at least 19 hours. Relays (Cleware GmbH, Schleswig, Germany) and software (AquaResp v.3, Denmark) were used to control the intermittent flushing of the respirometer with fresh water throughout the trial to ensure O₂ saturation of the respirometer water. MO₂ was calculated from the continuously recorded DO level (at 1 Hz) inside the respirometer chamber. The intermittent flow of water into the respirometer occurred over 930 s cycles with 30 s where water was flushed into the respirometer and 900 s where the pumps were off and the respirometer was a closed system. The first 240 s after each time the flushing pump was turned off were not used to measure MO₂ to allow O₂ levels inside the respirometer to stabilize. The remaining 660 s when the pumps were off during the cycle were used to measure MO₂. The in-line circulation pump for water in the O₂ measurement loop stayed on throughout the trial.

We characterize the aerobic costs for the swimming performance of fish using an established incremental step-wise critical swimming speed (U_crit) test (10). The first preliminary trial determined the U_crit of this population of Devario aequipinnatus as 8 BL s^-1. Characterizing the swimming performance curve required a second preliminary trial to strategically select 9 water velocities (0.3, 0.5, 0.8, 1.0, 1.3, 1.5, 1.8, 2.3, 2.8 BL s^-1) to bracket the hypothesized upward concave shaped aerobic metabolism-speed curve at the lower speed (< 40% U_crit). Additional five water velocities (3.8, 4.9, 5.9, 6.9, 8.0 BL s^-1) are used to characterize the exponentially increasing curve to the maximum and sustained swimming speed, U_crit (see Fig. S6). Altogether, 14 points provide a reliable resolution to characterize the swimming performance curve. The cross-section of the danio school is ∼10% of the cross-sectional area of the swim tunnel, hence the blocking effect is negligible (71). At each water velocity, fish swam for 10 mins (58) to reach a steady state in MO₂ at low speed (see Fig. S7). Above 40% U_crit, MO₂ can become more variable (32). Hence, in this protocol, we focus on measuring the sustained aerobic energy expenditure by calculating the average MO₂ for each 10-min velocity step using Eqn 1. At the 5^th min of each velocity step, both ventral and lateral-view cameras are triggered simultaneously to record 10-sec footage at 125 frames per second, at 1/1000 shutter speed and 1024 ×1024 pixel resolution. Thus, both data streams of MO₂ and high-speed videos are recorded simultaneously. The U_crit test is terminated when 12.5% of fish in the school or a solitary individual touches the back grid of the swimming section for more than 20 secs (43). The U_crit test lasted ∼140 mins and estimates the aerobic portion of TEE over the entire range of swimming performance.

To measure the contribution of non-aerobic O₂ cost, where the majority of the cost is related to substrate-level phosphorylation, and to calculate the TEE for swimming over the entire speed range, we measured EPOC after the U_crit test for the ensuing 19 hours, recorded by an automatic system. Most previous measurements of EPOC have used a duration of ∼5 hours (see review (42)), but our extended measurement period ensured that longer duration recovery O₂ consumption during EPOC was measured completely as fish were exercised to U_crit (see summary table in 16). The intermittent flow of water into the respirometer occurred over 30 s to replenish the DO level to ∼95% sat. For the following 900 s the flushing pump remained closed, and the respirometer becomes a closed system, with the first 240 s to allow O₂ saturation inside the respirometer to stabilize. The remaining 660 s when the flushing pump was off during the cycle were used to measure MO₂ (see Eqn. 1). The cycle is automated by computer software (AquaResp v.3) and provided 74 measurements of MO₂ to compute EPOC. Upon the completion of the three-day protocol, the school or individual fish are returned to the home aquarium for recovery. The fish condition was closely monitored during the first 48 hours after the experiment, during which no mortality was observed.

Bioenergetic measurement and modeling

To estimate the steady-rate whole-animal aerobic metabolic rate, MO₂ values were calculated from the sequential interval regression algorithm (Eqn. 1) using the DO points continuously sampled (∼1 Hz) from the respirometer.

Where d_DO/d_t is the change in O₂ saturation with time, V_r is the respirometer volume, V_f is the fish volume (1 g body mass = 1 ml water), S_o is the water solubility of O₂ (calculated by AquaResp v.3 software) at the experimental temperature, salinity and atmospheric pressure, t is a time constant of 3600 s h^-1, M_f is fish mass, and a is the sampling window duration, i is the next PO₂ sample after the preceding sampling window.

To account for allometric scaling, the MO₂ values of solitary fish were transformed to match the size of the individual fish in the school (see Table S4) using an allometric scaling exponent (b = 0.7546). The calculation of the scaling relationship [Log₁₀(MO₂) = b•Log10(M) + Log10(a), where M is the body mass & a is a constant] was performed by least squares linear regression analysis (y = 0.7546 • x + 0.2046; R² = 0.6727, p < 0.0001) on the 180 data points of metabolic rate and body mass from a closely related species (the best available dataset to our knowledge). (The mass scaling for tail beat frequency was not conducted because of the lack of data for D. aequipinnatus and its related species. Using the scaling exponent of distant species for mass scaling of tail beat frequency will introduce errors of unknown magnitude.) The allometrically scaled MO₂ values were used to derive other energetic metrics (listed below & such as aerobic scope) for the solitary fish. The energetic metrics of fish schools are calculated from the mass-specific MO₂.

The resting oxygen uptake (MO_2rest), the minimum resting metabolic demands of a group of fish or a solitary individual, is calculated from a quantile 20% algorithm (74) using the MO₂ estimated between the 10^th–18^th hour and beyond the 32^nd –51^st hour of the trial. These are the periods of quiescent state when fish completed the EPOC from handling and swimming test.

The MO₂ for the aggregation (Fig. 2B, MO_2aggregate) is calculated as the average value using the MO₂ estimated between the 10^th–18^th hour without the effects of any tests.

Minimum oxygen uptake (MO_2min) is the lowest MO₂ value recorded in the entire trial, which always occurred at the optimal speed when a school of fish collectively reached the lowest MO₂ value.

Active oxygen uptake (MO_2active) is the highest average MO₂ when fish are actively swimming (10).

The aerobic scope (the metric for aerobic capacity) is the numerical difference between MO_2active and MO_2min (i.e. MO_2active – MO_2min) (10).

The percentage of aerobic scope (% aerobic scope) is calculated by normalizing the MO₂ value at a water velocity as a % aerobic scope: [(MO₂ – MO_2min) / (MO_2active – MO_2min)]∗100%. The apportioning of aerobic scope to swimming performance is computed across the entire range of swim speeds.

The excess post-exercise oxygen consumption (EPOC) is an integral area of MO₂ measured during post-exercise recovery, from the end of U_crit until reached MO_2rest plus 10% or within 24 hours post-exercise, whichever endpoint occurred first (72). This approach reduces the likelihood of overestimating EPOC due to any potential spontaneous activities (72). To account for the allometric scaling effect, we used the total amount of O₂ consumed (mg O₂) by the standardized body mass of fish (1.66 g) for fish schools and solitary fish.

We model EPOC (i.e. non-aerobic O₂ cost) and MO₂ during U_crit to estimate a total O₂ cost over the duration of the swimming performance test. Our conceptual approach was pioneered by Brett (10) in fish (75) and is also used in sports science (9). Mathematical modeling of EPOC and MO₂ during swimming was applied to study the effects of temperature on the total cost of swimming for migratory salmon (43). We improved the mathematical modeling by applying the following physiological and physics criteria. The first criterion is that significant accumulation of glycolytic end-product occurred when fish swimming above 50% U_crit (31) which corresponds to > 40% MO_2max (or ∼ 50% aerobic scope) (9). This is also when fish start unsteady-state burst-&-glide swimming gait (31). The second criterion is that the integral area for the non-aerobic O₂ cost during swimming can only differ by ≤ 0.09% when compared to EPOC. The non-aerobic O₂ cost during swimming is the area bounded by modeled MO₂ and measured MO₂ as a function of time when fish swim > 50% U_crit (see Fig. 2A & Table S2). The third criterion is that total energy expenditure is expected to increase exponentially with swimming speed (Fig. S8). Specifically, these curves were fitted by power series or polynomial models, the same models that describe the relationship between water velocity and total power and energy cost of transport (Fig. S8). Following these criteria, the non-aerobic O₂ cost at each swimming speed is computed by a percentage (%) modifier based on the aerobic O₂ cost (Table S2). The exponential curve of total O₂ cost as swimming speed of each fish school or solitary individual was derived by an iterative process until the difference between non-aerobic O₂ cost and EPOC met the 2^nd criterion. The sum of non-aerobic O₂ cost and aerobic cost gives the total O₂ cost.

Total energy expenditure (TEE) is calculated by converting total O₂ cost to kJ × kg^-1 using an oxy-calorific equivalent of 3.25 cal per 1 mg O₂ (76).

Cost of transport (COT), in kJ × km^-1 × kg^-1 is calculated by dividing TEE by speed (in km × h^-1) (58).

Three-dimensional kinematic data extraction from high-speed videography

We used two synchronized 10-sec high-speed videos (lateral and ventral views, at each speed) for kinematic analyses. We calibrated the field of view of the high-speed cameras using a direct linear transformation for three-dimensional (3-D) kinematic reconstruction (DLTdv8)(77) by applying a stereo calibration to the swimming section of the respirometer (see Fig. S9). We digitized the anatomical landmarks of fish (see Fig. S10) to obtain the X, Y, and Z coordinates for each marker at the 1^st sec, 5^th sec and 10^th sec for videos recorded at each speed. These coordinates are used to calculate the following kinematic parameters. All the calculations are validated on the known length and angle of test objects inserted into the tank working section.

The 3-D distance between the tip of the nose of each fish in the school per frame is calculated in vector Eqn. 2

Where spatial coordinates of two neighbouring fish are (X_a, Y_a, Z_a) and (X_b, Y_b, Z_b) respectively.

The 3-D angle of each fish in the school to the frontal plane per frame is calculated by Eqn. 3

where the spatial coordinates of the caudal peduncle, nose of fish and right-angle crosshair between the peduncle and the nose are (X₁, Y₁, Z₁), (X₃, Y₃, Z₃) and (X₂, Y₂, Z₂) respectively (see Fig. S9).

The fish’s angle to water flow per frame is calculated using the arctangent function in Excel (Microsoft, United States) using the spatial coordinates of the caudal peduncle and nose of the fish.

The school length (in X-axis) is calculated as a 3-D distance (Eqn. 2) between the nose of the first fish of the school to the caudal peduncle of the last fish in the school.

The values (3-D distance of the individuals, 3-D body angle, 3-D school length) calculated above were averaged among the three frames at each speed as a representative kinematic feature of the schools (or solitary fish) for each speed. The standard deviation of the angle to water flow in fish schools is calculated from the angular values of the individuals.

Tail beat frequency (f_TB), in Hz, is calculated as the number of tail beats observed within one second. We sampled 10 tail beats from fish exhibiting steady-state swimming to obtain an average and representative f_TB for each solitary fish at each speed. We sampled 10 tail beats of steady-state swimming for three individuals in the center of the fish school for an average and representative f_TB for each fish school.

Turning frequency is the total number of events when fish made a 90° turn in the 10-sec video. To make the total number of turns per individual and fish schools comparable (multiple individuals inevitably have a larger absolute number of turns), we derived the average turning frequency per schooling fish by dividing the total number of turns of fish school by the number of fish in the school.

Total energy expenditure per tail beat (kJ kg^-1 beat^-1) is calculated by TEE • (f_TB • 60)^-1 to estimate the total metabolic energy spent to achieve one stride length.

Additional calculations of fluid dynamic metrics were:

Strouhal number = (f_TB • tail beat amplitude) • U ^-1 (8). Tail beat amplitude is measured as the average distance of five peak-to-peak oscillation amplitudes of the tip of the fish’s tail. The measurement is conducted on the calibrated high-speed video in video analysis software (Phontron FASTCAM Viewer 4, Photron USA, Inc.).

Reynolds number = (water density • U • fish fork length) • water dynamic viscosity ^-1 (8). Water density and dynamic viscosity are given at 28 °C.

Statistical analyses

Measurement points are presented as mean ± s.e.m. For the metrics that failed normality tests, logarithm transformations were applied to meet the assumptions of normality of residuals, homoscedasticity of the residuals, and no trend in the explanatory variables. Since fish schools exhibit the features of a coherent functional unit, the statistical model treated one school as one sample size, and the experiment measured five replicates of fish schools. The majority of statistical comparisons used the General Linear Model (Fixed factors: solitary fish vs. fish schools & swimming speeds, the label of solitary fish & fish schools as a random effect) with Holm–Šídák post-hoc tests. The few comparisons that used other statistical models are listed below. The comparison of MO_2min at U_opt, MO₂ of aggregating behaviours exhibited at the lowest speed (MO_2aggregate) and resting MO₂ (MO_2rest) in fish schools used one-way ANOVA with Holm– Šídák post-hoc tests. The comparison of EPOC between solitary fish and fish schools used a two-tailed Student’s t-test. The comparison of the duration of EPOC between solitary fish and fish schools used a two-tailed Student’s t-test. The overall difference in percentage aerobic scope (% aerobic scope) between fish schools and solitary individuals is compared by the Wilcoxon singed-rank test over the entire range of 14 swimming speeds. The statistical analyses were conducted in SPSS v.28 (SPSS Inc. Chicago, IL, USA). The best-fitting regression analyses were conducted using Prism v.9.4.1 (GraphPad Software, San Diego, CA, USA). 95% C.I. values were presented for all regression models as shaded areas around the regression or data points. Statistical significance is denoted by *, **, ***, **** for p-values of ≤ 0.05, ≤ 0.01, ≤ 0.001, ≤ 0.0001 respectively.

Acknowledgements

Many thanks to members of Lauder Laboratory for numerous discussions about fish schooling behaviour, for comments on the manuscript, and to Cory Hahn for fish care.

Funding

Funding provided by the National Science Foundation grant 1830881 (GVL), the Office of Naval Research grants N00014-21-1-2661 (GVL), N00014-16-1-2515 (GVL), 00014-22-1-2616 (GVL), and a Postdoctoral Fellowship of Natural Sciences and Engineering Research Council of Canada PDF - 557785 – 2021 (YZ).

Author contributions

Y.Z. and G.L. conceptualized the study. Y.Z. performed experiments and data analyses and wrote the manuscript. Y.Z. and G.L. provided manuscript edits and comments and approved the final version.

Competing interests

Authors declare that they have no competing interests.

Data and materials availability

All data generated or analysed during this study are included in this published article (and its supplementary information files).

Supplementary Materials

Introductory text

Figs. S1 to S9

Tables S1 to S4