Simple phalanx pattern leads to energy saving in cohesive fish schooling

Abstract

The question of how individuals in a population organize when living in groups arises for systems as different as a swarm of microorganisms or a flock of seagulls. The different patterns for moving collectively involve a wide spectrum of reasons, such as evading predators or optimizing food prospection. Also, the schooling pattern has often been associated with an advantage in terms of energy consumption. In this study, we use a popular aquarium fish, the red nose tetra fish, Hemigrammus bleheri, which is known to swim in highly cohesive groups, to analyze the schooling dynamics. In our experiments, fish swim in a shallow-water tunnel with controlled velocity, and stereoscopic video recordings are used to track the 3D positions of each individual in a school, as well as their tail-beating kinematics. Challenging the widespread idea of fish favoring a diamond pattern to swim more efficiently [Weihs D (1973) Nature 241:290-291], we observe that when fish are forced to swim fast-well above their free-swimming typical velocity, and hence in a situation where efficient swimming would be favored-the most frequent configuration is the "phalanx" or "soldier" formation, with all individuals swimming side by side. We explain this observation by considering the advantages of tail-beating synchronization between neighbors, which we have also characterized. Most importantly, we show that schooling is advantageous as compared with swimming alone from an energy-efficiency perspective.

Keywords: collective dynamics; energy efficiency; fish swimming; pattern formation; synchronization.

Fig. 1.

Characteristic swimming patterns for increasing fish group size at two different swimming speeds. (A–D) $U=0.77BL.s^{-1}$ (see also Movies S1 and S2). The school pattern is spread downstream with characteristic angles and distances to nearest neighbors. (E–H) $U=3.91BL.s^{-1}$ (see also Movies S3 and S4). As more effort is required to hold a high swimming regime, the fish reorganize in a compact in-line formation. In this configuration, fish within the group are synchronized with their nearest neighbors, corresponding to collaborative efficient swimming modes.

Fig. 2.

Statistical properties of the fish schools as a function of swimming speeds over the whole range of cases studied (see Tables S1 and S2). (A) Variation of the $z$-position of fish ($h_z$ is the depth normalized by the fish average body height) as a function of swimming speed. Small black dots represent the instantaneous $z$ position of each fish, whereas orange squares represent average $z$ position, averaged over all groups. (B and C) Probability density of the NND for low (B) and high (C) swimming speeds. B and C, Insets show the probability density of nearest-neighbor angles $\varphi$. (D) Probability density map of NND as a function of the swimming speed. (E) Percentage of occurrences of diamond-shaped (DS) [or T-shaped (TS)] and phalanx-shaped (PS) patterns. High-speed swimmers are mainly characterized by phalanx patterns and short NND in comparison with low speed regimes.

Fig. 3.

(A and B) Tail-flapping frequency $f$ (Hz) (A) and tail-flapping amplitude (nondimensionalized by fish body length) (B), as a function of the swimming velocity $U$ (in body lengths per second). (C) Transverse Reynolds number, $R{e}_t=⟨f⟩⟨A⟩⟨L⟩/⟨\nu ⟩$, as a function of the cruising Reynolds number, $R{e}_U=⟨U⟩⟨L⟩/⟨\nu ⟩$. (D) Evolution of the synchronization parameter $s^{\ast }=S/(S+NS)$ that represents the cumulative probability of a synchronized state between nearest neighbors, as a function of the swimming velocity $U$ (see Table S3).

Fig. S1.

Strouhal number as a function of the swimming velocity.

Fig. 4.

Examples of time series of the tail tip amplitude for one given individual (fish 2) within the school and its two nearest neighbors (fish 1 and 3). (A and B) Low (A) and high (B) swimming speed. The in-phase (IP) and out-of-phase (OP) swimming regimes are also shown. (C and D) Evolution of the phase difference $\mathrm{{\rm Y}}$ between fish 2 and its two nearest neighbors (fish 1 and 3) showing a nonsynchronized state ($NS$) at low swimming speed (C) and strongly synchronized state ($S$) at high swimming speeds (D). D, Insets show zooms in the case of synchronized swimming that are either around $0$ (for IP synchronization) or $\pi$ (for OP synchronization).

Fig. S2.

Schematic diagram of the swimming channel. An external pump drives the flow, which is directed toward the test section after a convergent ramp and through a honeycomb section to minimize swirling motions. A second honeycomb delimits the test section downstream.

Fig. S3.

(A) Turbulence intensity (TI) as a function of the average flow velocity in the channel. It can be seen that the turbulence in the flow remains fairly constant over the flow rate range explored and remains at <5%. (B) The velocity profile in the x direction at midheight and midsection of the channel, u(y), is rather flat and also does not change with the flow rate.