Fish School Behaviour

• Fish School Behaviour is the collective organization of fish into cohesive moving groups driven by local social interactions, sensory anisotropy, and environmental forces.
• Research employs data-driven stochastic models, hydrodynamic theory, and agent-based simulations to characterize regime transitions in alignment, milling, and swarming.
• Insights from FSB inform ecological studies, optimize bio-inspired robotics control, and inspire algorithms by quantifying order parameters and spacing scales.

Fish school behaviour (FSB) is the collective organization of fish into cohesive moving groups whose large-scale states emerge from local social interactions, sensory anisotropy, environmental forcing, and fluid-mediated coupling. In the research literature, FSB encompasses polarized schooling, swarming, milling, winding, avalanche-like collective turns, migratory consensus, obstacle-dependent reorganization, and non-clogging evacuation through bottlenecks. The subject is studied through laboratory tracking, data-driven stochastic models, hydrodynamic theory, active-matter analogies, and control-oriented frameworks, with recurrent emphasis on order parameters, characteristic spacing scales, and state transitions (Calovi et al., 2013, Puy et al., 2023, Larrieu et al., 2022).

1. Collective states and phase organization

A central line of work treats fish schools as stochastic self-propelled systems whose macroscopic state can be classified by polarization and rotational order. In the data-driven model explored beyond the experimentally fitted regime, four collective regimes are identified: swarming, schooling, milling, and winding or line states. The standard order parameters are the polarization

$$P=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec v_i}{v}\right|$$

and the milling order

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$

with $P\approx 1$ corresponding to strongly aligned schooling and $M\approx 1$ to coherent vortex-like milling (Calovi et al., 2013).

The same modeling tradition attributes these regimes to the balance between alignment, attraction, and noise. In the dimensionless formulation,

$$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$

so increasing swimming speed decreases $\alpha$ and increases both $\beta$ and $\gamma$. This is the mechanism behind the speed-induced transition from swarming to schooling. A modified version of the model introduces frontal preference through $\Omega_{ij}=1+\cos(\theta_{ij})$, and that anisotropic weighting is reported to enable a large milling region that is essentially absent in the original model (Calovi et al., 2013).

The transition between schooling and milling is not a sharp binary boundary. In perturbation studies using the related data-driven model, the transition region is characterized by multistability and irregular switching, and its fitted boundary is given by

$\beta = A\sqrt{\gamma}+B,$

with reported values $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$0 for $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$1 and $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$2 for $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$3. Susceptibility

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$4

and its milling analogue $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$5 peak near this region, which is also where a school responds most strongly to perturbing individuals (Calovi et al., 2014).

Group size also enters the phase structure. In the phase-diagram study, milling appears only above a minimum size around 60 fish, then weakens and disappears as groups become very large. The stated causes are the linear distance dependence of attraction and information-propagation delays that generate conflicting reactions in elongated large groups (Calovi et al., 2013).

2. Interaction rules, sensory anisotropy, and characteristic spacing

At the microscopic level, FSB is commonly modeled through avoidance, alignment, and attraction. In the vision-based “three-A” model, each fish moves at constant speed $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$6 and updates its heading according to neighbors in a repulsion zone $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$7, an alignment zone $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$8, and an attraction zone $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$9, together with a limited field of vision $P\approx 1$0, rotational noise, and a maximum turning angle $P\approx 1$1. If repulsion-zone neighbors exist, the fish turns away; otherwise it aligns with neighbors in the alignment zone and moves toward neighbors in the attraction zone (Chan et al., 21 Jul 2025).

A more explicitly data-driven formulation writes the target angular response as

$P\approx 1$2

where $P\approx 1$3 is the first Voronoi neighborhood, $P\approx 1$4 is the heading difference, and $P\approx 1$5 is the neighbor’s angular position relative to the focal fish. The factor $P\approx 1$6 breaks front–back symmetry and gives stronger influence to neighbors ahead (Calovi et al., 2014).

Preferred spacing is a recurring quantitative motif. In zebrafish moving in open water, the most probable inter-fish distance is

$P\approx 1$7

obtained from a log-normal fit of $P\approx 1$8. In the pillar-array experiments, this distance becomes the reference scale for the transition from natural social alignment to geometry-imposed orientation: the crossover occurs around $P\approx 1$9, corresponding to $M\approx 1$0, when the pillar spacing becomes comparable to the fish social distance (Ventéjou et al., 2024).

A related spacing scale appears in evacuation experiments on neon tetra. Measuring the local density in a virtual box of $M\approx 1$1 just before the opening, the study reports a nearly constant density

$M\approx 1$2

for openings larger than about $M\approx 1$3 cm, leading to the characteristic social spacing

$M\approx 1$4

A geometrical correction based on 91% occupancy gives $M\approx 1$5 (Larrieu et al., 2022).

Interaction strength and range also vary with ontogeny. In Pseudomugil signifer, Boltzmann inversion of pair statistics yields an effective potential of mean force,

$M\approx 1$6

and supports the conclusions that repulsion strength is essentially size-independent, attraction becomes stronger in larger fish, and the repulsion zone becomes larger with body size. The measured polar order parameter $M\approx 1$7 is reported as approximately $M\approx 1$8 for small fish and $M\approx 1$9 for medium fish, indicating stronger alignment in larger or more developed individuals (Romenskyy et al., 2015).

3. Cascades, avalanches, and collective memory

Beyond quasi-stationary phases, FSB exhibits intermittent collective rearrangements. One experimentally grounded formulation defines behavioral cascades as avalanches of consecutive large heading changes. Velocities are estimated from digitized trajectories by Richardson extrapolation, turning angles $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$0 are computed from successive velocity vectors, and a threshold $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$1 separates “small” from “large” turns. Avalanches are trains of consecutive frames with at least one active fish, characterized by duration $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$2 and size $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$3 (Mugica et al., 2022).

The empirical avalanche distributions in that framework show scale-free signatures. Reported exponents are $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$4, $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$5, $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$6, and $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$7, with

$$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$8

and an observed exponent $$\alpha=\frac{\xi^3\sigma^2}{v},\qquad \beta = \frac{k_V v}{\xi^2 \sigma^2},\qquad \gamma = \frac{k_P v}{\xi^4 \sigma^4},$$9, close to the theoretical estimate $\alpha$0. When turning angles are randomized within each trajectory, the avalanche distributions become exponential rather than scale-free, indicating that the power-law tails arise from temporal correlations in fish turning behavior. The same study reports leadership probabilities $\alpha$1 as high as $\alpha$2 for some individuals, interpreted as evidence of effective leaders that disproportionately initiate avalanches (Mugica et al., 2022).

A complementary experimental analysis of spontaneous turning avalanches in black neon tetra uses thresholded angular velocity

$\alpha$3

to define avalanches. For $\alpha$4, the size and duration distributions obey

$\alpha$5

with fitted exponents $\alpha$6 and $\alpha$7, while inter-event times satisfy

$\alpha$8

with $\alpha$9. The average avalanche size conditioned on duration scales as $\beta$0, and rescaled avalanche shapes collapse according to

$\beta$1

The same work reports boundary effects, “dragon king” tails associated with wall-induced turns, and an Omori-law-like clustering of aftershocks with $\beta$2, $\beta$3, and a characteristic clustering timescale $\beta$4 frames (Puy et al., 2023).

Collective memory has been reinterpreted in this context. In the “three-A” schooling model, the milling-to-schooling transition is described as a noisy transcritical bifurcation rather than structural bistability. The proposed phenomenological normal form for polarization is

$\beta$5

with $\beta$6. In this account, apparent hysteresis and transient milling arise from noise-mediated residence near the pre-bifurcation state, not from a persistent double-well landscape (Chan et al., 21 Jul 2025). This interpretation is consistent with perturbation studies showing that responsiveness is maximal near the schooling–milling transition and only below a threshold in the noise-to-social-interaction ratio (Calovi et al., 2014).

4. Hydrodynamics, wakes, and mechanical mediation

A persistent theme in FSB research is that schooling fish are embedded in a fluid and therefore cannot be reduced to vision-only point particles. One hydrodynamic perspective, based on aerial photographs, videos, and slender-body calculations, argues that several fish species often display random-shape patterns, no preferred orientation, and disordered swarm-like arrangements, especially under disturbance. In that framework, the mean total hydrodynamic force

$\beta$7

is reported to be 2 to 5 times larger in random patterns than in diamond patterns, with representative values $\beta$8 versus $\beta$9 for a $\gamma$0 school and $\gamma$1 versus $\gamma$2 for a $\gamma$3 school. The stated implication is enhanced transfer of hydrodynamic information, and hence enhanced alertness and maneuverability, in disordered schools (Kadri et al., 2016).

More mechanistic schooling models replace fish with self-propelled swimmers that both generate and respond to flow. In a vortex-dipole representation governed by Biot–Savart interactions, classical Aoki–Couzin-type attraction, alignment, and repulsion rules do not robustly maintain schooling once hydrodynamics is included: swimmers strain apart or collide. Reinforcement learning, specifically one-step $\gamma$4-learning with a shared policy, is then used to let swimmers adapt their gaits by changing vortex strengths. Within that framework, adaptive swimmers maintain relative positions in prescribed formations, and evolutionary optimization by CMA-ES identifies elongated, striated, hourglass-like schools as favorable configurations for minimizing the collective effort measure $\gamma$5 (Gazzola et al., 2015).

A minimal far-field model coupling social rules to hydrodynamic interactions predicts additional collective structure. In dimensionless form, it introduces alignment intensity $\gamma$6, noise intensity $\gamma$7, and hydrodynamic dipole intensity $\gamma$8. For a 10 cm fish with $\gamma$9 cm, $\Omega_{ij}=1+\cos(\theta_{ij})$0 m/s, and $\Omega_{ij}=1+\cos(\theta_{ij})$1, the estimate is $\Omega_{ij}=1+\cos(\theta_{ij})$2. Simulations recover swarming, schooling, and milling, but also predict a new turning phase that appears only when full hydrodynamics are included. The same model reports that hydrodynamics increases the average swimming speed $\Omega_{ij}=1+\cos(\theta_{ij})$3 above 1, destabilizes side-by-side arrangements, and biases schools toward in-line configurations (Filella et al., 2017).

A more recent hybrid agent-based and Newtonian model incorporates both body-induced potential flow and vortex wakes parameterized from DNS of a carangiform swimmer. In that model, fish obey surge, sway, and yaw dynamics, while the induced flow is decomposed as $\Omega_{ij}=1+\cos(\theta_{ij})$4. The body-induced potential flow decays rapidly, down to about 1% of swimming velocity within about half a body length, whereas the wake can extend several body lengths downstream and reach perturbation levels of about 7% of surge speed. The simulations indicate that wakes improve organization especially in high-alignment schools: the first principal component explains about 11.95% of the variance with wake versus 1.54% without wake, and followers tend to occupy the edge of the leading fish’s wake in diagonal or oblique patterns (Zhou et al., 2024).

5. Confinement, clutter, and bottleneck passage

Environmental geometry can reorganize FSB without abolishing cohesion. In zebrafish moving through a square lattice of opaque pillars of diameter $\Omega_{ij}=1+\cos(\theta_{ij})$5 cm, low pillar density preserves a typical inter-distance and strong alignment, but above a critical density the fish spread more randomly and orient along the free axes of the lattice. The relative-orientation distribution evolves from a peak near $\Omega_{ij}=1+\cos(\theta_{ij})$6 toward a $\Omega_{ij}=1+\cos(\theta_{ij})$7-periodic form,

$\Omega_{ij}=1+\cos(\theta_{ij})$8

and the order parameter

$\Omega_{ij}=1+\cos(\theta_{ij})$9

drops nonlinearly from about 0.9 without pillars to about 0.2 at the highest density. Model fitting attributes the transition primarily to a sharp increase in the tumbling rate $\beta = A\sqrt{\gamma}+B,$0, by nearly an order of magnitude around $\beta = A\sqrt{\gamma}+B,$1 (Ventéjou et al., 2024).

Spatial confinement alone also alters heterogeneity. In quasi-2D shoals of 25 cardinal tetra fish observed in cylindrical arenas of radii $\beta = A\sqrt{\gamma}+B,$2 and $\beta = A\sqrt{\gamma}+B,$3 cm, the convex hull area $\beta = A\sqrt{\gamma}+B,$4 and Voronoi cell areas $\beta = A\sqrt{\gamma}+B,$5 fluctuate strongly over time, but $\beta = A\sqrt{\gamma}+B,$6 remains roughly constant. The local-area distribution $\beta = A\sqrt{\gamma}+B,$7 changes from compact and nearly Gamma-like in small arenas to broad and strongly tailed in large arenas. Monte Carlo simulations using a Morse-like effective interaction

$\beta = A\sqrt{\gamma}+B,$8

reproduce the observed area distributions and reveal a non-monotonic behavioral preference transition between $\beta = A\sqrt{\gamma}+B,$9 cm and $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$00 cm. The study also reports a positive correlation between structural heterogeneity and dynamic activity, in contrast to short-range interacting soft-matter systems (Kuntz et al., 2023).

Visual confinement by light fields yields a different but related active-matter picture. In negatively phototactic rummy-nose tetra, static dark disks produce a maximum number density

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$01

that is independent of group size for $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$02. The swim pressure-like quantity

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$03

increases linearly with number density, and the fitted slope is interpreted as an isotherm with $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$04. Under dynamic compression $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$05, the density still saturates at the same $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$06, but the effective temperature depends on compression time according to

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$07

which the authors interpret as consistent with constant heat flux (Giannini et al., 2020).

Bottleneck passage supplies an especially sharp contrast with pedestrian or granular jamming. In neon tetra evacuation through circular openings from 4 cm down to 1.5 cm, the cumulative number of escaped fish grows almost linearly with time, with current increasing from about 1 fish/s for the smallest openings to roughly 3.5 fish/s for the 4 cm opening. Time lapses between exits show short-time inhibition but no power-law tail, and the complementary cumulative distribution is fit by

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$08

The discharge is modeled by a modified Beverloo-type relation

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$09

with reported values $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$10 and $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$11 for constrained $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$12, or $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$13 when fitted freely, and the flow-stopping diameter is given as $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$14. The interpretation is that fish pass as deformable 2D “social bubbles” and exhibit “wise queuing” rather than clogging (Larrieu et al., 2022).

6. Sensing, migration, stochastic coarse-graining, and control

FSB is also a problem of information integration. In noisy dynamic light gradients, golden shiners and rummy-nose tetra differ in how they combine social and environmental cues. The performance metric is

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$15

and tetras outperform shiners at all tested group sizes. Shiner accelerations correlate more strongly with social cues than environmental cues, whereas tetr a accelerations correlate more strongly with environmental cues. In the modified Berdahl–Couzin model,

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$16

and the nearest-neighbor distance is minimized at $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$17, with an empirically relevant range of approximately $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$18 to $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$19. This identifies a balance between gradient tracking and cohesion rather than dominance of either information source (Puckett et al., 2017).

Migration extends this informational problem across generations. In a stochastic adaptive-network model, each fish is a node with an internal state $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$20, links encode social interaction, and the control variable

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$21

measures sociality. Linked individuals follow a majority rule, while isolated individuals update according to a preference distribution biased by a memorized destination $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$22 with strength $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$23. The model shows thresholds, coexistence, and hysteresis in coordinated migration, and demonstrates that removal of knowledgeable individuals or reduction of preference strength can abruptly stop migration to a destination site (Luca et al., 2013).

Foraging models make the same point in bounded spaces with obstacles. In the SDE–PDE framework for schooling and food seeking,

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$24

with scent field $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$25 given by $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$26 under Neumann boundary conditions. In the one-obstacle configuration, the success probability increases with school size up to an optimal value $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$27, then decreases; in the two-obstacle configuration the reported optimum is $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$28. In all configurations, the school structure is preserved through the foraging process (Ta et al., 2015).

Coarse-grained stochastic modeling has recently separated the school into collective and internal motions. In that formulation,

$$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$29

where the center of mass $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$30 is modeled as an active Brownian pseudo-particle with Schienbein–Gruler friction and hard-wall interactions, while fish in the center-of-mass frame move approximately independently in an effective potential inferred from the radial density $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$31. The resulting framework reproduces CM wall-following, a crater-shaped CM velocity distribution, a flat core density with exponential outer decay, non-Gaussian speed and turning-rate statistics, burst-and-coast swimming, and the observed polarization distribution (Lamo et al., 10 Sep 2025).

Closed-loop control studies move from description to intervention. In experiments with real rummy-nose tetra and 2D virtual fish displayed on a screen, guidance is formulated as a discrete-state Q-learning problem in which the reward is based on the real school centroid’s $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$32-position relative to the target edge. After simulation pretraining and real-world learning, the proposed method yields mean centroid positions of 0.358 for left guidance and 0.525 for right guidance, compared with 0.461 and 0.562 for a “stay-at-edge” baseline and 0.501 and 0.495 without stimulus. Two-tailed Welch’s $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$33-tests give $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$34 for the proposed method, $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$35 for the baseline, and $$M=\frac{1}{N}\left|\sum_{i=1}^{N}\frac{\vec r_i\times \vec v_i}{|\vec r_i|v}\right|,$$36 without stimulus, indicating a statistically significant directional effect for the learned policy (Nishii et al., 17 Mar 2026).

In a separate engineering usage, “Fish School Behaviour” denotes a nature-inspired optimization algorithm rather than the biological phenomenon itself. On the Setonix platform, OpenMP experiments report that reduction outperforms critical, and that dynamic or guided scheduling outperform static scheduling at higher thread counts, with improvement as much as 50% once thread count exceeds about 20. This computational appropriation underscores the extent to which schooling has become a template for algorithmic design, but it is conceptually distinct from empirical FSB in animal groups (Wang et al., 27 Jul 2025).