Mathematical Models for Fish Schooling

Atsushi Yagi; Linh Thi Hoai Nguyen; Ton Viet Ta

REVIEW 3 major objections 3 minor 2 cited by

Mathematical Models for Fish Schooling

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This note reviews two stochastic models of fish schooling and reports a simulation in which a 25-fish school, using attraction, repulsion, velocity matching, obstacle reflection, and scent-gradient sensing, reaches a food source while keepi

desk verdict A clear review of the authors' own fish-schooling SDE models, but the one new simulation example is not reproducible and quietly simulates a modified model. read the letter →

arxiv 2508.08310 v1 pith:3NYEJTT4 submitted 2025-08-08 cond-mat.stat-mech math.PR

Linh Thi Hoai Nguyen , Ton Viet Ta , Atsushi Yagi This is my paper · ORCID

classification cond-mat.stat-mechmath.PR MSC 92A1860H1035R60

keywords FishschoolingCollectiveforagingObstacleavoidanceStochasticdifferentialequationsScent-gradientsensingAgent-basedmodelSwarmbehavior

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This note reviews two stochastic models the authors have built for fish schooling. The first, an SDE in free space, lets each fish balance attraction, repulsion, velocity matching, environmental forces, and noise. The second adds obstacle avoidance and food-scent tracking, coupling each fish's motion to a scent field obtained from an elliptic PDE. The paper's demonstration is a single simulation of 25 fish in a domain with two rectangular obstacles and one food source: the school reaches the food at (6, 0.1) while maintaining its structure. The paper's claim is that these local rules—each fish reacting only to nearby mates and to the scent gradient—are enough to produce collective foraging.

What carries the argument

The machinery is a five-component force balance in which the environment force carries all information about obstacles and food. In free space, the pairwise terms are the generalized inverse-square attraction $-\alpha r^p (x_i-x_j)/\|x_i-x_j\|^p$, the repulsion $\alpha r^q (x_i-x_j)/\|x_i-x_j\|^q$, and the distance-weighted velocity matching $-\beta(r^p/\|x_i-x_j\|^p + r^q/\|x_i-x_j\|^q)(v_i-v_j)$, with $r$ the critical distance separating attraction from repulsion. In the foraging model, the environment force adds an obstacle-reflection term $-\gamma(R^P/\|x_i-x_i^*\|^P + R^Q/\|x_i-x_i^*\|^Q)(v_i-v_i^*)$, where $x_i^*\in\partial\Omega$ is the predicted collision point and $v_i^*$ is the ref

What would settle it

Run model (3.3) with the Section 4 parameters but with several independent Brownian-motion realizations and a stated integrator and time step; if the school fails to reach the food in a majority of runs, or if any run blows up when two fish nearly coincide, then the displayed pattern is a single-trial artifact rather than a property of the model.

Watch

Extended reading notes

Core claim

The central claim is that model (3.3) reproduces collective foraging. The model writes each fish's position as an SDE with Brownian noise and its velocity as a deterministic balance of four pairwise forces—attraction, repulsion, velocity matching—plus an obstacle-reflection force and a food-scent gradient $\alpha_3\nabla X(x_i)$. The scent density $X$ is not prescribed by hand but solves $-\alpha_1\Delta X + \alpha_2 X = F$ with homogeneous Neumann boundary conditions, where $F$ is the food density. In the Section 4 example, with $N=25$ fish starting in $[0,2]\times[3.5,4]$ and parameters $\alpha=1$, $\beta=0.5$, $\gamma=1$, $\alpha_3=5.5$, $\sigma=0.001$, and obstacle rectangles $[2,2.5]\ti

Load-bearing premise

The example rests on a single random realization with hand-picked parameter values and an unspecified numerical integration scheme; if that one trajectory is not representative of model (3.3), or if the singular pairwise forces at $\|x_i-x_j\|\to 0$ break the simulation, then the claimed 'the school reaches to the food source' does not follow from the model.

Editorial extensions

If this is right

Leaderless fish can keep a school coherent from pairwise local rules alone, since model (2.1) with the five components has no global information.
Adding a scent-gradient force to the environment term is sufficient to route a school to a resource around obstacles in model (3.3).
The elliptic equation for scent turns any food-density function $F$ into a smooth potential field, so the same model can handle multiple food sources or moving resources by changing $F$.
The free-space model (2.1) is the special case of (3.3) in which obstacle and food forces vanish, so the reviewed SDE results for free space carry over to the foraging model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A robustness check not reported in the paper: run the same parameters with many independent noise realizations and measure the fraction of runs that reach the food and keep the school shape.
The scent potential $X$ solves a screened Poisson equation; the same force structure could represent chemical gradients, temperature plumes, or light fields, so the model may transfer to other taxis behaviors without changing the equations.
The singular $1/\|x_i-x_j\|^p$ forces require a numerical scheme that keeps fish separated or regularizes the potential; the paper does not state its integrator or time step, so a reproduction would need to fix those choices.
All parameters are stated in dimensionless units (e.g., $r=0.1$, $R=0.2$ on a $7\times4$ domain); applying the model to real fish would require rescaling distances, speeds, and noise to experimental data.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read this as a short review note rather than a research paper. The review of their five-component SDE model (2.1) and the foraging model (3.3) is concise and accurate, and for someone who hasn't read [8,10,16] it's a serviceable summary. The equations are laid out cleanly, and the local rules (R1)–(R5) are helpful framing. Credit where due: this is a competent recap of a line of work.

The new content is Section 4, one simulated trajectory of 25 fish. That's where the paper falls short. The simulation isn't a faithful numerical solution of (3.3). The speed-limiter in Section 4 is not in the SDE, so the figure actually shows a modified hybrid model. On top of that, the note gives no integrator, no time step, no seed, and the initial positions are 'taken randomly' with no distribution. One unseeded trajectory cannot support the caption's claim that 'the school reaches to the food source while maintaining its school structure.' Also, the interaction terms have 1/||xi−xj||^p singularities; the paper doesn't state any collision-avoidance or regularisation, so it's possible the simulation relied on repulsion keeping particles apart, but that isn't demonstrated. In the worst case, a near-collision could blow up the trajectory.

These are fixable reporting problems, but they are the difference between an illustrative simulation and a reproducible result. The review portion would pass without issue; the example needs code, an ensemble, and a statement of the actual simulated dynamics.

Who is this for? A reader who wants a compact index to Nguyen–Ta–Yagi's models. The example is not a new mechanism or a validation against data; it's a hand-picked parameter run. So treat it as a survey note with an illustrative figure, not as evidence.

I'd send this to peer review rather than desk reject, because the models deserve scrutiny and the reporting gaps are correctable. But I'd expect the referee to ask for the numerical details and a softened novelty claim.

Referee Report

3 major / 3 minor

Summary. This short note reviews two stochastic models for fish schooling developed by the authors. Section 2 presents an SDE model in free space, Eq. (2.1), combining attraction, repulsion, velocity alignment, environmental force, and noise. Section 3 extends it to domains with obstacles and a fixed food resource by coupling the fish SDEs to an elliptic PDE for scent density, Eqs. (3.1)-(3.3); fish are attracted along the scent gradient. Section 4 gives a two-dimensional simulation of 25 fish in an obstacle domain, with a food source at (6,0.1), and Figure 1 shows snapshots at t=0,50,100,150. The caption claims the school reaches the food source while maintaining school structure. The paper cites several earlier works, mostly by the same authors, for existence and more detailed results.

Significance. If the model and the numerical example were substantiated, the paper would contribute a useful stochastic agent-based formulation of collective foraging, with the interesting feature that food information is transmitted through a scalar scent field solving an elliptic PDE rather than through direct attraction to the food. The equations are stated explicitly and the potential-function approach is transparent, and the review of the two model families is concise. However, the central demonstration in Section 4 is a single unreproducible trajectory, and the SDEs have singular interaction terms whose well-posedness is not established. The paper therefore does not, in its current form, provide a reliable computational or mathematical validation of the claimed foraging behavior.

major comments (3)

[Section 4, Eq. (3.3) / speed-cap paragraph] The numerical example is not a simulation of the stated model (3.3). The text adds a speed-limiting projection: 'If the magnitude of vi exceeds ||v||max, our program would reset vi to a vector of magnitude ||v||max and same direction.' This reset is not part of the SDE (3.3), so Figure 1 illustrates a hybrid model. The captions and text claim the school reaches the food source 'for the model (3.3)', but no argument is given that solutions of (3.3) are unaffected by the reset, or that the reset is a negligible regularization. This is load-bearing because the plotted trajectories may owe their stability and obstacle avoidance precisely to this unmodeled projection.
[Section 3, Eq. (3.3), interaction terms] The interaction terms in (3.3) are singular at xi = xj: they contain (xi - xj)/||xi - xj||^p and (vi - vj)/||xi - xj||^q with p=3, q=4 in the example. At xi = xj the right-hand side is undefined, and for nearby particles the force diverges as a high power of the inverse distance. The paper gives no regularization, no collision-avoidance exclusion, no invariant preventing collisions, and no well-posedness theorem for (3.3) with these coefficients. A naive numerical integrator may encounter enormous forces during near-collisions, and the plotted trajectory may depend on an implicit or ad hoc regularization. This is a core mathematical gap, not a presentation issue, for any claim that (3.3) itself produces the shown behavior.
[Section 4, Figure 1 / numerical protocol] The example is a single random realization with no numerical scheme, time step, random seed, or precise initial configuration ('taken randomly'). Consequently the figure cannot be reproduped or checked, and a single trajectory cannot support a claim that the model produces collective foraging. With N=25 and the strong singular forces, different seeds or integrator choices may give qualitatively different outcomes. The lack of ensemble statistics, error bars, or comparison with a baseline (e.g., without scent gradient, or without velocity alignment) means the central statement 'The school reaches to the food source' is not a demonstrated property of (3.3) but an anecdote.

minor comments (3)

[Section 3, Eq. (3.3), x_i^* and v_i^*] The obstacle-avoidance term depends on a point x_i^* on ∂Ω and a reflection vector v_i^*, but the text does not say how x_i^* is determined from xi and vi for a general obstacle domain. For the non-convex domain in Section 4 the reflection law is not unique. A precise definition would be needed to make the model reproducible.
[Section 4, parameters] The list of parameters is given, but no rationale or sensitivity study is provided. In particular, alpha3=5.5, sigma=0.001, and the speed cap 0.8 are hand-picked. A short sensitivity discussion or at least a second realization would strengthen the presentation.
[General] There are several typographical slips and awkward phrasings, e.g., 'mathching' in Section 3, 'The sc hool reaches' in Figure 1 caption, and 'stochastic diﬀerentiation' where 'differential' seems intended. The paper would also benefit from a statement of which results are new here versus reviewed from [8], [10], [16].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: §4 is an illustrative simulation of a fully stated model; food-seeking is a modeling input, not a hidden prediction.

full rationale

The paper is a review of the authors' own stochastic models and presents a single illustrative simulation in §4. The model equations (3.3) are stated in full, so the simulation is self-contained rather than relying on an unstated cited theorem. The food-seeking term α3∇X(xi) is an explicit modeling assumption, and the §4 caption's observation that the school reaches the food is a consistency check of that assumption, not an independent prediction derived from something else. No parameter is fitted to data, and no quantity is predicted that was used as an input. The numerous self-citations identify the origins of the models but are not load-bearing for the example's claim; the example is presented directly with its own equations and figure. The speed reset and missing numerical details are reproducibility/correctness concerns, not circularity. Hence no circular step is identified.

Assumptions & free parameters 10 free parameters · 4 assumptions · 0 invented entities

The central example depends on a large set of hand-picked parameters and on unstated numerical/simulation assumptions; the models themselves are inherited from prior self-cited works, so the ledger shows the example is a backward demonstration rather than an independent validation.

free parameters (10)

alpha (attraction coefficient) = 1
Set by hand in Section 4; not fitted to data.
beta (velocity alignment coefficient) = 0.5
Set by hand in Section 4; not fitted to data.
gamma (obstacle avoidance strength) = 1
Set by hand in Section 4; not fitted to data.
r (attraction/repulsion critical distance) = 0.1
Chosen ad hoc; no calibration to empirical fish spacing.
R (obstacle avoidance distance) = 0.2
Chosen ad hoc in Section 4.
alpha_3 (scent sensitivity) = 5.5
Hand-picked; no fitting to behavioral data.
sigma (noise intensity) = 0.001
Hand-picked; small noise chosen to keep the school coherent.
v_max (speed cap) = 0.8
Introduced in Section 4 to restrict speed; chosen by hand.
alpha_1 (scent diffusion) = 0.1
In eq. (3.1); chosen for the example.
alpha_2 (scent decline rate) = 0.2
In eq. (3.1); chosen for the example.

assumptions (4)

domain assumption Behavioral rules (R1)-(R3) of Camazine et al. are a valid basis for fish schooling models.
Invoked in Section 2 to motivate the five-component model.
domain assumption The scent field X satisfies the elliptic Neumann problem (3.1), and fish move up its gradient (R5).
Introduced in Section 3 as the foraging mechanism.
standard math The SDE systems (2.1) and (3.3) have unique strong solutions.
Assumed implicitly; standard SDE existence theorems require Lipschitz coefficients, which the singular terms (xi-xj)/parallel xi-xj parallel^p do not satisfy at collision; no existence result is cited for (3.3).
ad hoc to paper The numerical simulation in Section 4 converges to the true solution of (3.3).
No numerical scheme or discretization parameters are given; the plot is presented as evidence.

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Cite this review

Pith. "Pith review of Mathematical Models for Fish Schooling." pith.science (2026). https://pith.science/paper/3NYEJTT4

@misc{pith2026250808310,
  author       = {Pith},
  title        = {Pith review of: Mathematical Models for Fish Schooling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NYEJTT4}},
  note         = {Machine review of arXiv:2508.08310}
}

read the original abstract

This note reviews our mathematical models for fish schooling, considered in free space, and in space with obstacle and food resource. These models are performed by stochastic differential equations or stochastic partial differential equations. We then present an example for the model in the last case.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Swarming behavior of a ﬁsh school consisting of a large number of individuals often surprises us

Introduction. Swarming behavior of a ﬁsh school consisting of a large number of individuals often surprises us. They swim co herently match- ing their velocity without collision and maintaining a cons tant scale of school, even though they have only moderate ability of infor mation pro- cessing and of execution of programming. Several mathematical models ...

arXiv 2010

[2] [2]

In [ 16], we introduced ﬁve components constructing a SDE model for ﬁsh schooling in free space

Fish schooling model based on ﬁve components in free spaces. In [ 16], we introduced ﬁve components constructing a SDE model for ﬁsh schooling in free space. These components are based on the fo llowing three local rules of Camazine-Deneubourg-Franks-Sneyd-Theraulaz-Bonabeau ([2]): (R1) The school has no leaders and each ﬁsh follows the same be havioral r...

work page

[3] [3]

When two particles i and j are far from each other, both would move toward each other

Attraction force. When two particles i and j are far from each other, both would move toward each other. In our model, this force is a generalization of the inverse-square law of universal grav itation: −αrp(xi − xj) ∥xi − xj∥p , where 1 < p < ∞ and r > 0 are constants, α a coeﬃcient of attraction among individuals

work page

[4] [4]

When two particles i and j are close enough, both would move far from each other

Repulsion force. When two particles i and j are close enough, both would move far from each other. This force is a generalizatio n of the Van der Waals forces: αrq(xi − xj) ∥xi − xj∥q , where 1 < p < q < ∞ is a ﬁxed exponent

work page

[5] [5]

Alignment or velocity matching. The velocity matching of the particle i to the particle j also has a similar weight depending on the distance ∥xi − xj∥: −β ( rp ∥xi − xj∥p + rq ∥xi − xj∥q ) (vi − vj), where β is a coeﬃcient of velocity matching among individuals

work page

[6] [6]

The individual i react to the environment a force Fi(xi, vi)

Reaction to the environment. The individual i react to the environment a force Fi(xi, vi). FISH SCHOOLING 3

work page

[7] [7]

All particles are subject to random factors or nois e

Noise. All particles are subject to random factors or nois e. These ﬁve components form our SDE model of the form (2.1)                    dxi(t) = vidt + σidwi(t), i = 1, 2, . . . N, dvi(t) = { − α N∑ j=1,j̸=i ( rp ∥xi−xj ∥p − rq ∥xi−xj ∥q ) (xi − xj) −β N∑ j=1,j̸=i ( rp ∥xi−xj∥p + rq ∥xi−xj∥q ) (vi − vj) +Fi(xi, vi) } dt, i = 1, 2, . ...

work page

[8] [8]

In [ 10], we introduced a ﬁsh schooling model in spaces with ob- stacle

Fish schooling model in spaces with obstacle and food re- source. In [ 10], we introduced a ﬁsh schooling model in spaces with ob- stacle. For this model, in addition to the three local rules ( a)–(c) in the previous section, we newly presented a local rule of obstacl e avoidance for individual ﬁsh: (R4) Each ﬁsh executes an action for avoiding obstacle a...

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