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Schooling fish exhibit giant rotating clusters such as balls, tori, andrings, among other collective patterns. In order to account for their giantnessand flexible shape change, we introduce an agent-based model that limits thenumber of agents that each agent can interact with (interaction capacity).Incorporating autonomous control of attractive interactions, we reproducerotating clusters (balls, tori, and rings) that are an order of magnitudelarger than the interaction range. We obtained a phase diagram of patternsincluding polarized schools and swarms. In our model, the scaling law betweenthe number of agents and the projected area of the cluster is in good agreementwith experimental results. The model indicates that giant rotating clusters areformed at low interaction capacity, without long-range interactions or inherentchirality of fish.

Emergence of a giant rotating cluster of fish in three dimensions by local interactions