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

Abstract

Schooling fish provide a spectacular example of self-organization in Nature. The most remarkable patterns they form are giant rotating clusters such as balls, tori, and rings, but the underlying mechanism remains largely unknown. Here we propose an agent-based model that limits the number of agents that can interact with each other. We incorporate the characteristic behaviors of fish by (i ) attraction that is weakened in a dense cluster of fish, and (ii ) acceleration with finite duration ("fast-start") when the fish is out of the cluster. By three-dimensional numerical simulations, we show emergence of giant rotating clusters (balls, tori, and rings) that are much larger than the radius of interaction. We present a phase diagram of patterns including polarized schools and swarms, and propose a physical mechanism that determines the cluster shape in terms of the interaction capacity and strength of attraction. Our model also indicates that each fish randomly moves back and forth between the inner and outer regions of a vortex on a large time-scale. These results show that fish without inherent chirality form giant rotating clusters spontaneously and only by short-ranged interactions.

Effect of gravitational field on collective motion of fish

“…On the other hand, when N u = 2 and 3, an increase in ζ tends to prevent formation of a rotating cluster and instead promotes formation of a polarized school. The tendency that rotating clusters are more easily obtained for smaller N u is explained by larger sensitivity of each agent to its neighbors [11], and is unchanged under the influence of gravity.…”

“…Theoretically, agent-based models [5][6][7][8][9][10] have been proposed and reproduced a torus and a ring. In addition, a ball-shaped rotating cluster (resembling a "bait-ball" [3]) is reproduced in a recent model by the present authors [11]. The model uses topological interaction [12]and some experimentally observed behaviors of fish, which facilitate formation of a giant rotating cluster.…”

“…Actually, the otolith ("the ear stone") allows the fish to perceive gravity and swim almost perpendicular to the water surface [14,15]. In the previous models [5,7,9,11], however, the direction of the vortex axis is not constrained in any particular direction. A systematic study of the effect of gravity on collective motion of fish has been lacking, although a previous study [16] introduced it as a fixed parameter and yielded a polarized school with high frontal density.…”

“…In this Letter, we introduce a gravitational factor into (a) E-mail: uchida@cmpt.phys.tohoku.ac.jp our previous model [11] and study its effect on collective motion. We find emergence of a giant tornado cluster and a polarized school flattened along the direction of movement.…”

“…We fixed the parameter values N = 3000, r e = 1, τ 0 = 1, r b = 2/3, r a = 5, v 0 = 1, v r = 1, v a = 5, τ = 0.1, and regarded N u , λ, ζ as the control parameters. We focus on the parameter region 1 ≤ N u ≤ 3 and 5.0 ≤ λ ≤ 9.0 where rotating clusters of various shapes emerge in the absence of gravity [11]. (See Ref.…”