E9 Collective Behavior

Topic

The collective behavior of fish schools is a paradigmatic example of emergent behavior in systems of self-propelled agents: coherent global patterns—such as group-wide polarization, collective rotation, or spreading waves—emerge from purely local interaction rules among individuals, without any central coordination directing them. Physically, the school can be described as an active fluid, or as a system of interacting active particles; this system bears a direct analogy to the phase transition occurring between a disordered state (where fish swim essentially at random) and an ordered state (where the entire school becomes polarized and swims in a unified direction).

The Vicsek model describes this schooling dynamic by representing each fish as a particle that self-propels at a fixed speed and constantly adjusts its direction of motion to match the average direction of its nearest neighbors—within a specific interaction radius—subject to a certain degree of angular noise that introduces randomness into this adjustment.

The transition from the disordered state to the ordered, polarized state occurs as a phase transition governed by both the intensity of the angular noise and the density of the school (i.e., the number of individuals per unit of space). The degree of alignment within the school is summarized by a polarization order parameter; at the transition point, this parameter shifts abruptly from a value near zero—characteristic of the disordered state—to a value near one, characteristic of the fully polarized state.

When a predator appears, the school responds by propagating alarm waves—a phenomenon known as "flash expansion" or "Selous waves"—which sweep through the group as waves of information at speeds ten to one hundred times faster than the swimming speed of an individual fish; The speed at which this alarm wave propagates is governed by the sensory response time of each fish and the interaction distance between neighbors introduced in the Vicsek model.

One of the benefits of this collective behavior is the dilution effect: the risk of a predator capturing a specific individual decreases as the school size increases, following an inverse relationship with the number of individuals comprising it.

The expected behavior of the Vicsek model describing this school is that there exists a critical level of angular noise above which the school completely loses its cohesion and disintegrates; furthermore, within the ordered phase, the previously introduced polarization order parameter approaches zero gradually and predictably as the angular noise nears that critical value from below, with a rate of approach following a power law characterized by a critical exponent close to 0.45.

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