From Lévy walks to Brownian motion in schools of zebrafish (Danio rerio)

Population density strongly affects the internal organization and movement dynamics of fish schools. Here, we investigate density-dependent collective motion in zebrafish (Danio rerio) schools with group sizes ranging from 2 to 30 individuals. Using trajectories measured relative to the school center of mass, we show that individual motion is superdiffusive at low densities and progressively approaches Brownian-like behavior as density increases. Step-length distributions are well described by truncated power laws: for $N\le 15$ , the fitted exponents are compatible with a Lévy-walk regime, whereas larger groups show a transition toward Brownian motion. In parallel, q-Gaussian fits of the signed radial displacement increments reveal heavy-tailed, non-Gaussian fluctuations at low densities and a gradual convergence toward Gaussian-like behavior at high densities. These results support a density-driven transition from a polarized, strongly correlated schooling state to a more disordered shoaling state, suggesting that increased density reorganizes local interactions within the group.