Chaos and fractals in fish school motion

The once abstract notions of fractal patterns and processes now appear naturally and inevitably in various chaotic dynamical systems. The examples range from Brownian motion [1], [2], [3], [4] and [5] to the dynamics of social relations [6]. In this paper, after introducing a certain hybrid mathematical model of the plankton–fish school interplay, we study the fractal properties of the model fish school walks. We show that the complex planktivorous fish school motion is dependent on the fish predation rate. A decrease in the rate is followed by a transition from low-persistent to high-persistent fish school walks, i.e., from a motion with frequent to a motion with few changes of direction. The low-persistent motion shows fractal properties for all time scales, whereas the high-persistent motion has pronounced multifractal properties for large-scale displacements.