Alignment in a fish school: a mixed Lagrangian–Eulerian approach

Social organization is one of the fundamental aspects of animal behavior, and has received attention both from experimental and theoretical perspectives. Examples of social groups appear at every size scale from the microscopic aggregates of mammalian cells (such as fibroblasts) to macroscopic herds of wildbeast, flocks of birds, and fish schools. There are two general frameworks when modeling such problems: the Lagrangian viewpoint and the Eulerian one. In this paper, we use both the approaches in the study of fish alignment. An individual-based model (IBM) (Lagrangian) provides a virtual world where fish forming a fish school try to adopt a common angular position. Fish are assumed to lie in horizontal planes, an individual angular position is the angle made by the oriented axis associated with the individual (tail to head) with a fixed direction. Two main forces are acting, a force of alignment, whose strength is assumed to be fixed in a given experiment but may be modified, and a force of dispersion, accounting for all disturbances. A transition from dispersion-dominant to alignment-dominant can be observed in the IBM experiments. A related PDE model (Eulerian) is used to determine the transition with sufficient accuracy.