KIT

Abstract

The Viscek model (1995) has been a prototype for many studies of self-organization and collective behavior in animal groups. The basic formulation consists of a system of N paricles moving on the plane (or in space) with constant speed and adjusting their velocities at discrete time instants to the local averages, while subject to random perturbations (noise). We consider a related purely deterministic system in which alignment effects are combined with local repulsive force interactions, and which has been used previously to model large scale behavior of pelagic fish schools. Scaling the parameters of the model in a particular way such that the discrete time step approaches zero and the number of particles N approaches infinity, we find nontrivial hydrodynamic regimes in which a reduced description of the system through the macroscopic density and the average angle becomes possible. We present a systematic procedure of deriving the hydrodynamic equations and numerical results on comparison of particle and continuum (PDE) solutions.