Asymptotic flocking for the three-zone model.

Abstract

We prove the asymptotic flocking behavior of a general model of swarming dynamics. The model describing interacting particles encompasses three types of behavior: repulsion, alignment and attraction. We refer to this dynamics as the three-zone model. Our result expands the analysis of the so-called Cucker-Smale model where only alignment rule is taken into account. Whereas in the Cucker-Smale model, the alignment should be strong enough at long distance to ensure flocking behavior, here we only require that the attraction is described by a confinement potential. The key for the proof is to use that the dynamics is dissipative thanks to the alignment term which plays the role of a friction term. Several numerical examples illustrate the result and we also extend the proof for the kinetic equation associated with the three-zone dynamics.