Kinetic Equations and Self-organized Band Formations

Self-organization is a ubiquitous phenomenon in nature which can be observed in a variety of different contexts and scales, with examples ranging from schools of fish, swarms of birds or locusts to flocks of bacteria. The observation of such global patterns can often be reproduced in models based on simple interactions between neighboring particles. In this paper we focus on two particular interaction dynamics closely related to the one described in the seminal paper of Vicsek and collaborators. After reviewing the current state of the art in the subject, we study a numerical scheme for the kinetic equation associated with the Vicsek models which has the specificity of reproducing many physical properties of the continuous models, like the preservation of energy and positivity and the diminution of an entropy functional. We describe a stable pattern of bands emerging in the dynamics proposed by Degond–Frouvelle–Liu dynamics and give some insights about their formation.