Continuum limit of self-driven particles with orientation interaction

TL;DR

This paper derives a macroscopic continuum model from a discrete self-driven particle algorithm describing animal group behavior, providing a formal limit that results in a hyperbolic system involving density and velocity direction.

Contribution

It introduces a kinetic mean-field version of the Couzin-Vicsek model and rigorously derives its macroscopic limit, including a novel concept of collisional invariant.

Findings

The macroscopic model is hyperbolic.

The derivation uses a new concept of collisional invariant.

The model captures collective animal movement dynamics.

Abstract

We consider the discrete Couzin-Vicsek algorithm (CVA), which describes the interactions of individuals among animal societies such as fish schools. In this article, we propose a kinetic (mean-field) version of the CVA model and provide its formal macroscopic limit. The final macroscopic model involves a conservation equation for the density of the individuals and a non conservative equation for the director of the mean velocity and is proved to be hyperbolic. The derivation is based on the introduction of a non-conventional concept of a collisional invariant of a collision operator.