A continuous model for alignment of self-propelled particles with anisotropy and density-dependent parameters

Amic Frouvelle (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In this article we propose a new macroscopic model derived from a time-continuous version of the Vicsek Algorithm, where we introduce the dependence of some local density and angle of vision in the orientation interaction. With some adaptations to the concept of generalized collisional invariants, we prove that the methodology developped by Degond and Motsch works to derive a macroscopic limit of this dynamical system. The system of PDE may present some zones of non-hyperbolicity, which is a new feature compared to their macroscopic model. We perform an asymptotic study of the coefficients obtained in this system of PDE's and show that there are indeed some cases where the system is not hyperbolic. This study is also useful to measure the influence of the angle of vision in the final continuum model.

Mots clés: Individual based model; Vicsek algorithm; orientation interaction; anisotropy; collisional invariants; non-hyperbolicity

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Amic Frouvelle. A continuous model for alignment of self-propelled particles with anisotropy and density-dependent parameters. 2009. ⟨hal-00438174v1⟩