Comptes Rendus
Partial Differential Equations
Macroscopic limit of self-driven particles with orientation interaction
Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 555-560.

The discrete Couzin–Vicsek algorithm (CVA) has been proposed to model the interactions of individuals among animal societies such as schools of fish. In this Note, 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. The result is based on the introduction of a non-conventional concept of a collisional invariant of the collision operator.

L'algorithme discret de Couzin–Vicsek (CVA) a été proposé pour modéliser l'interaction d'individus au sein de sociétés animales comme les bancs de poissons. Dans cette Note, nous proposons une version cinétique (champ-moyen) de l'algorithme CVA et en donnons la limite macroscopique formelle. Le modèle macroscopique final comprend une équation de conservation pour la densité des individus et une équation non-conservative pour le vecteur directeur de la vitesse moyenne. Ce résultat est basé sur l'introduction d'un concept non-conventionnel d'invariant collisionnel de l'opérateur de collision.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.10.024
Pierre Degond 1; Sébastien Motsch 1

1 Institut de mathématiques de Toulouse, UMR 5219 (CNRS-UPS-INSA-UT1-UT2), équipe MIP, Université P. Sabatier, 31062 Toulouse cedex 09, France
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Pierre Degond; Sébastien Motsch. Macroscopic limit of self-driven particles with orientation interaction. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 555-560. doi : 10.1016/j.crma.2007.10.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.024/

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