A Macroscopic Model for a System of Swarming Agents Using Curvature Control

Pierre Degond, Sébastien Motsch

Research output: Contribution to journalArticle

21 Scopus citations

Abstract

In this paper, we study the macroscopic limit of a new model of collective displacement. The model, called PTWA, is a combination of the Vicsek alignment model (Vicsek et al. in Phys. Rev. Lett. 75(6):1226-1229, 1995) and the Persistent Turning Walker (PTW) model of motion by curvature control (Degond and Motsch in J. Stat. Phys. 131(6):989-1021, 2008; Gautrais et al. in J. Math. Biol. 58(3):429-445, 2009). The PTW model was designed to fit measured trajectories of individual fish (Gautrais et al. in J. Math. Biol. 58(3):429-445, 2009). The PTWA model (Persistent Turning Walker with Alignment) describes the displacements of agents which modify their curvature in order to align with their neighbors. The derivation of its macroscopic limit uses the non-classical notion of generalized collisional invariant introduced in (Degond and Motsch in Math. Models Methods Appl. Sci. 18(1):1193-1215, 2008). The macroscopic limit of the PTWA model involves two physical quantities, the density and the mean velocity of individuals. It is a system of hyperbolic type but is non-conservative due to a geometric constraint on the velocity. This system has the same form as the macroscopic limit of the Vicsek model (Degond and Motsch in Math. Models Methods Appl. Sci. 18(1):1193-1215, 2008) (the 'Vicsek hydrodynamics') but for the expression of the model coefficients. The numerical computations show that the numerical values of the coefficients are very close. The 'Vicsek Hydrodynamic model' appears in this way as a more generic macroscopic model of swarming behavior as originally anticipated.

Original languageEnglish (US)
Pages (from-to)685-714
Number of pages30
JournalJournal of Statistical Physics
Volume143
Issue number4
DOIs
StatePublished - May 2011
Externally publishedYes

Keywords

  • Asymptotic analysis
  • Collision invariants
  • Fish behavior
  • Hydrodynamic limit
  • Individual based model
  • Orientation interaction
  • Persistent Turning Walker model
  • Vicsek model

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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