Journal of Statistical Physics

, Volume 143, Issue 4, pp 685–714

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


DOI: 10.1007/s10955-011-0201-3

Cite this article as:
Degond, P. & Motsch, S. J Stat Phys (2011) 143: 685. doi:10.1007/s10955-011-0201-3


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.


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

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.CNRSInstitut de Mathématiques de Toulouse UMR 5219ToulouseFrance
  2. 2.Institute of Mathematics of Toulouse UMR 5219 (CNRS-UPS-INSA-UT1-UT2)Université Paul SabatierToulouse cedexFrance
  3. 3.Center of Scientific Computation and Mathematical Modeling (CSCAMM)University of MarylandCollege ParkUSA