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

Authored by Sebastien Motsch, Pierre Degond

Date Published: 2011

DOI: 10.1007/s10955-011-0201-3

Sponsors: European Union United States National Science Foundation (NSF)

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

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.

Tags

movement collective behavior Phase-transition Fish schools Motion Self-driven particles Persistent turning walker Continuum-limit Flocking dynamics