### 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 language | English (US) |
---|---|

Pages (from-to) | 685-714 |

Number of pages | 30 |

Journal | Journal of Statistical Physics |

Volume | 143 |

Issue number | 4 |

DOIs | |

State | Published - May 2011 |

Externally published | Yes |

### Fingerprint

### 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

### Cite this

*Journal of Statistical Physics*,

*143*(4), 685-714. https://doi.org/10.1007/s10955-011-0201-3

**A Macroscopic Model for a System of Swarming Agents Using Curvature Control.** / Degond, Pierre; Motsch, Sebastien.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 143, no. 4, pp. 685-714. https://doi.org/10.1007/s10955-011-0201-3

}

TY - JOUR

T1 - A Macroscopic Model for a System of Swarming Agents Using Curvature Control

AU - Degond, Pierre

AU - Motsch, Sebastien

PY - 2011/5

Y1 - 2011/5

N2 - 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.

AB - 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.

KW - Asymptotic analysis

KW - Collision invariants

KW - Fish behavior

KW - Hydrodynamic limit

KW - Individual based model

KW - Orientation interaction

KW - Persistent Turning Walker model

KW - Vicsek model

UR - http://www.scopus.com/inward/record.url?scp=79956210310&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79956210310&partnerID=8YFLogxK

U2 - 10.1007/s10955-011-0201-3

DO - 10.1007/s10955-011-0201-3

M3 - Article

AN - SCOPUS:79956210310

VL - 143

SP - 685

EP - 714

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -