A Macroscopic Model for a System of Swarming Agents Using Curvature Control |
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Authors: | Pierre Degond Sébastien Motsch |
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Institution: | 1.CNRS,Institut de Mathématiques de Toulouse UMR 5219,Toulouse,France;2.Institute of Mathematics of Toulouse UMR 5219 (CNRS-UPS-INSA-UT1-UT2),Université Paul Sabatier,Toulouse cedex,France;3.Center of Scientific Computation and Mathematical Modeling (CSCAMM),University of Maryland,College Park,USA |
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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. |
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