Journal of Statistical Physics

, Volume 143, Issue 4, pp 685–714 | Cite as

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

  • Pierre Degond
  • Sébastien MotschEmail author


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 


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  1. 1.
    Bakry, D., Cattiaux, P., Guillin, A.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. (2007) Google Scholar
  2. 2.
    Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procaccini, A., et al.: Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Natl. Acad. Sci. 105(4), 1232 (2008) ADSCrossRefGoogle Scholar
  3. 3.
    Bellomo, N.: Modeling Complex Living Systems: A Kinetic Theory and Stochastic Game Approach. Birkhäuser, Basel (2008) zbMATHGoogle Scholar
  4. 4.
    Bertin, E., Droz, M., Grégoire, G.: Boltzmann and hydrodynamic description for self-propelled particles. Phys. Rev. E 74(2), 22101 (2006) ADSCrossRefGoogle Scholar
  5. 5.
    Bolley, F., Canizo, J.A., Carrillo, J.A.: Stochastic mean-field limit: non-Lipschitz forces & swarming. Arxiv preprint. arXiv:1009.5166 (2010)
  6. 6.
    Brézis, H.: Analyse Fonctionnelle. Théorie et Applications. Masson, Paris (1983) zbMATHGoogle Scholar
  7. 7.
    Camazine, S., Deneubourg, J.L., Franks, N.R., Sneyd, J., Theraulaz, G., Bonabeau, E.: Self-Organization in Biological Systems. Princeton University Press, Princeton (2001) Google Scholar
  8. 8.
    Canizo, J.A., Carrillo, J.A., Rosado, J.: A well-posedness theory in measures for some kinetic models of collective motion. Preprint (2009) Google Scholar
  9. 9.
    Carrillo, J.A., Fornasier, M., Rosado, J., Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42, 218–236 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cattiaux, P., Chafaï, D., Motsch, S.: Asymptotic analysis and diffusion limit of the persistent turning walker model. Asymptot. Anal. 67(1), 17–31 (2010) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, Berlin (1988) zbMATHCrossRefGoogle Scholar
  12. 12.
    Chaté, H., Ginelli, F., Grégoire, G., Peruani, F., Raynaud, F.: Modeling collective motion: variations on the Vicsek model. Eur. Phys. J. B 64, 451–456 (2008) ADSCrossRefGoogle Scholar
  13. 13.
    Chuang, Y., D’Orsogna, M.R., Marthaler, D., Bertozzi, A.L., Chayes, L.S.: State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Physica D 232(1), 33–47 (2007) MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Couzin, I.D., Franks, N.R.: Self-organized lane formation and optimized traffic flow in army ants. Proc. R. Soc. Lond. B, Biol. Sci. 270(1511), 139 (2003) CrossRefGoogle Scholar
  15. 15.
    Couzin, I.D., Krause, J.: Self-organization and collective behavior in vertebrates. Adv. Study Behav. 32(1) (2003) Google Scholar
  16. 16.
    Couzin, I.D., Krause, J., James, R., Ruxton, G.D., Franks, N.R.: Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218(1), 1–11 (2002) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852 (2007) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Czirók, A., Vicsek, T.: Collective behavior of interacting self-propelled particles. Physica A 281(1–4), 17–29 (2000) ADSCrossRefGoogle Scholar
  19. 19.
    Degond, P.: Macroscopic limits of the Boltzmann equation: a review. In: Degond, P., Russo, G., Pareschi, L. (eds.) Modeling and Computational Methods for Kinetic Equations. Birkhäuser, Basel (2004) CrossRefGoogle Scholar
  20. 20.
    Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18(1), 1193–1215 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Degond, P., Motsch, S.: Large scale dynamics of the persistent turning walker model of fish behavior. J. Stat. Phys. 131(6), 989–1021 (2008) MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Degond, P., Navoret, L., Bon, R., Sanchez, D.: Congestion in a macroscopic model of self-driven particles modeling gregariousness. J. Stat. Phys., 1–41 (2009) Google Scholar
  23. 23.
    Filbet, F., Laurençot, P., Perthame, B.: Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol. 50(2), 189–207 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Gautrais, J., Jost, C., Soria, M., Campo, A., Motsch, S., Fournier, R., Blanco, S., Theraulaz, G.: Analyzing fish movement as a persistent turning walker. J. Math. Biol. 58(3), 429–445 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Gautrais, J., Theraulaz, G.: In preparation Google Scholar
  26. 26.
    Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semigroups. In: Dirichlet Forms, pp. 54–88 (1993) CrossRefGoogle Scholar
  27. 27.
    Ha, S.Y., Liu, J.G.: A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7(2), 297–325 (2009) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ha, S.Y., Tadmor, E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models 1(3), 415–435 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Hemelrijk, C.K., Hildenbrandt, H.: Self-organized shape and frontal density of fish schools. Ethology 114(3), 245–254 (2008) CrossRefGoogle Scholar
  30. 30.
    Jeanson, R., Deneubourg, J.L., Grimal, A., Theraulaz, G.: Modulation of individual behavior and collective decision-making during aggregation site selection by the ant Messor barbarus. Behav. Ecol. Sociobiol. 55(4), 388–394 (2004) CrossRefGoogle Scholar
  31. 31.
    Lions, J.L.: Équations différentielles opérationnelles et problèmes aux limites. Springer, Berlin (1961) zbMATHGoogle Scholar
  32. 32.
    Meyn, S.P., Tweedie, R.L.: Stability of markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25(3), 518–548 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Motsch, S., Navoret, L.: Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior. Preprint (2010) Google Scholar
  34. 34.
    Nagy, M., Daruka, I., Vicsek, T.: New aspects of the continuous phase transition in the scalar noise model (SNM) of collective motion. Physica A 373, 445–454 (2007) ADSCrossRefGoogle Scholar
  35. 35.
    Oksendal, B.: Stochastic Differential Equations: An Introduction With Applications. Springer, New York (1992) Google Scholar
  36. 36.
    Parrish, J.K., Viscido, S.V., Grunbaum, D.: Self-organized fish schools: an examination of emergent properties. Biol. Bull. Marine Biol. Lab. Woods Hole 202(3), 296–305 (2002) CrossRefGoogle Scholar
  37. 37.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991) zbMATHGoogle Scholar
  38. 38.
    Szabo, P., Nagy, M., Vicsek, T.: Turning with the others: novel transitions in an SPP model with coupling of accelerations. In: Second IEEE International Conference on Self-Adaptive and Self-Organizing Systems, SASO’08, pp. 463–464 (2008) CrossRefGoogle Scholar
  39. 39.
    Sznitman, A.S.: Topics in propagation of chaos. In: École d’Été de Probabilités de Saint-Flour XIX-1989. Lecture Notes in Math., vol. 1464, pp. 165–251 (1989) Google Scholar
  40. 40.
    Theraulaz, G., Bonabeau, E., Nicolis, S.C., Sole, R.V., Fourcassie, V., Blanco, S., Fournier, R., Joly, J.L., Fernandez, P., Grimal, A., et al.: Spatial patterns in ant colonies. Proc. Natl. Acad. Sci. 99(15), 9645 (2002) ADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229 (1995) ADSCrossRefGoogle Scholar
  42. 42.
    Viscido, S.V., Parrish, J.K., Grünbaum, D.: Factors influencing the structure and maintenance of fish schools. Ecol. Model. 206(1–2), 153–165 (2007) CrossRefGoogle Scholar

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© 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

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