Particle, kinetic, and hydrodynamic models of swarming

  • José A. Carrillo
  • Massimo Fornasier
  • Giuseppe Toscani
  • Francesco Vecil
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Summary

We review the state-of-the-art in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individual-based models based on “particle”-like assumptions, we connect to hydrodynamic/macroscopic descriptions of collective motion via kinetic theory. We emphasize the role of the kinetic viewpoint in the modelling, in the derivation of continuum models and in the understanding of the complex behavior of the system.

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References

  1. 1.
    Aoki, I.: A Simulation Study on the Schooling Mechanism in Fish. Bull. Jpn. Soc. Scient. Fisher., 48, 1081–1088 (1982)Google Scholar
  2. 2.
    Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, L., Lecomte, L., Orlandi, A., Parisi, G., Procaccini, A., Viale, M., Zdravkovic, V.: Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proceedings of the National Academy of Sciences, 105(4), 1232–1237 (2008)CrossRefGoogle Scholar
  3. 3.
    Barbaro, A., Taylor, K., Trethewey, P.F., Youseff, L., Birnir, B.: Discrete and continuous models of the dynamics of pelagic fish: application to the capelin. Math. Comput. Simul., 79, 3397–3414 (2009)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barbaro, A., Einarsson, B., Birnir, B., Sigurthsson, S., Valdimarsson, H., Palsson, O.K., Sveinbjornsson, S., Sigurthsson, T.: Modelling and simulations of the migration of pelagic fish. ICES J. Mar. Sci., 66, 826–838 (2009)CrossRefGoogle Scholar
  5. 5.
    Benedetto, D., Caglioti, E., Pulvirenti, M.: A kinetic equation for granular media. RAIRO, Modélisation Math. Anal. Numér., 31, 615–641 (1997)MATHMathSciNetGoogle Scholar
  6. 6.
    Birnir, B.: An ODE model of the motion of pelagic fish. J. Stat. Phys., 128, 535–568 (2007)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Braun, W., Hepp, K.: The vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun. Math. Phys., 56, 101–113 (1977)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bonabeau, E., Dorigo, M., Theraulaz, G.: Swarm Intelligence: From Natural to Artificial Systems. Oxford University Press, New York (1999)MATHGoogle Scholar
  9. 9.
    Bolley, F., Canizo, J.A., Carrillo, J.A.: Propagation of chaos for some non-globally Lipschitz particle systems. Preprint UABGoogle Scholar
  10. 10.
    Burger, M., Capasso, V., Morale, D.: On an aggregation model with long and short range interactions. Nonlinear Analysis. Real World Applications. Int. Multidis. J., 8, 939–958 (2007)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cáceres, M.J., Toscani, G.: Kinetic approach to long time behavior of linearized fast diffusion equations. J. Stat. Phys., 128, 883–925 (2007)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Camazine, S., Deneubourg, J.-L., Franks, N.R., Sneyd, J., Theraulaz, G., Bonabeau, E.: Self-Organization in Biological Systems. Princeton University Press, Princeton, NJ (2003)MATHGoogle Scholar
  13. 13.
    Cañizo, J.A., Carrillo, J.A., Rosado, J.: A well-posedness theory in measures for some kinetic models of collective motion. Preprint UABGoogle Scholar
  14. 14.
    Carrillo, J.A., D’Orsogna, M.R., Panferov, V.: Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models, 2, 363–378 (2009)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Carrillo, J.A., Fornasier, M., Rosado, J., Toscani, G.: Asymptotic Flocking Dynamics for the kinetic Cucker-Smale model. Preprint UABGoogle Scholar
  16. 16.
    Carrillo, J.A., McCann, R., Villani, C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Matematica Iberoamericana, 19, 1–48 (2003)MathSciNetGoogle Scholar
  17. 17.
    Carrillo, J.A., McCann, R., Villani, C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal., 179, 217–263 (2006)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Carrillo, J.A., Toscani, G.: Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma, 6, 75–198 (2007)MathSciNetGoogle Scholar
  19. 19.
    Carrillo, J.A., Vecil, F.: Nonoscillatory interpolation methods applied to Vlasov-based models. SIAM J. Sci. Comput., 29, 1179–1206 (2007)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Springer series in Appl. Math. Sci., 106, Springer-Verlag (1994)Google Scholar
  21. 21.
    Chuang, Y.L., Huang, Y.R., D’Orsogna, M.R., Bertozzi, A.L.: Multi-vehicle flocking: scalability of cooperative control algorithms using pairwise potentials. IEEE Int. Conf. Robotics Automation, 2292–2299 (2007)Google Scholar
  22. 22.
    Chuang, Y.L., D’Orsogna, M.R., Marthaler, D., Bertozzi, A.L., Chayes, L.: State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Physica D, 232, 33–47 (2007)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Couzin, I.D., Krause, J., Franks, N.R., Levin, S.A.: Effective leadership and decision making in animal groups on the move. Nature, 433, 513–516 (2005)CrossRefGoogle Scholar
  24. 24.
    Couzin, I.D., Krause, J., James, R., Ruxton, G., Franks, N.: Collective memory and spatial sorting in animal groups. J. Theor. Biol., 218, 1–11 (2002)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Cucker, F., Smale, S.: On the mathematics of emergence. Jpn. J. Math., 2, 197–227 (2007)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Automat. Control, 52, 852–862 (2007)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci., 18, 1193–1215 (2008)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Degond, P., Motsch, S.: Large-scale dynamics of the Persistent Turing Walker model of fish behavior. J. Stat. Phys., 131, 989–1021 (2008)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Dobrushin, R.: Vlasov equations. Funct. Anal. Appl., 13, 115–123 (1979)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    D’Orsogna, M.R., Chuang, Y.L., Bertozzi, A.L., Chayes, L.: Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett., 96 (2006)Google Scholar
  31. 31.
    Duan, R., Fornasier, M., Toscani, G.: A kinetic flocking model with diffusion. Preprint (2009)Google Scholar
  32. 32.
    Eftimie, R., de Vries, G., Lewis, M.A.: Complex spatial group patterns result from different animal communication mechanisms. Proc. Natl. Acad. Sci., 104, 6974–6979 (2007)MATHCrossRefGoogle Scholar
  33. 33.
    Fellner, K., Raoul, G.: Stable stationary states of non-local interaction equations. Preprint CMLA/ENS-CachanGoogle Scholar
  34. 34.
    Fornasier, M., Haskovec, J., Toscani, G.: Fluid dynamic description of flocking via Povzner-Boltzmann equation. Preprint (2009)Google Scholar
  35. 35.
    Golse, F.: The mean-field limit for the dynamics of large particle systems. Journées équations aux dérivées partielles, 9, 1–47 (2003)MathSciNetGoogle Scholar
  36. 36.
    Grégoire, G., Chaté, H.: Onset of collective and cohesive motion. Phys. Rev. Lett., 92 (2004)Google Scholar
  37. 37.
    Ha, S.-Y., Liu, J.-G.: A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Comm. Math. Sci., 7, 297–325 (2009)MATHMathSciNetGoogle Scholar
  38. 38.
    Ha, S.-Y., Lee, K., Levy, D.: Emergence of time-asymptotic flocking in a stochastic cucker-smale system. Commun. Math. Sci., 7, 453–469 (2009)MATHMathSciNetGoogle Scholar
  39. 39.
    Ha, S.-Y., Tadmor, E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models, 1, 415–435 (2008)MATHMathSciNetGoogle Scholar
  40. 40.
    Ha, S.-Y., Ha, T., Kim, J.Ho: Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, to appear in IEEE Trans. Automatic Control (2010)Google Scholar
  41. 41.
    Hemelrijk, C.K. and Kunz, H.: Density distribution and size sorting in fish schools: an individual-based model. Behav. Ecol., 16, 178–187 (2005)CrossRefGoogle Scholar
  42. 42.
    Hemelrijk, C. K. and Hildenbrandt, H.: Self-organized shape and frontal density of fish schools. Ethology, 114 (2008)Google Scholar
  43. 43.
    Hildenbrandt, H, Carere, C., and Hemelrijk, C. K.: Self-organised complex aerial displays of thousands of starlings: a model.Google Scholar
  44. 44.
    Huth, A. and Wissel, C.: The Simulation of the Movement of Fish Schools. J. Theor. Biol. (1992)Google Scholar
  45. 45.
    Koch, A.L., White, D.: The social lifestyle of myxobacteria. Bioessays, 20, 1030–1038 (1998)CrossRefGoogle Scholar
  46. 46.
    Kunz, H. and Hemelrijk, C. K. 2003: Artificial fish schools: collective effects of school size, body size, and body form. Artificial Life, 9(3):237253CrossRefGoogle Scholar
  47. 47.
    Kunz, H., Zblin, T. and Hemelrijk, C. K. 2006: On prey grouping and predator confusion in artificial fish schools. In Proceedings of the Tenth International Conference of Artificial Life. MIT Press, Cambridge, MassachusettsGoogle Scholar
  48. 48.
    Lachowicz, M., Pulvirenti, M.: A stochastic system of particles modelling the Euler equations. Arch. Rat. Mech. Anal., 109, 81–93 (1990)MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Levine, H., Rappel, W.J., Cohen, I.: Self-organization in systems of self-propelled particles. Phys. Rev. E, 63 (2000)Google Scholar
  50. 50.
    Li, Y.X., Lukeman, R., Edelstein-Keshet, L.: Minimal mechanisms for school formation in self-propelled particles. Physica D, 237, 699–720 (2008)MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Li, Y.X., Lukeman, R., Edelstein-Keshet, L.: A conceptual model for milling formations in biological aggregates. Bull Math Biol., 71, 352–382 (2008)MathSciNetGoogle Scholar
  52. 52.
    McNamara, S., Young, W.R.: Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A, 5, 34–45 (1993)CrossRefMathSciNetGoogle Scholar
  53. 53.
    Mogilner, A., Edelstein-Keshet, L., Bent, L., Spiros, A., Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. Biol., 47, 353–389 (2003)MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Neunzert, H.: The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles. Trans. Fluid Dyn., 18, 663–678 (1977)Google Scholar
  55. 55.
    Neunzert, H.: An introduction to the nonlinear Boltzmann-Vlasov equation. Kinetic theories and the Boltzmann equation Lecture Notes in Math., 1048, Springer, Berlin, (1984)CrossRefGoogle Scholar
  56. 56.
    Parrish, J., Edelstein-Keshet, L.: Complexity, pattern, and evolutionary trade-offs in animal aggregation. Science, 294, 99–101 (1999)CrossRefGoogle Scholar
  57. 57.
    Perea, L., Gómez, G., Elosegui, P.: Extension of the Cucker–Smale control law to space flight formations. AIAA J. Guidance Contrl. Dyn., 32, 527–537 (2009)CrossRefGoogle Scholar
  58. 58.
    Povzner, A.Y.: The Boltzmann equation in kinetic theory of gases. Am. Math. Soc. Trans. Ser. 2, 47, 193–216 (1962)Google Scholar
  59. 59.
    Raoul, G.: Non-local interaction equations: Stationary states and stability analysis. Preprint CMLA/ENS-CachanGoogle Scholar
  60. 60.
    Shen, J.: Cucker-Smale Flocking under Hierarchical Leadership. SIAM J. Appl. Math., 68:3, 694–719 (2008)MATHCrossRefGoogle Scholar
  61. 61.
    Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Modern Phys., 52, 569–615 (1980)CrossRefMathSciNetGoogle Scholar
  62. 62.
    Spohn, H.: Large scale dynamics of interacting particles. Texts and Monographs in Physics, Springer, Berlin (1991)MATHGoogle Scholar
  63. 63.
    Toner, J., Tu, Y.: Long-range order in a two-dimensional dynamical xy model: How birds fly together. Phys. Rev. Lett., 75, 4326–4329 (1995)CrossRefGoogle Scholar
  64. 64.
    Topaz, C.M., Bertozzi, A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math., 65, 152–174 (2004)MATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    Topaz, C.M., Bertozzi, A.L., Lewis, M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Biol., 68, 1601–1623 (2006)CrossRefMathSciNetGoogle Scholar
  66. 66.
    Toscani, G.: Kinetic and hydrodynamic models of nearly elastic granular flow. Monatsh. Math., 142, 179–192 (2004)MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci., 4, 481–496 (2006)MATHMathSciNetGoogle Scholar
  68. 68.
    Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75, 1226–1229 (1995)CrossRefGoogle Scholar
  69. 69.
    Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics 58, Am. Math. Soc, Providence (2003)Google Scholar
  70. 70.
    Viscido, S.V., Parrish, J.K. and Grünbaum, D.: Individual behavior and emergent properties of fish schools: a comparison of observation and theory. Marine Ecol. Prog. Ser., 273, 239–249 (2004)CrossRefGoogle Scholar
  71. 71.
    Viscido, S.V., Parrish, J.K., Grünbaum, D.: The effect of population size and number of influential neighbors on the emergent properties of fish schools. Ecol. Model., 183, 347–363 (2005)CrossRefGoogle Scholar
  72. 72.
    Yates, C., Erban, R., Escudero, C., Couzin, L., Buhl, J., Kevrekidis, L., Maini, P., Sumpter, D.: Inherent noise can facilitate coherence in collective swarm motion. Proc. Natl. Acad. Sci., 106, 5464–5469 (2009)CrossRefGoogle Scholar
  73. 73.
    Youseff, L.M., Barbaro, A.B.T., Trethewey, P.F., Birnir, B. and Gilbert, J.: Parallel modeling of fish interaction. Parallel Modeling of Fish Interaction, 11th IEEE International Conference on Computational Science and Engineering (2008)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • José A. Carrillo
    • 1
  • Massimo Fornasier
    • 2
  • Giuseppe Toscani
    • 3
  • Francesco Vecil
    • 2
  1. 1.ICREA - Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)LinzAustria
  3. 3.Department of MathematicsUniversity of PaviaPaviaItaly

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