Skip to main content
SpringerLink
Log in

A New Model for Self-organized Dynamics and Its Flocking Behavior

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We introduce a model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The proposed model does not only take into account the distance between agents, but instead, the influence between agents is scaled in term of their relative distance. Consequently, our model does not involve any explicit dependence on the number of agents; only their geometry in phase space is taken into account. The use of relative distances destroys the symmetry property of the original C-S model, which was the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems, which covers systems with non-symmetric influence matrices. In particular, we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with “leaders”.

The methodology presented in this paper, based on the notion of active sets, carries over from the particle to kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic formulation of our proposed model, and prove its unconditional flocking for slowly decaying influence functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aoki, I.: A simulation study on the schooling mechanism in fish. Bull. Jpn. Soc. Sci. Fish. (Japan) 48(8), 1081 (1982)

    Article  Google Scholar 

  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. USA 105(4), 1232 (2008)

    Article  ADS  Google Scholar 

  3. Birnir, B.: An ODE model of the motion of pelagic fish. J. Stat. Phys. 128(1), 535–568 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Buhl, J., Sumpter, D.J.T., Couzin, I.D., Hale, J.J., Despland, E., Miller, E.R., Simpson, S.J.: From disorder to order in marching locusts. Science 312(5778), 1402–1406 (2006). American Association for the Advancement of Science

    Article  ADS  Google Scholar 

  5. 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 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    Article  MathSciNet  Google Scholar 

  8. Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852 (2007)

    Article  MathSciNet  Google Scholar 

  9. Cucker, F., Smale, S.: On the mathematics of emergence. Jpn. J. Math. 2(1), 197–227 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Methods Appl. Sci. 18(1), 1193–1215 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Degond, P., Motsch, S.: Large scale dynamics of the persistent turning walker model of fish behavior. J. Stat. Phys. 131(6), 989–1021 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Grimm, V., Railsback, S.F.: Individual-Based Modeling and Ecology. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  13. Grégoire, G., Chaté, H.: Onset of collective and cohesive motion. Phys. Rev. Lett. 92(2), 25702 (2004)

    Article  ADS  Google Scholar 

  14. 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)

    MathSciNet  MATH  Google Scholar 

  15. Ha, S.Y., Tadmor, E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinet. Relat. Models 1(3), 415–435 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ha, S.Y., Lee, K., Levy, D.: Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system. Commun. Math. Sci. 7(2), 453–469 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Hemelrijk, C.K., Hildenbrandt, H.: Self-organized shape and frontal density of fish schools. Ethology 114(3), 245–254 (2008)

    Article  Google Scholar 

  18. Hubbard, J.H., West, B.H.: Differential Equations: A Dynamical Systems Approach. Higher-Dimensional Systems. Springer, Berlin (1995)

    Book  Google Scholar 

  19. Huth, A., Wissel, C.: The simulation of the movement of fish schools. J. Theor. Biol. 156(3), 365–385 (1992)

    Article  Google Scholar 

  20. Huth, A., Wissel, C.: The simulation of fish schools in comparison with experimental data. Ecol. Model. 75, 135–146 (1994)

    Article  Google Scholar 

  21. Parrish, J.K., Viscido, S.V., Grunbaum, D.: Self-organized fish schools: an examination of emergent properties. Biol. Bull. 202(3), 296–305 (2002). Marine Biological Laboratory, Woods Hole

    Article  Google Scholar 

  22. Reynolds, C.W.: Flocks, herds and schools: a distributed behavioral model. In: ACM SIGGRAPH Computer Graphics, vol. 21, pp. 25–34 (1987)

    Google Scholar 

  23. Shen, J.: Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math. 68(3), 694–719 (2008)

    Article  MATH  Google Scholar 

  24. Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)

    MATH  Google Scholar 

  25. 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)

    Article  ADS  Google Scholar 

  26. Viscido, S.V., Parrish, J.K., Grunbaum, D.: Individual behavior and emergent properties of fish schools: a comparison of observation and theory. Mar. Ecol. Prog. Ser. 273, 239–249 (2004)

    Article  Google Scholar 

  27. Youseff, L., Barbaro, A., Trethewey, P., Birnir, B., Gilbert, J.G.: Parallel modeling of fish interaction. In: 11th IEEE International Conference on Computational Science and Engineering, CSE ’08, pp. 234–241. IEEE Comput. Soc., Los Alamitos (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD, 20742, USA

    Sebastien Motsch & Eitan Tadmor

  2. Department of Mathematics, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Eitan Tadmor

Authors
  1. Sebastien Motsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eitan Tadmor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eitan Tadmor.

Additional information

To Claude Bardos on his 70th birthday, with friendship and admiration.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Motsch, S., Tadmor, E. A New Model for Self-organized Dynamics and Its Flocking Behavior. J Stat Phys 144, 923 (2011). https://doi.org/10.1007/s10955-011-0285-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-011-0285-9

Keywords

Search

Navigation