Chases and Escapes: From Singles to Groups

  • Atsushi Kamimura
  • Shigenori Matsumoto
  • Toru Ohira


“Chases and escapes” is a traditional mathematical problem. The act of balancing a stick on human fingertips represents an experimental paradigm of typical “one-to-one” chase and escape. Recently, we have proposed a simple model where we extend the chaser and escape to a case a group of particles chasing another group, called “group chase and escape.” This extension connects the traditional problem with current interests on collective motions of animals, insects, vehicles, etc. In this chapter, we present our basic model and its rather complex behavior. In the model, each chaser approaches its nearest escapee, while each escapee steps away from its nearest chaser. Although there are no communications within each group, simulations show segregations of chasers and targets. Two order parameters are introduced to characterize the chasing and escaping in group. Further developments are reviewed to extend our basic model.


Chases and escapes Collective motion Traffic model Optimal velocity model Group chase and escape Pattern formation Order parameter 



A.K. is supported by the Japan Society for the Promotion of Science. T.O. has been supported by Kayamori Foundation of Information Science Advancement.


  1. 1.
    Angelani L (2012) Collective predation and escape strategies. Phys Rev Lett 109:118104CrossRefPubMedGoogle Scholar
  2. 2.
    Bando M, Hasebe K, Nakayama A, Shibata A, Sugiyama Y (1995) Dynamical model of traffic congestion and numerical simulation. Phys Rev E 51:1035CrossRefGoogle Scholar
  3. 3.
    Bulsara AR, Gammaitoni L (1996) Tuning in to noise. Phys Today 49:39CrossRefGoogle Scholar
  4. 4.
    Gammaitoni L, Hanngi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70:223CrossRefGoogle Scholar
  5. 5.
    Isaacs R (1965) Differential games. Wiley, New YorkGoogle Scholar
  6. 6.
    Iwama T, Sato M (2012) Group chase and escape with some fast chasers. Phys Rev E 86:067102CrossRefGoogle Scholar
  7. 7.
    Kamimura A, Ohira T (2010) Group chase and escape. New J Phys 12:053013CrossRefGoogle Scholar
  8. 8.
    Kamimura A, Matsumoto S, Nogawa T, Ito N, Ohira T (2011) Stochastic resonance with group chase and escape. In: 2011 21st international conference on noise and fluctuations (ICNF). Stochastic resonance with group chase and escape. Toronto, p 200Google Scholar
  9. 9.
    Matsumoto S, Nogawa T, Kamimura A, Ito N, Ohira T (2011) Dynamical aspect of group chase and escape. AIP Conf Proc 1332:226CrossRefGoogle Scholar
  10. 10.
    Nahin PJ (2007) Chases and escapes: the mathematics of pursuit and evasion. Princeton University Press, PrincetonGoogle Scholar
  11. 11.
    Nishi R, Kamimura A, Nishinari K, Ohira T (2012) Group chase and escape with conversion from targets to chasers. Physica A 391(1–2):337CrossRefGoogle Scholar
  12. 12.
    Ohira T, Sawatari R (1998) Phase transition in a computer network traffic model. Phys Rev E 58:193CrossRefGoogle Scholar
  13. 13.
    Ohira T, Kamimura A, Milton JG (2011) Pursuit-escape with distance-dependent delay. In: 7th European nonlinear dynamics conference (ENOC2011), Pursuit-escape with distance-dependent delay. Rome, MS-11Google Scholar
  14. 14.
    Reynolds CW (1987) Computer Graphics, 21(4)(SIGGRAPH ’87 conference proceedings), 25Google Scholar
  15. 15.
    Rucle WH (1991) A discrete search game. In: Raghavan TES et al (eds) Stochastic games and related topics. Kluwer Academic, Dordrecht, pp 29–43CrossRefGoogle Scholar
  16. 16.
    Sato M (2012) Chasing and escaping by three groups of species. Phys Rev E 85:066102CrossRefGoogle Scholar
  17. 17.
    Sengupta A, Kruppa T, Lowen H (2011) Chemotactic predator-prey dynamics. Phys Rev E 83:031914CrossRefGoogle Scholar
  18. 18.
    Sugiyama Y, Fukui M, Kikuchi M, Hasebe K, Nakayama A, Nishinari K, Tadaki S, Yukawa S (2008) Traffic jams without bottlenecks–experimental evidence for the physical mechanism of the formation of a jam. New J Phys 10(3):033001CrossRefGoogle Scholar
  19. 19.
    Vicsek T (2010) Statistical physics: closing in on evaders. Nature 466:43CrossRefPubMedGoogle Scholar
  20. 20.
    Vicsek T, Zafeiris A (2012) Collective motion. Phys Rep 517(3–4):71CrossRefGoogle Scholar
  21. 21.
    Vicsek T, Czirok A, Ben-Jacob E, Cohen I, Shochet O (1995) Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 75:1226CrossRefPubMedGoogle Scholar
  22. 22.
    Wiesenfeld K, Moss F (1995) Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs. Nature 373:33CrossRefPubMedGoogle Scholar
  23. 23.
    Yamamoto K, Yamamoto S (2013) Analysis of group chase and escape by honeycomb grid cellular automata. In: Proceedings of SICE annual conference (SICE), IEEE. Analysis of group chase and escape by honeycomb grid cellular automata. 2013, p 1004–1009Google Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Atsushi Kamimura
    • 1
  • Shigenori Matsumoto
    • 2
  • Toru Ohira
    • 3
  1. 1.Department of Basic ScienceThe University of TokyoKomaba, TokyoJapan
  2. 2.Department of Applied PhysicsThe University of TokyoHongo, TokyoJapan
  3. 3.Graduate School of MathematicsNagoya UniversityFurocho, NagoyaJapan

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