Population Ecology

, Volume 58, Issue 3, pp 357–370 | Cite as

Phase diagram of a multiple forces model for animal group formation: marches versus circles determined by the relative strength of alignment and cohesion

  • Yuki Kubo
  • Yoh Iwasa
Original article


Many species of fish, bird, and insect form groups of individuals that move together, called schools, flocks, or swarms, of characteristic shape and speed. Here we study a model that traces movements of many individuals, in which each individual moves at a constant speed, and changes its movement angle in response to its neighbors within a radius of interaction. Outside of a short range of separation (or repulsion), each individual changes moving direction to achieve a similar moving direction as its neighbors (alignment) and to move toward them (cohesion). Between each pair of individuals within an interaction range, both alignment and cohesion are at work simultaneously (multiple forces model). This is different from many other models for animal group formation in which only one of the two forces is at work (single force model), different forces operating in different zones of between-individual distance. Depending on the relative strength of alignment and cohesion, our model produces groups of two distinct patterns: marches and circles. We showed the phase diagram of group patterns depending on the relative strength of alignment and cohesion. As the strength of alignment relative to cohesion increases, the shapes of groups change gradually in the following order: (1) circles, (2) mixture of circles and marches, (3) short marches, (4) long marches, (5) wide marches. We derived a formula for the spatial size of circles, which explains that the radius of circles does not change with the number of individuals, but it increases with moving speed and decreases with the sensitivity of moving direction to neighbors.


Animal movement Collective animal behavior Individual-based simulation Self-organization 



This work was supported by Japan Society for the Promotion of Science Pre-doctoral Fellowship and a Grant-in-Aid for JSPS Fellows 15J02857 to Y. K., and another Grant-in-Aid for Basic Scientific Research (B) 15H004423 to Y. I. We would like to thank the following people for their useful comments: Atsushi Yamauchi, Hiro-Sato Niwa and Masayuki Nakamura.


  1. Aoki I (1982) A simulation study on the schooling mechanism in fish. Bull Jpn Soc Sci Fish 48:1081–1088CrossRefGoogle Scholar
  2. Bonabeau E, Theraulaz G, Deneubourg JL, Aron S, Camazine S (1997) Self-organization in social insects. Trends Ecol Evol 12:188–193CrossRefPubMedGoogle Scholar
  3. Breder CM (1954) Equations descriptive of fish schools and other animal aggregations. Ecology 35:361–370CrossRefGoogle Scholar
  4. Buhl J, Sumpter DJT, Couzin ID, Hale JJ, Despland E, Miller ER, Simpson SJ (2006) From disorder to order in marching locusts. Science 312:1402–1406CrossRefPubMedGoogle Scholar
  5. Burstedde C, Klauck K, Schadschneider A, Zittartz J (2001) Simulation of pedestrian dynamics using a two-dimensional cellular automaton. Phys A 295:507–525CrossRefGoogle Scholar
  6. Cao YU, Fukunaga AS, Kahng AB (1997) Cooperative mobile robotics: antecedents and directions. Auton Robots 4:7–27CrossRefGoogle Scholar
  7. Couzin ID, Krause J (2003) Self-organization and collective behavior in vertebrates. In: Slater PJB, Rosenblatt JS, Snowdon CT, Roper TJ (eds) Advances in the study of behavior, vol 32, pp 1–75Google Scholar
  8. Couzin ID, Krause J, James R, Ruxton GD, Franks NR (2002) Collective memory and spatial sorting in animal groups. J Theor Biol 218:1–11CrossRefPubMedGoogle Scholar
  9. Couzin ID, Krause J, Franks NR, Levin SA (2005) Effective leadership and decision-making in animal groups on the move. Nature 433:513–516CrossRefPubMedGoogle Scholar
  10. Czirok A, Stanley HE, Vicsek T (1997) Spontaneously ordered motion of self-propelled particles. J Phys A-Math Gen 30:1375–1385CrossRefGoogle Scholar
  11. Czirok A, Vicsek M, Vicsek T (1999) Collective motion of organisms in three dimensions. Phys A-Stat Mech Appl 264:299–304CrossRefGoogle Scholar
  12. Dorigo M, Bonabeau E, Theraulaz G (2000) Ant algorithms and stigmergy. FuturE Gen Comput Syst Int J Esci 16:851–871CrossRefGoogle Scholar
  13. Faria JJ, Dyer JRG, Clement RO, Couzin ID, Holt N, Ward AJW, Waters D, Krause J (2010) A novel method for investigating the collective behaviour of fish: introducing “Robofish”. Behav Ecol Sociobiol 64:1211–1218CrossRefGoogle Scholar
  14. Freon P, Gerlotto F, Soria M (1992) Changes in school structure according to external stimuli: description and influence on acoustic assessment. Fish Res 15:45–66CrossRefGoogle Scholar
  15. Gregoire G, Chate H (2004) Onset of collective and cohesive motion. Phys Rev Lett 92:4CrossRefGoogle Scholar
  16. Grünbaum D, Okubo A (1994) Modelling social animal aggregations. In: Levin S (ed) Frontiers in mathematical biology. Springer, Berlin, Heidelberg, pp 296–325CrossRefGoogle Scholar
  17. Gueron S, Levin SA (1995) The dynamics of group formation. Math Biosci 128:243–264CrossRefPubMedGoogle Scholar
  18. Gueron S, Levin SA, Rubenstein DI (1996) The dynamics of herds: from individuals to aggregations. J Theor Biol 182:85–98CrossRefGoogle Scholar
  19. Helbing D, Molnar P (1995) Social force model for pedestrian dynamics. Phys Rev E 51:4282–4286CrossRefGoogle Scholar
  20. Helbing D, Farkas I, Vicsek T (2000) Simulating dynamical features of escape panic. Nature 407:487–490CrossRefPubMedGoogle Scholar
  21. Helbing D, Buzna L, Johansson A, Werner T (2005) Self-organized pedestrian crowd dynamics: experiments, simulations, and design solutions. Transp Sci 39:1–24CrossRefGoogle Scholar
  22. Hemelrijk CK, Hildenbrandt H (2008) Self-organized shape and frontal density of fish schools. Ethology 114:245–254CrossRefGoogle Scholar
  23. Hemelrijk CK, Hildenbrandt H (2011) Some causes of the variable shape of flocks of birds. PLoS One 6:13CrossRefGoogle Scholar
  24. Hemelrijk CK, Hildenbrandt H (2012) Schools of fish and flocks of birds: their shape and internal structure by self-organization. Interface Focus 2:726–737CrossRefPubMedPubMedCentralGoogle Scholar
  25. Hemelrijk CK, Kunz H (2005) Density distribution and size sorting in fish schools: an individual-based model. Behav Ecol 16:178–187CrossRefGoogle Scholar
  26. Hemelrijk CK, Hildenbrandt H, Reinders J, Stamhuis EJ (2010) Emergence of oblong school shape: models and empirical data of fish. Ethology 116:1099–1112CrossRefGoogle Scholar
  27. Hildenbrandt H, Carere C, Hemelrijk CK (2010) Self-organized aerial displays of thousands of starlings: a model. Behav Ecol 21:1349–1359CrossRefGoogle Scholar
  28. Huth A, Wissel C (1992) The simulation of the movement of fish schools. J Theor Biol 156:365–385CrossRefGoogle Scholar
  29. Huth A, Wissel C (1994) The simulation of fish schools in comparison with experimental data. Ecol Model 75:135–146CrossRefGoogle Scholar
  30. Jadbabaie A, Lin J, Morse AS (2003) Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Autom Control 48:988–1001CrossRefGoogle Scholar
  31. Katz Y, Tunstrom K, Ioannou CC, Huepe C, Couzin ID (2011) Inferring the structure and dynamics of interactions in schooling fish. Proc Natl Acad Sci USA 108:18720–18725CrossRefPubMedPubMedCentralGoogle Scholar
  32. Kunz H, Hemelrijk CK (2003) Artificial fish schools: collective effects of school size, body size, and body form. Artif Life 9:237–253CrossRefPubMedGoogle Scholar
  33. Kunz H, Hemelrijk CK (2012) Simulations of the social organization of large schools of fish whose perception is obstructed. Appl Anim Behav Sci 138:142–151CrossRefGoogle Scholar
  34. Li YX, Lukeman R, Edelstein-Keshet L (2008) Minimal mechanisms for school formation in self-propelled particles. Phys D-Nonlinear Phenom 237:699–720CrossRefGoogle Scholar
  35. Lukeman R, Li YX, Edelstein-Keshet L (2009) A conceptual model for milling formations in biological aggregates. Bull Math Biol 71:352–382CrossRefPubMedGoogle Scholar
  36. Mirabet V, Auger P, Lett C (2007) Spatial structures in simulations of animal grouping. Ecol Model 201:468–476CrossRefGoogle Scholar
  37. Niwa HS (1994) Self-organizing dynamic model of fish schooling. J Theor Biol 171:123–136CrossRefGoogle Scholar
  38. Niwa HS (1996) Newtonian dynamical approach to fish schooling. J Theor Biol 181:47–63CrossRefGoogle Scholar
  39. Okubo A (1986) Dynamical aspects of animal grouping: swarms, schools, flocks, and herds. Adv Biophys 22:1–94CrossRefPubMedGoogle Scholar
  40. Parrish JK, Viscido SV, Grunbaum D (2002) Self-organized fish schools: an examination of emergent properties. Biol Bull 202:296–305CrossRefPubMedGoogle Scholar
  41. Partridge BL, Pitcher TJ (1980) The sensory basis of fish schools: relative roles of lateral line and vision. J Comput Physiol 135:315–325CrossRefGoogle Scholar
  42. Reuter H, Breckling B (1994) Selforganization of fish schools: an object-oriented model. Ecol Model 75:147–159CrossRefGoogle Scholar
  43. Reynolds CW (1987) Flocks, herds and schools: a distributed behavioral model. SIGGRAPH Comput Graph 21:25–34CrossRefGoogle Scholar
  44. Romey WL (1996) Individual differences make a difference in the trajectories of simulated schools of fish. Ecol Model 92:65–77CrossRefGoogle Scholar
  45. Sakai S (1973) A model for group structure and its behavior. Biophys Jpn 13:82–90CrossRefGoogle Scholar
  46. Shaw E (1978) Schooling fishes. Am Sci 66:166–175Google Scholar
  47. Shimoyama N, Sugawara K, Mizuguchi T, Yoshinori H, Masaki S (1996) Collective motion in a system of motile elements. Phys Rev Lett 76:3870–3873CrossRefPubMedGoogle Scholar
  48. Strombom D (2011) Collective motion from local attraction. J Theor Biol 283:145–151CrossRefPubMedGoogle Scholar
  49. Sumpter DJT (2006) The principles of collective animal behaviour. Philos Trans R Soc B-Biol Sci 361:5–22CrossRefGoogle Scholar
  50. Tu X, Terzopoulos D (1994) Artificial fishes: physics, locomotion, perception, behavior. In: Proceedings of the 21st annual conference on computer graphics and interactive techniques, pp 43–50Google Scholar
  51. Tunstrom K, Katz Y, Ioannou CC, Huepe C, Lutz MJ, Couzin ID (2013) Collective states, multistability and transitional behavior in schooling fish. PLoS Comput Biol 9:11CrossRefGoogle Scholar
  52. Vicsek T, Zafeiris A (2012) Collective motion. Phys Rep-Rev Sect Phys Lett 517:71–140Google Scholar
  53. Vicsek T, Czirok A, Benjacob E, Cohen I, Shochet O (1995) Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 75:1226–1229CrossRefPubMedGoogle Scholar

Copyright information

© The Society of Population Ecology and Springer Japan 2016

Authors and Affiliations

  1. 1.Department of Biology, Faculty of ScienceKyushu UniversityFukuokaJapan

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