From Flocs to Flocks

  • Shannon Dee Algar
  • Thomas Stemler
  • Michael Small
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 22)


In this chapter we present the different ways in which mathematicians think about group formation and group behaviour. We provide an overview of the frameworks that have been borrowed from Physics and discuss their appropriateness and limitations when applied to groups such as a flock of birds. With a focus at the the level of local behaviour and global results, we explore the origin of flocking models that paved the way for studying some of the most fascinating phenomena and hottest topics of the last few decades self organisation, emergence and coherent collective motion. We then outline the standard approach for model formulation and analysis with two key stages in mind: grouping and flocking - usually treated quite separately in the literature. That is, how do individual motivations drive aggregation of particles to groups, and then how do these groups generate coordinated collective motion. Finally we hint at future work that aims to amalgamate these two stages with a single model.


  1. 1.
    Aldana, M., Larralde, H., & Vazquez, B. (2009). On the emergence of collective order in swarming systems: A recent debate. International Journal of Modern Physics B, 23(18), 3661–3685.CrossRefGoogle Scholar
  2. 2.
    Ariel, G., & Ayali, A. (2015). Locust collective motion and its modeling. PLoS Computational Biology, 11(12), 1–25.CrossRefGoogle Scholar
  3. 3.
    Attanasi, A., Cavagna, A., Del Castello, L., Giardina, I., Melillo, S., Parisi, L., et al. (2014). Collective behaviour without collective order in wild swarms of midges. PLOS Computational Biology, 10(7), 1–10.CrossRefGoogle Scholar
  4. 4.
    Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., et al. (2008). 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.CrossRefGoogle Scholar
  5. 5.
    Bazazi, S., Buhl, J., Hale, J. J., Anstey, M. L., Sword, G. A., Simpson, S. J., et al. (2008). Collective motion and cannibalism in locust migratory bands. Current Biology, 18(10), 735–739.CrossRefGoogle Scholar
  6. 6.
    Bialek, W., Cavagna, A., Giardina, I., Mora, T., Pohl, O., Silvestri, E., et al. (2014). Social interactions dominate speed control in poising natural flocks near criticality. Proceedings of the National Academy of Sciences, 111(20), 7212–7217.CrossRefGoogle Scholar
  7. 7.
    Bialek, W., Cavagna, A., Giardina, I., Mora, T., Silvestri, E., Viale, M., et al. (2012). Statistical mechanics for natural flocks of birds. Proceedings of the National Academy of Sciences, 109(13), 4786–4791.CrossRefGoogle Scholar
  8. 8.
    Bode, N. W. F., Wood, A. J., & Franks, D. W. (2011). Social networks and models for collective motion in animals. Behavioral Ecology and Sociobiology, 65(2), 117–130.CrossRefGoogle Scholar
  9. 9.
    Bricard, A., Caussin, J. B., Desreumaux, N., Dauchot, O., & Bartolo, D. (2013). Emergence of macroscopic directed motion in populations of motile colloids. Nature, 503, 95–98.CrossRefGoogle Scholar
  10. 10.
    Camazine, S., Franks, N. R., Sneyd, J., Bonabeau, E., Deneubourg, J. L., & Theraula, G. (2001). Self-organization in biological systems. Princeton, NJ: Princeton University Press.zbMATHGoogle Scholar
  11. 11.
    Cardy, J. L. (1996). Scaling and renormalization in statistical physics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  12. 12.
    Carrillo, J. A., Klar, A., Martin, S., & Tiwari, S. (2010). Self-propelled interacting particle systems with roosting force. Mathematical Models and Methods in Applied Sciences, 20(supp01), 1533–1552.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cavagna, A., Del Castello, L., Giardina, I., Grigera, T., Jelic, A., Melillo, S., et al. (2015). Flocking and turning: A new model for self-organized collective motion. Journal of Statistical Physics, 158(3), 601–627.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cavagna, A., Giardina, I., Ginelli, F., Mora, T., Piovani, D., Tavarone, R., et al. (2014). Dynamical maximum entropy approach to flocking. Physical Review E, 89, 042707.CrossRefGoogle Scholar
  15. 15.
    Cavagna, A., Giardina, I., Orlandi, A., Parisi, G., & Procaccini, A. (2008). The starflag handbook on collective animal behaviour: 2. Three-dimensional analysis. Animal Behaviour, 76(1), 237–248.CrossRefGoogle Scholar
  16. 16.
    Cavagna, A., Giardina, I., Orlandi, A., Parisi, G., Procaccini, A., Viale, M., et al. (2008). The starflag handbook on collective animal behaviour: 1. Empirical methods. Animal Behaviour, 76(1), 217–236.CrossRefGoogle Scholar
  17. 17.
    Chaté, H., Ginelli, F., Grégoire, G., Peruani, F., & Raynaud, F. (2008). Modeling collective motion: Variations on the Vicsek model. The European Physical Journal B, 64(3), 451–456.CrossRefGoogle Scholar
  18. 18.
    Couzin, I. (2007). Collective minds. Nature, 445, 715.CrossRefGoogle Scholar
  19. 19.
    Couzin, I., Krause, J., James, R., Ruxton, G., & Franks, N. (2002). Collective memory and spatial sorting in animal groups. Journal of Theoretical Biology, 218(1), 1–11.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Couzin, I. D., & Krause, J. (2003). Self-organization and collective behavior in vertebrates. Advances in the Study of Behavior, 32, 1–75.CrossRefGoogle Scholar
  21. 21.
    Cucker, F., & Smale, S. (2007). Emergent behavior in flocks. IEEE Transactions on Automatic Control, 52(5), 852–862.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Danchin, E., Wagner, R. H., Parrish, J. K., & Edelstein-Keshet, L. (2000). Benefits of membership. Science, 287(5454), 804–807.CrossRefGoogle Scholar
  23. 23.
    De Vos, A., & O’Riain, M.J. (2013). Movement in a selfish seal herd: Do seals follow simple or complex movement rules? Behavioral Ecology, 24(1), 190–197.CrossRefGoogle Scholar
  24. 24.
    D’Orsogna, M. R., Chuang, Y. L., Bertozzi, A. L., & Chayes, L. S. (2006). Self-propelled particles with soft-core interactions: Patterns, stability, and collapse. Physical Review Letters, 96, 104302.CrossRefGoogle Scholar
  25. 25.
    Fels, D., Rhisiart, A. A., & Vollrath, F. (1995). The selfish crouton. Behaviour, 132(1/2), 49–55.CrossRefGoogle Scholar
  26. 26.
    Feng, J., & He, Y. (2017). Collective motion of bacteria and their dynamic assembly behavior. Science China Materials, 60(11), 1079–1092.CrossRefGoogle Scholar
  27. 27.
    Giardina, I. (2008). Collective behavior in animal groups: Theoretical models and empirical studies. HFSP Journal, 2, 205–19.CrossRefGoogle Scholar
  28. 28.
    Grünbaum, D. (1994). Translating stochastic density-dependent individual behavior with sensory constraints to an Eulerian model of animal swarming. Journal of Mathematical Biology, 33(2), 139–161.CrossRefGoogle Scholar
  29. 29.
    Grünbaum, D., & Okubo, A. (1994). Modelling social animal aggregations. In S. A. Levin (Ed.) Frontiers in mathematical biology (pp. 296–325). Berlin: Springer.CrossRefGoogle Scholar
  30. 30.
    Haken, H. (2012). Synergetics: An introduction nonequilibrium phase transitions and self-organization in physics, chemistry and biology. Berlin: Springer.zbMATHGoogle Scholar
  31. 31.
    Hamilton, W. (1971). Geometry for the selfish herd. Journal of Theoretical Biology, 31(2), 295–311.CrossRefGoogle Scholar
  32. 32.
    Hauert, S., Leven, S., Varga, M., Ruini, F., Cangelosi, A., Zufferey, J. C., et al. (2011). Reynolds flocking in reality with fixed-wing robots: Communication range vs. maximum turning rate. In 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems (pp. 5015–5020).Google Scholar
  33. 33.
    Heinz, S. (2011). Mathematical modeling. Berlin: Springer.CrossRefGoogle Scholar
  34. 34.
    Helbing, D. (2012). Social self-organization: Agent-based simulations and experiments to study emergent social behavior. Understanding complex systems. Berlin: Springer.CrossRefGoogle Scholar
  35. 35.
    Helbing, D., & Molnár, P. (1995). Social force model for pedestrian dynamics. Physical Review E, 51, 4282–4286.CrossRefGoogle Scholar
  36. 36.
    Hildenbrandt, H., Carere, C., & Hemelrijk, C. (2010). Self-organized aerial displays of thousands of starlings: A model. Behavioral Ecology, 21(6), 1349–1359.CrossRefGoogle Scholar
  37. 37.
    Javarone, M. A., & Marinazzo, D. (2017). Evolutionary dynamics of group formation. PLoS One, 12(11), 1–10.CrossRefGoogle Scholar
  38. 38.
    Jeanson, R., Rivault, C., Deneubourg, J. L., Blanco, S., Fournier, R., Jost, C., et al. (2005). Self-organized aggregation in cockroaches. Animal Behaviour, 69, 169–180.CrossRefGoogle Scholar
  39. 39.
    Jeldres, R. I., Fawell, P. D., & Florio, B. J. (2018). Population balance modelling to describe the particle aggregation process: A review. Powder Technology, 326, 190–207.CrossRefGoogle Scholar
  40. 40.
    Katz, Y., Tunstrom, K., Ioannou, C. C., Huepe, C., & Couzin, I. D. (2011). Inferring the structure and dynamics of interactions in schooling fish. Proceedings of the National Academy of Sciences, 108(46), 18720–18725.CrossRefGoogle Scholar
  41. 41.
    Krause, J., & Ruxton, G. (2002). Living in groups. Oxford series in ecology and evolution. Oxford: Oxford University Press.Google Scholar
  42. 42.
    Ledder, G. (2013). Mathematics for the life sciences: Calculus, modeling, probability, and dynamical systems. Springer undergraduate texts in mathematics and technology. New York: Springer.CrossRefGoogle Scholar
  43. 43.
    Lerman, K., & Shehory, O. (2000). Coalition formation for large-scale electronic markets. In Proceedings Fourth International Conference on Multiagent Systems (pp. 167–174).Google Scholar
  44. 44.
    Levin, S. (2010). Crossing scales, crossing disciplines: Collective motion and collective action in the global commons. Philosophical Transactions B, 365(1537), 13–18.CrossRefGoogle Scholar
  45. 45.
    Lewis, C. T., Short, C., & Andrews, E. A. (1879). Harpers’ latin dictionary: A new latin dictionary founded on the translation of Freund’s Latin-German Lexicon. Oxford: Clarendon Press.Google Scholar
  46. 46.
    Lopez, U., Gautrais, J., Couzin, I. D., & Theraulaz, G. (2012). From behavioural analyses to models of collective motion in fish schools. Interface Focus, 2(6), 693–707.CrossRefGoogle Scholar
  47. 47.
    Lukeman, R., Li, Y. X., & Edelstein-Keshet, L. (2010). Inferring individual rules from collective behavior. Proceedings of the National Academy of Sciences, 107(28), 12576–12580.CrossRefGoogle Scholar
  48. 48.
    Meakin, P. (1999). A historical introduction to computer models for fractal aggregates. Journal of Sol-Gel Science and Technology, 15(2), 97–117.CrossRefGoogle Scholar
  49. 49.
    Meyer, K., Hall, G., & Offin, D. (2008). Introduction to hamiltonian dynamical systems and the N-body problem. Applied mathematical sciences. New York: Springer.Google Scholar
  50. 50.
    Miller, R. C. (1921). The mind of the flock. The Condor, 23(6), 183–186.CrossRefGoogle Scholar
  51. 51.
    Mishra, S., Tunstrøm, K., Couzin, I. D., & Huepe, C. (2012). Collective dynamics of self-propelled particles with variable speed. Physical Review E, 86, 011901.CrossRefGoogle Scholar
  52. 52.
    Mogilner, A., Edelstein-Keshet, L., Bent, L., & Spiros, A. (2003). Mutual interactions, potentials, and individual distance in a social aggregation. Journal of Mathematical Biology, 47(4), 353–389.MathSciNetCrossRefGoogle Scholar
  53. 53.
    Mora, T., & Bialek, W. (2011). Are biological systems poised at criticality? Journal of Statistical Physics, 144(2), 268–302.MathSciNetCrossRefGoogle Scholar
  54. 54.
    Morrell, L. J., Ruxton, G. D., & James, R. (2011). Spatial positioning in the selfish herd. Behavioral Ecology, 22(1), 16–22.CrossRefGoogle Scholar
  55. 55.
    Morton, T. L., Haefner, J. W., Nugala, V., Decino, R. D., & Mendes, L. (1994). The selfish herd revisited: Do simple movement rules reduce relative predation risk? Journal of Theoretical Biology, 167(1), 73–79.CrossRefGoogle Scholar
  56. 56.
    Nagy, M., Ákos, Z., Biro, D., & Vicsek, T. (2010). Hierarchical group dynamics in pigeon flocks. Nature, 464, 890–893.CrossRefGoogle Scholar
  57. 57.
    Passino, K. (2005). Biomimicry for optimization, control, and automation. London: Springer.zbMATHGoogle Scholar
  58. 58.
    Perc, M., & Grigolini, P.: Collective behavior and evolutionary games – An introduction. Chaos, Solitons and Fractals, 56, 1–5 (2013).MathSciNetCrossRefGoogle Scholar
  59. 59.
    Poduri, S., & Sukhatme, G. S. (2007). Latency analysis of coalescence for robot groups. In Proceedings 2007 IEEE International Conference on Robotics and Automation (pp. 3295–3300).Google Scholar
  60. 60.
    Popkin, G. (2016). The physics of life. Nature News, 529, 16–18.CrossRefGoogle Scholar
  61. 61.
    Ranft, J., Basan, M., Elgeti, J., Joanny, J. F., Prost, J., & Jülicher, F. (2010). Fluidization of tissues by cell division and apoptosis. Proceedings of the National Academy of Sciences, 107(49), 20863–20868.CrossRefGoogle Scholar
  62. 62.
    Reynolds, C. W. (1987). Flocks, herds, and schools: A distributed behavioral model. In Computer graphics (pp. 25–34). New York: ACM.Google Scholar
  63. 63.
    Risken, H.(1996). Fokker-Planck equation (pp. 63–95). Berlin: Springer.CrossRefGoogle Scholar
  64. 64.
    Romanczuk, P. (2011). Active motion and swarming: From individual to collective dynamics. Nichtlineare Und Stochastische Physik. Berlin: Logos Verlag.Google Scholar
  65. 65.
    Romanczuk, P., Bär, M., Ebeling, W., Lindner, B., & Schimansky-Geier, L. (2012). Active Brownian particles. The European Physical Journal Special Topics, 202(1), 1–162.CrossRefGoogle Scholar
  66. 66.
    Romanczuk, P., Couzin, I. D., & Schimansky-Geier, L.: Collective motion due to individual escape and pursuit response. Physical Review Letters, 102, 010602 (2009).CrossRefGoogle Scholar
  67. 67.
    Schelling, T. C. (1971). Dynamic models of segregation. The Journal of Mathematical Sociology, 1(2), 143–186.CrossRefGoogle Scholar
  68. 68.
    Strömbom, D. (2011). Collective motion from local attraction. Journal of Theoretical Biology, 283(1), 145.MathSciNetCrossRefGoogle Scholar
  69. 69.
    Sumpter, D. (2010). Collective animal behavior. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
  70. 70.
    Sumpter, D. J. T., Mann, R. P., & Perna, A. (2012). The modelling cycle for collective animal behaviour. Interface Focus, 2(6), 764–773.CrossRefGoogle Scholar
  71. 71.
    Toner, J., & Tu, Y. (1998). Flocks, herds, and schools: A quantitative theory of flocking. Physical Review E, 58, 4828–4858.MathSciNetCrossRefGoogle Scholar
  72. 72.
    Topaz, C. M., Bertozzi, A. L. (2004). Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM Journal on Applied Mathematics, 65(1), 152–174.MathSciNetCrossRefGoogle Scholar
  73. 73.
    Van Dyke Parunak, H., Savit, R., & Riolo, R. L. (1998). Agent-based modeling vs. equation-based modeling: A case study and users’ guide. In J. S. Sichman, R. Conte, & N. Gilbert (Eds.) Multi-agent systems and agent-based simulation (pp. 10–25). Berlin: Springer.CrossRefGoogle Scholar
  74. 74.
    Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., & Shochet, O. (1995). Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75, 1226–1229.MathSciNetCrossRefGoogle Scholar
  75. 75.
    Vicsek, T., & Zafeiris, A. (2012). Collective motion. Physics Reports, 517(3), 71–140. Collective motionCrossRefGoogle Scholar
  76. 76.
    Viscido, S. V., Miller, M., & Wethey, D. S. (2002). The dilemma of the selfish herd: The search for a realistic movement rule. Journal of Theoretical Biology, 217(2), 183–194.MathSciNetCrossRefGoogle Scholar
  77. 77.
    Ward, A., & Webster, M. (2016). Sociality: The behaviour of group-living animals. Cham: Springer International Publishing.CrossRefGoogle Scholar
  78. 78.
    Waters, A., Blanchette, F., & Kim, A. D. (2012). Modeling huddling penguins. PLoS One, 7(11), 1–8.Google Scholar
  79. 79.
    Whitesides, G. M., & Boncheva, M. (2002). Beyond molecules: Self-assembly of mesoscopic and macroscopic components. Proceedings of the National Academy of Sciences, 99(8), 4769–4774.CrossRefGoogle Scholar
  80. 80.
    Witten, T. A., & Sander, L. M. (1981). Diffusion-limited aggregation, a kinetic critical phenomenon. Physical Review Letters, 47, 1400–1403.CrossRefGoogle Scholar
  81. 81.
    Wood, A. J. (2010). Strategy selection under predation; evolutionary analysis of the emergence of cohesive aggregations. Journal of Theoretical Biology, 264(4), 1102–1110.MathSciNetCrossRefGoogle Scholar
  82. 82.
    Yates, C. A., Erban, R., Escudero, C., Couzin, I. D., Buhl, J., Kevrekidis, I. G., et al. (2009). Inherent noise can facilitate coherence in collective swarm motion. Proceedings of the National Academy of Sciences, 106(14), 5464–5469.CrossRefGoogle Scholar
  83. 83.
    Zaitouny, A., Stemler, T., & Small, M. (2017). Modelling and tracking the flight dynamics of flocking pigeons based on real GPS data (small flock). Ecological Modelling, 344, 62–72.CrossRefGoogle Scholar
  84. 84.
    Zaitouny, A., Stemler, T., & Small, M. (2017). Tracking a single pigeon using a shadowing filter algorithm. Ecology and Evolution, 7(12), 4419–4431.CrossRefGoogle Scholar
  85. 85.
    Zhang, H. P., Be’er, A., Florin, E. L., & Swinney, H. L. (2010). Collective motion and density fluctuations in bacterial colonies. Proceedings of the National Academy of Sciences, 107(31), 13626–13630.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Shannon Dee Algar
    • 1
  • Thomas Stemler
    • 1
  • Michael Small
    • 1
  1. 1.University of Western AustraliaCrawleyAustralia

Personalised recommendations