Journal of Statistical Physics

, Volume 158, Issue 3, pp 563–578 | Cite as

Scale-Free Correlations, Influential Neighbours and Speed Control in Flocks of Birds

  • Charlotte K. HemelrijkEmail author
  • Hanno Hildenbrandt


Coordination of birds in large flocks is amazing, especially, since individual birds only interact with a few neighbors (the so-called ‘influential neighbours’). Yet, empirical data show that fluctuations of velocity and speed of different birds are correlated beyond the influential neighbours and are correlated over a larger distance in a larger flock. This correlation between the correlation length of velocity or speed and flock size was found to be linear, called a scale-free correlation. It depends on the way individuals interact in the flock, for instance, on the number of influential neighbours and speed control. It is unknown however, how exactly the number of influential neighbours affects this scale-free correlation. Recent empirical data show that different degrees of control of speed affect the scale-free correlation for speed fluctuations. Theoretically, based on statistical mechanics, it is predicted that at very high speed control, the correlation is no longer scale-free but saturates at a certain correlation length and this hampers coordination in flocks. We study these issues in a model, called StarDisplay, because its behavioural rules are biologically inspired and many of its flocking patterns resemble empirical data. Our results show that the correlation length of fluctuations of velocity as well as speed correlate with flock size in a scale-free manner. A higher number of influential neighbours causes a diminishing increase of the slope of the scale-free correlation with velocity, resulting thus in flocks that coordinate more uniformly. Similar to recent empirical data higher speed control reduces the correlation length of speed fluctuations in our model. As predicted theoretically, at very high speed control the model generates a non-scale free correlation, and although there are still flocks, they are in the process of disintegrating.


Flocks of birds Self-organization Number of influential neighbours Spatial dynamics Scale-free correlation Speed control Information transmission 



We are grateful for the grants that have been supporting this research, Hemelrijk’s Startup Grant of her Rosalind Franklin Fellowship and her Grant in the European Project of the 7th framework, StarFlag for the work by Hanno Hildenbrandt. We thank the self-organization group for regular discussions.

Supplementary material

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  1. 1.
    Davis, M.J.: The coordinated aerobatics of dunlin flocks. Anim. Behav. 28(3), 668–673 (1980)CrossRefGoogle Scholar
  2. 2.
    Carere, C., Montanino, S., Moreschini, F., et al.: Aerial flocking patterns of wintering starlings, Sturnus vulgaris, under different predation risk. Anim. Behav. 77, 101–107 (2009)CrossRefGoogle Scholar
  3. 3.
    Clergeau, P.: Flocking behavior of starlings (sturnus vulgaris) during the day: a gradual gathering to the roost. J. Ornithol. 131(4), 458–460 (1990)CrossRefGoogle Scholar
  4. 4.
    Feare, C.J.: The Starling. Oxford University Press, Oxford (1984)Google Scholar
  5. 5.
    Brodie, J.: The flight behaviour of starlings at a winter roost. Br. Birds 69, 51–60 (1976)Google Scholar
  6. 6.
    Eastwood, E., Isted, G.A., Rider, G.C.: Radar ring angles and the roosting behaviour of starlings. Proc. R. Soc. Lond. B 156(963), 242–267 (1962)ADSCrossRefGoogle Scholar
  7. 7.
    Ballerini, M., Cabibbo, N., Candelier, R., et al.: Empirical investigation of starling flocks: a benchmark study in collective animal behaviour. Anim. Behav. 76(1), 201–215 (2008)CrossRefGoogle Scholar
  8. 8.
    Hildenbrandt, H., Carere, C., Hemelrijk, C.K.: Self-organized aerial displays of thousands of starlings: a model. Behav. Ecol. 21(6), 1349–1359 (2010). doi: 10.1093/beheco/arq149 CrossRefGoogle Scholar
  9. 9.
    Cavagna, A., Cimarelli, A., Giardina, I., et al.: Scale-free correlations in starling flocks. PNAS 107(26), 11865–11870 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Ballerini, M., Cabibbo, N., Candelier, R., et al.: Interaction ruling animal collective behaviour depends on topological rather than metric distance: evidence from a field study. PNAS 105(4), 1232–1237 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    Lukeman, R., Li, Y., Edelstein-Keshet, L.: Inferring individual rules from collective behavior. PNAS 107(28), 12576–12580 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Nagy, M., Ákos, Z., Biro, D., Vicsek, T.: Hierarchical group dynamics in pigeon flocks. Nature 464(7290), 890–893 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    Inada, Y., Kawachi, K.: Order and flexibility in the motion of fish schools. J. Theor. Biol. 214(3), 371–387 (2002)CrossRefGoogle Scholar
  14. 14.
    Mirabet, V., Pierre, F., Christophe, L.: Factors affecting information transfer from knowledgeable to naive individuals in groups. Behav. Ecol. Sociobiol. 63(2), 159–171 (2008)CrossRefGoogle Scholar
  15. 15.
    Lemasson, B.H., Anderson, J.J., Goodwin, R.A.: Collective motion in animal groups from a neurobiological perspective: the adaptive benefits of dynamic sensory loads and selective attention. J. Theor. Biol. 261(4), 501–510 (2009)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Bialek, W., Cavagna, A., Giardina, I., et al.: Social interactions dominate speed control in poising natural flocks near criticality. Proc. Natl. Acad. Sci. USA 111(20), 7212–7217 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    Hemelrijk, C.K., Hildenbrandt, H.: Some causes of the variable shape of flocks of birds. PLoS One 6(8), e22479 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    Hemelrijk, C.K., Hildenbrandt, H.: Schools of fish and flocks of birds: their shape and internal structure by self-organization. Interface Focus 2(6), 726–737 (2012)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Huth, A., Wissel, C.: The simulation of the movement of fish schools. J. Theor. Biol. 156(3), 365–385 (1992)CrossRefGoogle Scholar
  21. 21.
    Huth, A., Wissel, C.: The simulation of fish schools in comparison with experimental data. Ecol. Model. 75(76), 135–145 (1994)CrossRefGoogle Scholar
  22. 22.
    Kunz, H., Hemelrijk, C.K.: Artificial fish schools: collective effects of school size, body size, and body form. Artif. Life 9(3), 237–253 (2003)CrossRefGoogle Scholar
  23. 23.
    Hemelrijk, C.K., Hildenbrandt, H.: Self-organized shape and frontal density of fish schools. Ethology 114, 245–254 (2008)CrossRefGoogle Scholar
  24. 24.
    Reuter, H., Breckling, B.: Selforganization of fish schools: an object-oriented model. Ecol. Model. 75, 147–159 (1994)CrossRefGoogle Scholar
  25. 25.
    Norberg, U.M.: Vertebrate Flight: Mechanics, Physiology, Morphology, Ecology and Evolution, vol. 27. Springer, New York (1990)Google Scholar
  26. 26.
    Videler, J.J.: Avian Flight. Oxford University Press, Oxford (2005)Google Scholar
  27. 27.
    Ward, S., Mueller, U., Rayner, J.M.V., Jackson, D.M., Nachtigall, W., Speakman, J.R.: Metabolic power of european starlings sturnus vulgaris during flight in a wind tunnel, estimated from heat transfer modelling, doubly labelled water and mask respirometry. J. Exp. Biol. 207(Pt 24), 4291–4298 (2004)CrossRefGoogle Scholar
  28. 28.
    Pomeroy, H., Heppner, F.: Structure of turning in airborne rock dove (Columba livia) flocks. Auk 109(2), 256–267 (1992)CrossRefGoogle Scholar
  29. 29.
    Gillies, J.A., Thomas, A.L.R., Taylor, G.K.: Soaring and manoeuvring flight of a steppe eagle aquila nipalensis. J. Avian Biol. 42(5), 377–386 (2011)CrossRefGoogle Scholar
  30. 30.
    Selous, E.: Thought Transference (or what?) in Birds. Constanble and Company Ltd., London (1931)Google Scholar
  31. 31.
    Reynolds, C.W.: Flocks, herds and schools: a distributed behavioral model. In: Proceedings of the 14th annual conference on computer graphics and interactive techniques. Vol 21. ACM, New York, pp. 25–34 (1987)Google Scholar
  32. 32.
    Gillies, J.A., Bacic, M., Yuan, F.G., Thomas, A.L.R., Taylor, G.K.: Modeling and identification of steppe eagle (aquila nipalensis) dynamics. AIAA Modeling and Simulations Technologies Conference and Exhibit. American Institute of Aeronautics and Astronautics, Honolulu, Hawaii, pp. 18–21 (2008)Google Scholar
  33. 33.
    Taylor, G.K., Bacic, M., Carruthers, A.C., Gillies, J., Ozawa, Y., Thomas, A.L.R.: Flight control mechanisms in birds of prey. 45th AIAA Aerospace Sciences Meeting and Exhibit. American Institute of Aeronautics and Astronautics, Reno, Nevada, pp. 8–11 (2007)Google Scholar
  34. 34.
    Viscido, S.V., Parrish, J.K., Grunbaum, D.: The effect of population size and number of influential neighbors on the emergent properties of fish schools. Ecol. Model. 183(2–3), 347–363 (2005)CrossRefGoogle Scholar
  35. 35.
    Niizato, T., Gunji, Y.: Fluctuation-driven flocking movement in three dimensions and scale-free correlation. PloS One 7(5), e35615 (2012)ADSCrossRefGoogle Scholar
  36. 36.
    Kunz, H., Hemelrijk, C.K.: Simulations of the social organization of large schools of fish whose perception is obstructed. Appl. Anim. Behav. Sci. 138(3–4), 142–151 (2012)CrossRefGoogle Scholar
  37. 37.
    Kunz, H., Hemelrijk, C.K.: Simulations of the social organization of large schools of fish whose perception is obstructed. Appl. Anim. Behav. Sci. 138(3–4), 142–151 (2012)CrossRefGoogle Scholar
  38. 38.
    Zheng, M., Kashimori, Y., Hoshino, O., Fujita, K., Kambara, T.: Behavior pattern (innate action) of individuals in fish schools generating efficient collective evasion from predation. J. Theor. Biol. 235(2), 153–167 (2005)CrossRefMathSciNetGoogle Scholar
  39. 39.
    Hemelrijk, C.K., Kunz, H.: Density distribution and size sorting in fish schools: an individual-based model. Behav. Ecol. 16(1), 178–187 (2005)CrossRefGoogle Scholar
  40. 40.
    Hamilton, W.D.: Geometry for the selfish herd. J. Theor. Biol. 31, 295–311 (1971)CrossRefGoogle Scholar
  41. 41.
    Hemelrijk, C.K., Wantia, J.: Individual variation by self-organisation: a model. Neurosci. Biobehav. Rev. 29(1), 125–136 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Behavioural Ecology and Self-Organisation, Centre for Ecological and Evolutionary StudiesUniversity of GroningenGroningenThe Netherlands

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