Bulletin of Mathematical Biology

, Volume 58, Issue 4, pp 661–717 | Cite as

Mathematical model of self-organizing hierarchies in animal societies

  • Eric Bonabeau
  • Guy Theraulaz
  • Jean-Louis Deneubourg


We propose a mathematical approach to the modelling of self-organizing hierarchies in animal societies. This approach relies on a basic positive feedback mechanism that reinforces the ability of a given individual to win or to lose in a hierarchical interaction, depending on how many times it won or lost in previous interactions. Motivated by experiments carried out on primitively eusocial waspsPolistes, the model, is based on coupled differential equations supplemented with a small stochastic term. Numerical integrations allow many different hierarchical profiles to be obtained depending on the model parameters: (1) the particular form of the probability for an individual to win or lose a fight given its history, (2) the probability of interaction between two individuals, (3) the forgetting strength, which determines the rate at which events in the past are forgotten and no longer influence the force of an individual and (4) two individual recognition parameters, which set the contribution of individual recognition in the process of hierarchical genesis. We compare the results, expressed in terms of a hierarchical index or of the Landau number that describes the degree of linearity of the hierarchy, with various experimental results.


Dominance Hierarchy Dominance Index Individual Recognition Dominance Order Flat Profile 
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  1. Alexander, R. D. 1961. Aggressiveness, territoriality, and sexual behaviour in field crickets (Orthoptera: Gryllidae).Behaviour 17, 130–223.Google Scholar
  2. Allee, W. C. 1942. Social dominance and subordination among vertebrates. Levels of integration in biological and social systems.Biol. Symp. 8, 139–162.Google Scholar
  3. Allee, W. C. 1951.Cooperation Among Animals. New York: Henry Schulman.Google Scholar
  4. Allee, W. C. 1952. Dominance and hierarchy in societies of vertebrates. Structure et physiologie des sociétés animales.Colloq. Internat. CNRS 34, 157–181.Google Scholar
  5. Baldwin, J. D. 1971. The social organization of a semifree-ranging troop of squirrel monkeys (Saimiri sciureus).Folia Primatologica 14, 23–50.CrossRefGoogle Scholar
  6. Barchas, P. R. and S. D. Mendoza 1984. Emergent hierarchical relationships in rhesus macaques: an application of Chase's model. InSocial Hierarchies: Essays Towards a Sociophysiological Perspective. P. R. Barchas (Ed.), pp. 81–95. Westport, CT: Greenwood Press.Google Scholar
  7. Barnard, C. J. and T. E. Burk. 1979. Dominance hierarchies and the evolution of “individual recognition”.J. Theor. Biol. 81, 65–73.CrossRefGoogle Scholar
  8. Barton, E. P., S. L. Donaldson, M. A. Ross and J. L. Albright. 1974. Social rank and social index as related to age, body weight and milk production in dairy cows.Proc. Indiana Acad. Sci. 83, 473–477.Google Scholar
  9. Beaugrand, J. P. and R. C. Zayan. 1984 An experimental model of aggressive dominance inXiphophorus helleri (Pisces: Poeciliidae).Behav. Proc. 10, 1–53.CrossRefGoogle Scholar
  10. Black-Cleworth, P. 1970. The role of electrical discharges in the non-reproductive social behaviour ofGymnotus carapo (Gymnotidae, Pisces).Anim. Behav. Monogr. 3, 1–77.Google Scholar
  11. Boice, R. and D. W. Witter. 1969. Hierarchical feeding behaviour in the leopard frog (Rana pipiens).Anim. Behav. 17, 474–479.CrossRefGoogle Scholar
  12. Bonabeau, E., G. Theraulaz and J.-L. Deneubourg. 1995. Phase diagram of a model of self-organizing hierarchies.Physica A 217, 373–392.CrossRefGoogle Scholar
  13. Bourke, A. F. G. 1988. Dominance orders, worker reproduction, and queen-worker conflict in the slave-making antHarpagoxenus sublaevis.Behav. Ecol. Sociobiol. 23, 323–333.CrossRefGoogle Scholar
  14. Bovbjerg, R. V. 1956. Some factors affecting aggressive behaviour in crayfish.Physiol. Zool. 29, 127–136.Google Scholar
  15. Bovbjerg, R. V. and S. L. Stephen. 1971. Behavioural changes in crayfish with increased population density.Bull. Ecol. Soc. Am. 52, 37–38.Google Scholar
  16. Broom, D. M. 1981.Biology of Behaviour. Cambridge: Cambridge University Press.Google Scholar
  17. Burk, T. E. 1979. An analysis of social behaviour in crickets. Ph.D. thesis, University of Oxford.Google Scholar
  18. Candland, D. K. and A. I. Leshner. 1971. Formation of squirrel monkey dominance order is correlated with endocrine output.Bull. Ecol. Soc. Am. 52, 54.Google Scholar
  19. Chase, I. D. 1974. Models of hierarchy formation in animal societies.Behav. Sci. 19, 374–382.Google Scholar
  20. Chase, I. D. 1980. Social process and hierarchy formation in small groups: a comparative perspective.Am. Sociol. Rev. 45, 905–924.CrossRefGoogle Scholar
  21. Chase, I. D. 1982a. Dynamics of hierarchy formation: the sequential development of dominance relationships.Behaviour 80, 218–240.Google Scholar
  22. Chase, I. D. 1982b. Behavioral sequences during dominance hierarchy formation in chickens.Science 216, 439–440.Google Scholar
  23. Chase, I. D. 1985. The sequential analysis of aggressive acts during hierarchy formation: an application of the “jigsaw” puzzle approach.Anim. Behav. 33, 86–100.CrossRefGoogle Scholar
  24. Chase, I. D. 1986. Explanations of hierarchy structure.Anim. Behav. 34, 1265–1267.CrossRefGoogle Scholar
  25. Chase, I. D. and S. Rohwer. 1987. Two methods for quantifying the development of dominance hierarchies in large groups with applications to Harris' sparrows.Anim. Behav.,35, 1113–1128.CrossRefGoogle Scholar
  26. Cole, B. J. 1981. Dominance hierarchies inLeptothorax ants.Science 212, 83–84.Google Scholar
  27. Deneubourg, J.-L. 1977. Application de l'ordre par fluctuations à la description de certaines étapes de la construction du nid chez les termites.Ins. Soc. 24, 117–130.CrossRefGoogle Scholar
  28. Deneubourg, J.-L., S. Goss, J.-M. Pasteels, D. Fresneau and J.-P. Lachaud. 1987. Selforganization mechanisms in ant societies (II): learning in foraging and divison of labour. InFrom Individual to Collective Behaviour in Social Insects. J.-M. Pasteels and J.-L. Deneubourg (Eds).Experientia Suppl. 54, 177–196.Google Scholar
  29. Dugatkin, L. A., M. S. Alfieri and A. J. Moore. 1994. Can dominance hierarchies be replicated? Form—re-form experiments using the cockroach (Nauphoeta cinerea).Ethology 97, 94–102.Google Scholar
  30. Eberhard, M. J. W. 1969. The social biology of polistine wasps. Miscellaneous publications 140. Museum of Zoology, University of Michigan, Ann Arbor, pp. 1–101.Google Scholar
  31. Etkin, W. 1965. Co-operation and competition in social behaviour. InSocial Behaviour and Organization among Vertebrates. W. Etkin (Ed). Chicago: The University of Chicago Press.Google Scholar
  32. Evans, L. T. 1951. Field study of the social behaviour of the black lizard,Ctenosaura pectinata. Am. Museum Novitates 1943.Google Scholar
  33. Evans, L. T. 1953. Tail display in an iguanid lizard,Liocephalus carinatus coryi.Copeia 1, 50–54.CrossRefGoogle Scholar
  34. Evans, H. E. and M. J. W. Eberhard. 1970.The Wasps. Chicago: University of Michigan Press.Google Scholar
  35. Francis, R. C. 1983. Experimental effects on agonistic behaviour in the paradise fish,Macropodus opercularis.Behaviour 85, 292–313.Google Scholar
  36. Franks, N. R. and E. Scovell. 1983. Dominance and reproductive success among slave-making worker ants.Nature 304, 724–725.CrossRefGoogle Scholar
  37. Franks, N. R. and A. B. Sendova-Franks. 1992. Brood sorting by ants: distributing the workload over the work-surface.Behav. Ecol. Sociobiol. 30, 109–123.CrossRefGoogle Scholar
  38. Gervet, J. 1962. Etudes de l'effet de groupe sur la ponte dans la société polygyne dePolistes gallicus L. (hymen. vesp).Ins. Soc. 9, 231–263.CrossRefGoogle Scholar
  39. Gervet, J. 1964. La ponte et sa régulation dans la société polygyne dePolistes gallicus L.Ann. Sci. Nat. Zool. 6, 601–778.Google Scholar
  40. Gervet, J., L. Blanc, M. Pratte and S. Semenoff-Tian-Chansky. 1993. Experimentally induced circular dominance relationships in a polygynous Polistes (Polistes dominulus Christ) wasp colony.Experientia 49, 599–604.CrossRefGoogle Scholar
  41. Ginsburg, B. and W. C. Allee. 1975. Some effects of conditioning on social dominance and subordination in inbred strains of mice. InSocial Hierarchy and Dominance. M. W. Schein (Ed), pp. 282–303. Stroudsburg, PA: Dowden, Hutchinson & Ross.Google Scholar
  42. Guhl, A. M. 1968. Social stability and social inertia in chickens.Anim. Behav. 16, 219–232.CrossRefGoogle Scholar
  43. Haubrich, R. 1961. Hierarchical behaviour in the South African clawed frog,Xenopus laevis Daudin.Anim. Behav. 9, 71–76.CrossRefGoogle Scholar
  44. Heinze, J. 1990. Dominace behaviour among ant females.Naturwissenschaften 77, 41–43.CrossRefGoogle Scholar
  45. Heinze, J., B. Hölldobler and C. Peeters. 1994. Conflict and cooperation in ants societies.Naturwissenschaften 81, 489–497.Google Scholar
  46. Hogeweg, P. and B. Hesper. 1983. The ontogeny of the interaction structure in bumblebee colonies: a MIRROR model.Behav. Ecol. Sociobiol. 12, 271–283.CrossRefGoogle Scholar
  47. Hogeweg, P. and B. Hesper. 1985. Socioinformatic processes: MIRROR modelling methodology.J. Theor. Biol. 113, 311–330.CrossRefGoogle Scholar
  48. Jäger, E. and L. A. Segel. 1992. On the distribution of dominance in populations of social organisms.SIAM J. Appl. Math. 52, 1442–1468.MATHMathSciNetCrossRefGoogle Scholar
  49. Kummer, H. 1968.Social Organization of Hamadrayas Baboons. Chicago: University of Chicago Press.Google Scholar
  50. Landau, H. G. 1951. On dominance relations and the structure of animal societies: I. Effect of inherent characteristics.Bull. Math. Biophys. 13, 1–19.MathSciNetGoogle Scholar
  51. Lowe, M. E. 1956. Dominance-subordinance relationships in the crayfishCambarellus shufeldti.Tulane Studies in Zool.4, 139–170.Google Scholar
  52. Maynard-Smith, J. 1974.J. Theor. Biol. 47, 223.MathSciNetCrossRefGoogle Scholar
  53. Maynard-Smith, J. and G. R. Price. 1973.Nature (London) 246, 15.CrossRefGoogle Scholar
  54. McBride, G. 1958. The measurement of aggressiveness in the domestic hen.Anim. Behav. 6, 87–91.CrossRefGoogle Scholar
  55. Medeiros, F. N. S., L. E. Lopes, P. R. S. Moutinho, P. S. Oliveira and B. Hölldobler. 1992. Functional polygyny, agonistic interactions and reproductive dominance in the neotropical antOdontomachus chelifer (Hymenoptera, Formicidae, Ponerinae).Ethology 91, 134–146.Google Scholar
  56. Mendoza, S. D. and P. R. Barchas. 1983. Behavioral processes leading to linear status hierarchies following group formation in rhesus macaques.J. Hum. Evol. 12, 185–192.CrossRefGoogle Scholar
  57. Nicolis, G. and I. Prigogine. 1977.Self-Organization in Non-Equilibrium Systems. New York: Wiley.Google Scholar
  58. Nicolis, G. and I. Prigogine. 1989.Exploring Complexity: An Introduction. Munich: R. Piper GmbH & Co. KG Verlag.Google Scholar
  59. Oliveira, P. S. and B. Hölldobler. 1990. Dominance orders in the ponerine antPachycondyla apicalis (Hymenoptera, Formicidae).Behav. Ecol. Sociobiol. 27, 385–393.CrossRefGoogle Scholar
  60. Pardi, L. 1942. Ricerche sui Polistini. V. La poliginia initiale diPolistes gallicus (L.)Boll. Ist. Entom. Univ. Bologna 14, 1–106.Google Scholar
  61. Pardi, L. 1946. Richerche sui Polistini. VII. La “dominazione” e il ciclo ovario annuale inPolistes gallicus (L.).Boll. Ist. Entom. Univ. Bologna 15, 25–84.Google Scholar
  62. Pardi, L. 1948. Dominance order inPolistes wasps.Physiol. Zool. 21, 1–13.Google Scholar
  63. Pasteels, J. M., J.-L. Deneubourg and S. Goss. 1987. Self-organization in ant societies (I): trail recruitment to newly discovered food sources. InFrom Individual to Collective Behaviour in Social Insects. J.-M. Pasteels and J.-L. Deneubourg (Eds).Experientia Suppl. 54, 155–175.Google Scholar
  64. Röseler, P.-F., I. Röseler, A. Strambi and R. Augier. 1984. Influence of insect hormones on the establishment of dominance hierarchies among foundresses of the paper waspPolistes gallicus.Behav. Ecol. Sociobiol. 15, 133–142.CrossRefGoogle Scholar
  65. Röseler, P.-F. 1991. Reproductive competition during colony establishment. InThe Social Biology of Wasps. K. G. Ross and R. G. Matthews (Eds.), pp. 309–335. Ithaca, NY: Cornell University Press.Google Scholar
  66. Schein, M. W. and M. H. Fohrman. 1955. Social dominance relationships in a herd of daily cattle.Brit. J. Anim. Behav. 3, 45–55.CrossRefGoogle Scholar
  67. Schjelderup-Ebbe, T. 1913. Hönsenes stemme. Bidrag til hönsenes psykologi.Naturen 37, 262–276.Google Scholar
  68. Schjelderup-Ebbe, T. 1922. Beiträge zur sozialpsychologie des haushuhns.Z. Psychol. 88, 225–252.Google Scholar
  69. Sendova-Franks, A. and N. R. Franks. 1992. Task allocation in ant colonies within variable environments (a study of temporal polyethism).Bull. Math. Biol. 55, 75–96.CrossRefGoogle Scholar
  70. Theraulaz, G., M. Pratte and J. Gervet. 1989. Effects of removal of α-individuals from aPolistes dominulus Christ wasp society: changes in behavioural patterns resulting from hierarchical changes.Actes. Colloq. Ins. Soc. 5, 169–179.Google Scholar
  71. Theraulaz, G., M. Pratte and J. Gervet. 1990. Behavioural profiles inPolistes dominulus (Christ) wasp societies: a quantitative study.Behaviour 113, 223–250.Google Scholar
  72. Theraulaz, G., J. Gervet and S. Semenoff-Tian-Chansky. 1991a. Social regulation of foraging activities inPolistes dominulus Christ: a systemic approach to behavioural organization.Behaviour 116, 292–320.Google Scholar
  73. Theraulaz, G., S. Goss, J. Gervet and J.-L. Deneubourg. 1991b. Task differentiation in Polistes wasp colonies: a model for self-organizing groups of robots. In:From Animals to Animats: Proceedings of the First International Conference on Simulation of Adaptive Behaviour. J.-A. Meyer and S. W. Wilson (Eds.), pp. 346–355. Cambridge, MA: MIT Press.Google Scholar
  74. Theraulaz, G., J. Gervet, B. Thon, M. Pratte and S. Semenoff-Tian-Chansky. 1992. The dynamics of colony organization in the primitively eusocial waspPolistes dominulus (Christ).Ethology 91, 177–202.CrossRefGoogle Scholar
  75. Theraulaz, G., E. Bonabeau and J.-L. Deneubourg. 1995. Self-organization of hierarchies in animal societies: the case of the primitively eusocial waspPolistes dominulus Christ.J. Theor. Biol. 174, 313–323.CrossRefGoogle Scholar
  76. Tyler, S. J. 1972. The behaviour and social organisation of the new forest ponies.Anim. Behav. Monogr. 48, 223–233.Google Scholar
  77. Van de Poll, N. E., F. De Jonge, H. G. Van Oyen and J. Van Pett. 1982. Aggressive behaviour in rats: effects of winning and losing on subsequent aggressive interactions.Behav. Proc. 7, 143–155.CrossRefGoogle Scholar
  78. Van Honk, C. and P. Hogeweg. 1981. The ontogeny of the social structure in a captiveBombus terrestris colony.Behav. Ecol. Sociobiol. 9, 111–119.CrossRefGoogle Scholar
  79. Van Kampen, N. G. 1992.Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland.Google Scholar
  80. Wilson, E. O. 1971.The Insect Societies. Cambridge, MA: Belknap Press.Google Scholar
  81. Wilson, E. O. 1975.Sociobiology: The New Synthesis. Cambridge, MA: Harvard University Press.Google Scholar
  82. Zayan, R. 1994. Special issue on individual recognition: editor's preface.Behav. Proc. 33, 1–2; Mental representations in the recognition of conspecific individuals.Behav. Proc.,33, 233–246.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1996

Authors and Affiliations

  • Eric Bonabeau
    • 1
  • Guy Theraulaz
    • 2
  • Jean-Louis Deneubourg
    • 3
  1. 1.France Telecom-CNET LannionB RIO/TNTLannion CédexFrance
  2. 2.CNRS-URA 664, Centre de Recherches en Biologie du ComportementUniversité Paul SabatierToulouseFrance
  3. 3.Unit of Theoretical Behavioural Ecology, Service de Chimie-PhysiqueUniversité Libre de BruxellesBruxellesBelgique

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