Skip to main content
Log in

Mathematical model of self-organizing hierarchies in animal societies

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Alexander, R. D. 1961. Aggressiveness, territoriality, and sexual behaviour in field crickets (Orthoptera: Gryllidae).Behaviour 17, 130–223.

    Google Scholar 

  • 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 

  • Allee, W. C. 1951.Cooperation Among Animals. New York: Henry Schulman.

    Google Scholar 

  • 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 

  • Baldwin, J. D. 1971. The social organization of a semifree-ranging troop of squirrel monkeys (Saimiri sciureus).Folia Primatologica 14, 23–50.

    Article  Google Scholar 

  • 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 

  • Barnard, C. J. and T. E. Burk. 1979. Dominance hierarchies and the evolution of “individual recognition”.J. Theor. Biol. 81, 65–73.

    Article  Google Scholar 

  • 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 

  • Beaugrand, J. P. and R. C. Zayan. 1984 An experimental model of aggressive dominance inXiphophorus helleri (Pisces: Poeciliidae).Behav. Proc. 10, 1–53.

    Article  Google Scholar 

  • 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 

  • Boice, R. and D. W. Witter. 1969. Hierarchical feeding behaviour in the leopard frog (Rana pipiens).Anim. Behav. 17, 474–479.

    Article  Google Scholar 

  • Bonabeau, E., G. Theraulaz and J.-L. Deneubourg. 1995. Phase diagram of a model of self-organizing hierarchies.Physica A 217, 373–392.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Bovbjerg, R. V. 1956. Some factors affecting aggressive behaviour in crayfish.Physiol. Zool. 29, 127–136.

    Google Scholar 

  • 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 

  • Broom, D. M. 1981.Biology of Behaviour. Cambridge: Cambridge University Press.

    Google Scholar 

  • Burk, T. E. 1979. An analysis of social behaviour in crickets. Ph.D. thesis, University of Oxford.

  • 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 

  • Chase, I. D. 1974. Models of hierarchy formation in animal societies.Behav. Sci. 19, 374–382.

    Google Scholar 

  • Chase, I. D. 1980. Social process and hierarchy formation in small groups: a comparative perspective.Am. Sociol. Rev. 45, 905–924.

    Article  Google Scholar 

  • Chase, I. D. 1982a. Dynamics of hierarchy formation: the sequential development of dominance relationships.Behaviour 80, 218–240.

    Google Scholar 

  • Chase, I. D. 1982b. Behavioral sequences during dominance hierarchy formation in chickens.Science 216, 439–440.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Chase, I. D. 1986. Explanations of hierarchy structure.Anim. Behav. 34, 1265–1267.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Cole, B. J. 1981. Dominance hierarchies inLeptothorax ants.Science 212, 83–84.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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 

  • 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 

  • 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 

  • Evans, L. T. 1951. Field study of the social behaviour of the black lizard,Ctenosaura pectinata. Am. Museum Novitates 1943.

  • Evans, L. T. 1953. Tail display in an iguanid lizard,Liocephalus carinatus coryi.Copeia 1, 50–54.

    Article  Google Scholar 

  • Evans, H. E. and M. J. W. Eberhard. 1970.The Wasps. Chicago: University of Michigan Press.

    Google Scholar 

  • Francis, R. C. 1983. Experimental effects on agonistic behaviour in the paradise fish,Macropodus opercularis.Behaviour 85, 292–313.

    Google Scholar 

  • Franks, N. R. and E. Scovell. 1983. Dominance and reproductive success among slave-making worker ants.Nature 304, 724–725.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Gervet, J. 1964. La ponte et sa régulation dans la société polygyne dePolistes gallicus L.Ann. Sci. Nat. Zool. 6, 601–778.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Guhl, A. M. 1968. Social stability and social inertia in chickens.Anim. Behav. 16, 219–232.

    Article  Google Scholar 

  • Haubrich, R. 1961. Hierarchical behaviour in the South African clawed frog,Xenopus laevis Daudin.Anim. Behav. 9, 71–76.

    Article  Google Scholar 

  • Heinze, J. 1990. Dominace behaviour among ant females.Naturwissenschaften 77, 41–43.

    Article  Google Scholar 

  • Heinze, J., B. Hölldobler and C. Peeters. 1994. Conflict and cooperation in ants societies.Naturwissenschaften 81, 489–497.

    Google Scholar 

  • Hogeweg, P. and B. Hesper. 1983. The ontogeny of the interaction structure in bumblebee colonies: a MIRROR model.Behav. Ecol. Sociobiol. 12, 271–283.

    Article  Google Scholar 

  • Hogeweg, P. and B. Hesper. 1985. Socioinformatic processes: MIRROR modelling methodology.J. Theor. Biol. 113, 311–330.

    Article  Google Scholar 

  • Jäger, E. and L. A. Segel. 1992. On the distribution of dominance in populations of social organisms.SIAM J. Appl. Math. 52, 1442–1468.

    Article  MATH  MathSciNet  Google Scholar 

  • Kummer, H. 1968.Social Organization of Hamadrayas Baboons. Chicago: University of Chicago Press.

    Google Scholar 

  • Landau, H. G. 1951. On dominance relations and the structure of animal societies: I. Effect of inherent characteristics.Bull. Math. Biophys. 13, 1–19.

    MathSciNet  Google Scholar 

  • Lowe, M. E. 1956. Dominance-subordinance relationships in the crayfishCambarellus shufeldti.Tulane Studies in Zool.4, 139–170.

    Google Scholar 

  • Maynard-Smith, J. 1974.J. Theor. Biol. 47, 223.

    Article  MathSciNet  Google Scholar 

  • Maynard-Smith, J. and G. R. Price. 1973.Nature (London) 246, 15.

    Article  Google Scholar 

  • McBride, G. 1958. The measurement of aggressiveness in the domestic hen.Anim. Behav. 6, 87–91.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Nicolis, G. and I. Prigogine. 1977.Self-Organization in Non-Equilibrium Systems. New York: Wiley.

    Google Scholar 

  • Nicolis, G. and I. Prigogine. 1989.Exploring Complexity: An Introduction. Munich: R. Piper GmbH & Co. KG Verlag.

    Google Scholar 

  • Oliveira, P. S. and B. Hölldobler. 1990. Dominance orders in the ponerine antPachycondyla apicalis (Hymenoptera, Formicidae).Behav. Ecol. Sociobiol. 27, 385–393.

    Article  Google Scholar 

  • Pardi, L. 1942. Ricerche sui Polistini. V. La poliginia initiale diPolistes gallicus (L.)Boll. Ist. Entom. Univ. Bologna 14, 1–106.

    Google Scholar 

  • 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 

  • Pardi, L. 1948. Dominance order inPolistes wasps.Physiol. Zool. 21, 1–13.

    Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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 

  • Schein, M. W. and M. H. Fohrman. 1955. Social dominance relationships in a herd of daily cattle.Brit. J. Anim. Behav. 3, 45–55.

    Article  Google Scholar 

  • Schjelderup-Ebbe, T. 1913. Hönsenes stemme. Bidrag til hönsenes psykologi.Naturen 37, 262–276.

    Google Scholar 

  • Schjelderup-Ebbe, T. 1922. Beiträge zur sozialpsychologie des haushuhns.Z. Psychol. 88, 225–252.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Theraulaz, G., M. Pratte and J. Gervet. 1990. Behavioural profiles inPolistes dominulus (Christ) wasp societies: a quantitative study.Behaviour 113, 223–250.

    Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Tyler, S. J. 1972. The behaviour and social organisation of the new forest ponies.Anim. Behav. Monogr. 48, 223–233.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Van Honk, C. and P. Hogeweg. 1981. The ontogeny of the social structure in a captiveBombus terrestris colony.Behav. Ecol. Sociobiol. 9, 111–119.

    Article  Google Scholar 

  • Van Kampen, N. G. 1992.Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland.

    Google Scholar 

  • Wilson, E. O. 1971.The Insect Societies. Cambridge, MA: Belknap Press.

    Google Scholar 

  • Wilson, E. O. 1975.Sociobiology: The New Synthesis. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bonabeau, E., Theraulaz, G. & Deneubourg, JL. Mathematical model of self-organizing hierarchies in animal societies. Bltn Mathcal Biology 58, 661–717 (1996). https://doi.org/10.1007/BF02459478

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02459478

Keywords

Navigation