Lotka Volterra coevolution at the edge of chaos

  • Paul Bourgine
  • Dominique Snyers
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1063)


In this paper, we study the coevolution of species by combining a theoretical approach with a computer simulation in order to show how a discrete distribution of viable species emerges. Coevolution is modelled as a replicator system which, with an additional diffusion term representing the mutation, leads to a Schrödinger equation. This system dynamics can be interpreted as a survival race between species on a multimodal sinking and drifting landscape whose modes correspond to the eigen modes of the Schrödinger equation. This coevolution dynamics is further illustrated by a simulation based on a continuous phenotypic model due to Kaneko in which the interactions between species are interpreted through a Lotka-Volterra model. This simulated coevolution is seen to converge to viable species associated with a dynamics at the edge of chaos (i.e with a null Lyapounov exponent). The transition from such a viable species to another results from some kind of tunnel effects characteristic of the punctuated equilibrium classically observed in biology in which rapid changes in the species distribution follow long plateaus of stable distribution.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.P. Aubin. Viability Theory. Birkhauser, 1991.Google Scholar
  2. 2.
    P. Bak. Self-organized criticality. Physica A, 1990.Google Scholar
  3. 3.
    P. Bak and K. Chen. Self-organized criticality. Scientific American, pages 26–33, January 1991.Google Scholar
  4. 4.
    P. Bak, H. Flyvbjerg, and B. Lautrup. Coevolution in a rugged fitness landscape. Phys. rev A, 46:6724–6730, November 1992.PubMedGoogle Scholar
  5. 5.
    R. Feistel and W. Ebeling. Evolution of Complex Systems: Selforganization, Entropy and Development. Kluwer Academic,Dordrecht, 1989.Google Scholar
  6. 6.
    F.Hoffmeister and T. Báck. Genetic algorithms and evolution strategies: Similarities and differences. Technical report, University of Dortmund, 1992.Google Scholar
  7. 7.
    H. Flyvbjerg and B. Lautrup. Evolution in a rugged fitness landscape. Phys. rev A, 46:6714–6723, November 1992.PubMedGoogle Scholar
  8. 8.
    K. Kaneko and J. Suzuki. Evolution to the edge of chaos in an imitation game. In Artificial Life III, 1994.Google Scholar
  9. 9.
    S. A. Kauffman and S. Johnsen. Co-evolution to the edge of chaos: coupled fitness landscape, poised states and co-evolutionary avalanches. In J. D. Farmer C. G. Langton, C. Taylor and S. Rasmussen, editors, Artificial Life II, Santa Fe, New Mexico, 1991. Addison Wesley.Google Scholar
  10. 10.
    S.G. Mikhlin. Integral equations. Pergamon Press, 1957.Google Scholar
  11. 11.
    H.-O. Peitgen, H. Jürgens, and D. Saupe. Chaos and Fractals: New Frontiers of Science. Springer-Verlag, New York, 1992.Google Scholar
  12. 12.
    M. Peschel and W. Mende. The Predator-Prey Model. Springer-Verlag, Wien, 1986.Google Scholar
  13. 13.
    H. G. Schuster. Deterministic Chaos: An Introduction. VCH Verlagsgesellschaft, 1989.Google Scholar
  14. 14.
    V. Volterra. Leçons sur la théorie mathématique de la lutte pour la vie. Herrmann & Cie, Paris, 1931.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Paul Bourgine
    • 1
  • Dominique Snyers
    • 2
  1. 1.Ecole PolytechniqueCREAParisFrance
  2. 2.Télécom BretagneLIASCBrest CédexFrance

Personalised recommendations