Advertisement

Permanence for Replicator Equations

  • J. Hofbauer
  • K. Sigmund
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 287)

Abstract

Many dynamical systems display strange attractors and hence orbits that are so sensitive to initial conditions as to make any long-term prediction (except on a statistical basis) a hopeless task. Such a lack of Ljapunov stability is not always crucial, however: Lagrange stability may be more relevant. Thus, for some models the precise asymptotic behavior — whether it settles down to an equilibrium or keeps oscillating in a regular or irregular fashion — is less important than the fact that all orbits wind up in some preassigned bounded set. The former problem can be impossibly hard to solve and the latter one easy to handle.

Keywords

Nash Equilibrium Interior Equilibrium Rest Point Replicator Equation Lagrange Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amann, E. (1986), Permanence of Catalytic Networks (to appear).Google Scholar
  2. Amann, E. and Hofbauer, J. (1985), Permanence in Lotka-Volterra and replicator equations, in E. Ebeling and M. Peschel (Eds), Lotka-Volterra Approaches to Cooperation and Competition in Dynamic Systems (Akademie Verlag, Berlin).Google Scholar
  3. Butler, G. and Waltman, P.E. (1984), Persistence in Three-Dimensional Lotka-Volterra Systems (to appear).Google Scholar
  4. Butler, G., Freedman, H.I., and Waltman, P.E. (1985), Uniformity persistent systems, to appear in Proceedings AMS.Google Scholar
  5. Freedman, H.I. and Waltman, P.E. (1985), Persistence in a model of three competitive populations, Math. Biosciences, 73, 89–101.CrossRefGoogle Scholar
  6. Hallam, T., Svoboda, L., and Gard, T. (1979), Persistence and extinction in three species Lotka-Volterra competitive systems, Math. Biosciences, 46, 117–124.CrossRefGoogle Scholar
  7. Hofbauer, J. (1981a), On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Analysis, TMA, 5, 1003–1007.CrossRefGoogle Scholar
  8. Hofbauer, J. (1981b), A general cooperation theorem for hypercycles, Monatsh. Math., 91, 233–240.CrossRefGoogle Scholar
  9. Hofbauer, J. (1984), A difference equation model for the hypercycle, SIAM J. Appl. Math., 44, 762–772.CrossRefGoogle Scholar
  10. Hofbauer, J. (1986), Permanence and Persistence of Lotka-Volterra Systems (to appear).Google Scholar
  11. Hofbauer, J. and Sigmund, K. (1984), Evolutionstheorie und dynamische Systeme (Paul Parey Verlag, Hamburg, Berlin).Google Scholar
  12. Hofbauer, J.f Schuster, P., and Sigmund, K. (1981), Competition and cooperation in catalytic self-replication, J. Math. Biol., 11, 155–168.CrossRefGoogle Scholar
  13. Hutson, V. (1984), Predator mediated coexistence with a switching predator, Math. Biosci., 63, 293–269.Google Scholar
  14. Hutson, V. (1986), A theorem on average Ljapunov functions, Monatsh. Math., 98, 267–275.CrossRefGoogle Scholar
  15. Hutson, V. and Law, R. (1985), Permanent coexistence in general models of three interacting species, J. Math. Biol., 21, 289–298.CrossRefGoogle Scholar
  16. Hutson, V. and Moran, W. (1982), Persistence of species obeying difference equations, 3, Math. Biol., 15, 203–213.CrossRefGoogle Scholar
  17. Hutson, V. and Vickers, G.T. (1983), A criterion for permanent co-existence of species, with an application to a two prey-one predator system, Math. Biosciences, 63, 253–269.CrossRefGoogle Scholar
  18. Jansen, W. (1986), A Permanence Theorem on Replicator Systems (to appear).Google Scholar
  19. Kirlinger, G. (1986), Permanence in Lotka-Volterra equations: linked predator prey systems (to appear).Google Scholar
  20. Schuster, P. and Sigmund, K. (1983), Replicator dynamics, J. Math. Biol., 100, 533–538.Google Scholar
  21. Schuster, P. and Sigmund, K. (1984), Permanence and uninvadability for deterministic population models, in P. Schuster (Ed), Stochastic Phenomena and Chaotic Behaviour in Complex Systems, Synergetics 21 (Springer, Berlin, Heidelberg, New York).Google Scholar
  22. Schuster, P., Sigmund, K., and Wolff, R. (1979), Dynamical systems under constant organization 3: Cooperative and competitive behaviour of hypercycles, J. Diff. Eqs., 32, 357–368.CrossRefGoogle Scholar
  23. Sieveking, G. (1983), Unpublished lectures on dynamical systems.Google Scholar
  24. Sigmund, K. (1985), A survey on replicator equations, in J. Casti and A. Karlquist (Eds), Complexity, Language and Life: Mathematical Approaches, Biomathematics 16 (Springer, Berlin, Heidelberg, New York).Google Scholar
  25. Svirezev, Y.M. and Logofet, D.D. (1983), Stability of Biological Communities (Mir Publishers, Moscow).Google Scholar
  26. Taylor, P. and Jonker, L. (1978), Evolutionarily stable strategies and game dynamics, Math. Bioscience, 40, 145–156.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • J. Hofbauer
    • 1
  • K. Sigmund
    • 2
  1. 1.Institute for MathematicsUniversity of ViennaAustria
  2. 2.International Institute for Applied Systems AnalysisLaxenburgAustria

Personalised recommendations