Bulletin of Mathematical Biology

, Volume 53, Issue 3, pp 469–485 | Cite as

Small autocatalytic reaction networks—III. Monotone growth functions

  • Bärbel M. R. Stadler
  • Peter F. Stadler
Article
  • 54 Downloads

Abstract

The classification of the dynamical behaviour of first order replicator equations is extended to models with monotonical growth rates. It is shown that for two species there is a general classification independent of the particular form of the growth function. For three species a common dynamical behaviour for all power laws can be found and the existence of limit cycles is disproved. For more general growth functions, however, limit cycles may occur.

Keywords

Periodic Orbit Hopf Bifurcation Phase Portrait Growth Function Evolutionary Stable Strategy 

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Literature

  1. Andronov, W. W., E. Leontovich, I. Gordon and A. Maier. 1973.Qualitative Theory of Second Order Dynamic Systems. Halsted Press, New York.MATHGoogle Scholar
  2. Biebricher, C. K., M. Eigen and W. C. Gardiner. 1983. Kinetics of RNA replication.Biochemistry 22, 2544–2559.CrossRefGoogle Scholar
  3. Biebricher, C. K., M. Eigen and W. C. Gardiner. 1984. Kinetics of RNA replication: plus-minus asymmetry and double strand formation.Biochemistry 23, 3186–3194.CrossRefGoogle Scholar
  4. Biebricher, C. K., M. Eigen and W. C. Gardiner. 1985. Kinetics of RNA replication: Competition and selection among self-replicating RNA species.Biochemistry 24, 6550–6560.CrossRefGoogle Scholar
  5. Biebricher, C. K. and M. Eigen. 1988. Kinetics of RNA Replication byQβ-replicase. In:RNA Genetics (Vol. 1), E. Domingo, J. J. Holland and P. Ahlquist (eds), pp. 1–22. CRC Press, Boca Raton, Florida.Google Scholar
  6. Bomze, I. M. 1983. Lotka Volterra equation and replicator dynamics. A two-dimensional classification.Biol. Cybern. 48, 201–211.MATHCrossRefGoogle Scholar
  7. Cech, T. R. 1986. RNA as an enzyme.Sci. Am. 255(5), 76–84.CrossRefGoogle Scholar
  8. Cech, T. R. 1988. Conserved sequences and structures of group I introns: building an active site for RNA catalysis.Gene 73, 259–271.CrossRefGoogle Scholar
  9. Coppel, W. 1966. A survey of quadratic systems.J. Diff. Eqns. 2, 293–304.MATHMathSciNetCrossRefGoogle Scholar
  10. Doudna, J. A. and J. W. Szostak. 1989. RNA catalysed synthesis of complementary-strand RNA.Nature 339, 519–522.CrossRefGoogle Scholar
  11. Eigen, M. 1971. Self-organization of matter and the evolution of biological macromolecules.Naturwissenschaften 58, 465–523.CrossRefGoogle Scholar
  12. Eigen, M. and P. Schuster. 1978. The Hypercycle. A principle of natural self-organisation. Part B: The abstract hypercycle.Naturwissenschaften 65, 7–41.CrossRefGoogle Scholar
  13. Guckenheimer, J. and P. Holmes, 1986. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields.Applied Mathematical Sciences Vol. 42. Berlin: Springer.Google Scholar
  14. Hofbauer, J., P. Schuster and K. Sigmund. 1979. A note on evolutionary stable strategies and game dynamics.J. theor. Biol. 81, 609–612.MathSciNetCrossRefGoogle Scholar
  15. Hofbauer, J., P. Schuster, K. Sigmund and R. Wolff. 1980. Dynamical systems under constant organization II: homogeneous growth functions of degreep=2.SIAM J. appl. Math. 38, 282–304.MATHMathSciNetCrossRefGoogle Scholar
  16. Hofbauer, J. 1981. On the occurrence of limit cycles in the Lotka-Volterra equation.Nonlinear Analysis 5, 1003–1007.MATHMathSciNetCrossRefGoogle Scholar
  17. Hofbauer, J. and K. Sigmund. 1988.The Theory of Evolution and Dynamical Systems. Cambridge University Press.Google Scholar
  18. Maynard-Smith, J. 1982.Evolution and the Theory of Games. Cambridge University Press.Google Scholar
  19. Perelson, A. S. (ed.) 1988.Theoretical Immunology. Vols. I and II. Redwood City, CA: Addison-Wesley.Google Scholar
  20. Schuster, P., K. Sigmund and R. Wolff. 1978. Dynamical systems under constant organisation—I. Topological analysis of a family of non-linear differential equations—a model for catalytic hypercycles.Bull. math. Biol. 40, 743–769.MathSciNetCrossRefGoogle Scholar
  21. Schuster, P., K. Sigmund and R. Wolff. 1981. Mass action kinetics of self-replication in flow reactors.J. Math. Anal. Appl. 78, 88–112.MathSciNetCrossRefGoogle Scholar
  22. Schuster, P. 1981. Selection and Evolution in Molecular Systems. InNonlinear Phenomena in Physics and Biology, R. H. Enns, B. L. Jones, R. M. Mimura and S. S. Rangnekar (eds), pp. 485–543. New York: Plenum Press.Google Scholar
  23. Schuster, P. and K. Sigmund. 1983. Replicator Dynamics.J. theor. Biol. 100, 533–538.MathSciNetCrossRefGoogle Scholar
  24. Schuster, P. and K. Sigmund. 1985. Towards a dynamics of social behaviour: strategic and genetic models for the evolution of animal conflicts.J. social. biol. Struct. 8, 255–277.CrossRefGoogle Scholar
  25. Stadler, P. F. and P. Schuster. 1990. Dynamics of small autocatalytic reaction networks—I. Bifurcations, permanence and exclusion.Bull Math. Biol. 52, 485–508.MATHGoogle Scholar
  26. Taylor, P. D. and L. B. Jonker. 1978. Evolutionary stable strategies and game dynamics.Math. Biosci. 40, 145–156.MATHMathSciNetCrossRefGoogle Scholar
  27. Weissmann Ch. 1974. The making of a phage.FEBS Letters 40, S10-S18.CrossRefGoogle Scholar
  28. Zeeman, E. C. 1981. Dynamics of the evolution of animal conflicts.J. theor. Biol. 89, 249–270.MathSciNetCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1991

Authors and Affiliations

  • Bärbel M. R. Stadler
    • 1
  • Peter F. Stadler
    • 1
  1. 1.Institut für theoretische Chemie der Universität WienWienAustria

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