Skip to main content
SpringerLink
Log in

Dynamical systems under constant organization I. Topological analysis of a family of non-linear differential equations —A model for catalytic hypercycles

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

Abstract

The paper presents a qualitative analysis of the following systems ofn differential equations:\(\dot x_i = x_i x_j - x_i \sum\nolimits_r^n { = 1} x_r x_s {\mathbf{ }}(j = i - 1 + n\delta _{i1} {\mathbf{ }}and{\mathbf{ }}s = r - 1 + n\delta _{r1} )\), which show cyclic symmetry. These dynamical systems are of particular interest in the theory of selforganization and biological evolution as well as for application to other fields.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Coddington, E. A. and N. Levinson. 1955.Theory of Ordinary Differential Equations, pp. 371–388. New York: McGraw-Hill.

    Google Scholar 

  • Crow, J. F. and M. Kimura. 1970.An Introduction to Population Genetics Theory. Harper & Row.

  • Eigen, M. 1971. “Selforganization of Matter and the Evolution of Biological Macromolecules.”Die Naturwissenschaften,58, 465–523.

    Article  Google Scholar 

  • Eigen, M. and P. Schuster. 1977. “The Catalytic Hypercycle—a Principle of Precellular Organization.”Die Naturwissenschaften,64, 541–565.

    Article  Google Scholar 

  • Hadeler, K. P. 1974.Mathematik für Biologen, pp. 96–97, 147–149. Berlin: Springer.

    Google Scholar 

  • Jones, B. L. 1977. “A Solvable Selfproductive Hypercycle Model for the Selection of Biological Molecules.”J. Math. Biol.,4, 187–193.

    Article  MATH  Google Scholar 

  • Marsden, J. E. and M. McCracken. 1976. “The Hopf Bifurcation and its Applications.”Springer Series in Applied Mathematical Sciences 19. New York: Springer.

    Google Scholar 

  • McWeeny, R. 1963.Symmetry; pp. 96–100. Oxford: Pergamon Press.

    Google Scholar 

  • Nemytskii, V. V. and V. V. Stepanov. 1960. “Qualitative Theory of Differential Equations.”Princeton Mathematical Series 22. Princeton University Press.

  • Schuster, P., K. Sigmund and R. Wolff. 1978. “Dynamical Systems Under Constant Organization III: Cooperative and Competitive Behaviour of Hypercycles.” Submitted for publication.

Download references

Author information

Authors and Affiliations

  1. Institute of Theoretical and Radiation Chemistry, University of Vienna, Austria

    P. Schuster, K. Sigmund & R. Wolff

  2. Mathematical Institute, University of Vienna, Austria

    P. Schuster, K. Sigmund & R. Wolff

Authors
  1. P. Schuster
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Sigmund
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Wolff
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work has been supported financially by the Austrian Fonds zur Förderung der Wissenschaftlichen Forschung. Project N. 2261.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schuster, P., Sigmund, K. & Wolff, R. Dynamical systems under constant organization I. Topological analysis of a family of non-linear differential equations —A model for catalytic hypercycles. Bltn Mathcal Biology 40, 743–769 (1978). https://doi.org/10.1007/BF02460605

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02460605

Keywords

Search

Navigation