Skip to main content
SpringerLink
Log in

Invariant probabilities for systems in a random environment—With applications to the Brusselator

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

Abstract

In this paper we are concerned with problems of the long-term behavior for nonlinear systems in random environment. The general model is assumed to be given by an ordinary differential equation with random parameters or random input. The disturbance process can be taken from a fairly general class of Markov processes having a bounded state space. In terms of the system’s dynamics we give sufficient conditions for the existence and uniqueness of invariant probabilities. Finally, we apply these results to the two-dimensional biochemical model which is known as the Brusselator.

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

  • Arnold, L. and W. Kliemann. 1983. “Qualitative Theory of Stochastic Systems.” Report, Universität Bremen. In:Probabilistic Analysis and Related Topics, Ed. A. T. Bharucha-Reid, Vol. 3, in press.

  • —, W. Horsthemke and R. Lefever. 1978. “White and Coloured External Noise and and Transition Phenomena in Nonlinear Systems.”Z. Phys. (B) 29, 367–373.

    Google Scholar 

  • Has’minskii, R. Z. 1980.Stochastic Stability of Differential Equations. Alphen ann den Rijn: Sijthoff and Noordhoff.

    Google Scholar 

  • Kepper, P. de and W. Horsthemke. 1979. “Experimental Evidence of Noise Induced Transitions in Open Chemical Systems.” In:Synergetics—Far from Equilibrium, Ed. A. Pacault and C. Vidal, pp. 61–63. Berlin: Springer-Verlag.

    Google Scholar 

  • Kliemann, W. 1983. “Qualitative Theory of Stochastic Dynamical Systems—Applications to Life Sciences.”Bull. math. Biol. 45, 483–506.

    Article  MATH  MathSciNet  Google Scholar 

  • Lefever, R. and G. Nicolis. 1971. “Chemical Instabilities and Sustained Oscillations.”J. theor. Biol. 30, 267–284.

    Article  Google Scholar 

  • Nicolis, G. and I. Prigogine. 1977.Self-organization in Nonequilibrium Systems. New York: Wiley and Sons.

    Google Scholar 

  • Ponzo, P. J. and N. Wax. 1978. “Note on a Model of Biochemical Reaction.”J. math. Analysis Applic. 66, 354–357.

    Article  MATH  MathSciNet  Google Scholar 

  • Prigogine, I. and P. Glansdorff, 1971.Thermodynamic Theory of Structure, Stability and Fluctuations. New York: Wiley-Interscience.

    Google Scholar 

  • Sussmann H. J. and V. Jurdjevic. 1972. “Controllability of Nonlinear Systems.”J. diff. Equations 12, 95–116.

    Article  MATH  MathSciNet  Google Scholar 

  • Tyson, J. J. 1973. “Some Further Studies of Nonlinear Oscillations in Chemical Systems.”J. chem. Phys. 58, 3919–3930.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, Bibliothekstraße, Postfach 330 440, 2800, Bremen 33, West Germany

    Manfred Ehrhardt

Authors
  1. Manfred Ehrhardt
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work is part of a research project supported by the ‘Stiftung Volkswagenwerk’.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ehrhardt, M. Invariant probabilities for systems in a random environment—With applications to the Brusselator. Bltn Mathcal Biology 45, 579–590 (1983). https://doi.org/10.1007/BF02459589

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02459589

Keywords

Search

Navigation