Skip to main content
SpringerLink
Log in

Dynamics of the stochastic low concentration trimolecular chemical reaction model

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

In this paper, we will introduce a low concentration trimolecular stochastic chemical reaction model. We will prove that the solution of the system is positive and global. And then we draw a conclusion that there is a stationary distribution for the stochastic system and it has ergodicity under appropriate conditions. Finally, we test our theory conclusion by simulations. It is interesting that no matter what states the unique equilibrium of ordinary differential equation model appears, and regardless of whether the limit cycle exists, our stochastic model is always ergodic.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. J.J. Tyson, J.C. Light, Properties of two-component bimolecular and trimolecular chemical reaction systems. J. Chem. Phys. 59, 4164 (1973)

    Article  CAS  Google Scholar 

  2. S.A. Kauffman, J.J. Wille, The mitotic oscillator in Physarum polycephalum. J. Theor. Biol. 55, 47–93 (1974)

  3. L.S. Chen, D.D. Wang, Acta Math. Sci. 5, 261–266 (1985)

    CAS  Google Scholar 

  4. I. Prigogine, R. Lefever, Symmetry-breaking instabilities in dissipative systems. J. Chem. Phys. 48, 1665–1700 (1968)

    Article  Google Scholar 

  5. Y. Yang, Y. Zhao, D. Jiang, The dynamics of the stochastic multi-molecule biochemical reaction model. J. Math. Chem. 52, 1477–1495 (2014)

    Article  CAS  Google Scholar 

  6. X. Mao, Stochastic Differential Equations and Applications (Horwood, Chichester, 1997)

    Google Scholar 

  7. A. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71, 876–902 (2011)

    Article  Google Scholar 

  8. N. Dalal, D. Greenhalgh, X. Mao, A stochastic model of AIDS and condom use. J. Math. Anal. 325, 36–53 (2007)

    Article  Google Scholar 

  9. Y. Zhao, D. Jiang, D. O’Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination. Phys. A 392, 4916–4927 (2013)

    Article  Google Scholar 

  10. C. Ji, D. Jiang, N. Shi, Multiple SIR epidemic model with stochastic perturbation. Phys. A 390, 1747–1762 (2011)

    Article  Google Scholar 

  11. R.Z. Hasminskii, Stochastic Stability of Differential Equations (Sijthoff Noordhoff, Alphen aan den Rijn, 1980)

    Book  Google Scholar 

  12. C. Ji, D. Jiang, N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation. Automatica 48, 121–131 (2012)

    Article  Google Scholar 

  13. T.C. Gard, Introduction to Stochastic Differential Equations, vol. 270 (Madison Avenue, New York, 1988)

    Google Scholar 

  14. G. Strang, Linear Algebra and Its Applications (Thomson Learning Inc., London, 1988)

    Google Scholar 

  15. C. Zhu, G. Yin, Asymptotic properties of hybrid diffusion systems. SIAM J. Control Optim. 46, 1155–1179 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Ying Yang & Daqing Jiang

  2. School of Mathematics and Statistics, Jilin Normal University, Siping, 136000, People’s Republic of China

    Ying Yang

  3. College of Science, China University of Petroleum (East China), Qingdao, 266580, China

    Daqing Jiang

Authors
  1. Ying Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daqing Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daqing Jiang.

Additional information

The work was supported by NSFC of China (No. 11371085 ), and the Ph.D. Programs Foundation of Ministry of China (No. 200918).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Y., Jiang, D. Dynamics of the stochastic low concentration trimolecular chemical reaction model. J Math Chem 52, 2532–2545 (2014). https://doi.org/10.1007/s10910-014-0398-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-014-0398-x

Keywords

Search

Navigation