Skip to main content
SpringerLink
Log in

Study of a Monod–Haldene type food chain chemostat with pulsed substrate

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

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.

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

References

  1. Hale J.K., Somolinos A.S.,(1983). J. Math. Biol. 18, 255–280

    Article  Google Scholar 

  2. Hsu S.B., (1980). J. Math. Biol. 18: 115–132

    Article  Google Scholar 

  3. Alessandra G., Oscar D.F., Sergio R., (1998). Bull. Math. Biol. 60, 703–719

    Article  Google Scholar 

  4. Mark K., Sayler G.S., Waltman T.W., (1992). Bull. Math. Biol. 54, 619–648

    Google Scholar 

  5. Funaski E., Kor M., (1993). Theoret. Popul. Biol. 44, 203–224

    Article  Google Scholar 

  6. Butler G.J., Hsu S.B., Waltman P., (1985). SIAM J. Appl. Math. 45, 435–449

    Article  Google Scholar 

  7. Lenas P., Pavlou S., (1995). Math. Biosci. 129, 111–142

    Article  CAS  Google Scholar 

  8. DeAnglis D.L., Goldstein R.A., O’Neill R.V., (1975). Ecology 56, 881–892

    Article  Google Scholar 

  9. Chen L., Chen J., (1993). Nonlinear Biodynamical Systems (Chinese copies). Science Publisher, Mascow

    Google Scholar 

  10. Bainov D., Simeonor P., (1993). Pitman Monogr Surreys Pure Appl. Math. 66, 22–23

    Google Scholar 

  11. Laksmikantham V., Bainov D.D., Simeonov P.S., (1989). Theory Of Impulsive Differential Equations. World Scientific, Singapore

    Google Scholar 

  12. Tang S.Y., Chen L.S., (2002). J. Math. Biol. 44, 185–199

    Article  Google Scholar 

  13. Ballinger G., Liu X., (1997). Math. Compute. Mode. 26, 59–72

    Article  Google Scholar 

  14. Shulgin B., Stone L., Agur I., (1998). Bull. Math. Biol. 60, 1–26

    Article  Google Scholar 

  15. D’Onofrio A., (2002). Bull. Math. Biol. 60, 1–26

    Google Scholar 

  16. Panetta J.C., (1996). Bull Math. Biol. 58, 425–447

    Article  CAS  Google Scholar 

  17. Wang F., Zhang S., Chen L., Sun L., (2005). Nonlinear J. Sci. Num. Simul. 6(2): 169–180

    Google Scholar 

  18. Cushing J.M., (1977). SIAM J. Appl. Math.10, 384–400

    Google Scholar 

  19. May R.M., (1974). Science 186: 645–647

    Article  CAS  Google Scholar 

  20. Gakkhar S., Naji M.A.,(2003). Chaos Solitons Fractal 18, 229–239

    Article  Google Scholar 

  21. Klebanoff A., Hastings A., (1994). J. Math. Biol. 32, 427–451

    Article  Google Scholar 

  22. Neubert M.G., Caswell H., (2000). J. Math. Biol. 41, 103–121

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information and Computer Science, North University for Minorities, Yinchuan, Ningxia, 750021, People’s Republic of China

    Fengyan Wang

  2. College of Science, Jimei University Xiamen, Fujian, 361021, People’s Republic of China

    Fengyan Wang

  3. Department of Mathematics and Computer Science, Yulin Normal University, Yulin, Guangxi, 537000, People’s Republic of China

    Guoping Pang

  4. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Lansun Chen

Authors
  1. Fengyan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoping Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lansun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fengyan Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, F., Pang, G. & Chen, L. Study of a Monod–Haldene type food chain chemostat with pulsed substrate. J Math Chem 43, 210–226 (2008). https://doi.org/10.1007/s10910-006-9189-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-006-9189-3

Keywords

Search

Navigation