Journal of Mathematical Chemistry

, Volume 42, Issue 4, pp 837–847 | Cite as

Dynamic behaviors of Monod type chemostat model with impulsive perturbation on the nutrient concentration


In this paper, the dynamic behaviors of a Monod type chemostat model with impulsive perturbation are investigated. Using Floquet theory and small amplitude perturbation method, we prove that the microorganism-eradication periodic solution is asymptotically stable if the impulsive period satisfies some conditions. Moreover, the permanence of the system is discussed in detail. Finally, we verify the main results by numerical simulation.


chemostat impulsive input extinction permanence 

AMS subject classification

34K45 34K60 92D25 92D40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Simth H.L., Waltman P., (1995). The Theory of the Chemostat. Cambridge University Press, CambridgeGoogle Scholar
  2. 2.
    L.S. Chen and J. Chen, Nonlinear Biological Dynamic Systems (Science Press, Beijing, 1993) (Chinese).Google Scholar
  3. 3.
    Hsu S.B., Hubbell S.P., Waltman P., (1977) A mathematical theory for single nutrient competition in continuous cultures of micro-organisms. SIAM J. Appl. Math. 32: 366–383CrossRefGoogle Scholar
  4. 4.
    Hale J.K., Somolinas A.S., (1983) Competition for fluctuating nutrient. J. Math. Biol. 18: 255–280CrossRefGoogle Scholar
  5. 5.
    Buler G.J., Hsu S.B., Waltman P., (1985) A mathematical model of the chemostat with periodic washout rate. SIAM J. Appl. Math. 45: 435–449CrossRefGoogle Scholar
  6. 6.
    Wolkowicz G.S.K., Zhao X.Q., (1998) N-spicies competition in a periodic chemostat. Diff. Integr. Eq. 11: 465–491Google Scholar
  7. 7.
    Lakshmikantham V., Bainov D.D., Simeonov P.S., (1989). Theory of Impulsive Differential Equations. World Scientific, SingaporeGoogle Scholar
  8. 8.
    Bainov D.D., Simeonov P.S., (1993). Impulsive Differential Eqations: Periodic Solutions and Applications. Longman Scientific and Technical, Burnt MillGoogle Scholar
  9. 9.
    Liu X.N., Chen L.S., (2003) Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator. Chaos Soliton. Fract. 16: 311–320CrossRefGoogle Scholar
  10. 10.
    Roberts M.G., Kao R.R., (1998) The dynamics of an infectious disease in a population with birth pulses. Math. Biosci. 149: 23–36CrossRefGoogle Scholar
  11. 11.
    Ballinger G., Liu X., (1997) Permanence of population growth models with impulsive effects. Math. Comput. Modell. 26: 59–72CrossRefGoogle Scholar
  12. 12.
    Liu X., Rohlf K., (1998) Impulsive control of a Lotka–Voterra system. IMA J. Math. Control Inform. 15: 269–284CrossRefGoogle Scholar
  13. 13.
    Funasaki E., Kot M, (1993) Invasion and chaos in a periodically pulsed mass-action chemostat. Theor. Popul. Biol. 44: 203–224CrossRefGoogle Scholar
  14. 14.
    Smith R.J., Wolkowicz G.S.K., (2004) Analysis of a model of the nutrient driven self-cycling fermentation process, Dyn. Contin. Discrete Impul. Syst. Ser. B, 11: 239–265Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Applied MathematicsDalian University of TechnologyDalianP.R.China
  2. 2.School of Mathematics and Computer ScienceShanxi Normal UniversityShanxiP.R. China

Personalised recommendations