Global behaviors of Monod type chemostat model with nutrient recycling and impulsive input

Original Paper

Abstract

In this paper, we consider the global behaviors of Monod type chemostat model with nutrient recycling and impulsive input. By introducing a new study method, the sufficient and necessary conditions on the permanence and extinction of the microorganisms are obtained. Furthermore, by using the Liapunov function method, the sufficient condition on the global attractivity of the system is established. Lastly, an example is given, the numerical simulation shows that if only the system is permanent, then it also is globally attractive.

Keywords

Chemostat Nutrient recycling Impulsive input Permanence Extinction Global attractivity 

References

  1. 1.
    Beretta E., Takeuchi Y.: Global stability for chemostat equations with delayed nutrient recycling. Nonlinear World 1, 191–306 (1994)Google Scholar
  2. 2.
    Bulter G.J., Hsu S.B., Waltman P.: A mathematical model of the chemostat with periodic washout rate. SIAM J. Appl. Math. 45, 435–449 (1985)CrossRefGoogle Scholar
  3. 3.
    Freedman H.I., Xu Y.: Models of competition in the chemostat with instantaneous and delayed nutrient recycling. J. Math. Biol. 31, 513–527 (1993)CrossRefGoogle Scholar
  4. 4.
    Hale J.K., Somolinas A.S.: Competition for fluctuating nutrient. J. Math. Biol. 18, 255–280 (1983)CrossRefGoogle Scholar
  5. 5.
    He X., Ruan S., Xia H.: Global stability in chemostat-type equatinos with distributed delays. SIAM J. Math. Anal. 29, 681–696 (1998)CrossRefGoogle Scholar
  6. 6.
    He X., Ruan S.: Global stability in chemostat-type plankton models with delayed nutrient recycling. J. Math. Biol. 37, 253–271 (1998)CrossRefGoogle Scholar
  7. 7.
    Jang S.: Dynamics of variable-yield nutrient-phytoplankton-zooplankton models with nutient recycling and self-shading. J. Math. Biol. 40, 229–250 (2000)CrossRefGoogle Scholar
  8. 8.
    Jiao J., Chen L.: Dynamical analysis of a chemostat model with delayed response in growth and pulse input in polluted environment. J. Math. Chem. 46, 502–513 (2009) doi: 10.1007/s10910-008-9474-4 CrossRefGoogle Scholar
  9. 9.
    Meng X., Zhao Q., Chen L.: Global qualitative analysis of new Monod type chemostat model with delayed growth response and pulsed input in polluted environment. Appl. Math. Mech. 29, 75–87 (2008)CrossRefGoogle Scholar
  10. 10.
    Pang G., Liang Y., Wang F.: Analysis of Monod type food chain chemostat with k-times periodically pulsed input. J. Math. Chem. 43, 1371–1388 (2008)CrossRefGoogle Scholar
  11. 11.
    Pilyugin S., Waltman P.: Competition in the unstirred chemostat with periodic input and washout. SIAM J. Appl. Math. 59, 1157–1177 (1999)CrossRefGoogle Scholar
  12. 12.
    Ruan S.: Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling. J. Math. Biol. 31, 633–654 (1993)CrossRefGoogle Scholar
  13. 13.
    Ruan S.: A three-trophic-level model of plankton dynamics with nutrient recycling. Canad. Appl. Math. Quart. 1, 529–553 (1993)Google Scholar
  14. 14.
    Ruan S.: The effect of delays on stability and persistence in plankton models. Nonlinear Anal. 24, 575–585 (1995)CrossRefGoogle Scholar
  15. 15.
    Ruan S., He X.: Global stability in chemostat-type competition models with nutrient recycling. SIAM J. Appl. Math. 58, 170–192 (1998)CrossRefGoogle Scholar
  16. 16.
    Smith H.L.: Competitive coexistence in an oscillating chemostat chemostat. SIAM J. Appl. Math. 40, 498–522 (1981)CrossRefGoogle Scholar
  17. 17.
    Simth H.L., Waltman P.: The theory of the chemostat. Cambrige University Press, Cambridge (1995)CrossRefGoogle Scholar
  18. 18.
    Sree Hari Rao V., Raja Sekhara Rao P.: Global stability in chemostat models involving time delays and wall growth. Nonlinear Anal RWA 5, 141–158 (2004)Google Scholar
  19. 19.
    Sun S., Chen L., Sun S.: Dynamic behaviors of Monod type chemostat model with impulsive perturbation on the nutrient concentration. J. Math. Chem. 42, 837–847 (2007)CrossRefGoogle Scholar
  20. 20.
    Wang F., Hao C., Chen L.: Bifurcation and chaos in a Tessiet type food chain chemostat with pulsed input and washout. Chaos Solitons Fractals 32, 1547–1561 (2007)CrossRefGoogle Scholar
  21. 21.
    Wang F., Pang G.: Competition in a chemostat with Beddington-DEAngelis growth rates and periodic pulsed nutrient. J. Math. Chem. 44, 691–710 (2008)CrossRefGoogle Scholar
  22. 22.
    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)CrossRefGoogle Scholar
  23. 23.
    Xiang Z., Song X.: A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with periodic input. Chaos Solitons Fractals 32, 1419–1428 (2007)CrossRefGoogle Scholar
  24. 24.
    Zhang S., Tan D.: Study of a chemostat model with Beddington-DeAngelis functional response and pulsed input and washout at different times. J. Math. Chem. 44, 217–227 (2008)CrossRefGoogle Scholar
  25. 25.
    Zhao Z., Chen L., Song X.: Extinction and permanence of chemostat model with pulsed input in a polluted environment. Commun. Nonlinear Sci. Numer. Simul. 14, 1737–1745 (2009)CrossRefGoogle Scholar
  26. 26.
    Zhou X., Song X., Shi X.: Analysis of competitive chemostat models with the Beddington-DeAngelis functional response and impulsive effect. Appl. Math. Model. 31, 2299–2312 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Zhidong Teng
    • 1
  • Rong Gao
    • 1
  • Mehbuba Rehim
    • 1
  • Kai Wang
    • 1
  1. 1.College of Mathematics and System SciencesXinjiang UniversityUrumqiPeople’s Republic of China

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