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

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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.

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References

  1. Beretta E., Takeuchi Y.: Global stability for chemostat equations with delayed nutrient recycling. Nonlinear World 1, 191–306 (1994)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  4. Hale J.K., Somolinas A.S.: Competition for fluctuating nutrient. J. Math. Biol. 18, 255–280 (1983)

    Article  Google Scholar 

  5. He X., Ruan S., Xia H.: Global stability in chemostat-type equatinos with distributed delays. SIAM J. Math. Anal. 29, 681–696 (1998)

    Article  Google Scholar 

  6. He X., Ruan S.: Global stability in chemostat-type plankton models with delayed nutrient recycling. J. Math. Biol. 37, 253–271 (1998)

    Article  Google Scholar 

  7. Jang S.: Dynamics of variable-yield nutrient-phytoplankton-zooplankton models with nutient recycling and self-shading. J. Math. Biol. 40, 229–250 (2000)

    Article  CAS  Google Scholar 

  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

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  11. Pilyugin S., Waltman P.: Competition in the unstirred chemostat with periodic input and washout. SIAM J. Appl. Math. 59, 1157–1177 (1999)

    Article  Google Scholar 

  12. Ruan S.: Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling. J. Math. Biol. 31, 633–654 (1993)

    Article  Google Scholar 

  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. Ruan S.: The effect of delays on stability and persistence in plankton models. Nonlinear Anal. 24, 575–585 (1995)

    Article  Google Scholar 

  15. Ruan S., He X.: Global stability in chemostat-type competition models with nutrient recycling. SIAM J. Appl. Math. 58, 170–192 (1998)

    Article  Google Scholar 

  16. Smith H.L.: Competitive coexistence in an oscillating chemostat chemostat. SIAM J. Appl. Math. 40, 498–522 (1981)

    Article  Google Scholar 

  17. Simth H.L., Waltman P.: The theory of the chemostat. Cambrige University Press, Cambridge (1995)

    Book  Google Scholar 

  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. 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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, 830046, Urumqi, People’s Republic of China

    Zhidong Teng, Rong Gao, Mehbuba Rehim & Kai Wang

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  1. Zhidong Teng
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  2. Rong Gao
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  3. Mehbuba Rehim
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  4. Kai Wang
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Correspondence to Zhidong Teng.

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Teng, Z., Gao, R., Rehim, M. et al. Global behaviors of Monod type chemostat model with nutrient recycling and impulsive input. J Math Chem 47, 276–294 (2010). https://doi.org/10.1007/s10910-009-9567-8

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  • DOI: https://doi.org/10.1007/s10910-009-9567-8

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