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Study of a chemostat model with Beddington–DeAngelis functional response and pulsed input and washout at different times

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Abstract

In the paper, we considered a food chain chemostat model with a Beddington–DeAngelis functional response of predator, with periodical input and washout occurring at different fixed times. We obtained exact periodic solutions for the model with substrate and prey only. The stability analysis for this periodic solutions yields an invasion threshold for period of pulses of the predator. When the impulsive period is greater than the threshold, there are periodic oscillations in substrate, prey, and predator. If the impulsive period is increased further, the system undergoes the complex dynamic process. By analyzing bifurcation diagrams, we can see that the impulsive system shows two kinds of bifurcations; period-doubling and period-halving.

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References

  1. Hsu S.B., Hubbell, Waltman P. (1977). SIAM J. Appl. Math. 32(2): 206–218

    Article  Google Scholar 

  2. Hsu S.B. (1978). SIAM J. Appl. Math. 34(4): 760–775

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Beddington J.R. (1975). J. Anim. Ecol. 44, 331–340

    Article  Google Scholar 

  7. Ruxton G., Gurney WSC., DeRoos A. (1992). Theoret. Popul. Biol. 42, 235–253

    Article  Google Scholar 

  8. Cosner C., DeAngelis D.L., Ault J.S., Olson J.S. (1999). Theret. Popul. Biol. 56, 65–75

    Article  CAS  Google Scholar 

  9. Harrision G.W. (1995). Ecology 76, 357–369

    Article  Google Scholar 

  10. Hsu S.B. (1980). J. Math. Biol. 18, 255–280

    Google Scholar 

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

    Article  Google Scholar 

  12. Pavlou S., Muratori I.G. (1992). Math. Biosci. 108, 1–55

    Article  CAS  Google Scholar 

  13. Pilyugin S.S., Waltman P. (1999). SIAM J. Appl. Math. 59, 1157–1177

    Article  Google Scholar 

  14. Li B. (1999). SIAM J. Appl. Math. 59, 411–422

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Roberts M.G., Kao R.R. (1998). Math. Biosci. 149, 23–36

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  19. Lakmeche A., Arino O. (2000). Dyn. Cont. Discrete Impul. Syst. 7, 165–187

    Google Scholar 

  20. Ballinger G., Liu X. (1997). Math. Comput. Model. 26, 59–72

    Article  Google Scholar 

  21. Tang S.Y., Chen L.S. (2002). Chaos Solitons Fractals 13, 875–884

    Article  Google Scholar 

  22. Sabin G.C.W., Summer D. (1993). Math. Biosci. 113, 91–113

    Article  CAS  Google Scholar 

  23. Gakkhar S., Naji R.K. (2003). Chaos Solitons Fractals 18, 1075–1083

    Article  Google Scholar 

  24. Venkatesan A., Parthasarathy S., Lakshmanan M. (2003). Chaos Solitons Fractals 18, 891–898

    Article  Google Scholar 

  25. Zhang S.W., Tan D.J., Chen L.S. (2006). Chaos Solitons Fractals 28, 367–376

    Article  Google Scholar 

  26. Funasaki E., Kot M. (1997). Theoret. Popul. Biol. 44, 203–224

    Article  Google Scholar 

  27. Bainov D.D., Simeonov D.D. (1993). Impulsive Differential Equations: Periodic Solutions and Applications. Longman, England

    Google Scholar 

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

    Article  CAS  Google Scholar 

  29. Klebanoff A., Hasting A. (1994). J. Math. Biol. 32, 229–239

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

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Authors and Affiliations

  1. College of Science, Jimei University, Xiamen, Fujian, 361021, P.R. China

    Shuwen Zhang & Dejun Tan

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  1. Shuwen Zhang
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  2. Dejun Tan
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Correspondence to Shuwen Zhang.

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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). https://doi.org/10.1007/s10910-007-9304-0

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  • DOI: https://doi.org/10.1007/s10910-007-9304-0

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