Dynamic analysis of Michaelis–Menten chemostat-type competition models with time delay and pulse in a polluted environment

Original Paper


In this paper, a new Michaelis–Menten type chemostat model with time delay and pulsed input nutrient concentration in a polluted environment is considered. We obtain a ‘microorganism-extinction’ semi-trivial periodic solution and establish the sufficient conditions for the global attractivity of the semi-trivial periodic solution. By use of new computational techniques for impulsive differential equations with delay, we prove and support with numerical calculations that the system is permanent. Our results show that time delays and the polluted environment can lead the microorganism species to be extinct.


Permanence Impulsive input Michaelis–Menten type chemostat model Time delay for growth response Extinction 


  1. 1.
    Novick A., Szilard L.: Description of the chemostat. Science 112, 715–716 (1950)CrossRefGoogle Scholar
  2. 2.
    Dykhuizen D.E., Hartl D.L.: Selection in chemostats. Microbiol. Rev. 47, 150–168 (1983)Google Scholar
  3. 3.
    Hsu S.B.: A competition model for a seasonally fluctuating nutrient. J. Math. Biol. 9(2), 115–132 (1980)CrossRefGoogle Scholar
  4. 4.
    Butler G.J., Hsu S.B., Waltman P.: A mathematical model of the chemostat with periodic washout rate. SIAM J. Appl. Math. 45(3), 435–449 (1985)CrossRefGoogle Scholar
  5. 5.
    Pilyugin S.S., Waltman P.: Competition in the unstirred chemostat with periodic input and washout. SIAM J. Appl. Math. 59(4), 1157–1177 (1999)CrossRefGoogle Scholar
  6. 6.
    Smith H.L., Waltman P.: The Theory of the Chemostat. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  7. 7.
    Monod J.: La technique de culture continue; théorie et applications. Ann. Inst. Pasteur 79(19), 390–401 (1950)Google Scholar
  8. 8.
    Hsu S.B., Li C.C.: A discrete-delayed model with plasmid-bearing, plasmid-free competition in a chemostat. Discrete Contin. Dyn. Syst. Ser. B 5, 699–718 (2005)CrossRefGoogle Scholar
  9. 9.
    Hsu S.B., Hubbell S.P., Waltman P.: A mathematical theory for single-nutrient competition in the continuous cultures of micro-organisms. SIAM J. Appl. Math. 32(2), 366–383 (1977)CrossRefGoogle Scholar
  10. 10.
    Hsu S.B., Tzeng Y.H.: Plasmid-bearing, plasmid-free organisms competing for two complementary nutrients in a chemostat. Math. Biosci. 179, 183–206 (2002)CrossRefGoogle Scholar
  11. 11.
    Wolkowicz G.S.K., Xia H.Y.: Global asymptotic behavior of a chemostat model with discrete delays. SIAM J. Appl. Math. 57(4), 1019–1043 (1997)CrossRefGoogle Scholar
  12. 12.
    Caperon J.: Time lag in population growth response of Isochrysis galbana to a variable nitrate environment. Ecology 50(2), 188–192 (1969)CrossRefGoogle Scholar
  13. 13.
    Freedman H.I., So J.W-H., Waltman P.: Coexistence in a model of competition in the chemostat incorporating discrete delays. SIAM J. Appl. Math. 49(3), 859–870 (1989)CrossRefGoogle Scholar
  14. 14.
    Ruan S.G., Wolkowicz G.S.K.: Bifurcation analysis of a chemostat model with a distributed delay. J. Math. Anal. Appl. 204(3), 786–812 (1996)CrossRefGoogle Scholar
  15. 15.
    Xia H.Y., Wolkowicz G.S.K., Wang L.: Transient oscillations induced by delayed growth response in the chemostat. J. Math. Biol. 50(5), 489–530 (2005)CrossRefGoogle Scholar
  16. 16.
    Hale J.K., Somolinas A.S.: Competition for fluctuating nutrient. J. Math. Biol. 18(3), 255–280 (1983)CrossRefGoogle Scholar
  17. 17.
    Wolkowicz G.S.K., Zhao X.Q.: N-species competition in a periodic chemostat. Differential Integral Equations. 11(3), 465–491 (1998)Google Scholar
  18. 18.
    Lakshmikantham V., Bainov D., Simeonov P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)Google Scholar
  19. 19.
    Haddad W.M., Chellaboina V., Nersesov S.G.: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, Princeton (2006)Google Scholar
  20. 20.
    Zavalishchin S.T., Sesekin A.N.: Dynamic Impulse Systems. Theory and Applications. Mathematics and Its Applications, vol 394. Kluwer Academic Publishers, Dordrecht (1997)Google Scholar
  21. 21.
    W.B. Wang, J.H. Shen, J.J.Nieto, Permanence and periodic solution of predator prey system with Holling type functional response and impulses. Discrete Dyn. Nat. Soc. Art. ID 81756, 15 pp (2007). doi:10.1155/2007/81756
  22. 22.
    Dai B., Su H., Hu D.: Periodic solution of a delayed ratio-dependent predator–prey model with monotonic functional response and impulse. Nonlinear Anal. 70(1), 126–134 (2009). doi:10.1016/j.na.2007.11.036 CrossRefGoogle Scholar
  23. 23.
    Nieto J.J., Rodriguez-Lopez R.: New comparison results for impulsive integro-differential equations and applications. J. Math. Anal. Appl. 328, 1343–1368 (2007)CrossRefGoogle Scholar
  24. 24.
    Liu B., Teng Z.D., Liu W.B.: Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations. Chaos Solitons Fractals 31(2), 356–370 (2007)CrossRefGoogle Scholar
  25. 25.
    Liu B., Zhang Y.J., Chen L.S.: The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management. Nonlinear Anal. Real World Appl. 6(2), 227–243 (2005)CrossRefGoogle Scholar
  26. 26.
    J.H. Shen, J.L. Li: Existence and global attractivity of positive periodic solutions for impulsive predator–prey model with dispersion and time delays. Nonlinear Anal. Real World Appl. 10(1), 227–243 (2009). doi:110.1016/j.nonrwa CrossRefGoogle Scholar
  27. 27.
    Chu J., Nieto J.J.: Impulsive periodic solutions of first order singular differential equations. Bull. London Math. Soc. 40, 143–150 (2008)CrossRefGoogle Scholar
  28. 28.
    Georgescu P., Morosanu G.: Pest regulation by means of impulsive controls. Appl. Math. Comput. 190, 790–803 (2007)CrossRefGoogle Scholar
  29. 29.
    Jianng G., Lu Q., Qian L.: Chaos and its control in an impulsive differential system. Chaos Solitons Fractals 34, 1135–1147 (2007)CrossRefGoogle Scholar
  30. 30.
    Meng X., Jiao J., Chen L.: The dynamics of an age structured predator–prey model with disturbing pulse and time delays. Nonlinear Anal. Real World Appl. 9, 547–561 (2008)CrossRefGoogle Scholar
  31. 31.
    Meng X., Chen L.: Global dynamical behaviors for an SIR epidemic model with time delay and pulse vaccination. Taiwanese J. Math. 12(5), 1107–1122 (2008)Google Scholar
  32. 32.
    Zhou J., Xiang L., Liu Z.: Synchronization in complex delayed dynamical networks with impulsive effects. Physica A Stat. Mech. Appl. 384, 684–692 (2007)CrossRefGoogle Scholar
  33. 33.
    Sun S.L., Chen L.S.: Dynamic behaviors of Monod type chemostat model with impulsive perturbation on the nutrient concentration. J. Math. Chem. 42(4), 837–848 (2007)CrossRefGoogle Scholar
  34. 34.
    Sun M.J., Chen L.S.: Analysis of the dynamical behavior for enzyme-catalyzed reactions with impulsive input. J. Math. Chem. 43(2), 447–456 (2008)CrossRefGoogle Scholar
  35. 35.
    Pang G.P., Liang Y.L., Wang F.Y.: Analysis of Monod type food chain chemostat with k-times periodically pulsed input. J. Math. Chem. 43(4), 1371–1388 (2008). doi:10.1007/s10910-007-9258-2 CrossRefGoogle Scholar
  36. 36.
    Song X., Zhao Z., Extinction and permanence of two-nutrient and one-microorganism chemostat model with pulsed input. Discrete Dyn. Nat. Soc. Art. ID 38310, 14 pp (2006). doi:10.1155/DDNS/2006/38310
  37. 37.
    Wang F., Hao C., Chen L.: Bifurcation and chaos in a Monod-Haldene type food chain chemostat with pulsed input and washout. Chaos Solitons Fractals 32, 181–194 (2007)CrossRefGoogle Scholar
  38. 38.
    Kuang Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego (1993)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Information SchoolShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  2. 2.State Key Laboratory of Vegetation and Environmental Change, Institute of BotanyChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de Santiago de CompostelaSantiago de CampostelaSpain

Personalised recommendations