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Journal of Mathematical Chemistry

, Volume 47, Issue 4, pp 1224–1239 | Cite as

Impulsive state feedback control of the microorganism culture in a turbidostat

  • Zhong Zhao
  • Li Yang
  • Lansun Chen
Original Paper

Abstract

In this paper, a mathematical model with the impulsive state feedback control is proposed for turbidostat system. The sufficient conditions of existence of order-1 and order-2 periodic solutions are obtained by the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state and the initial concentration of microorganism and substrate. Finally, some discussions and numerical simulations are given.

Keywords

Turbidostat Impulsive state feedback control Period-1 solution Period-2 solution 

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References

  1. 1.
    Armstrong R.A., McGehee R.: Competitive exclusion. Am. Nat. 115, 151–170 (1980)CrossRefGoogle Scholar
  2. 2.
    Butler G.J., Wolkowicz G.S.K.: A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM J. Appl. Math. 45, 138–151 (1985)CrossRefGoogle Scholar
  3. 3.
    Hsu S.B.: Limiting behavior for competing species. SIAM J. Appl. Math. 34, 760–763 (1978)CrossRefGoogle Scholar
  4. 4.
    Hsu S.B., Hubbell S., Waltman P.: A mathematical theory of single-nutrient competition in continuous cultures of micro-organisms. SIAM J. Appl. Math. 32, 366–383 (1977)CrossRefGoogle Scholar
  5. 5.
    Levins R.: Coexistence in a variable environment. Am. Nat. 114, 765–783 (1979)CrossRefGoogle Scholar
  6. 6.
    Wolkowicz G.S.K., Lu Z.: Global dynamics of a mathematical model of competition in the chemostat: general response function and differential death rates. SIAM J. Appl. Math. 52, 222–233 (1992)CrossRefGoogle Scholar
  7. 7.
    Grover J.P.: Resource Competition. Chapman and Hall, London (1997)Google Scholar
  8. 8.
    Hansen S.R., Hubbell S.P.: Single-nutrient microbial competition: agreement between experimental and theoretical forecast outcomes. Science 207, 1491–1493 (1980)CrossRefGoogle Scholar
  9. 9.
    Tilman D.: Resource competition and Community Structure. Princeton U.P., Princeton, N.J (1982)Google Scholar
  10. 10.
    De Leenheer P., Smith H.L.: Feedback control for the chemostat. J. Math. Biol. 46, 48–70 (2003)CrossRefGoogle Scholar
  11. 11.
    Flegr J.: Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems. J. Theor. Biol. 188, 121–126 (1997)CrossRefGoogle Scholar
  12. 12.
    H. Guo, L. Chen, Periodic solution of a turbidostat system with impulsive state feedback control. J. Math. Chem. doi: 10.1007/s10910-008-9492-2
  13. 13.
    Jiang G.R., Lu Q.S.: Impulsive state feedback control of a predator-prey model. J. Comput. Appl. Math. 34(2), 1135–1147 (2007)Google Scholar
  14. 14.
    Jiang G.R., Lu Q.S., Qian L.N.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Solitons Fractals 31(2), 448–461 (2007)CrossRefGoogle Scholar
  15. 15.
    Zeng G.Z., Chen L.S., Sun L.H.: Existence of periodic solution of order one of planar impulsive autonomous system. J. Comp. Appl. Math. 186, 466–481 (2006)CrossRefGoogle Scholar
  16. 16.
    Yang T.: Impulsive Control Theory, pp. 307–333. Springer, Berlin Heidelberg (2001)Google Scholar
  17. 17.
    Kamel O.M., Soliman A.S.: On the optimization of the generalized coplanar Hohmann impulsive transfer adopting energy change concept. Acta Astronaut. 56, 431–438 (2005)CrossRefGoogle Scholar
  18. 18.
    Lakshmikantham V., Bainov D.D., Simeonov P.: Theory of impulsive differential equations. World Scientific, Singapore (1989)Google Scholar
  19. 19.
    Zeng G.Z., Chen L.S., Sun L.H.: Existence of periodic solution of order one of planar impulsive autonomous system. J. Comp. Appl. Math. 186, 466–481 (2006)CrossRefGoogle Scholar
  20. 20.
    Ayaaki Ishizaki, Tomomi Ohta: Batch culture kinetics of -lactate fermentation employing Strep-tococcus sp. IO-1. J. Ferment. Bioeng. 68, 123–130 (1989)CrossRefGoogle Scholar
  21. 21.
    Simeonov P.E., Bainov D.D.: Orbital stability of periodic solutions of autonomous systems with impulse effect. Int. J. Syst. Sci. 19, 2562–2585 (1988)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsHuanghuai UniversityZhumadianPeople’s Republic of China
  2. 2.Department of Applied MathematicsDalian University of TechnologyDalianPeople’s Republic of China

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