Skip to main content
Log in

Global analysis of continuous flow bioreactor and membrane reactor models with death and maintenance

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

The aim of the paper is to investigate the global dynamics of mathematical models for a continuous flow bioreactor and a membrane reactor. It assumes that the models include terms representing death of the microorganism and maintenance energy, respectively. By carrying out a qualitative analysis of the models, we give the classification of the equilibria and show that an unstable limit cycle can exist when the non-washout equilibrium is a focus.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. Fan, G.S.K. Wolkowicz, A predator-prey model in the chemostat with time delay, Int. J. Differ. Equ. 2010, Article ID 287969

  2. Huang X., Zhu L.: A three-dimensional chemostat with quadratic yields. J. Math. Chem. 38, 412 (2005)

    Google Scholar 

  3. Huang X., Wang Y., Zhu L.: Competition in the bioreactor with general quadratic yields when one competitor produces a toxin. J. Math. Chem. 39, 294 (2006)

    Google Scholar 

  4. Huang X., Zhu L.: A note on competition in the bioreactor with toxin. J. Math. Chem. 42, 659 (2007)

    Article  Google Scholar 

  5. Imhofa L., Walcherb S.: Exclusion and persistence in deterministic and stochastic chemostat models. J. Differ. Equ. 217, 26–53 (2005)

    Article  Google Scholar 

  6. Koga S., Humphrey A.E: Study of the dynamic behavior of the chemostat system. Biotechnol. Bioeng. 9, 375–386 (1967)

    Article  Google Scholar 

  7. Han M.: Properties in the large of quadratic systems in the plane. Chin. Ann. Math. Ser. B 10, 312–322 (1989)

    Google Scholar 

  8. McCarty, Kinetics of waste assimilation in anaerobic treatment, in Developments in Industrial Microbiology, vol.7, pp 144–155, Society of Industrial Microbiology (1966)

  9. Monod J.: La technique de culture continue. Théorie et applications. Ann. Inst. Pasteur (Paris) 79, 390–410 (1950)

    CAS  Google Scholar 

  10. Nelson M.I., Kerr T.B., Chen X.D.: A fundamental analysis of continuous flow bioreactor and membrane reactor models with death and maintenance included, Asis-Pac. J. Chem. Eng. 3, 70–80 (2008)

    CAS  Google Scholar 

  11. Nelson M.I., Sidhu H.S.: Analysis of a chemostat model with variable yield coefficient: Tessier kinetics. J. Math. Chem. 46, 303–321 (2009)

    Article  CAS  Google Scholar 

  12. Nelson M.I., Holder A.: A fundamental analysis of continuous flow bioreactor models governed by Contois kinetics II. Reactor cascades. Chem. Eng. J. 149, 406–416 (2009)

    Article  CAS  Google Scholar 

  13. Pirt S.J.: The maintenance energy of bacteria in growing cultures. Proc. R. Soc. Lond. Ser-B 163, 224–231 (1965)

    Article  CAS  Google Scholar 

  14. Ruan S.G.: Bifurcation analysis of a chemostat model with a distributed delay. J. Math. Anal. Appl. 204, 786–812 (1996)

    Article  Google Scholar 

  15. Smith H.L., Waltman P.: The Theory of the Chemostat. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  16. Tessier G.: Les lois quantitatives de la croissance. Ann. de Physiol. et de Phys. Biol. 12, 527–573 (1936)

    Google Scholar 

  17. Wolkowicz G.S.K., Xia H.: Global asymptotic behaviour of a chemostat model with discrete delays. SIAM J. Appl. Math. 57, 1019–1043 (1997)

    Article  Google Scholar 

  18. Y.Q. Ye, Theory of limit cycles, Trans. Math. Monogr., vol. 66, (AMS, Providence, 1986)

  19. Ye Y.Q.: Qualitative theory of differential polynomial systems. Shanghai Science Technology Publishing House, Shanghai (1995)

    Google Scholar 

  20. Yuan S., Zhang T.: Dynamics of a plasmid chemostat model with periodic nutrient input and delayed nutrient recycling. Nonlinear Anal. Real World Appl. 13, 2104–2119 (2012)

    Article  CAS  Google Scholar 

  21. Zhu L., Huang X., Su H.: Bifurcation for a functional yield chemostat when one competitor produces a toxin. J. Math. Anal. Appl. 329, 891–903 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tonghua Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, T. Global analysis of continuous flow bioreactor and membrane reactor models with death and maintenance. J Math Chem 50, 2239–2247 (2012). https://doi.org/10.1007/s10910-012-0027-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-012-0027-5

Keywords

Navigation