Advanced Search
Article Contents
Article Contents

Global dynamics of the chemostat with different removal rates and variable yields

Abstract Related Papers Cited by
  • In this paper, we consider a competition model between $n$ species in a chemostat including both monotone and non-monotone growth functions, distinct removal rates and variable yields. We show that only the species with the lowest break-even concentration survives, provided that additional technical conditions on the growth functions and yields are satisfied. We construct a Lyapunov function which reduces to the Lyapunov function used by S. B. Hsu [SIAM J. Appl. Math., 34 (1978), pp. 760-763] in the Monod case when the growth functions are of Michaelis-Menten type and the yields are constant. Various applications are given including linear, quadratic and cubic yields.
    Mathematics Subject Classification: Primary: 92A15, 92A17; Secondary: 34C15, 34C35.


    \begin{equation} \\ \end{equation}
  • [1]

    J. Arino, S. S. Pilyugin and G. S. K. Wolkowicz, Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models, Canadian Applied Mathematics Quarterly, 11 (2003), 107-142.


    R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.doi: 10.1086/283553.


    G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.doi: 10.1137/0145006.


    P. Gajardo, F. Mazenc and H. C. Ramírez, Competitive exclusion principle in a model of chemostat with delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 16 (2009), 253-272.


    S. B. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763.doi: 10.1137/0134064.


    S. B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.doi: 10.1137/0132030.


    X. Huang, L. Zhu and E. H. C. Chang, Limit cycles in a chemostat with variable yields and growth rates, Nonlinear Analysis Real World Applications, 8 (2007), 165-173.doi: 10.1016/j.nonrwa.2005.06.007.


    P. de Leenheer, B. Li and H. L. Smith, Competition in the chemostat: Some remarks, Can. Appl. Math. Q., 11 (2003), 229-248.


    B. Li, Global asymptotic behavior of the chemostat: General response functions and differential removal rates, SIAM Journal on Applied Mathematics, 59 (1999), 411-422.doi: 10.1137/S003613999631100X.


    C. Lobry and F. MazencEffect on persistence of intra-specific competition in competition models, Electron. J. Differential Equations, 2007, 10.


    M. Malisoff and F. Mazenc, "Constructions of Strict Lyapunov Functions," Communications and Control Engineering Series, Springer-Verlag London, Ltd, London, 2009.doi: 10.1007/978-1-84882-535-2.


    F. Mazenc, M. Malisoff and J. Harmand, Stabilization in a two-species chemostat with Monod growth functions, IEEE Trans. Automat. Control, 54 (2009), 855-861.doi: 10.1109/TAC.2008.2010964.


    F. Mazenc, M. Malisoff and J. HarmandFurther results on stabilization of periodic trajectories for a chemostat with two species, IEEE Trans. Circuits Syst. I. Regul. Pap., 2008, Special issue on systems biology, 66-74.


    F. Mazenc, M. Malisoff and P. De Leenheer, On the stability of periodic solutions in the perturbed chemostat, Math. Biosci. Eng., 4 (2007), 319-338.


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


    S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield, Mathematical Biosciences, 182 (2003), 151-166.doi: 10.1016/S0025-5564(02)00214-6.


    A. Rapaport and J. Harmand, Biological control of the chemostat with nonmonotone response and different removal rates, Math. Biosci. Eng., 5 (2008), 539-547.


    T. Sari, A Lyapunov function for the chemostat with variable yields, C. R. Math. Acad. Sci. Paris, 348 (2010), 747-751.


    H. L. Smith, P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition," Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995.


    G. S. K. Wolkowicz, M. M Ballyk and Z. Lu, Microbial dynamics in a chemostat: Competition, growth, implication of enrichment, Lecture Notes in Pure and Appl. Math., 176, Dekker, New-York, 1996.


    G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM Journal on Applied Mathematics, 52 (1992), 222-233.doi: 10.1137/0152012.


    G. S. K. Wolkowicz and H. Xia, Global asymptotic behavior of a chemostat model with discrete delays, SIAM Journal on Applied Mathematics, 57 (1997), 1019-1043.doi: 10.1137/S0036139995287314.

  • 加载中

Article Metrics

HTML views() PDF downloads(60) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint