Mathematical Biosciences and Engineering (MBE)

Global dynamics of the chemostat with different removal rates and variable yields
Pages: 827 - 840, Volume 8, Issue 3, July 2011

doi:10.3934/mbe.2011.8.827      Abstract        References        Full text (480.3K)           Related Articles

Tewfik Sari - Université de Haute Alsace, Mulhouse, France (email)
Frederic Mazenc - Projet INRIA DISCO, CNRS-SUPELEC, 3 Rue Joliot Curie, 91192, Gif-sur-Yvette, France (email)

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.       
2 R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.       
3 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.       
4 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.       
5 S. B. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763.       
6 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.       
7 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.       
8 P. de Leenheer, B. Li and H. L. Smith, Competition in the chemostat: Some remarks, Can. Appl. Math. Q., 11 (2003), 229-248.       
9 B. Li, Global asymptotic behavior of the chemostat: General response functions and differential removal rates, SIAM Journal on Applied Mathematics, 59 (1999), 411-422.       
10 C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models, Electron. J. Differential Equations, 2007, 10.       
11 M. Malisoff and F. Mazenc, "Constructions of Strict Lyapunov Functions," Communications and Control Engineering Series, Springer-Verlag London, Ltd, London, 2009.       
12 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.       
13 F. Mazenc, M. Malisoff and J. Harmand, Further 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.       
14 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.       
15 J. Monod, La technique de culture continue. Théorie et applications, Ann. Inst. Pasteur, 79 (1950), 390-410.
16 S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield, Mathematical Biosciences, 182 (2003), 151-166.       
17 A. Rapaport and J. Harmand, Biological control of the chemostat with nonmonotone response and different removal rates, Math. Biosci. Eng., 5 (2008), 539-547.       
18 T. Sari, A Lyapunov function for the chemostat with variable yields, C. R. Math. Acad. Sci. Paris, 348 (2010), 747-751.       
19 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.       
20 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.       
21 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.       
22 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.       

Go to top