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

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2011, 8(3): 827-840. doi: 10.3934/mbe.2011.8.827

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

Tewfik Sari ^1, and Frederic Mazenc ^2,

1.	Université de Haute Alsace, Mulhouse, France
2.	Projet INRIA DISCO, CNRS-SUPELEC, 3 Rue Joliot Curie, 91192, Gif-sur-Yvette, France

Received April 2010 Revised November 2010 Published June 2011

Abstract
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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.

Keywords: Lyapunov function, competitive exclusion principle, Chemostat, global asymptotic stability, variable yield model..

Mathematics Subject Classification: Primary: 92A15, 92A17; Secondary: 34C15, 34C3.

Citation: Tewfik Sari, Frederic Mazenc. Global dynamics of the chemostat with different removal rates and variable yields. Mathematical Biosciences & Engineering, 2011, 8 (3) : 827-840. doi: 10.3934/mbe.2011.8.827

References:

[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. doi: 10.1086/283553.
[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. doi: 10.1137/0145006.
[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. doi: 10.1137/0134064.
[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. doi: 10.1137/0132030.
[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. doi: 10.1016/j.nonrwa.2005.06.007.
[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. doi: 10.1137/S003613999631100X.
[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. doi: 10.1007/978-1-84882-535-2.
[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. doi: 10.1109/TAC.2008.2010964.
[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. doi: 10.1016/S0025-5564(02)00214-6.
[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. doi: 10.1137/0152012.
[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. doi: 10.1137/S0036139995287314.

show all references

References:

[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. doi: 10.1086/283553.
[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. doi: 10.1137/0145006.
[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. doi: 10.1137/0134064.
[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. doi: 10.1137/0132030.
[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. doi: 10.1016/j.nonrwa.2005.06.007.
[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. doi: 10.1137/S003613999631100X.
[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. doi: 10.1007/978-1-84882-535-2.
[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. doi: 10.1109/TAC.2008.2010964.
[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. doi: 10.1016/S0025-5564(02)00214-6.
[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. doi: 10.1137/0152012.
[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. doi: 10.1137/S0036139995287314.

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