American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models,, Electron. J. Differential Equations, 2007 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 (): 66.   Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

C. Lobry and F. Mazenc, Effect on persistence of intra-specific competition in competition models,, Electron. J. Differential Equations, 2007 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 (): 66.   Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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