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

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

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.
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.   Google Scholar

[2]

R. A. Armstrong and R. McGehee, Competitive exclusion,, Amer. Natur., 115 (1980), 151.  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.  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.   Google Scholar

[5]

S. B. Hsu, Limiting behavior for competing species,, SIAM Journal on Applied Mathematics, 34 (1978), 760.  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.  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.  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.   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.  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, (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.  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.   Google Scholar

[15]

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

[16]

S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield,, Mathematical Biosciences, 182 (2003), 151.  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.   Google Scholar

[18]

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

[19]

H. L. Smith, P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition,", Cambridge Studies in Mathematical Biology, 13 (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 (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.  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.  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.   Google Scholar

[2]

R. A. Armstrong and R. McGehee, Competitive exclusion,, Amer. Natur., 115 (1980), 151.  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.  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.   Google Scholar

[5]

S. B. Hsu, Limiting behavior for competing species,, SIAM Journal on Applied Mathematics, 34 (1978), 760.  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.  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.  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.   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.  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, (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.  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.   Google Scholar

[15]

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

[16]

S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield,, Mathematical Biosciences, 182 (2003), 151.  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.   Google Scholar

[18]

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

[19]

H. L. Smith, P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition,", Cambridge Studies in Mathematical Biology, 13 (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 (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.  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.  doi: 10.1137/S0036139995287314.  Google Scholar

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