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Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents
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 |
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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