Comments on “Limit cycles in the chemostat with constant yields” Mathematical and Computer Modelling 45 (2007) 927-932. (English) Zbl 1205.34056
Summary: Several elements of the paper “Limit cycles in the chemostat with constant yields” [Math. Comput. Modelling 45, No. 7-8, 927–932 (2007; Zbl 1144.34340)] are incorrect and in contradiction to well established results of the literature. In particular the claim that limit cycles can exist in the chemostat with two competitors for a single nutrient and constant yields is utterly false. It is well-known that in this model the competitive exclusion principle holds.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 92D25 | Population dynamics (general) |
Citations:
Zbl 1144.34340References:
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| [2] | Hsu, S. B.; Hubbell, S. P.; Waltman, P., A mathematical theory for single nutrient competition in continuous culture of micro-organisms, SIAM Journal on Applied Mathematics, 32, 366-383 (1977) · Zbl 0354.92033 |
| [3] | Butler, G. J.; Wolkowicz, G. S.K., A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal of Applied Mathematics, 45, 137-151 (1985) · Zbl 0569.92020 |
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| [8] | T. Sari, Global dynamics of the chemostat with variable yields, preprint arXiv1002.4807v1; T. Sari, Global dynamics of the chemostat with variable yields, preprint arXiv1002.4807v1 |
| [9] | Wolkowicz, G. S.K.; Ballyk, M. M.; Lu, Z., Microbial dynamics in a chemostat: competition, growth, implication of enrichment, (Differential Equations and Control Theory, Wuhan, 1994. Differential Equations and Control Theory, Wuhan, 1994, Lecture Notes in Pure and Appl. Math., vol. 176 (1996), Dekker: Dekker New York), 389-406 · Zbl 0837.92031 |
| [10] | Zhu, L., The Limit cycles in chemostat with constant yields, Mathematical and Computer Modelling, 45, 927-932 (2007) · Zbl 1144.34340 |
| [11] | El Hajji, M.; Rapaport, A., Practical coexistence of two species in the chemostat—a slow-fast characterization, Mathematical Biosciences, 218, 1, 33-39 (2009) · Zbl 1157.92032 |
| [12] | Pilyugin, S. S.; Waltman, P., Multiple limit cycles in the chemostat with variable yields, Mathematical Biosciences, 182, 151-166 (2003) · Zbl 1012.92044 |
| [13] | Nelson, M. I.; Sidhu, H. S., Analysis of a chemostat model with variable yield coefficient: Tessier kinetics, Journal of Mathematical Chemistry, 46, 2, 303-321 (2009) · Zbl 1196.92044 |
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