Microorganisms produce toxins against its competitors sometimes, and variable yields are useful to explain the observed oscillatory behavior in the reactor. In this paper, a model with general quadric yields of competition in the bioreactor of two competitors for a single nutrient where one of the competitors can produce toxin against its opponent, is proposed. We analyze the asymptotic behavior of the model in terms of the relevant parameters. The conditions of the three dimensional Hopf bifurcation, and the existence of limit cycles in the nutrient-organism phase plane are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chao L., Levin B.R. (1981). Structured habitats and the evolution of anti-competitor toxins in bacteria. Proc. Nat. Acad. Sci. 75: 6324–6328
Hsu S.B., Waltman P. (1992). Analysis of a model of two competitors in a chemostat with an external inhibitor. SIAM J. Appl. Math. 52: 528–540
Hsu S.B., Waltman P. (1997). Competition between plasmid-bearing and plasmid-free organisms in selective media. Chem. Eng. Sci. 52: 23–35
Lenski R.E., Hattingh S. (1986). Coexistence of two competitors on one resource and one inhibitor: a chemostat model based on bacteria and antibiotics. J. Theoret. Biol. 122: 83–93
Hsu S.B., Luo T.K. (1995). Global analysis of a model of plasmid-bearing plasmid-free competition in a chemostat with inhibition. J. Math. Biol. 34: 41–76
Levin B.R. (1988). Frequency-dependent selection in bacterial populations. Phil. Trans. Roy. Soc. London 319: 459–472
Hsu S.B., Waltman P. (1998). Competition in the chemostat when one competitor produces toxin. Jpn. J. Indust. Appl. Math. 15: 471–490
Huang X.C. (1990). Limit cycles in a continuous fermentation model. J. Math. Chem. 5: 287–296
Pilyugin S.S., Waltman P. (2003). Multiple limit cycles in the chemostat with variable yield. Math. Biosci. 182: 151–166
Dorofeev A.G., Glagolev M.V., Bondarenko T.F., Panikov N.S. (1992). Observation and explanation of the unusual growth kinetics of Arthrobacter globiforms. Microbiology 61: 33–42
Crooke P.S., Wei C.-J., Tanner R.D. (1980). The effect of the specific growth rate and yield expressions on the existence of oscillatory behavior of a continuous fermentation model. Chem. Eng. Commun. 6: 333–339
Crooke P.S., Tanner R.D. (1982). Hopf bifurcations for a variable yield continuous fermentation model. Int. J. Eng. Sci. 20: 439–443
Zhu L.M., Huang X.C. (2005). Relative positions of limit cycles in the continuous culture vessel with variable yield. J. Math. Chem. 38(2): 119–128
Wolkowicz G.S.K., Lu Z. (19992). Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates. SIAM J. Appl. Math. 32: 222–233
Zhang J. (1987). The Geometric Theory and Bifurcation Problem of Ordinary Differential Equation. Peking University Press, Beijing
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, X., Wang, Y. & Zhu, L. Competition in the bioreactor with general quadratic yields when one competitor produces a toxin. J Math Chem 39, 281–294 (2006). https://doi.org/10.1007/s10910-005-9040-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-005-9040-2