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Qualitative analysis of the chemostat model with variable yield and a time delay

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Abstract

In this paper, we consider the chemostat model with inhibitory exponential substrate, variable yield and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. The Hopf bifurcation of solutions to the system is studied. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Based on some known techniques on limit sets of differential dynamical systems, we show that, for any time delay, the chemostat model is permanent if and only if only one positive equilibrium exits.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Yanan University, Yan’an, 716000, China

    Qinglai Dong

  2. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Wanbiao Ma

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  1. Qinglai Dong
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  2. Wanbiao Ma
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Correspondence to Qinglai Dong.

Additional information

The research is partially supported by NSFC(11071013), the Funds of the construction of high-level university in Shaanxi province (2012SXTS06) and the Funds of Yanan University (YD2012-03).

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Dong, Q., Ma, W. Qualitative analysis of the chemostat model with variable yield and a time delay. J Math Chem 51, 1274–1292 (2013). https://doi.org/10.1007/s10910-013-0144-9

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  • DOI: https://doi.org/10.1007/s10910-013-0144-9

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