Qualitative analysis of the chemostat model with variable yield and a time delay
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Qualitative analysis of the chemostat model with variable yield and a time delay Qinglai Dong · Wanbiao Ma
Received: 10 October 2012 / Accepted: 3 January 2013 / Published online: 11 January 2013 © Springer Science+Business Media New York 2013
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. Keywords Chemostat · Time delay · Stability · Lyapunov–LaSalle invariance principle · Hopf bifurcation · Permanence 1 Introduction and statement of improved model The chemostat is an important laboratory apparatus used to culture microorganisms [1–3]. It is assumed that species grow in continuously stirred-tank fermenters which are fed continuously by a nutrient and the cells are drawn off continuously. Therefore, the chemostat is of both ecological and mathematical interest since its applicability
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). Q. Dong (B) School of Mathematics and Computer Science, Yanan University, Yan’an 716000, China e-mail: [email protected] W. Ma School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
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J Math Chem (2013) 51:1274–1292
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in many areas, for example, waste water treatment and the operation of industrial fermenters etc [2,3]. It is well known that the basic chemostat model with single microorganism and nutrient takes the following form [2,3]
˙ = D(S 0 − S) − δ −1 μ(S)X , S(t) X˙ (t) = (μ(S) − D)X,
(1.1)
where S(t) and X (t) denote concentrations of the nutrient and the microorganism at time t respectively, S0 denotes the input concentration of nutrient, D denotes the volumetric dilution rate (flow rate/volume), δ is yield term, the function μ(S) denotes the microbial growth rate and a typical choice for μ(S) is Monod kinetics function (Michaelis-Menten or Holling type II), which takes the form of μ(S) = μm S/(km +S), and satisfies the following conditions: μ(0) = 0, μ (S) > 0, lim S→+∞ μ(S) = μm < +∞. Here μm > 0 is called the maximal specific growth rate; km > 0 is the halfsaturation constant, such that μ(km ) = μm /2. Clearly, μ(S) is an increasing function of S over the entire interval [0, +∞). So far, the chemostat models with Monod kinetics for nutrient uptake and its various modified versions have been studied sufficiently, because they could be used to account for variou
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