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Global analysis of a delayed Monod type chemostat model with impulsive input on two substrates
 Jianzhi Cao^{1}Email author,
 Junyan Bao^{1} and
 Peiguang Wang^{2}
https://doi.org/10.1186/s1366201506233
© Cao et al. 2015
Received: 27 March 2015
Accepted: 26 August 2015
Published: 9 October 2015
Abstract
In this paper, a new Monod type chemostat model with delay and impulsive input on two substrates is considered. By using the global attractivity of a k times periodically pulsed input chemostat model, we obtain the bound of the system. By the means of a fixed point in a Poincaré map for the discrete dynamical system, we obtain a semitrivial periodic solution; further, we establish the sufficient conditions for the global attractivity of the semitrivial periodic solution. Using the theory on delay functional and impulsive differential equations, we obtain a sufficient condition with time delay for the permanence of the system.
Keywords
 Monod type
 globally attractivity
 nutrient recycling
 chemostat model
MSC
 34K45
 34K20
1 Introduction and the model
The chemostat is an important laboratory apparatus to study the growth of microorganism in a continuous environment. It has begun to occupy an increasing central role in ecological studies. As a tool in biotechnology, the chemostat plays an important role in bioprocessing and the chemostat has many applications in waste water treatment, production by genetically altered organisms, etc. Chemostats with periodic inputs are studied in [1–7], those with periodic washout rate in [8, 9], and those with periodic input and washout in [10]. The structure assigned to the organisms in the model accounts for the dependence of the growth on the past history of the cells, and hence it is capable of predicting the lag phases and transient oscillations observed in experiments. Many authors have directly incorporated time delays in the modeling equations and, as a result, the models take the form of delay differential equations [11–22].
Many scholars pointed out that it was necessary and important to consider models with periodic perturbations, since these models might be quite naturally exposed in many real world phenomena (for instance, food supply, mating habits, harvesting). In fact, almost perturbations occur in a moreorless periodic fashion. However, there are some other perturbations such as fires, floods, and drainage of sewage which are not suitable to be considered continuous. These perturbations bring sudden changes to the system. A chemostat model with time delays was first studied by Caperon [23] based on some experimental data. Unfortunately, the model proposed by Caperon created the possibility of a negative concentration of the substrate (nutrient). To correct this, Bush and Cook [24] investigated a model of the growth of one microorganism in the chemostat with a delay in the intrinsic growth rate of the organism but with no delay in the nutrient equation. They have also established that oscillations are possible in their model. Systems with sudden perturbations are involved in the impulsive differential equation, which have been studied intensively and systematically in [25, 26].
2 Preliminaries
In this section, we will give some notations and lemmas which will be used for our main results.
Let \(R_{+}=[0,\infty)\), \(R_{+}^{3}=\{(x_{1}, x_{2}, x_{3})\in R^{3}:x_{1}>0, x_{2}>0, x_{3}>0\}\). \(S_{1}(nT^{+})=\lim_{t\rightarrow nT^{+}}S_{1}(t)\), \(S_{2}(nT^{+})=\lim_{t\rightarrow nT^{+}}S_{2}(t)\), \(x(nT^{+})=\lim_{t\rightarrow nT^{+}}x(t)\), \(S_{1}(t)\), \(S_{2}(t)\) are leftcontinuous at \(t=nT\), \(x(t)\) is continuous at \(t=nT\).
Lemma 2.1
 (A_{0}):

the sequence \({t_{k}}\) satisfies \(0\leq t_{0}< t_{1}< t_{2}<\cdots\), with \(\lim_{t\to\infty}t_{k}=\infty\);
 (A_{1}):

\(\omega\in PC'(R_{+},R)\) and \(\omega(t)\) is leftcontinuous at \(t_{k}\), \(k\in N\).
Lemma 2.2
System (2.2) has a positive periodic solution \(u_{i}^{*}(t)=\frac{p_{i}\exp(D(tnT))}{1\exp(DT)}\) for all \(t\in(nT,(n+1)T]\), \(n\in Z_{+}\), which is globally uniformly attractive.
The proofs of Lemmas 2.1 and 2.2 are simple, we omit them here.
Lemma 2.3
([27])
 (i)
If \(r_{1}+r_{2}< r_{3}\), then \(\lim_{t\to\infty}x(t)=0\).
 (ii)
If \(r_{1}+r_{2}>r_{3}\), then \(\lim_{t\to \infty}x(t)=+\infty\).
Lemma 2.4
Lemma 2.5
Let \((S_{1}(t), S_{2}(t), x(t))\) be any solution of system (1.1) with initial values \((S_{1}(0^{+}), S_{2}(0^{+}), x(0))\in R_{+}^{3}\). There exists a constant \(L>0\) such that \(S_{1}(t)< L\), \(S_{2}(t)< L\), \(x(t)< L\).
Proof
3 Main results
Theorem 3.1
Proof
By a similar argument to the above, we know \(S_{2}(t)\to u_{2}^{*}(t)\) as \(t\to\infty\). This completes the proof. □
Theorem 3.2
Proof
Since the interval \([\bar{t},\bar{t}+\omega]\) is arbitrarily chosen (we only need t̄ to be large), we get \(x(t)\geq m_{2}\) for t large enough. In view of our arguments above, the choice of \(m_{2}\) is independent of the positive solution of (1.1) which satisfies \(x(t)\geq m_{2}\) for sufficiently large t.
4 Numerical analysis and discussion
In this paper, we introduce a growth time delay and pulse input nutrient into the Monod type chemostat model, and theoretically analyze the influence of them on the extinction of the population of the microorganism and the permanence of the system. In Section 3, we give the conditions for extinction and permanence of the microorganisms. Our main results show that if the impulsive periodic nutrient concentration inputs \(p_{1}\) and \(p_{2}\) are under a certain value, then the population of microorganisms will be eventually extinct. Contrarily, if the impulsive periodic nutrient concentration input \(p_{1}\) or \(p_{2}\) is over a certain value, it will be permanent. In this case, the microorganism is kept.
Declarations
Acknowledgements
We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This work was supported by the National Natural Science Foundation of China (No. 11271106), Natural Science Foundation of Hebei Province of China (No. A2013201232) and the Youth Foundation of Hebei University (No. 2014295).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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