Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input
 Tongqian Zhang^{1, 2, 3},
 Wanbiao Ma^{1}Email author and
 Xinzhu Meng^{2, 3}
https://doi.org/10.1186/s1366201711639
© The Author(s) 2017
Received: 26 December 2016
Accepted: 29 March 2017
Published: 20 April 2017
Abstract
A mathematical model describing continuous microbial culture and harvest in a chemostat, incorporating a control strategy and defined by impulsive differential equations, is presented and investigated. Theoretical results indicate that the model has a microbeextinction periodic solution, which is globally attractive if the threshold \(R_{1}\) is less than unity, and the model is permanent if the threshold \(R_{2}\) is greater than unity. Further, we consider the control strategy under time delay and periodical impulsive effect. Analysis shows that continuous microbial culture and harvest process can be implemented by adjusting time delay, impulsive period or input amount of flocculant. Finally, we give an example with numerical simulations to illustrate the control strategy.
Keywords
MSC
1 Introduction and model formulation
The rest of the paper is organized as follows. In Section 2, we briefly introduce some concepts and fundamental results, which are necessary for future discussion. In Section 3, we focus our attention on the global property of system (2), including the existence, global attractivity of the microbeextinction periodic solution and the permanence of system (2). In Section 4, we give the threshold of key parameters of system (2) and discuss the control strategy. We finally give a conclusion and numerical simulations in Section 5, from which it can be seen that all simulations agree with the theoretical results.
2 Preliminaries
In this section, we give some useful lemmas.
Let \(f=(f_{1}, f_{2}, f_{3})^{T}\) be the map defined by the righthand side of the anterior three equations of system (2). Let \(R_{+} = [0,\infty)\), \(R^{3}_{+}=\{x\in R^{3}: x \geq0\}\), \(\Omega=\operatorname{int}R^{3}_{+}\). Let \(U:R_{+}\times R_{+}^{3}\rightarrow R_{+}\). If U satisfies the following conditions: (1) V is continuous in \(((n1)T,nT]\times R_{+}^{3}\), \(n\in N\), and for each \(x\in R_{+}^{3}\), \(\lim _{(t,z)\to((n1)T^{+},x)}U(t,z)=U((n1)T,x)\) and \(\lim _{(t,z)\to(nT^{+},x)}U(t,z)=U(nT^{+},x)\) exists; (2) U is locally Lipschitzian in x. Then U is said to belong to class \(U_{0}\).
Lemma 2.1
Lemma 2.2
[54]
 (i)
if \(q_{1}< q_{2}\), then \(\lim _{t\to\infty}z(t)=0\);
 (ii)
if \(q_{1}>q_{2}\), then \(\lim _{t\to\infty }z(t)=\infty\).
Lemma 2.3
 \((A_{0})\) :

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

\(u\in PC'(R_{+},R)\) and \(u(t)\) is leftcontinuous at \(t_{k}\), \(k\in N\). Then$$\begin{aligned} u(t) \leq (\geq)&u(t_{0}) \prod_{t_{0}< t_{k}< t}d_{k} \exp \biggl( \int_{t_{0}}^{t}a(s)\,ds \biggr)+ \sum _{t_{0}< t_{k}< t} \biggl(\prod_{t_{k}< t_{j}< t}d_{j} \exp \biggl( \int_{t_{k}}^{t}a(s)\,ds \biggr) \biggr)d_{k} \\ &{}+ \int_{t_{0}}^{t}\prod_{s< t_{k}< t}b_{k} \exp \biggl( \int_{s}^{t}a(\theta )\,d\theta \biggr)c(s)\,ds, \quad t \geq t_{0}. \end{aligned}$$
Lemma 2.4
[43]
Lemma 2.5
There exist constants \(M_{1},M_{2},M_{3}>0\) such that \(S(t)\leq M_{1}\), \(x(t)\leq M_{2}\), \(F(t)\leq M_{3}\) for each solution of (2) with all t large enough.
Proof
3 Global dynamical analysis for system (2)
In this section, we discuss the global dynamics of model (2), including the existence and global attractivity of the microbeextinction periodic solution and the permanence.
3.1 Existence and global attractivity of the microbeextinction periodic solution
Theorem 3.1
System (2) has a microbeextinction periodic solution \((S_{0}, 0, F^{*}(t))\).
Theorem 3.2
If \(R_{1}<1\), then the microbeextinction periodic solution \((S_{0}, 0, F^{*}(t))\) of system (2) is globally attractive.
Proof
3.2 Permanence
Lemma 3.1
Proof
 (i)
\(x(t)>m_{4}\) for all t large enough;
 (ii)
\(x(t)\) oscillates about \(m_{4}\) for all large t.
Case ii(a): \(\varphi\leq T_{5}\), obviously our aim is obtained.
Case ii(c): \(\varphi\geq\tau\). We have proved \(x(t)\geq m_{2}\) for \(\bar{t}< t \leq\bar{t}+\tau\). For \(\bar{t}+\tau\leq t\leq\bar{t}+\varphi \), we can analyze and prove \(x(t)\geq m_{2}\) as the proof for the above claim. Because of the arbitrariness of interval \([\bar{t},\bar {t}+\varphi]\) and because t̄ is an arbitrarily big constant, we have that \(x(t)\geq m_{2}\) holds for t large enough. Finally, notice that the choice of \(m_{2}\) is independent of the positive solution of (2), which satisfies that \(x(t)\geq m_{2}\) for t large enough. This completes the proof of Lemma 2.4. □
Theorem 3.3
For \(R_{2}>1\), then system (2) will be permanent.
Proof
4 Control strategy of continuous microbial culture and harvest
In Section 3, we obtain the threshold values \(R_{1}\) and \(R_{2}\) associated with microbial extinction and existence. Next, we discuss the control strategy of continuous microbial culture and harvest by analyzing the key parameters of the threshold.
5 Discussion and numerical simulations
In this paper, to achieve the continuous microbial culture and harvest, we improve the classic chemostat model and propose a new chemostat model with time delay and periodical flocculant input. Our main aim is to investigate the control strategy of continuous microbial culture and harvest. By using the theory of impulsive delayed differential equations, global properties of the system are discussed. We prove that if \(R_{1}<1\), then the microbe will be eventually extinct, and if \(R_{2}>1\), the microbe species is permanent. Based on the threshold values associated with microbial extinction and existence, we consider the control strategy. Results show that we can culture microbia continuously and harvest microbia many times by adjusting the time interval (T) or the input amount of flocculant (\(\gamma F_{0}\)), or the time delay (τ).
 (i)
We can reduce the input of flocculant \(F_{0}\) (from 1 to 0.4). By calculating, we have \(R_{2}=1.0647>1\). According to Theorem 3.3, system (26) is permanent. A lower amount of flocculant can increase population microbia in the medium so that microbe can be cultured continuously in the chemostat system. Figure 4 shows that system (26) is permanent.
 (ii)
We can increase the time travail T (from 2 to 10). Longer time travail can decrease input mount of flocculant indirectly and increase population microbia in the medium, which makes microbia cultured and harvested continuously. By calculating, we have \(R_{2}=1.0069>1\). According to Theorem 3.3, system (26) is permanent (see Figure 5). Figure 5 also shows that system (26) has an asymptotically stable periodic solution.
 (iii)We can decrease time delay τ (from 2 to 0.5) by some biotechnology and biological engineering. Reduction of growth time delay makes the microbial growth cycle shorter, which is beneficial for microbial continuous cultivation. Figure 6 shows that system (26) is permanent. Moreover, system (26) has an asymptotically stable periodic solution (where \(R_{2}=1.1432>1\)). Detailed parameter values, thresholds and states of system (26), please see Table 1.Table 1
Values of parameters, threshold and state of the system
τ
D
\(\boldsymbol{S_{0}}\)
\(\boldsymbol{h_{1}}\)
h
\(\boldsymbol{h_{2}}\)
\(\boldsymbol{h_{3}}\)
T
γ
\(\boldsymbol{F_{0}}\)
\(\boldsymbol{R_{i}}\) ( i = 1,2)
Microbe
Figure
2
0.3
2.8
0.4
0.3
0.15
0.02
2
1
1
\(R_{1}=0.9910<1\)
Eradication
Figure 3
2
0.3
2.8
0.4
0.3
0.15
0.02
2
1
0.4
\(R_{2}=1.0647>1\)
Permanence
Figure 4
2
0.3
2.8
0.4
0.3
0.15
0.02
10
1
1
\(R_{2}=1.0069>1\)
Permanence
Figure 5
0.5
0.3
2.8
0.4
0.3
0.15
0.02
2
1
1
\(R_{2}=1.1432>1\)
Permanence
Figure 6
Declarations
Acknowledgements
The first author and the third author were supported by the National Natural Science Foundation of China (No. 11371230), Shandong Provincial Natural Science Foundation, China (No. ZR2015AQ001), the Project for Higher Educational Science and Technology Program of Shandong Province of China (No. J13LI05), SDUST Research Fund (2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China. The second author was supported by the National Natural Science Foundation of China (No. 11471034).
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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