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Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input
Advances in Difference Equations volumeÂ 2017, ArticleÂ number:Â 115 (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.
1 Introduction and model formulation
A chemostat is a classical bioreactor for microbes culture and has been widely applied in the field of microbiology and bioengineering [1, 2]. The chemostat model has attracted the attention of many scholars since it was introduced by Monod in 1942 [3]. These models include mathematical models [4â€“16] and experimental models [17â€“19]. A simple chemostat can be designed by using a pump or an overflow system (see FigureÂ 1), by which the volume of the chemostat can be controlled either [20].
It was found that some microbes can produce flocculation under the action of flocculating agents. This phenomenon makes it possible to harvest microbes by flocculating agents [21â€“32]. Recently, based on a classical simple chemostat model in which a microbial species consumes a single growth limiting substrate [3, 8], Tai et al. [33] have proposed a delayed differential equations (DDEs) model to describe the process of microbial continuous culture and harvest as follows:
where \(S(t)\), \(x(t)\) and \(F(t)\) represent the concentration of the substrate, microbia and flocculating agent at time t, respectively. D represents the velocity of medium, \(S_{0}\) and \(F_{0}\) represent the input concentration of substrate and flocculating agent, respectively. \(h_{1}\) and h represent the consumption of medium and the yield of microbia, respectively. \(h_{3}\) is the loss rate of flocculant. Ï„ represents the time involved converting nutrient into the microbia [34â€“45]. The authors found that the model can produce backward bifurcation and complex dynamics. By establishing analytic thresholds for the existence of backward bifurcation, they analyzed the local stability of the equilibria.
In model (1), the flocculating agent is assumed to be added into the chemostat continuously. While in practice, by considering resource savings and the growth cycle of microorganisms, the flocculating agent can be periodically added into the chemostat at some fixed moment. Thus we perfect the chemostat system by adding input channel of flocculants and output channel of flocculation by two pumps, respectively (see FigureÂ 2). It can be regulated through inputting flocculant to flocculate microbia according to the concentration of microbia. This process can be described by impulsive differential equations (IDEs). Impulsive differential equations, on the one hand, can fully reflect the actual control situation; on the other hand, they can guide the operator to implement the impulsive control strategy conveniently and accurately [46â€“51]. Thus, we propose a new continuous culture chemostat model with time delay and impulsive harvest as follows:
where T is the period of the impulsive effect, \(\gamma F_{0}\) is the input amount of flocculant at every impulsive period T. \(F(nT^{+})= \lim _{t\to nT^{+}}F(t)\), and \(F(t)\) is left continuous at \(t=nT\), i.e., \(F(nT)= \lim _{t\to nT^{}}F(t)\), \(S(t)\), \(x(t)\) are continuous for all \(t\geq0\), the details can be seen in [52, 53].
Let \(C_{+}=C([\tau,0],R^{3}_{+})\) be the Banach space, \(\psi=(\psi _{1}(s),\psi_{2}(s),\psi_{3}(s))^{T}\), \(\psi_{i}(\theta)\geq0\) (\(\tau\leq \theta\leq0\), \(i=1,2,3\)) the initial conditions are given as
The rest of the paper is organized as follows. In SectionÂ 2, we briefly introduce some concepts and fundamental results, which are necessary for future discussion. In SectionÂ 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 SectionÂ 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 SectionÂ 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
Let \(U: R_{+}\times R^{3}_{+}\rightarrow R_{+}\), \(H: R_{+}\times R_{+}\rightarrow R\) and \(U\in U_{0}\). Assume that
here H is continuous in \((n\omega, (n+1)\omega] \times R_{+}\) and \(\forall x\in R_{+}\), \(n \in N\), \(\lim _{(t,y)\to((n\omega )^{+},x)} H(t, y) = H((n\omega)^{+}, x)\) exist; \(\Upsilon_{n}: R_{+}\to R_{+}\) is nondecreasing. Let \(r(t)=r(t, 0, u_{0})\) be the maximal (minimal) solution of the scalar impulsive differential equation
existing on \([0, \infty)\). Then \(U(0^{+}, w_{0})\leq(\geq)\ u_{0}\) implies that \(U(t, w(t)) \leq(\geq)\ r(t)\), \(t \geq0\), where \(\omega(t)=\omega(t, 0, w_{0})\) is any solution of (4) existing on \([0, \infty)\).
Lemma 2.2
[54]
Let \(q_{1}\), \(q_{2}\), Ï„ be all positive constants and \(z(t)>0\) for \(t\in[\tau,0]\). Consider the following delay differential equation:
then

(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
Consider the following impulse differential inequalities:
where \(a(t),c(t)\in C(R_{+},R)\), \(b_{k}\geq0\), and \(d_{k}\) are constants. Assume
 \((A_{0})\) :

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]
Consider the following impulsive differential system:
for each solution \(\hbar(t)\) of (6), \(\hbar(t) \rightarrow\hbar^{*}(t)\) as \(t\rightarrow\infty\), where \(\hbar^{*}(t)=\frac{r_{1}}{r_{2}}+\frac{\mu e^{r_{2}(tnT)}}{1e^{r_{2}T}}\) for \(t\in(nT,(n+1)T]\).
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
Firstly, from the third and sixth equation of system (2), we have
By Lemma 2.3, we have \(F(t)\le\gamma F_{0}\frac {e^{DT}}{e^{DT}1}+\varepsilon_{2}\) for t large enough.
Let \(V(t)=e^{D\tau}\frac{h}{h_{1}}S(t\tau)+x(t)\). It is clear that \(V\in U_{0}\). Calculating the upper right derivative of \(V(t)\) along a solution of system (2), one can get
then by Lemma 2.3, we have \(\limsup _{t\to\infty} V(t)\le e^{D\tau}\frac{hS_{0}}{h_{1}}\), so \(V(t)\) is ultimately bounded. Thus, \(S(t)\) and \(x(t)\) are ultimately bounded and \(\limsup _{t\to\infty}S(t)\le S_{0}\), \(\limsup _{t\to\infty}x(t)\le e^{D\tau}\frac{hS_{0}}{h_{1}}\). Let \(M_{1}=S_{0}+\varepsilon\), \(M_{2}=e^{D\tau}\frac{hS_{0}}{h_{1}}+\varepsilon\), \(M_{3}=\gamma F_{0}\frac{e^{DT}}{e^{DT}1}+\varepsilon\), we have \(S(t)\leq M_{1}\), \(x(t)\leq M_{2}\), \(F(t)\le M_{3}\) for t large enough. The proof is completed.â€ƒâ–¡
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
Microbeextinction solution describes that microbes are eventually absent from system (2), thus we let \(x(t)=0\) in system (2), then system (2) changes to the following system:
Note that the variates \(S(t)\) and \(F(t)\) are independent of each other in system (7). Thus, by Lemma 2.4, we obtain that system (7) has a unique positive Tperiodic solution \((S^{*}(t), F^{*}(t))\) and for each solution \((S(t), F(t))\) of system (7), \(S(t) \rightarrow S^{*}(t)\) and \(F(t) \rightarrow F^{*}(t)\) as \(t\rightarrow\infty\), where
Therefore, we have the existence theorem for system (2).
Theorem 3.1
System (2) has a microbeextinction periodic solution \((S_{0}, 0, F^{*}(t))\).
Denote
We have the following theorem about the attractivity of the microbeextinction periodic solution of system (2).
Theorem 3.2
If \(R_{1}<1\), then the microbeextinction periodic solution \((S_{0}, 0, F^{*}(t))\) of system (2) is globally attractive.
Proof
Let \((S(t), x(t), F(t))\) be any solution of system (2) satisfying initial condition (3). Since \(R_{1}<1\), one can choose \(\varepsilon_{1}, \varepsilon_{2}>0\) such that
By the first equation of system (2), we have
According to Lemma 2.3, we have
Hence, there exists \(n_{1}\in N^{+}\) such that
for all \(t\geq n_{1}T\), where \(\varepsilon_{1}\) is an arbitrarily small positive constant.
By the third and sixth equations of system (2), we have
then consider the following impulsive differential system:
Then, by using Lemma 2.1, we have \(F(t)\geq q(t)\) and \(q(t)\rightarrow q^{*}(t)\) as \(t\rightarrow\infty\), \(q^{*}(t)\) is the periodic solution of (12), where \(q^{*}(t)=\frac {\gamma F_{0} e^{(D+h_{3}M_{2})(tnT)}}{1e^{(D+h_{3}M_{2})T}}\), \(nT< t\leq (n+1)T\). By Lemma 2.4, we have that \(q^{*}(t)\) is globally asymptotically stable. Hence there exists \(n_{2}\in N^{+}\) such that
for all \(t\geq n_{2}T\), where \(\varepsilon_{2}\) is an arbitrarily small positive constant.
From the second equation, (10) and (13), there exists a positive integer \(n_{3}>\max\{n_{1},n_{2}\}\), for \(t>n_{3}T+\tau\), we have
Consider the following delay differential equation:
Since (9) holds, by Lemma 2.2, we get \({\lim } _{t\to\infty}y(t)= 0\). Notice that for all \(\theta\in[\tau, 0]\), \(x(\theta) = y(\theta) = \psi_{2}(\theta)>0\) holds. By the comparison theorem in differential equation and the positivity of solution (with \(x(t)\geq0\)), we obtain \(\lim _{t\to\infty}x(t)= 0\).
Next, we will prove \(\lim _{t\to\infty}S(t)=S_{0}\) and \(\lim _{t\to\infty}F(t)=F^{*}(t)\). We assume that \(0< x(t)<\varepsilon\) holds for all \(t\geq0\) in the following discussion without loss of generality. One the one hand, by the first equation of system (2), one gets
Then, we have \(\liminf _{t\to\infty} S(t)\geq S_{0}\frac {D}{D+\varepsilon h_{1}}\). Thus, there exists \(T>0\) such that for any \(\varepsilon_{1}>0\),
for \(t>T\). Let \(\varepsilon\rightarrow0\), from (10) and (14) we have
for t large enough, then we have \(\lim _{t\to\infty}S(t)=S_{0}\).
On the other hand, from the third and sixth equations of system (2), we have
Then we get the following comparison system:
By Lemma 2.4, system (15) has a globally asymptotically stable positive periodic solution \(w_{1}^{*}(t)\), where \(w_{1}^{*}(t)=\frac{\gamma F_{0} e^{(D+\varepsilon h_{3})(tnT)}}{1e^{(D+\varepsilon h_{3})T}}\). Thus, by Lemma 2.1, we have \(F(t)\geq w_{1}(t)\) and \(w_{1}(t)\rightarrow w_{1}^{*}(t)\) as \(t\rightarrow\infty\). Therefore, there exists \(T'>0\) such that for any \(\varepsilon_{2}>0\),
for \(t>T'\). From the third and sixth equations of (2), one gets
Consider the following comparison system:
then we have \(F(t)\leq w_{2}(t)\) and \(w_{2}(t)\rightarrow F^{*}(t)\). Then, for any \(\varepsilon_{2} >0\), there exists \(T''>0\) such that
for \(t>T''\). Thus, by (16) and (18), for \(t>\max\{T',T''\}\), we have
Let \(\varepsilon\rightarrow0\), we have
for t large enough, thus we get \(\lim _{t\to\infty }F(t)=F_{0}\). This completes the proof.â€ƒâ–¡
3.2 Permanence
In this section, we prove that system (2) is persistent for \(R_{2}>1\). Firstly, we give the following lemma supporting our main conclusion. Denote
Lemma 3.1
If \(R_{2}>1\), then there exists a constant \(m_{4}>0\) such that
Proof
Let \(X(t)=(S(t), x(t), F(t))\) be any positive solution of system (2) with initial condition (3). We rewrite the second equation of system (2) as follows:
Define
Calculating the derivative of \(G(t)\) along the solution of (2) yields
Let \({m_{4}=\frac{(R_{2}1)D}{(R_{2}+1)h_{1}}}\). Since \(R_{2}>1\), it is clear that \(m_{4}>0\). For \(m_{4}\), one can choose \(\varepsilon _{1},\varepsilon_{2}>0\) small enough such that
where \(\varrho_{1}=\frac{DS_{0}}{D+h_{1}m_{4}}\varepsilon_{1}\), \(\varrho_{2}=\frac {\gamma F_{0}}{1e^{DT}}+\varepsilon_{2}\). Then, for any positive constant \(t_{0}\) and for all \(t\geq t_{0}\), we claim that the inequality \(x(t)< m_{4}\) cannot hold. Otherwise, there must exist a positive constant \(t_{0}\) such that \(x(t)< m_{4}\) for all \(t\geq t_{0}\). The first equation of system (2) leads to
By Lemma 2.3, there exists such \(T_{1} >t_{0}+\tau\) for \(t>T_{1}\) that
From (18), there exists such \(T_{2} >t_{0}+\tau\) for \(t\geq T_{2}\) that
Thus by (21), (22) and (19), for \(t>T_{3}=\max \{T_{1},T_{2}\}\), one gets
Let
We can prove that \(x(t)\geq x_{l}\) for all \(t\geq T_{3}\). Otherwise, there exists a constant \(T_{4}\geq0\) such that \(x(t)\geq x_{l}\) for \(t\in[T_{3},T_{3}+\tau+T_{4}]\), \(x(T_{3}+\tau +T_{4})=x_{l}\) and \(\dot{x}(T_{3}+\tau+T_{4})\leq0\). Then from the second equation of (2), (20) and (23), we have
which is a contradiction. Hence \(x(t)\geq x_{l}>0\) for all \(t\geq T_{3}\). Inequality (23) implies
which implies \(G(t)\rightarrow+\infty\) as \(t\rightarrow+\infty\). This is a contradiction to \(G(t)\leq M(1+M\tau e^{DM})\) for t large enough. Therefore, for any positive constant \(t_{0}\), the inequality \(x(t)< m_{4}\) cannot hold for all \(t\geq t_{0}\).
Step II: From Step I, we only need to consider:

(i)
\(x(t)>m_{4}\) for all t large enough;

(ii)
\(x(t)\) oscillates about \(m_{4}\) for all large t.
However, case (i) is obviously the result of this theorem, so we only need to consider case (ii), in which we shall show that \(x(t)>m_{2}\) for all large t, where
First, we notice that there would be two positive arbitrarily big constants tÌ„, Ï† such that \(x(t)< m_{4}\) for \(\bar{t}< t<\bar {t}+\varphi\) and \(x(\bar{t})=x(\bar{t}+\varphi)=m_{4}\). Second, there exists a constant \(0< T_{5}<\tau\) such that \(x(t)>\frac {m_{4}}{2}\) for all \(\bar{t}\leq t\leq\bar{t}+T_{5}\). Because \(x(t)\) is not affected by impulses and, moreover, \(x(t)\) is bound and continuous, then we conclude that \(T_{5}\) is independent of the choice of tÌ„. Next, according to the position of Ï†, \(T_{5}\), Ï„, there will be three cases we should discuss.
Case ii(a): \(\varphi\leq T_{5}\), obviously our aim is obtained.
Case ii(b): \(T_{5}<\varphi\leq\tau\). By (22), the second equation of (2) implies
for \(\bar{t}< t \leq\bar{t}+\varphi<\bar{t}+\tau\). Then we have
for \(\bar{t}< t \leq\bar{t}+\varphi \leq\bar{t}+\tau\), notice \(x(\bar{t})=m_{4}\), one can get
for \(\bar{t}< t \leq\bar{t}+\varphi \leq\bar{t}+\tau\). Thus we have \(x(t)\geq m_{2}\) for \(\bar{t}< t \leq\bar{t}+\varphi\).
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 tÌ„ 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.â€ƒâ–¡
Theorem 3.3
For \(R_{2}>1\), then system (2) will be permanent.
Proof
Let \(X(t)=(S(t), x(t), F(t))\) be any positive solution of system (2) with initial condition (3). From the first equation of system (2), we get
then we have
Thus there exists a constant Îµ small enough such that
for t large enough. And from (13) we have
for t large enough.
Set
Thus D is a bounded compact region and every solution of system (2) will eventually enter and remain in region D, then system (2) is permanent. The proof of Theorem 3.3 is completed.â€ƒâ–¡
4 Control strategy of continuous microbial culture and harvest
In SectionÂ 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.
Denote
According to Theorem 3.2, we have that if \(T< T^{*}\) or \(F_{0}>F_{0}^{*}\) or \(\tau>\tau^{*}\), the microbeextinction periodic solution \((S_{0}, 0, F^{*}(t))\) is globally attractive. That means the microbial continuous cultivation and harvest have failed. And from Theorem 3.3, we know that if \(T>T_{*}\) or \(F_{0}< F_{0*}\) or \(\tau<\tau_{*}\), then system (2) is permanent. That means we can achieve the process of microbial cultivation by increasing the time interval, or reducing the input amount of flocculant, or shortening the growth delay.
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 (Ï„).
Next we will verify the effectiveness of control strategy by an example and some numerical simulations. Let \(D=0.3\), \(S_{0}=2.8\), \(h_{1}=0.4\), \(h=0.3\), \(h_{2}=0.15\), \(h_{3}=0.02\), \(\gamma=1\), and let the initial value be \((1,1,0)\). We get the following system:
To investigate the effects of key parameters on the system, we assume \(\tau=2\), \(T=2\), \(F_{0}=1\). By simple calculation, we have \(R_{1}=0.9910<1\). FigureÂ 3 shows that the microbeeradication periodic solution \((S_{0}, 0, F^{*}(t))\) is globally attractive. That means microbial continuous cultivation and harvest have failed because the microbe will be eventually extinct. To achieve microbial continuous cultivation and harvest, we can take three kinds of control strategies.

(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. FigureÂ 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 FigureÂ 5). FigureÂ 5 also shows that system (26) has an asymptotically stable periodic solution.

(iii)
We can decrease time delay Ï„ (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. FigureÂ 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 TableÂ 1.
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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).
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TZ and WM designed the study and carried out the analysis. TZ, WM and XM contributed to writing the paper. TZ performed numerical simulations. All authors read and approved the final manuscript.
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Zhang, T., Ma, W. & Meng, X. Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input. Adv Differ Equ 2017, 115 (2017). https://doi.org/10.1186/s1366201711639
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DOI: https://doi.org/10.1186/s1366201711639