- Research
- Open Access
A class of dynamic models describing microbial flocculant with nutrient competition and metabolic products in wastewater treatment
- Keying Song^{1},
- Wanbiao Ma^{1}Email author,
- Songbai Guo^{2} and
- Hai Yan^{3}
https://doi.org/10.1186/s13662-018-1473-6
© The Author(s) 2018
- Received: 19 September 2017
- Accepted: 4 January 2018
- Published: 23 January 2018
Abstract
In this paper, based on the related theories of microbial continuous culture, fermentation dynamics, and microbial flocculant, a class of dynamic models which describe microbial flocculant with resource competition and metabolic products in wastewater treatment is proposed. By analyzing the global dynamic properties of the model, the feasibility of employing microbial metabolites as flocculant to remove harmful microorganisms is considered.
Keywords
- wastewater treatment
- microbial flocculant
- dynamic model
- global stability
MSC
- 92B05
- 34D20
1 Introduction and statement of model formulation
With the increase in global population and the rapid development of industry and agriculture, water consumption increases sharply. Wastewater containing harmful substances, such as harmful microbes and heavy metal elements, is discharged into lakes, rivers, etc., and the ecological environment is seriously polluted. Water pollution threatens human health and the sustainable development of the whole society. In order to control water pollution, a series of methods of wastewater treatment have been proposed [1]. The methods of wastewater treatment typically involve physical treatment (such as filtering), chemical treatment (such as electrolysis), and biological treatment (such as aerobic/anaerobic method). Biological treatment is to remove harmful microorganisms in wastewater by using the metabolism of organisms such as bacteria, molds, or protozoa. Biological treatment largely depends on the process of adsorption degradation to organic material solids, harmful microbes, etc. [2, 3]. Because of its cost effectiveness, superior performance, and environmental friendliness, biological treatment has been generally employed in wastewater treatment [3, 4].
In the past few years, the flocculation sedimentation method has been widely used in wastewater treatment and collecting microorganisms [5, 6]. Harmful substances in wastewater can be degraded by using flocculants. There are many kinds of flocculants, which mainly include two categories: inorganic flocculants and organic flocculants [7]. Inorganic flocculants mainly include inorganic coagulants and inorganic high molecular flocculants such as iron salt, aluminum salt, etc. [8]. Related studies have shown that inorganic flocculants are likely to cause secondary pollution which has a negative impact on human health [9]. Organic flocculants include synthetic organic polymer flocculants, natural organic high molecular flocculants, and microbial flocculants such as polyacrylamide [4]. With the progress of science and technology, the research and development of different kinds of microbial flocculants have been carried out rapidly.
As a new type of flocculants, microbial flocculants have been widely used in wastewater treatment [3, 10, 11]. Microbial flocculants are mainly divided into three types according to their compositions [12]: (1) microbial cells such as some bacteria; (2) microbial cell extracts such as yeast cell wall glucan; and (3) microbial metabolites. Microbial flocculants have many advantages such as biodegradation, no secondary pollution, non-toxicity, high security, etc. A lot of experimental studies have been carried out on microbial flocculants [13].
In the research of microbial flocculants, flocculant-producing bacteria have received extensive attention. As early as 1935, Butterfield successfully isolated flocculant-producing bacteria from activated sludge. Since 1970s, researchers have isolated a variety of flocculant-producing bacteria from the environment. Kurane and Tomizuka isolated Rhodococcus erythropolis with flocculant activity from nature and first got microbial flocculant NOC-1 [14]. The flocculant has a very good effectiveness on the livestock wastewater treatment, expansion of sludge treatment, and so on.
The chemostat is an important experimental device which has been used extensively in microbial continuous culture and environmental pollution treatment [15–19]. The chemostat can be used to model nutrient consumption and microbial competition [20–27]. In some cases, microorganisms can produce toxins or inhibitors against its competitors. Chao and Levin performed a basic experiment about antibiotic inhibitors [28]. In the evolutive chemostat model, in order to make it more biologically significant, inhibitors or toxins are introduced to competitive models [29–33]. For example, in [29, 33], and [34], the chemostat models with internal and external inhibitors were established, respectively.
In this paper, motivated by microbial flocculants and model (1), we propose a model which may have potential applications of microbial metabolites as flocculants in the control of harmful microorganisms in wastewater treatment.
If \(\beta -D\alpha =d_{3}=0\), model (2) has a similar structural form to model (1). In [32], model (1) has been completely analyzed by reducing the dimension.
The remaining part of this paper is organized as follows. In Section 2, global existence, uniqueness, nonnegativity, and boundedness of the solutions of model (3) with the initial condition (4) are investigated. The existence of the equilibria of model (3) and their stability are considered in Section 3. Finally, some numerical simulations and conclusions are given in Section 4.
2 Global existence, uniqueness, nonnegativity, and boundedness of solutions
In this section, the existence, uniqueness, nonnegativity, and boundedness of the solutions of model (3) with the initial condition (4) are studied.
Theorem 2.1
Proof
From local existence and uniqueness theorems of solutions [38], we have that the solution \((S(t), X(t), Y(t), P(t))\) of model (3) with the initial condition (4) is existent and unique on \([0,\delta)\) for some constant \(\delta >0\). Furthermore, it is easy to show that the solution \((S(t), X(t), Y(t), P(t))\) is also nonnegative on \([0,\delta)\).
From (5), it easily follows that \(\limsup_{t\rightarrow +\infty }H(t) \leq N(d_{2}+1)\). Hence, from the first equation of model (3) and \(\limsup_{t\rightarrow +\infty }(X(t)+Y(t))\leq N(d_{2}+1)\), we easily get that \(\liminf_{t\rightarrow +\infty }S(t)\geq Q_{0}\).
The proof is complete. □
Furthermore, we can easily show the following result.
Corollary 2.1
The set \(G= \{U=(S,X,Y,P)\in \mathbb{R}^{+}_{4}: H=NS+X+(d_{2}+1)Y+P \leq N(d_{2}+1), Q_{0} \leq S \leq 1\} \) attracts all of solutions of model (3) and is positively invariant with respect to model (3).
The subsequent discussion will be confined to the closed set G.
3 The existence of equilibria and their stability analysis
3.1 The existence of equilibria
- (a)
Model (3) always has the boundary equilibrium \(E_{0}=(S _{0},X_{0},Y_{0},P_{0})=(1,0,0,0)\) without harmful microorganism, microbial flocculant-producing bacterium, and microbial flocculant.
- (b)
If \(b_{1}>1\), then model (3) has the second boundary equilibrium \(E_{1}=(S_{1},X_{1},Y_{1},P_{1})=(\frac{1}{b_{1}},\frac{b _{1}-1}{a_{1}},0,0)\) without microbial flocculant-producing bacterium and microbial flocculant.
- (c)
If \(c_{1}>1\), then model (3) has the third boundary equilibrium that is harmful microorganism free equilibrium \(E_{2}=(S _{2},X_{2},Y_{2},P_{2})=(\frac{1}{c_{1}},0,\frac{c_{1}-1}{a_{2}}, \frac{(d _{1}+d_{2})(c_{1}-1)}{a_{2}})\).
- (d)If \(b_{1}>c_{1}\) and \(b_{2}c_{1}(d_{1}+d_{2})(c_{1}-1)>a_{2}(b _{1}-c_{1})\), then model (3) has a positive equilibrium \(E^{*}=(S^{*},X^{*},Y^{*},P^{*})\), where$$\begin{aligned}& S^{*}=\frac{1}{c_{1}},\qquad X^{*}=\frac{b_{2}c_{1}(c_{1}-1)(d_{1}+d_{2})-a _{2}(b_{1}-c_{1})}{a_{1}b_{2}c_{1}(d_{1}+d_{2})+a_{2}d_{3}(b_{1}-c _{1})}, \\& Y^{*}=\frac{P^{*}(1+d_{3}X^{*})}{d_{1}+d_{2}},\qquad P^{*}=\frac{b_{1}-c _{1}}{b_{2}c_{1}}. \end{aligned}$$
The biological meanings of the conditions for the existence of the equilibria will be given in Remarks 3.1, 3.2, 3.3, and 3.4 in Subsection 3.2.
3.2 Stability of the boundary equilibria
Stability of the boundary equilibrium \(E_{0}\) is given as follows.
Theorem 3.1
If \(b_{1}<1\) and \(c_{1}<1\), then the boundary equilibrium \(E_{0}\) is globally asymptotically stable with respect to G.
Proof
By calculating, the characteristic equation at the boundary equilibrium \(E_{0}\) is given by \((\lambda +1)^{2}(\lambda -b_{1}+1)(\lambda -c _{1}+1)=0\). Clearly, the boundary equilibrium \(E_{0}\) is locally asymptotically stable since \(b_{1}<1\) and \(c_{1}<1\). Thus, we only need to show that \(E_{0}\) is globally attractive.
The proof is completed. □
Remark 3.1
The conditions \(b_{1}<1\) and \(c_{1}<1\) in Theorem 3.1 are equivalent to \(b_{1}s^{0}< D\) and \(c_{1}s^{0}< D\) for model (2). From biological points of view, stability of the boundary equilibrium \({E_{0}}=(1,0,0,0)\) indicates that, for the fixed input concentration of nutrient \(s^{0}\), when both the growth rate of harmful microorganism \(b_{1}\) and microbial flocculant-producing bacterium \(c_{1}\) are smaller comparing with the washout rate D, as the time t goes on, the concentrations of harmful microorganism, microbial flocculant-producing bacterium, and microbial flocculant, \(x(t)\), \(y(t)\), and \(p(t)\), tend to zero, and the concentration of nutrient \(s(t)\) tends to some constant value.
Stability of the boundary equilibrium \(E_{1}\) is given as follows.
Theorem 3.2
If \(b_{1}>1\) and \(c_{1}< b_{1}\), then the boundary equilibrium \(E_{1}\) is locally asymptotically stable. In addition, if \(a_{1}>0\), \(b_{2}>0\), and \(b_{2}c_{1}(d_{1}+d_{2})(b_{1}-1) < a_{2}(b_{1}-c_{1})\), then the boundary equilibrium \(E_{1}\) is globally asymptotically stable with respect to \(G_{1}\), where \(G_{1}=\{(S,X,Y,P)\mid (S,X,Y,P)\in G, X>0\}\).
Proof
By calculating, the characteristic equation of model (3) at the boundary equilibrium \(E_{1}\) is given by \((\lambda +1-\frac{c _{1}}{b_{1}} ) (\lambda +1 +\frac{d_{3}(b_{1}-1)}{a_{1}} )( \lambda +1)(\lambda +b_{1}-1)=0\). Clearly, the boundary equilibrium \(E_{1}\) is locally asymptotically stable since \(b_{1}>1\) and \(c_{1}< b_{1}\). Thus, we only need to show that \(E_{1}\) is globally attractive.
The proof is completed. □
Remark 3.2
The conditions \(c_{1}< b_{1}\) and \(b_{1}>1\) in Theorem 3.2 are equivalent to \(c_{1}< b_{1}\) and \(D< b_{1}s^{0}\) for model (2). From biological points of view, stability of the boundary equilibrium \({E_{1}}= (\frac{1}{b_{1}},\frac{b_{1}-1}{a_{1}},0,0)\) implies that, for the fixed input concentration of nutrient \(s^{0}\), when (i) the growth rate of harmful microorganism \(b_{1}\) is larger comparing with the growth rate of microbial flocculant-producing bacterium \(c_{1}\), and (ii) the growth rate of harmful microorganism \(b_{1}\) is larger comparing with the washout rate D, as the time t goes on, the concentrations of microbial flocculant-producing bacterium and microbial flocculant, \(y(t)\) and \(p(t)\), tend to zero, and the concentrations of nutrient and harmful microorganism, \(s(t)\) and \(x(t)\), tend to some constant values.
Stability of the boundary equilibrium \(E_{2}\) is given as follows.
Theorem 3.3
If \(b_{2}c_{1}(d_{1}+d_{2})(c_{1}-1)> a_{2}(b_{1}-c_{1})\) and \(c_{1}>1\), then the boundary equilibrium \(E_{2}\) is locally asymptotically stable. In addition, if \(a_{1}>0\) and \(c_{1}>b_{1}\), then the boundary equilibrium \(E_{2}\) is globally asymptotically stable with respect to \(G_{2}\), where \(G_{2}=\{(S,X,Y,P)\mid (S,X,Y,P)\in G, Y>0\}\).
Proof
The characteristic equation of model (3) at the boundary equilibrium \(E_{2}\) is given by \((\lambda +1) (\lambda +1-\frac{b_{1}}{c_{1}} +\frac{b_{2}(d_{1}+d_{2})(c_{1}-1)}{a_{2}} )(\lambda +1)(\lambda +c _{1}-1)=0\). Clearly, the boundary equilibrium \(E_{2}\) is locally asymptotically stable since \(c_{1}>1\) and \(b_{2}c_{1}(d_{1}+d_{2})(c _{1}-1)> a_{2}(b_{1}-c_{1})\). Thus, we only need to show that the boundary equilibrium \(E_{2}\) is globally attractive.
The proof is completed. □
Remark 3.3
The conditions \(b_{2}c_{1}(d_{1}+d_{2})(c_{1}-1) > a_{2}(b_{1}-c_{1})\) and \(c_{1}>1\) in Theorem 3.3 are equivalent to \(b_{2}c_{1} \beta (c_{1}s^{0}-D) > D^{2}a_{2}(b_{1}-c_{1})\) and \(c_{1}s^{0}>D\) for model (2). From biological points of view, stability of the boundary equilibrium \({E_{2}}=(\frac{1}{c_{1}},0, \frac{c_{1}-1}{a_{2}},\frac{(d_{1}+d_{2})(c_{1}-1)}{a_{2}})\) indicates that, for the fixed input concentration of nutrient \(s^{0}\), when (i) the growth rate of microbial flocculant-producing bacterium \(c_{1}\) or the removing rate of microbial flocculant-producing bacterium \(b_{2}\) is larger comparing with the washout rate D, and (ii) the growth rate of harmful microorganism \(b_{1}\) is sufficiently small, as the time t goes on, the concentration of harmful microorganism \(x(t)\) tends to zero (i.e., harmful microorganism can be removed successfully), and the concentrations of nutrient, microbial flocculant-producing bacterium, and microbial flocculant, \(s(t)\), \(y(t)\), and \(p(t)\), tend to some constant values.
3.3 Stability of the positive equilibrium
For stability of the positive equilibrium \(E^{*}\), we have the following result.
Theorem 3.4
If \(b_{1}>c_{1}\) and \(b_{2}c_{1}(d_{1}+d_{2})(c_{1}-1)>a_{2}(b_{1}-c _{1})\), then the positive equilibrium \(E^{*}\) is unstable.
Proof
The proof is completed. □
Remark 3.4
Instability of the positive equilibrium \(E^{*}\) in Theorem 3.4 indicates that, under certain conditions, the evolution among nutrient, harmful microorganism, microbial flocculant-producing bacterium, and microbial flocculant may become more complicated. Whether or not harmful microorganism can be removed will depend on the initial state \((S_{0},X_{0},Y_{0},P_{0})\).
4 Numerical simulations and discussions
4.1 Numerical simulations
However, the numerical simulations strongly suggest that, in the regions \(D_{1}\) and \(D_{2}\) in Figures 7 and 8, respectively, the boundary equilibrium \(E_{1}\) should also be globally asymptotically stable.
4.2 Discussions
Firstly, it should be pointed out that we cannot get global asymptotic stability of the boundary equilibrium \(E_{1}\) in the regions \(D_{1}\) and \(D_{2}\) in Figures 7 and 8 because of the difficulties in constructing suitable Lyapunov functions.
Secondly, in biology, when model (3) is applied to some wastewater treatment, and the state variable \(x(t)\) represents harmful microorganism to be removed, the local/global asymptotic stability of the boundary equilibrium \(E_{1}=(S_{1},X_{1},Y_{1},P_{1})=\) \((\frac{1}{b_{1}},\frac{b_{1}-1}{a_{1}},0,0)\) in Theorem 3.2 implies that harmful microorganism cannot be removed by using the microbial flocculant because of insufficiency of the growth of microbial flocculant-producing bacterium and microbial flocculant.
On the other hand, the local/global asymptotic stability of the boundary equilibrium \(E_{2}=(S_{2},X_{2},Y_{2},P_{2})=\) \((\frac{1}{c_{1}},0,\frac{c _{1}-1}{a_{2}}, \frac{(d_{1}+d_{2})(c_{1}-1)}{a_{2}})\) in Theorem 3.3 implies that harmful microorganism can be removed successfully by using microbial flocculant under suitable assumptions. Hence, Theorem 3.3 gives some feasible strategy to control harmful microorganism in applications. However, from Theorem 3.4, we have that the positive equilibrium \(E^{*}\) is always unstable if it exists, and there exists a bistable region (i.e., both the boundary equilibrium \(E_{1}\) and the boundary equilibrium \(E_{2}\) are asymptotically stable). Thus, Theorem 3.4 implies that the controlling of harmful microorganism will become more complicated, and the initial value \((S_{0},X_{0},Y_{0},P_{0})\) will play an important role.
Finally, model (2) can be also considered as a class of chemostat competition models. It is well known that the general chemostat competition models have been studied systematically [15] and ‘competitive exclusion principle’ holds in general chemostat competition models [41–44]. The so-called competitive exclusion principle refers to the phenomenon that different species compete in a shortage of resources, making a species excluded or replaced in competition. In 1980, Hansen and Hubbel [43] observed the competitive exclusion principle in the biological chemostat experiments. Some important research shows that, if inhibitors are considered (for example, model (1)), the competition results may be changed [32]. In model (2), microbial flocculant-producing bacterium and microbial flocculant are considered and Theorems 3.2 and 3.3 show similar properties.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11471034).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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