Abstract
The rhizosphere microbe plays an important role in removing the pollutant generated from industrial and agricultural production. To investigate the dynamics of the microbial degradation, a nonlinear mathematical model of the rhizosphere microbial degradation is proposed based on impulsive state feedback control. The sufficient conditions for existence of the positive order-1 or order-2 periodic solution are obtained by using the geometrical theory of the semicontinuous dynamical system. We show the impulsive control system tends to an order-1 periodic solution or order-2 periodic solution if the control measures are achieved during the process of the microbial degradation. Furthermore, mathematical results are justified by some numerical simulations.
1. Introduction
The constructed wetland is regarded as the most efficient and cost-effective system that involves microbes consuming pollutant generated from industrial and agricultural production. However, degradation ability of the microbe in the constructed wetland is affected by many factors such as temperature, pH, dissolved oxygen, and substrate concentration [1]. Studies have shown that excessive pollutant strongly affects the dynamic behavior of microbe, which can lead to the incidence of the sustained oscillation and decrease degradation ability of the microbe under certain conditions [2]. For the purpose of improving degradation ability of the microbe and decreasing the inhibition effect of the substrate, it is necessary to control the pollutant concentration lower than a certain level.
The removal of the sewage from a water body by the biodegradation has been investigated by many researchers [3–15]. In [14], pollutant removal process of a constructed wetland is simulated by the following model: where denotes the mangrove growth rate which is chosen to follow Monod kinetics . is the mangrove biomass concentration, denotes the total nitrogen concentration in wastewater, and shows the total nitrogen concentration in soil solution. The meaning of other parameters is also given in [14]. Scott et al. [15] propose a mathematical model for dispersal of bacterial inoculants colonizing the water rhizosphere to describe bacterial growth and movement in the rhizosphere.
Recently, there are many studies on the geometric theory of semicontinuous dynamical system which are applied into the chemostat model [16–20]. Considering the product inhibition during the microorganism culture, Guo and Chen [16] develop a mathematical model concerning a chemostat with impulsive state feedback control to investigate the periodicity of bioprocess and an order-1 periodic solution is obtained by the existence criteria of periodic solution. The mathematical model of a continuous bioprocess with impulsive state feedback control is formulated [18] and the analysis of the bioprocess stability is presented. As far as I know, little information is available concerning the dynamical research on the microbial degradation of the rhizosphere. The main purpose of this paper is to construct a mathematical model of the microbial degradation with impulsive state feedback control and understand the dynamics of the microbial degradation in the constructed wetland.
The paper is organized as follows: a mathematical model of the microbial degradation with impulsive state feedback control is proposed in Section 2. In Section 3, the qualitative analysis of system without impulsive control is given. Furthermore, the existence of order-1 and order-2 periodic solutions is investigated in Section 4. Finally, we give some numerical simulations and a brief discussion.
2. Model Description and Preliminaries
In the constructed wetland, it is showed that the rhizosphere microbe makes an important contribution to the degradation of pollutant [1, 21]. Based on references [1, 16–21], we give some assumptions to investigate the dynamics of the microbial degradation of the rhizosphere.(a)Suppose the pollutant and rhizosphere microbe are uniformly distributed inside the rhizosphere. denotes the pollutant concentration discharged from the household and industrial sources. is the concentration of the rhizosphere microbe.(b)It is assumed the pollutant is discharged into the plant rhizosphere from outside with a constant .(c) is the rate of decrease due to biochemical and other factors, which is proportional to the pollutant concentration .(d)The growth of the rhizosphere microbe is assumed to follow the Monod equation involving the pollutant concentration as well as the microbial concentration (i.e., ), where is the maximum specific growth rate and the constant is yield term and is a half-saturation constant.(e)The mortality of the rhizosphere microbe is denoted as . Considering the nutrient recycling, we suppose the fraction of the death microbe is converted into the substrate and () is the fraction of the conversion.(f)Let show a predetermined threshold, which is a minimum allowed value harmless to human health. When pollutant concentration is higher than the critical threshold (which can be measured in advance), control measures are taken to reduce the pollutant concentration below the critical threshold; that is, we will inoculate the microbe into the plant rhizosphere so as to improve microbial ability of degradation and denotes the amount of releasing the microbe into the plant rhizosphere. At the same time, we will control the emissions of pollutant. () is the effect of controlling the pollutant sources.
According to the above assumption, we formulate the following model:where and .
Next, we will discuss the existence of periodic solution of (2) by using the geometry theory of semicontinuous dynamic system. For convenience, we give some definitions and lemmas.
Definition 1 (see [22]). Let denote the impulsive set and be the phase set. Suppose is a mapping. For any point , there exists such that , , and then is called the successor function of point and the point is called the successor point of .
To investigate the dynamics of system (2), we construct the following Pioncaré map. Choose two sections and as the Poincaré sections, where and . Assume the point is on the Pioncaré section , then lies on the section due to the impulsive effect, and the trajectory starting from the initial value intersects the section at the point , where is determined by and the parameter . Therefore, we get the Pioncaré map as follows:
3. Qualitative Analysis for System (2)
Before discussing the periodic solution of system (2), we should consider the qualitative property of (2) without the impulsive effect.
Lemma 2. Suppose is a solution of (2) subject to , then for all , and furthermore , if .
Theorem 3. The system (2) is ultimately bounded.
Proof. Define a function . We calculate the upper right derivative of along a solution of the first and second equations of system (2) and get the following differential equation: where . We have According to the positivity of and , we obtain and . The proof is completed.
Next, we explore the asymptotical behavior of the system (2) without impulsive effect.An equilibrium point of system (6) satisfies the systemIt can be seen that system (7) has a microbe-free equilibrium of the form . We start by analyzing the behavior of the system (7) near . The characteristic equation of the linearization of (6) near isTwo eigenvalues are and , respectively. We obtain the microbe-free equilibrium of system (6) is unique and locally asymptotically stable if the condition holds.
Define .
Theorem 4. The microbe-free equilibrium is asymptotically stable if . is unstable if .
From (7), it follows that when the trivial equilibrium of system (6) is asymptotically stable then positive equilibrium does not exist. When , system (6) has a unique positive equilibrium , where , , . We can easily obtain and for . Next, we analyze the stability of the positive equilibrium . The characteristic equation at isWe know . After a few computations, we have , for . Therefore, we have.
Theorem 5. If holds, the positive equilibrium of system (6) is locally asymptotically stable.
Now, let us discuss the global stability of system (2). Firstly, we will give the following lemma.
Lemma 6. Suppose is a periodic orbit of the system (2) with the period . The bounded region consists of all the points in phase plane . Denote where and . Then we obtain
Proof. From system (2), we obtain . Obviously, we have because of the periodicity of . Therefore, we obtain .
Theorem 7. The positive equilibrium is globally asymptotically stable if , where is defined above.
Proof. From Theorem 5, we know that is locally asymptotically stable. According to Lemma 6, we obtain the periodic solution is stable if it exists around . However, it is impossible according to Poincare-Bendixson Theorem, which implies that is globally asymptotically stable.
4. Existence of the Periodic Solution
When , the trajectory from the initial value with will intersect the Poincaré section infinite times; while , the trajectory from the initial value with will tend to the equilibrium after the impulsive effect of the finite times. In order to investigate the impact of the impulsive effect on the system (2), we have two cases: (a) and (b) .
4.1. Existence of the Order-1 Periodic Solution for
The impulsive set intersects the isoclinal line at the point , where is satisfied . The phase set intersects the isoclinal line at the point , where is satisfied .
Choose , where is small enough and . According to the qualitative analysis, the trajectory from the point reaches the point on the Poincaré section , next jumps onto the point , and then reaches the point again. Obviously, the point is above the point due to impulsive effect and . Therefore, The point is the successor point of the point , and in view of the property of the successor function, we have
Suppose the trajectory from the point interacts the Poincaré section at the point and next jumps onto the Poincaré section at the point . There are two possible cases between the point and . One is that the point is below the point , and the other case is that point is above the point .
For the point being below the point (see Figure 1(a)), the point is the successor point of the point , we haveHence, there must be a point such that ; that is, system (2) has an order-1 periodic solution.
(a)
(b)
(c)
When the point is above the point , the trajectory from the point interacts the Poincaré section at the point and next jumps onto the point . Obviously, the point is below the point according to the impulsive effect and qualitative property of system (2) (see Figure 1(b)). The point is the successor point of the point ; we obtainFrom (12) and (14), we obtain system (2) exists an order-1 periodic solution.
4.2. Existence of the Periodic Solution for
Suppose the Poincaré section interacts the isoclinal line at the point and the Poincaré section intersects the isoclinal line at the point . There must exist a trajectory passing through the point and tangent to the Poincaré section at the point with (see Figure 1(c)). Furthermore, the trajectory crosses the Poincaré section at the point and the point with . For , the trajectory passing through the point will not interact with the Poincaré section and will ultimately tend to the positive equilibrium . For any trajectory of (2) will pass through the point after some impulsive effects on the Poincaré section , they ultimately tend to the positive equilibrium and there is not any periodic solution. For , a trajectory of system (2) will intersect the Pioncaré section infinite times due to the impulsive effect , . When the condition is satisfied, we have for any two points and . The corresponding impulsive points and are above the point . According to the property of the vector field, we obtain that ; that is,For and , we consider the Poincaré map , . If , then system (2) has an order-1 periodic solution. If , , we obtain that system (2) has an order-2 periodic solution.
For convenience, we investigate the more general thing in the following, that is, ().(a)For , it follows from , which leads to the relation of to be one of the following cases according to the qualitative property of system (2). , in the case, it follows from (15) that . Repeating the above process, the following circumstances are obtained:When , similarly, we have (b)When , again from (15), we have . The result leads to the relation of , and to be one of the following cases:If , similarly, we also have Furthermore, in case of (a), we obtain and , where . Hence we obtain and ; therefore, system (2) has an order-2 periodic solution, which is orbitally asymptotically stable. Similarly, for case of (a) and of (b), system (2) has an orbitally asymptotically stable order-2 periodic solution. In case () of (b), system (2) also has an order-2 periodic solution, which is orbitally asymptotically stable.
5. Numerical Simulations and Discussion
In this paper, we have formulated the mathematical model of microbial degradation with the impulsive state feedback control and obtained the existence of order-1 periodic solution and order-2 periodic solution by using the geometry theory of semicontinuous dynamic system. System (2) without the impulsive effect has two equilibria, and ; we show the microbe-free equilibrium is asymptotically stable if . The positive equilibrium is globally asymptotically stable if , which means that pollutant concentration will reach unacceptable level if no measurement is adopted. In order to reduce the pollutant concentration to the predetermined threshold, which is less harmful to human health, the model of the microbial degradation is proposed. By using the geometric theory of semicontinuous dynamical system, we obtain system (2) exists an order-1 periodic solution for , which implies the control measurement is effective even if the initial concentration of pollutant is very high and the example is given as follows:
System (20) can be simulated in Figure 2. We see the trajectory starting through the initial value is attracted by the order-1 periodic solution (see Figure 2). When , system (2) exists an order-2 periodic solution starting from the initial value with the parameters , , , , , , , , and (see Figure 3).
(a)
(b)
(c)
(a)
(b)
(c)
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 11371164), NSFC-Talent Training Fund of Henan (U1304104), Innovative talents of science and technology plan in Henan province (15HASTIT014), and the young backbone teachers of Henan (no. 2013GGJS-214).