# Impulsive State Feedback Control of the Rhizosphere Microbial Degradation in the Wetland Plant

Authors: Zhong Zhao; Liuyong Pang; Zhanping Zhao; Chengguang Luo
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## 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.


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## Body

## 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: (1)dTdt=fST-σT,dWdt=Q-γ+αW,dSdt=γW-fSTY-ϕS,where f(S) denotes the mangrove growth rate which is chosen to follow Monod kinetics (f(S)=βS/(k+S),β>0,k>0). T(t) is the mangrove biomass concentration, W(t) denotes the total nitrogen concentration in wastewater, and S(t) 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 Section2. 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.S(t) denotes the pollutant concentration discharged from the household and industrial sources. x(t) is the concentration of the rhizosphere microbe.(b)
It is assumed the pollutant is discharged into the plant rhizosphere from outside with a constantQ.(c)
α is the rate of decrease due to biochemical and other factors, which is proportional to the pollutant concentration S(t).(d)
The growth of the rhizosphere microbe is assumed to follow the Monod equation involving the pollutant concentrationS(t) as well as the microbial concentration x(t) (i.e., μSx/δ(K+S)), where μ is the maximum specific growth rate and the constant δ is yield term and K is a half-saturation constant.(e)
The mortality of the rhizosphere microbe is denoted asD. Considering the nutrient recycling, we suppose the fraction of the death microbe is converted into the substrate and r (0<r≤1) is the fraction of the conversion.(f)
Leth show a predetermined threshold, which is a minimum allowed value harmless to human health. When pollutant concentration is higher than the critical threshold h (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 τ denotes the amount of releasing the microbe into the plant rhizosphere. At the same time, we will control the emissions of pollutant. E (0<E<1) is the effect of controlling the pollutant sources.According to the above assumption, we formulate the following model:(2)dSdt=Q-αS-μSxδK+S+rDx,dxdt=μSxK+S-Dx,S<h,ΔS=-ES,Δx=τ,S=h,where ΔS=S(t+)-s(t) and Δx=x(t+)-x(t).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]).
LetM denote the impulsive set and N be the phase set. Suppose g:N→N is a mapping. For any point P∈N, there exists t1>0 such that F(P)=f(P,t1)=P1∈M, P1+=φ(P1)∈N, and then g(P)=l(P1+)-l(P) is called the successor function of point P and the point P1+ is called the successor point of P.To investigate the dynamics of system (2), we construct the following Pioncaré map. Choose two sections P0 and P1 as the Poincaré sections, where P0={(S,x)∣S=(1-E)h,x≥τ} and P1={(S,x)∣S=h,x≥0}. Assume the point Bk(h,xk) is on the Pioncaré section P1, then Bk+((1-E)h,xk+1) lies on the section P0 due to the impulsive effect, and the trajectory starting from the initial value Bk+ intersects the section P1 at the point Bk+1=(h,xk+1), where xk+1 is determined by xk and the parameter τ. Therefore, we get the Pioncaré map as follows: (3)xk+1=Fτ,xk.


## 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.
Supposeω(t)=(S(t),x(t)) is a solution of (2) subject to ω(0+)≥0, then ω(t)≥0 for all t≥0, and furthermore ω(t)>0, t≥0 if ω(0+)>0.Theorem 3.
The system (2) is ultimately bounded.Proof.
Define a functionV(t)=S(t)+xt/δ. We calculate the upper right derivative of V(t) along a solution of the first and second equations of system (2) and get the following differential equation: (4)dVdt=Q-αS+r-1δDx≤Q-βVt,where β=min⁡⁡{α,(1/δ-r)Dδ}. We have (5)St+xtδ≤limt→∞⁡Vt=Qβ.According to the positivity of S(t) and x(t), we obtain S(t)≤Q/β and x(t)≤Q/β. The proof is completed.Next, we explore the asymptotical behavior of the system (2) without impulsive effect.(6)dSdt=Q-αS-μSxδ1K+S+rDx,dxdt=μSxK+S-Dx.An equilibrium point of system (6) satisfies the system(7)Q-αS-μSxδ1K+S+rDx=0,μSxK+S-Dx=0.It can be seen that system (7) has a microbe-free equilibrium of the form E0=(Q/α,0). We start by analyzing the behavior of the system (7) near E0. The characteristic equation of the linearization of (6) near E0 is(8)det⁡⁡-α-λrD0μQkα+Q-D-λ=0.Two eigenvalues are λ1=-α and λ2=μQ/(kα+Q)-D, respectively. We obtain the microbe-free equilibrium of system (6) is unique and locally asymptotically stable if the condition μQ/(kα+Q)<D holds.DefineR1=μQ/D(kα+Q).Theorem 4.
The microbe-free equilibriumE0 is asymptotically stable if R1<1. E0 is unstable if R1>1.From (7), it follows that when the trivial equilibrium E0 of system (6) is asymptotically stable then positive equilibrium does not exist. When R1>1, system (6) has a unique positive equilibrium E1(S∗,x∗), where S∗=Dk/(μ-D), x∗=δ(Q-αS∗)(k+S∗)/((μ-rDδ)S∗-rDδk)=δ(Qμ-QD-αDk)/μDk(μ-D)(1-rδ), μ>D. We can easily obtain S∗>0 and x∗>0 for R1>1. Next, we analyze the stability of the positive equilibrium E∗(S∗,x∗). The characteristic equation at E∗(S∗,P∗) is(9)det⁡⁡λ+α+μkx∗δk+S∗2-rD+μS∗δk+S∗-μkx∗k+S∗2λ+D-μS∗k+S∗=0.We know λ2+(α+μkx∗/δ(k+S∗)2)λ+(μkx∗/(k+S∗)2)(μS∗/δk+S∗-rD)=0. After a few computations, we have λ1+λ2<0, λ1λ2>0 for R1>1. Therefore, we have.Theorem 5.
IfR1>1 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.
SupposeΓ(t)=(S(t),x(t)) is a periodic orbit of the system (2) with the period T. The bounded region R consists of all the points in phase plane Γ. Denote (10)Δ=∫0T∂f1∂SSt,xt+∂f2∂xSt,xtdt,where dS/dt≜f1(S,x) and dx/dt≜f2(S,x). Then we obtain(11)Δ=-∫0Tα+μkxtδ1k+St2dt+∫0TμStk+St-Ddt≤0.Proof.
From system (2), we obtain ∫0T‍[μSt/(k+S(t)-D]dt=∫0T‍d(ln⁡x(t)). Obviously, we have ∫0T‍(1/x)(dx/dt)=0 because of the periodicity of x(t). Therefore, we obtain Δ=-∫0T‍[α+μkx(t)/δ1(k+St)2]dt≤0.Theorem 7.
The positive equilibriumE∗(S∗,x∗) is globally asymptotically stable if R1>1, where R1 is defined above.Proof.
From Theorem5, we know that E∗(S∗,x∗) is locally asymptotically stable. According to Lemma 6, we obtain the periodic solution is stable if it exists around E∗(S∗,x∗). However, it is impossible according to Poincare-Bendixson Theorem, which implies that E∗(S∗,x∗) is globally asymptotically stable.


## 4. Existence of the Periodic Solution

Whenh<kD/(μ-D), the trajectory from the initial value (S(0),x(0)) with S(0)<h will intersect the Poincaré section P1 infinite times; while h≥kD/(μ-D), the trajectory from the initial value (S(0),x(0)) with S(0)<h will tend to the equilibrium (S∗,x∗) 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) h<kD/(μ-D) and (b) h≥kD/(μ-D).


### 4.1. Existence of the Order-1 Periodic Solution forh<kD/(μ-D)

The impulsive setS=h intersects the isoclinal line dS/dt=0 at the point D(h,xD), where xD is satisfied xD=δ(Q-αh)(k+h)/((μ-rDδ)h-rDδk). The phase set S=(1-E)h intersects the isoclinal line dS/dt=0 at the point H((1-E)h,xH), where  xH is satisfied xH=δ(Q-α(1-E)h)(k+(1-E)h)/((μ-rDδ)(1-E)h-rDδk).ChooseA1((1-E)h,ε), where ε is small enough and ε<τ. According to the qualitative analysis, the trajectory from the point A1 reaches the point B1(h,ε1) on the Poincaré section P1, next jumps onto the point A2((1-E)h,ε1+τ), and then reaches the point B2(h,ε2) again. Obviously, the point A2 is above the point A1 due to impulsive effect Δx=τ and τ+ε1>ε. Therefore, The point A2 is the successor point of the point A1, and in view of the property of the successor function, we have (12)fxA2-fxA1>0.Suppose the trajectory from the pointH((1-E)h,xH) interacts the Poincaré section P1 at the point G(h,l) and next jumps onto the Poincaré section P0 at the point H1((1-E)h,γ1). There are two possible cases between the point H and H1. One is that the point H1 is below the point H, and the other case is that point H1 is above the point H.For the pointH1 being below the point H (see Figure 1(a)), the point H1 is the successor point of the point H, we have(13)fxH1-fxH<0.Hence, there must be a point F((1-E)h,xF) such that f(xF)=0; that is, system (2) has an order-1 periodic solution.Figure 1
(a) and (b) The existence of order-1 periodic solution forh<kD/(μ-D). (c) The existence of order-1 or order-2 periodic solution for h≥kD/(μ-D).
(a)
(b)
(c)When the pointH1 is above the point H, the trajectory from the point H1 interacts the Poincaré section at the point G1(h,l1) and next jumps onto the point H2((1-E)h,γ2). Obviously, the point H2 is below the point H1 according to the impulsive effect and qualitative property of system (2) (see Figure 1(b)). The point H2 is the successor point of the point H1; we obtain(14)fxH2-fxH1<0.From (12) and (14), we obtain system (2) exists an order-1 periodic solution.


### 4.2. Existence of the Periodic Solution forh≥kD/(μ-D)

Suppose the Poincaré sectionS=h interacts the isoclinal line dS/dt=0 at the point D(h,xD) and the Poincaré section S=(1-E)h intersects the isoclinal line dS/dt=0 at the point H((1-E)h,xH). There must exist a trajectory Γ passing through the point D and tangent to the Poincaré section S=h at the point D(h,xD) with xD=δ(Q-αh)(k+h)/((mu-rDδ)h-rDδk) (see Figure 1(c)). Furthermore, the trajectory Γ crosses the Poincaré section S=(1-E)h at the point E((1-E)h,x¯1) and the point F((1-E)h,x¯2) with x¯1<δ(Q-α(1-E)h)(k+(1-E)h)/((mu-rDδ)(1-E)h-rDδk)<x¯2. For x∈(x¯1,x¯2), the trajectory passing through the point ((1-E)h,x) will not interact with the Poincaré section S=h and will ultimately tend to the positive equilibrium E∗(S∗,x∗). For any trajectory of (2) will pass through the point ((1-E)h,x),x∈(x¯1,x¯2) after some impulsive effects on the Poincaré section P1, they ultimately tend to the positive equilibrium E∗(S∗,x∗) and there is not any periodic solution. For τ>x¯2, a trajectory of system (2) will intersect the Pioncaré section S=(1-E)h infinite times due to the impulsive effect ΔS=-ES, Δx=τ. When the condition τ>x¯2 is satisfied, we have 0<Bi(h,xi)<Bj(h,xj)<δQ-αhk+h/((mu-rDδ)h-rDδk) for any two points Bi(h,xi) and Bj(h,xj). The corresponding impulsive points Bi+(h,xi+τ) and Bj+(h,xj+τ) are above the point F. According to the property of the vector field, we obtain that 0<xj+1<xi+1<δ(Q-αh)(k+h)/((mu-rDδ)h-rDδk); that is,(15)0<xj+1<xi+1<δQ-αhk+hmu-rDδh-rDδk,for0<xi<xj<δQ-αhk+hmu-rDδh-rDδk,τ>x¯2.For x0∈(0,δQ-αhk+h/((mu-rDδ)h-rDδk)) and τ>x¯2, we consider the Poincaré map xi=F(τ,xi-1), i=1,2,…,n. If x1=x0, then system (2) has an order-1 periodic solution. If x0≠x1, x0=x2, we obtain that system (2) has an order-2 periodic solution.For convenience, we investigate the more general thing in the following, that is,x0≠x1≠x2≠⋯≠xn (n≥3).(a)
Forx0<x1, it follows from x2<x1, which leads to the relation of x0,x1,x2 to be one of the following cases according to the qualitative property of system (2).
(1a)
x2<x0<x1, in the case, it follows from (15) that x2<x1<x3. Repeating the above process, the following circumstances are obtained:(16)0<⋯<x2n<⋯<x2<x0<Z1<⋯<x2n+1<⋯<δQ-αhk+hmu-rDδh-rDδk.(2a)
Whenx0<x2<x1, similarly, we have (17)x0<x2<⋯<x2n<⋯<x2n+1<⋯<x3<x1<δQ-αhk+hmu-rDδh-rDδk.(b)
Whenx0>x1, again from (15), we have x1<x2. The result leads to the relation of x0,x1, and x2 to be one of the following cases:
(1b)
(18)0<⋯<x2n+1<⋯<x1<x0<x2<⋯<x2n<⋯<δQ-αhk+hmu-rDδh-rDδk.(2b)
Ifx1<x2<x0, similarly, we also have (19)0<x1<⋯<x2n+1<⋯<x2n<⋯<x2<x0<δQ-αhk+hmu-rDδh-rDδk.Furthermore, in case (1a) of (a), we obtain limn→∞x2n=ξ1 and limn→∞x2n+1=ξ2, where 0<ξ1<ξ2<δ(Q-αh)(k+h)/((mu-rDδ)h-rDδk). Hence we obtain ξ1=F(τ,ξ2) and ξ2=F(τ,ξ1); therefore, system (2) has an order-2 periodic solution, which is orbitally asymptotically stable. Similarly, for case (2a) of (a) and (2b) of (b), system (2) has an orbitally asymptotically stable order-2 periodic solution. In case (1b) of (b), system (2) also has an order-2 periodic solution, which is orbitally asymptotically stable.


## 4.1. Existence of the Order-1 Periodic Solution forh<kD/(μ-D)

The impulsive setS=h intersects the isoclinal line dS/dt=0 at the point D(h,xD), where xD is satisfied xD=δ(Q-αh)(k+h)/((μ-rDδ)h-rDδk). The phase set S=(1-E)h intersects the isoclinal line dS/dt=0 at the point H((1-E)h,xH), where  xH is satisfied xH=δ(Q-α(1-E)h)(k+(1-E)h)/((μ-rDδ)(1-E)h-rDδk).ChooseA1((1-E)h,ε), where ε is small enough and ε<τ. According to the qualitative analysis, the trajectory from the point A1 reaches the point B1(h,ε1) on the Poincaré section P1, next jumps onto the point A2((1-E)h,ε1+τ), and then reaches the point B2(h,ε2) again. Obviously, the point A2 is above the point A1 due to impulsive effect Δx=τ and τ+ε1>ε. Therefore, The point A2 is the successor point of the point A1, and in view of the property of the successor function, we have (12)fxA2-fxA1>0.Suppose the trajectory from the pointH((1-E)h,xH) interacts the Poincaré section P1 at the point G(h,l) and next jumps onto the Poincaré section P0 at the point H1((1-E)h,γ1). There are two possible cases between the point H and H1. One is that the point H1 is below the point H, and the other case is that point H1 is above the point H.For the pointH1 being below the point H (see Figure 1(a)), the point H1 is the successor point of the point H, we have(13)fxH1-fxH<0.Hence, there must be a point F((1-E)h,xF) such that f(xF)=0; that is, system (2) has an order-1 periodic solution.Figure 1
(a) and (b) The existence of order-1 periodic solution forh<kD/(μ-D). (c) The existence of order-1 or order-2 periodic solution for h≥kD/(μ-D).
(a)
(b)
(c)When the pointH1 is above the point H, the trajectory from the point H1 interacts the Poincaré section at the point G1(h,l1) and next jumps onto the point H2((1-E)h,γ2). Obviously, the point H2 is below the point H1 according to the impulsive effect and qualitative property of system (2) (see Figure 1(b)). The point H2 is the successor point of the point H1; we obtain(14)fxH2-fxH1<0.From (12) and (14), we obtain system (2) exists an order-1 periodic solution.


## 4.2. Existence of the Periodic Solution forh≥kD/(μ-D)

Suppose the Poincaré sectionS=h interacts the isoclinal line dS/dt=0 at the point D(h,xD) and the Poincaré section S=(1-E)h intersects the isoclinal line dS/dt=0 at the point H((1-E)h,xH). There must exist a trajectory Γ passing through the point D and tangent to the Poincaré section S=h at the point D(h,xD) with xD=δ(Q-αh)(k+h)/((mu-rDδ)h-rDδk) (see Figure 1(c)). Furthermore, the trajectory Γ crosses the Poincaré section S=(1-E)h at the point E((1-E)h,x¯1) and the point F((1-E)h,x¯2) with x¯1<δ(Q-α(1-E)h)(k+(1-E)h)/((mu-rDδ)(1-E)h-rDδk)<x¯2. For x∈(x¯1,x¯2), the trajectory passing through the point ((1-E)h,x) will not interact with the Poincaré section S=h and will ultimately tend to the positive equilibrium E∗(S∗,x∗). For any trajectory of (2) will pass through the point ((1-E)h,x),x∈(x¯1,x¯2) after some impulsive effects on the Poincaré section P1, they ultimately tend to the positive equilibrium E∗(S∗,x∗) and there is not any periodic solution. For τ>x¯2, a trajectory of system (2) will intersect the Pioncaré section S=(1-E)h infinite times due to the impulsive effect ΔS=-ES, Δx=τ. When the condition τ>x¯2 is satisfied, we have 0<Bi(h,xi)<Bj(h,xj)<δQ-αhk+h/((mu-rDδ)h-rDδk) for any two points Bi(h,xi) and Bj(h,xj). The corresponding impulsive points Bi+(h,xi+τ) and Bj+(h,xj+τ) are above the point F. According to the property of the vector field, we obtain that 0<xj+1<xi+1<δ(Q-αh)(k+h)/((mu-rDδ)h-rDδk); that is,(15)0<xj+1<xi+1<δQ-αhk+hmu-rDδh-rDδk,for0<xi<xj<δQ-αhk+hmu-rDδh-rDδk,τ>x¯2.For x0∈(0,δQ-αhk+h/((mu-rDδ)h-rDδk)) and τ>x¯2, we consider the Poincaré map xi=F(τ,xi-1), i=1,2,…,n. If x1=x0, then system (2) has an order-1 periodic solution. If x0≠x1, x0=x2, we obtain that system (2) has an order-2 periodic solution.For convenience, we investigate the more general thing in the following, that is,x0≠x1≠x2≠⋯≠xn (n≥3).(a)
Forx0<x1, it follows from x2<x1, which leads to the relation of x0,x1,x2 to be one of the following cases according to the qualitative property of system (2).
(1a)
x2<x0<x1, in the case, it follows from (15) that x2<x1<x3. Repeating the above process, the following circumstances are obtained:(16)0<⋯<x2n<⋯<x2<x0<Z1<⋯<x2n+1<⋯<δQ-αhk+hmu-rDδh-rDδk.(2a)
Whenx0<x2<x1, similarly, we have (17)x0<x2<⋯<x2n<⋯<x2n+1<⋯<x3<x1<δQ-αhk+hmu-rDδh-rDδk.(b)
Whenx0>x1, again from (15), we have x1<x2. The result leads to the relation of x0,x1, and x2 to be one of the following cases:
(1b)
(18)0<⋯<x2n+1<⋯<x1<x0<x2<⋯<x2n<⋯<δQ-αhk+hmu-rDδh-rDδk.(2b)
Ifx1<x2<x0, similarly, we also have (19)0<x1<⋯<x2n+1<⋯<x2n<⋯<x2<x0<δQ-αhk+hmu-rDδh-rDδk.Furthermore, in case (1a) of (a), we obtain limn→∞x2n=ξ1 and limn→∞x2n+1=ξ2, where 0<ξ1<ξ2<δ(Q-αh)(k+h)/((mu-rDδ)h-rDδk). Hence we obtain ξ1=F(τ,ξ2) and ξ2=F(τ,ξ1); therefore, system (2) has an order-2 periodic solution, which is orbitally asymptotically stable. Similarly, for case (2a) of (a) and (2b) of (b), system (2) has an orbitally asymptotically stable order-2 periodic solution. In case (1b) 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, E0(Q/α,0) and E∗(S∗,x∗); we show the microbe-free equilibrium E0(Q/α,0) is asymptotically stable if R1<1. The positive equilibrium E∗(S∗,x∗) is globally asymptotically stable if R1>1, 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 h<kD/(μ-D), which implies the control measurement is effective even if the initial concentration of pollutant is very high and the example is given as follows:(20)dSdt=6-αS-4Sx0.014+S+0.0003x,dxdt=4Sx4+S-0.03x,S<0.2,ΔS=-0.1S,Δx=4,S=0.2.System (20) can be simulated in Figure 2. We see the trajectory starting through the initial value (0.6,0.3) is attracted by the order-1 periodic solution (see Figure 2). When h≥kD/(μ-D), system (2) exists an order-2 periodic solution starting from the initial value (0.13,0.36) with the parameters μ=0.4, Q=5, r=0.02, D=0.03, E=0.071, τ=7.6, α=0.2, δ=0.02, and h=0.6233 (see Figure 3).Figure 2
Forh<kD/(μ-D), the existence of order-1 periodic solution with the parameters μ=0.4, Q=6, k=4, r=0.01, D=0.03, E=0.1, τ=4, h=0.2, α=0.2, and δ=0.01.
(a)
(b)
(c)Figure 3
Forh≥kD/(μ-D), the existence of order-1 periodic solution with the parameters μ=0.4, Q=5, r=0.02, D=0.03, E=0.071, τ=7.6, α=0.2, δ=0.02, and h=0.6233.
(a)
(b)
(c)



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