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

The rhizosphere microbe plays an important role in removing the pollutant generated from industrial and agricultural production. To investigate the dynamics of themicrobial degradation, a nonlinearmathematicalmodel of the rhizospheremicrobial 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.


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 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][17][18][19][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.

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][17][18][19][20][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  (0 <  ≤ 1) is the fraction of the conversion.
(f) Let ℎ 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  denotes the amount of releasing the microbe into the plant rhizosphere.At the same time, we will control the emissions of pollutant. (0 <  < 1) is the effect of controlling the pollutant sources.
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.

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.
Next, we explore the asymptotical behavior of the system (2) without impulsive effect.
An equilibrium point of system (6) satisfies the system It can be seen that system (7) has a microbe-free equilibrium of the form  0 = (/, 0).We start by analyzing the behavior of the system (7) near  0 .The characteristic equation of the linearization of ( 6) near  0 is det ( Two eigenvalues are  1 = − and  2 = /( + ) − , respectively.We obtain the microbe-free equilibrium of system ( 6) is unique and locally asymptotically stable if the condition /( + ) <  holds.

Existence of the
Suppose the trajectory from the point ((1 − )ℎ,   ) interacts the Poincaré section  1 at the point (ℎ, ) and next jumps onto the Poincaré section  0 at the point  1 ((1 − )ℎ,  1 ).There are two possible cases between the point  and  1 .One is that the point  1 is below the point , and the other case is that point  1 is above the point .
For the point  1 being below the point  (see Figure 1(a)), the point  1 is the successor point of the point , we have Hence, there must be a point ((1−)ℎ,   ) such that (  ) = 0; that is, system (2) has an order-1 periodic solution.
When the point  1 is above the point , the trajectory from the point  1 interacts the Poincaré section at the point  1 (ℎ,  1 ) and next jumps onto the point  2 ((1 − )ℎ,  2 ).Obviously, the point  2 is below the point  1 according to the impulsive effect and qualitative property of system (2) (see Figure 1(b)).The point  2 is the successor point of the point  1 ; we obtain From ( 12) and ( 14), we obtain system (2) exists an order-1 periodic solution.