Discrete Dynamics in Nature and Society

Volume 2018, Article ID 6401059, 10 pages

https://doi.org/10.1155/2018/6401059

## Chemostat Model of Competition between Plasmid-Bearing and Plasmid-Free Organism with the Impulsive State Feedback Control

^{1}School of Mathematics and Information Science, Anshan Normal University, Anshan 114007, Liaoning, China^{2}Department of Mathematics, Huanghuai University, Zhumadian 463000, Henan, China^{3}Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

Correspondence should be addressed to Zhong Zhao; moc.361@9988gnohzoahz

Received 27 June 2018; Accepted 16 September 2018; Published 23 October 2018

Academic Editor: Josef Diblík

Copyright © 2018 Fengmei Tao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we propose a chemostat model of competition between plasmid-bearing and plasmid-free organism with the impulsive state feedback control. The sufficient condition for existence of the positive period-1 solution is obtained by means of successor function and the qualitative properties of the corresponding continuous system. We show that the impulsive control system is more effective than the corresponding continuous system if we choose a suitable threshold value of the state feedback control in the process of manufacturing the desired products through genetically modified techniques. Furthermore, a new method of proving the stability of the order-1 periodic solution is given based on the theory of the limit cycle of the continuous dynamical system. Finally, mathematical results are justified by some numerical simulations.

#### 1. Introduction

With the rapid development of biotechnology, manufacturing the desired products through genetically modified techniques has been widely applied in many fields, such as agriculture, industrial biotechnology, and medicine. In general, genetical alteration is carried out by the introduction of a recombinant DNA into the cell in the form of a plasmid [1]. However, the plasmid may be lost in the reproductive process, which will introduce the plasmid-free organism into the process [2]. The genetically altered organism (plasmid-bearing organism) is less competitor than the plasmid-free organism because of the burden imposed by production [3]. At the same time, plasmid-free organism not only consumes the nutrient but also reduces the efficiency of the desired products. Therefore, the key problem is how to improve the efficiency of the desired products and to reduce the negative effect of the plasmid-free organism on the plasmid-bearing organism. Researchers [2–8] have done a lot of work to reduce the negative effect of the plasmid-free organism on the production efficiency of the desired products. In [2], a selective media was added from an external source and authors investigated the global stability of the equilibria by using the Dulac criterion and the Poincaré-Bendixson Theorem. Ai [5] introduced an inhibitor into the model of competition between plasmid-bearing and plasmid-free organisms to reduce the effect of the plasmid-free organism on the plasmid-bearing organism. Yuan et al. [6] proposed a model for competition between plasmid-bearing and plasmid-free organisms in the chemostat with an external inhibitor and obtained some sufficient conditions of global attractivity to the extinction equilibria by constructing appropriate Lyapunov-like functionals. Based on a feedback control, Dimitrova [9] formulated a dynamical model of plasmid-bearing, plasmid-free competition in the chemostat with general specific growth rates, and global stabilization of the dynamics towards a practically important coexistence equilibrium point is obtained. Shi et al. [10] investigated a chemostat model with plasmid-bearing, plasmid-free competition and impulsive effect, and they obtained the invasion threshold of the plasmid-free organism and plasmid-bearing organism.

Recently, impulsive state feedback control has been widely used in chemostat model [11–14]. Tian et al. [12] studied the dynamics of the bioprocess with the impulsive state feedback control and obtained the existence and stability of period-1 solution of the bioprocess. In [13], authors proposed a turbidostat model with the feedback control, and the sufficient conditions of existence of positive order-1 periodic solution were obtained by using the existence criteria of periodic solution of a general planar impulsive autonomous system. As far as the authors know, little information is given about the introduction of the impulsive state feedback control into chemostat model of competition between plasmid-bearing and plasmid-free organism. In this paper, we will formulate a mathematical model of competition between plasmid-bearing and plasmid-free organism with the impulsive state feedback control so as to further improve the efficiency of the desired products in the process of the genetic alteration.

The paper is organized as follows: a mathematical model of competition between plasmid-bearing and plasmid-free organism with the 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 and stability of order-1 periodic solution are investigated in Section 4. Finally, we give some numerical simulations and a brief discussion.

#### 2. Model Description and Preliminaries

In the process of manufacturing the desired products through genetically modified techniques, an important way to control the negative effect is to reduce the competitive advantage of the plasmid-free organism.

The process of the genetic alteration is complex because it includes a series of biological and chemical reactions. Therefore, understanding the dynamic mechanism of genetic alteration is the most important factor for improving the efficiency of the desired products. Luo et al. [15] formulated a model of plasmid-bearing and plasmid-free competition in a chemostat as follows: where is the nutrient concentration at time . denotes the concentration of plasmid-bearing organisms at time , and is the concentration of plasmid-free organisms at time The consumption rates and the specific growth rates of plasmid-bearing and plasmid-free organisms are and (i=1,2), respectively. The probability that a plasmid is lost in reproduction is represented by

Although the reason why plasmid is lost in the reproductive process is not completely understood by theoretical biologists even at present time, it has been experimentally validated that the plasmid-bearing cells have at least for some substrate concentrations a lower maximum specific growth rate than the plasmid-free counterpart [9, 16, 17]. To enhance the efficiency of the desired products and prevent the negative effect of the plasmid-free organism on the plasmid-bearing organism, we try to find the critical threshold value (where plasmid-bearing cells reach the maximum specific growth rate), which can be obtained by the experiment. When the growth rate of the plasmid-free organism reaches the critical threshold value, we begin to extract the plasmid-free organism so as to control the concentration of the plasmid-free organism to be lower than the critical threshold value. Based on [11–14], we introduce the impulsive state feedback control into system (1): where and denote the maximal growth rate of the plasmid-bearing and plasmid-free organisms. is called a yield constant. is a critical threshold value which can be obtained by experiment. and are the ratios of the plasmid-bearing and plasmid-free organisms extracted from the chemostat.

We firstly consider the qualitative property of (2) without the impulsive effect. The variables in the above system may be rescaled by measuring , and then system (2) becomes where .

Lemma 1. *Suppose is a solution of (3) subject to , and then for all , and further if *

The proof is obvious. Hence we omit it.

From the first three equations, we have which shows and we obtain

From system (3) and (5), we obtain that the dynamical behavior of system (2) can be determined by the following system: where ,

*Definition 2 (see [18]). *Let denote the impulsive set and be the phase set. Suppose that is a mapping. For any point , there exists a such that , ; then is called the successor function of point and the point is called the successor point of .

#### 3. Qualitative Analysis

We firstly consider the qualitative property of (6) without the impulsive effect. System (7) has three equilibria: , , and a positive equilibrium , where

Lemma 3. *System (7) has no periodic trajectories in the positive quadrant.*

*Proof. *We construct the Dulac function , and then we have According to the Bendixson-Dulac Theorem [19], there is no periodic orbit in the positive quadrant.

Next, we discuss the stability of the equilibria. The Jacobian matrix evaluated at the point is and the eigenvalues are Hence, the equilibrium is saddle for

Theorem 4. *The equilibrium is globally asymptotically stable if *

*Proof. *We can compute Jacobian matrix of the equilibrium as follows: and the eigenvalues are , . Obviously, the equilibrium is asymptotically stable if Again from Lemma 3, the equilibrium is globally asymptotically stable for

Theorem 5. *The positive equilibrium is globally asymptotically stable if *

*Proof. *Firstly, we show the positive equilibrium is locally asymptotically stable. Jacobian matrix of the equilibrium is given as follows: We perform a basic elementary column transformation for the matrix and obtain Hence, the eigenvalues of matrix have the same eigenvalues as the matrix , and the eigenvalues are , . In view of Lemma 3, the equilibrium is globally asymptotically stable if

#### 4. Existence and Stability of the Order-1 Periodic Solution

*Case 1 (existence and stability of semitrivial periodic solution). *Suppose , and we have the following subsystem: System (15) possesses a periodic solution defined on for , where If we let and , which shows that system (6) has the following semitrivial periodic solution , where for Therefore, we obtain that is the semitrivial periodic solution of system (6).

Next we give one lemma firstly to discuss the stability of this semitrivial periodic solution of system (6).

Lemma 6. *The -periodic solution of the system is orbitally asymptotically stable if the Floquet multiplier satisfies the condition , where with and , , , and are calculated at the point , , is a sufficiently smooth function with grad and is the time of th jump.*

The proof of this lemma is referred to in [20].

Theorem 7. *The semitrivial periodic solution of system (6) is orbitally asymptotically stable if , where *

*Proof. *Since and , , , Using Lemma 6, we obtain and Hence Therefore, if holds, the semitrivial periodic solution is orbitally asymptotically stable.

*Remark*. Defining , we obtain that a fold bifurcation occurs at due to , and a positive periodic solution may emerge for .

*Case 2 (existence and stability of the positive periodic solution). *In this section, we denote the impulsive set , the impulsive function , and the phase set In the following, we will prove the existence and stability of the order-1 periodic solution of system (6).

Obviously, the trajectories starting from the region will tend to the positive equilibrium after impulsive effect of at most finite times.

Theorem 8. *If , then system (6) has an order-1 periodic solution.*

*Proof. *Suppose the impulsive set intersects the axis at the point . The phase set intersects the axis at the point and the isoclinal line at the point . Suppose the point is close enough to the point and the trajectory starting from the point intersects the impulsive set at the point and reaches the point due to the impulsive effect , . The point is the successor point of Thus the successor function of satisfies Similarly, the trajectory from the point inevitably intersects the impulsive set at the point The point is mapped into the point after pulses. Furthermore, point is surely on the left of point from the property of the vector field; hence the successor function of the point becomes According to the continuity of the successor function, we obtain that there exists a point such that ; that is, system (6) presents an order-1 periodic solution (see Figure 1).

In the following, we will investigate the stability of the order-1 periodic solution by means of stability of the limit cycle of continuous dynamic system. We consider the general continuous system as follows: where functions and are continuous and derivative about