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Mathematical Approaches in Advanced Control Theories

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Volume 2012 |Article ID 715497 | https://doi.org/10.1155/2012/715497

Yuangan Wang, "Stability Analysis of Predator-Prey System with Fuzzy Impulsive Control", Journal of Applied Mathematics, vol. 2012, Article ID 715497, 9 pages, 2012. https://doi.org/10.1155/2012/715497

Stability Analysis of Predator-Prey System with Fuzzy Impulsive Control

Academic Editor: Xianxia Zhang
Received14 Dec 2011
Revised22 Mar 2012
Accepted11 Apr 2012
Published06 Jun 2012

Abstract

Having attracted much attention in the past few years, predator-prey system provides a good mathematical model to present the correlation between predators and preys. This paper focuses on the robust stability of Lotka-Volterra predator-prey system with the fuzzy impulsive control model, and Takagi-Sugeno (T-S) fuzzy impulsive control model as well. Via the T-S model and the Lyapunov method, the controlling conditions of the asymptotical stability and exponential stability are established. Furthermore, the numerical simulation for the Lotka-Volterra predator-prey system with impulsive effects verifies the effectiveness of the proposed methods.

1. Introduction

Since Volterra presented the differential equation to solve the issue of the sharp change of the population of the sharks (predator) and the minions (prey) in 1925, the predator-prey system has been applied into many areas and played an important role in the biomathematics. Much attention has been attracted to the stability of the predator-prey system. Brauer and Soudack studied the global behavior of a predator-prey system under constant-rate prey harvesting with a pair of nonlinear ordinary differential equations [1]. Xu and his workmates concluded that a short-time delay could ensure the stability of the predator-prey system [2]. After analyzing the different capability between the mature and immature predator, Wang and his workmates obtained the global stability with the small time-delay system [3]. Li and his partners studied the impulsive control of Lotka-Volterra predator-prey system and established sufficient conditions of the asymptotic stability with the method of Lyapunov functions [4]. Liu and Zhang studied the coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure [5]. Li did some work on the predator-prey system with Holling II functional response and obtained the existence, uniqueness and global asymptotic stability of the in random perturbation [6]. Furthermore, Ko and Ryu studied the qualitative behavior of nonconstant positive solutions on a general Gauss-type predator-prey model with constant diffusion rates under homogenous Neumann boundary condition [7]. Additionally, many papers discussed the predator-prey system with other different methods, such as LaSalle’s invariance principle method [8], Liu and Chen’s impulsive perturbations method [9], and Moghadas and Alexander’s generalized Gauss-type predator–prey model [10].

In recent years, fuzzy impulsive theory has been applied to the stability analysis of the non-linear differential equations [1115]. However, it should be admitted that the stability of fuzzy logic controller (FLC) is still an open problem. It is well-known that the parallel distributed compensation technique has been the most popular controller design approach and belongs to a continuous input control way. It is important to point out that there exist many systems, like the predator-prey system, which cannot commonly endure continuous control inputs, or they have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes. In this sense, it is the same with communication networks, biological population management, chemical control, and so forth [1623]. Hence, it is necessary to extend FLC and reflect these impulsive jump phenomena in the predator-prey system. Until recently, few papers talk about the stability of Lotka-Volterra predator-prey system with fuzzy impulsive control. In this paper, the writer will study the robustness of the predator-prey system by the fuzzy impulsive control based on the T-S mathematical model.

The rest of this paper is organized as follows. Section 2 describes the Lotka-Volterra predator-prey system and T-S fuzzy system with impulsive control. In Section 3, the theoretic analysis and design algorithm on stability of the impulsive fuzzy system are performed. Numerical simulations for the predator-prey system with impulsive effects are carried out with respect to the proposed method in Section 4. Finally, some conclusions are made in Section 5.

2. Problem Equation

The Lotka-Volterra predator-prey system is expressed with the following differential equation: ̇𝑥1(𝑡)=𝑥1𝜇(𝑡)1𝑟12𝑥2,(𝑡)̇𝑥2(𝑡)=𝑥2(𝑡)𝜇2+𝑟21𝑥1(,𝑡)(2.1) where 𝑥1(𝑡),𝑥2(𝑡)(𝑥1(𝑡)>0,𝑥2(𝑡)>0) denote the species density of the preys and the predators in the group at time 𝑡 respectively. The coefficient 𝜇1>0 denotes the birth rate of the preys, and 𝜇2>0 denotes the death rate of the predators. The other two coefficients 𝑟12 and 𝑟21 (both positive) describe interactions between the species.

In order to discuss the stability of the system, a matrix differential equation is presented as follows:𝜇̇𝑥=𝐴𝑥+Φ(𝑥),where𝐴=100𝜇2,Φ(𝑥)=𝑟12𝑥1𝑥2𝑟21𝑥1𝑥2.(2.2)

Lemma 2.1. ̇𝑥=𝑓(𝑥(𝑡)), where 𝑥(𝑡)𝑅𝑛 is the state variable, and 𝑓𝐶[𝑅𝑛,𝑅𝑛] satisfies𝑓(0)=0, is a vector field defined over a compact region 𝑊𝑅𝑛. By using the methods introduced in [24], one can construct fuzzy model for system (2.1) as follows.
Control Rule 𝑖(𝑖=1,2,,𝑟): IF 𝑧1(𝑡) is 𝑀𝑖1,𝑧2(𝑡) is 𝑀𝑖2, and 𝑧𝑝(𝑡) is 𝑀𝑖𝑝, THEN ̇𝑥(𝑡)=𝐴𝑖𝑥(𝑡), where 𝑟 is the number of T-S fuzzy rules, and 𝑧1(𝑡),𝑧2(𝑡),,𝑧𝑝(𝑡) are the premise variables, each 𝑀𝑖𝑗(𝑗=1,2,,𝑝) is a fuzzy set, and 𝐴𝑖𝑅𝑛×𝑛 is a constant matrix.
Thus, the nonlinear equation can be transformed to the following linear equation.
If 𝑥2(𝑡) is 𝑀𝑖̇𝑥=𝐴𝑖𝑥(𝑡),𝑡𝜏𝑗,||Δ𝑥𝑡=𝜏𝑗=𝐾𝑖,𝑗𝑥(𝑡),𝑡=𝜏𝑗,𝑖=1,2,,𝑟,𝑗=1,2,(2.3) where 𝐴𝑖=𝜇1𝑑𝑖𝑟120𝑑𝑖𝑟21𝜇2,(2.4) and 𝑑𝑖 is related to the value of 𝑥2(𝑡) (here, 𝑑𝑖=𝑥2(𝑡)). 𝑀𝑖,𝑥(𝑡),𝐴𝑖𝑅2×2,𝑟 is the number of the IF-THEN rules, 𝐾𝑖,𝑗𝑅2×2 denotes the control of the j th impulsive instant, Δ𝑥|𝑡=𝜏𝑗𝑥(𝜏+𝑗𝜏𝑗).
Correspondently, with center-average defuzzifier, the overall T-S fuzzy impulsive system can be represented as follows:̇𝑥(𝑡)=𝑟𝑖=1𝑖𝑥2𝐴(𝑡)𝑖𝑥(𝑡),𝑡𝜏𝑗,||Δ𝑥𝑡=𝜏𝑗=𝑟𝑖=1𝑖𝑥2𝐾(𝑡)𝑖,𝑗𝑥,𝑡=𝜏𝑗,(2.5) where 𝑖(𝑥2(𝑡))=𝜔𝑖(𝑥2(𝑡))/𝑟𝑖=1𝜔𝑖(𝑥2(𝑡)) and 𝜔𝑖(𝑥2(𝑡))=𝑝𝑗=1𝑀𝑖,𝑗(𝑥2(𝑡)).

Obviously, 𝑖(𝑥2(𝑡))0,𝑟𝑖=1𝑖(𝑥2(𝑡))=1,𝑖=1,2,,𝑟.

Lemma 2.2. If P is a real semipositive matrix, then a real matrix C exists, making 𝑃=𝐶𝑇𝐶.

3. Stability Analysis

Theorem 3.1. Assume that 𝜆𝑖 is the maximum eigenvalue of [𝐴𝑇𝑖+𝐴𝑖](𝑖=1,2,,𝑟), let 𝜆(𝛼)=max𝑖{𝜆𝑖}, 0<𝛿𝑗=𝜏𝑗𝜏𝑗1< is impulsive distance [25]. If 𝜆(𝛼)0 and there exists a constant scalar 𝜀>1 and a semipositive matrix P, such that ln𝜀𝛽𝑗+𝜆(𝛼)𝛿𝑗0,𝑃𝐴𝑖=𝐴𝑖𝑃,(3.1)
where𝑃=𝐶𝑇𝐶,𝛽𝑗=max𝑖𝐶𝐼+𝐾𝑖,𝑗.(3.2)
Then the system (2.5) is stable globally and asymptotically.

Proof. Let the candidate Lyapunov function be in the form of 1𝑉(𝑥)=2𝑥𝑇𝑃𝑥.(3.3) Clearly, for 𝑡𝜏𝑗, 𝑉1(𝑥)=2𝑟𝑖=1𝑖𝑥2𝑥(𝑡)𝑇𝐴𝑇𝑖𝑃+𝑃𝐴𝑖𝑥=12𝑟𝑖=1𝑖𝑥2𝑥(𝑡)𝑇𝑃𝑃1𝐴𝑇𝑖𝑃+𝐴𝑖𝑥12𝜆(𝛼)𝑥𝑇𝑃𝑟𝑖=1𝑖𝑥2𝑥=1(𝑡)2𝜆(𝛼)𝑥𝑇𝑃𝑥=𝜆(𝛼)𝑉(𝑥(𝑡)),(3.4) where 𝑡(𝜏𝑗1,𝜏𝑗](𝑗=1,2,).
For 𝑡=𝜏𝑗, we have𝑉𝑥𝜏+𝑗=12𝑟𝑖=1𝑖𝑥2(𝑡)𝐼+𝐾𝑖,𝑗𝑥𝜏𝑗𝑇𝑃𝐼+𝐾𝑖,𝑗𝑥𝜏𝑗=12𝑟𝑖=1𝑖𝑥2(𝑡)𝐼+𝐾𝑖,𝑗𝑥𝜏𝑗𝑇𝐶𝑇𝐶𝐼+𝐾𝑖,𝑗𝑥𝜏𝑗=12𝑟𝑖=1𝑖𝑥2𝐶(𝑡)𝐼+𝐾𝑖,𝑗𝑥𝜏𝑗12𝑟𝑖=1𝑖𝑥2𝐶(𝑡)𝐼+𝐾𝑖,𝑗𝑥𝜏𝑗12𝑟𝑖=1𝑖𝑥2𝛽(𝑡)𝑗𝑥𝜏𝑗=𝛽𝑗𝑉𝑥𝜏𝑗,𝑗𝑁.(3.5) Let 𝑗=1, for any 𝑡(𝜏0,𝜏1], by (3.4), we obtain 𝑉𝑥𝜏(𝑥(𝑡))𝑉0𝜆exp(𝛼)𝑡𝜏0.(3.6) Then 𝑉𝑥𝜏1𝑥𝜏𝑉0𝜆𝜏exp(𝛼)1𝜏0.(3.7) From (3.5) and (3.7), we obtain 𝑉𝑥𝜏+1𝛽1𝑉𝑥𝜏1𝛽1𝑉𝑥𝜏0𝜆𝜏exp(𝛼)1𝜏0.(3.8) In the same way, for any 𝑡(𝜏1,𝜏2], we have 𝑉𝜏(𝑡,𝑥)𝑉+1𝜆,𝑥exp(𝛼)𝑡𝜏1𝛽1𝑉𝜏0𝜆,𝑥exp(𝛼)𝑡𝜏0.(3.9) Similarly, for all 𝑘 and 𝑡(𝜏𝑘,𝜏𝑘+1], we obtain 𝑉(𝑡,𝑥)𝛽𝑘𝛽2𝛽1𝑉𝜏0𝜆,𝑥exp(𝛼)𝑡𝜏0.(3.10) From (3.2), we obtain 𝛽𝑘exp𝜆(𝛼)𝛿𝑘1𝜀,𝑘𝑁.(3.11) Thus, for 𝑡(𝜏𝑘,𝜏𝑘+1],𝑘𝑁, we have 𝑉𝑥𝜏(𝑥(𝑡))𝑉0𝛽1𝛽2𝛽𝑘𝜆exp(𝛼)𝑡𝜏0𝑥𝜏=𝑉0𝛽1exp𝜆(𝛼)𝛿1𝛽𝑘exp𝜆(𝛼)𝛿𝑘exp𝜆(𝛼)𝑡𝜏𝑘𝑥𝜏𝑉01𝜀𝑘exp𝜆(𝛼)𝑡𝜏𝑘.(3.12) So, if 𝑡, then 𝑘 and 𝑉(𝑡,𝑥)0. So the system (2.5) is stable globally and asymptotically.

Theorem 3.2. Assume that 𝜆𝑖 is the maximum eigenvalue of [𝐴𝑖+𝐴𝑇𝑖](𝑖=1,2,,𝑟), let 𝜆(𝛼)=max𝑖{𝜆𝑖}, 0<𝛿𝑗=𝜏𝑗𝜏𝑗1< is impulsive distance. If 𝜆(𝛼)<0 and a constant scalar 0𝜀<𝜆(𝛼) exists, such that ln(𝛽)𝜀𝛿𝑗0,𝑃𝐴𝑖=𝐴𝑖𝑃,(3.13) where 𝑃=𝐶𝑇𝐶 and 𝛽𝑗=max𝑖𝐶(𝐼+𝐾𝑖,𝑗).
Then the system (2.5) is stable globally and exponentially.

Proof. Let the candidate Lyapunov function be in the form of 1𝑉(𝑥)=2𝑥𝑇𝑃𝑥.(3.14)
Firstly, (3.4)–(3.10) hold.
From (3.13), we obtain𝛽𝑘exp𝜀𝜎𝑘1,𝑘𝑁.(3.15) Thus, for 𝑡(𝜏𝑘,𝜏𝑘+1],𝑘𝑁, 𝑉𝑥𝑡(𝑥(𝑡))𝑉0𝛽1𝛽2𝛽𝑘𝜆exp(𝛼)𝑡𝑡0𝑥𝑡=𝑉0𝛽1𝛽2𝛽𝑘(exp𝜀)𝑡𝑡0(exp𝜆(𝛼)+𝜀)𝑡𝑡0𝑥𝑡=𝑉0𝛽1𝑡exp𝜀1𝑡0𝛽𝑗exp𝜀𝑡𝑡𝑘exp(𝜆(𝛼)+𝜀)𝑡𝑡0𝑥𝑡𝑉0exp(𝜆(𝛼)+𝜀)𝑡𝑡0.(3.16)
Note that 0𝜀<𝜆(𝛼), thus 𝜆(𝛼)+𝜀<0. So the system (2.5) is stable globally and exponentially.
Next, we consider some special cases of the two theorems. Assume that 𝐾=𝐾𝑖,𝑗 and 𝜎=𝜎𝑗 in the two theorems above, so we can have the following corollary.

Corollary 3.3. Let 𝜆𝑖 be the largest eigenvalue of [𝐴+𝐴𝑇],(𝑖=1,2,,r),𝜆(𝛼)=max𝑖{𝜆𝑖}>0. If there exists a constant 𝜀>1 and a real semi-positive P such that ln(𝜀𝛽)+𝜆(𝑎)𝛿0,𝑃𝐴𝑖=𝐴𝑖𝑃,(3.17) where 𝑃=𝐶𝑇𝐶,𝛽𝑗=max𝑖𝐶(𝐼+𝐾𝑖,𝑗), and 0<𝛿=𝜏𝑗𝜏𝑗1<(𝑗𝑁)  is impulsive distance. Then the system (2.5) is stable globally and asymptotically.

Corollary 3.4. Let 𝜆𝑖 be the largest eigenvalue of [𝐴+𝐴𝑇](𝑖=1,2,,𝑟),𝜆(𝛼)=max𝑖{𝜆𝑖}<0. If there exists a constant 0𝜀<𝜆(𝛼) and a real semi-positive P such that ln(𝛽)𝜀𝛿0,𝑃𝐴𝑖=𝐴𝑖𝑃,(3.18) where 𝑃=𝐶𝑇𝐶,𝛽𝑗=max𝑖𝐶(𝐼+𝐾𝑖,𝑗), and 0<𝛿=𝜏𝑗𝜏𝑗1<(𝑗𝑁) is impulsive distance. Then the system (2.5) is stable globally and exponentially.

4. Numerical Simulation

In this section, we present a design example to show how to perform the impulsive fuzzy control on the Lotka-Volterra predator-prey systems with impulsive effects. Especially, the biological systems are very complex, nonlinear, and uncertain. As a result, they should be represented by fuzzy logical method with linguistic description.

Now, consider a predator-prey system with impulsive effects as follows: ̇𝑥=𝐴𝑥+Φ(𝑥),(4.1) where, 𝜇𝐴=100𝜇2,Φ(𝑥)=𝑟12𝑥1𝑥2𝑟21𝑥1𝑥2.(4.2)

Solving
From (2.3), we have the following impulsive fuzzy control for the above predator-prey model.

Rule i
IF 𝑥2(𝑡) is 𝑀𝑖, then ̇𝑥(𝑡)=𝐴𝑖𝑥(𝑡)𝑡𝜏𝑗,Δ𝑥=𝐾𝑖,𝑗𝑥(𝑡)𝑡=𝜏𝑗,𝑖=1,2,𝑗𝑁,(4.3)
where, 𝐴1=𝜇1𝑑𝑟120𝑑𝑟21𝜇2,𝐴2=𝜇1(1/2)𝑑𝑟120(1/2)𝑑𝑟21𝜇2,(4.4)
due to 𝑥2(𝑡)[0,𝑑]=[0,0.12], and 𝑀1(𝑥2(𝑡))=𝑥2(𝑡)/𝑑, 𝑀2(𝑥2(𝑡))=𝑥2(𝑡)/𝑑.
Let 𝜀=1.2, 𝛿=0.05,𝑃=𝐼,𝜇1=0.2,𝜇2=0.16,𝑟12=0.10,𝑟21=0.31.
From Theorem 3.1 and Corollary 3.3, we can get that 𝜆(𝛼)=0.194.
Thus, we have chosen diag ([0.82,0.82]) as impulsive control matrix, such that𝛽=𝐼+𝐾=0.18,ln(𝜀𝛽)+𝜆(𝛼)𝛿=1.3160.(4.5)
Thus, from Theorem 3.1 and Corollary 3.3, we can conclude that the numerical example is globally stable. The phase portrait of the system with impulsive control is shown in Figure 1.

5. Conclusions

The impulsive control technique, which was proved to be suitable for complex and nonlinear system with impulsive effects, was applied to analyzing the framework of the fuzzy systems based on T-S model and the proposed design approach. First, the robustness of the Lotka-Volterra predator-prey system based on the fuzzy impulsive control was carefully analyzed. Then, the overall impulsive fuzzy system was obtained by blending local linear impulsive system. Meanwhile, the asymptotical stability and exponential stability of the impulsive fuzzy system were derived by Lyapunov method. Finally, a numerical example for predator-prey systems with impulsive effects was given to illustrate the application of impulsive fuzzy control. The simulation results show that the proposed method was effective.

Acknowledgments

This work was supported by the National Natural Science Foundation of China, Project no. 50975300 and the Foundation of Education Department of Guangxi, China, Project no. 200808MS079.

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Copyright © 2012 Yuangan Wang. 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.


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