- Email: [email protected]
Impulsive control of uncertain Lotka–Volterra predator–prey system
Chaos, Solitons and Fractals 41 (2009) 1572–1577
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Impulsive control of uncertain Lotka–Volterra predator–prey system Dong Li a,b,*, Shilong Wang b, Xiaohong Zhang c, Dan Yang c a b c
College of Mathematics and Physics Science, Chongqing University, Chongqing 400030, PR China College of Mechanical Engineering, Chongqing University, Chongqing 400030, PR China College of Software Engineering, Chongqing University, Chongqing 400030, PR China
a r t i c l e
i n f o
Article history: Accepted 23 June 2008
Communicated by Prof. Ji-Huan He
a b s t r a c t In this letter, we investigate the impulsive control of Lotka–Volterra predator–prey system. The uncertainties in the system are considered. The model parameters which character the uncertainties are formulated through matrix analysis. The sufficient conditions of the asymptotic stability are established by employing the method of Lyapunov functions. Finally, the validity of the results is demonstrated by a numerical example. Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved.
1. Introduction The predator–prey system is an important model in population dynamics. The stability of predator–prey system has been studied extensively since the theoretical work of Lotka (1926), Volterra (1931), Nicholson and Bailey (1935) and the experimental work of Gause (1934). On the other hand, impulsive effects exist in a variety of evolutionary processes where states are changed abruptly at certain moments. This complex dynamical behavior can be modeled by impulsive differential equations. The theory of impulsive differential systems has been developed by numerous mathematicians [1–3,18]. As for the application of impulsive differential equations, systems with impulsive effects are used for practical problems in most cases [4–17]. Especially, there are many results about the application of impulsive differential equations to ecology. For examples, Ballinger [8] established some criteria for permanence of populations which undergo impulsive effects at fixed times. Zhang [9] investigated the dynamic complexity of a two-prey one-predator system with impulsive perturbation at fixed moments. Song [10] discussed the dynamic behaviors of a Holling II two-prey and one-predator system with impulsive effects. Liu [11] investigated the dynamic behaviors of a classical periodic Lotka–Volterra competing system with impulsive perturbations, and gives the conditions for global stability of trivial and semi-trivial periodic solutions and the conditions for persistence. Jiang [12] investigated the dynamics of a predator–prey model with impulsive state feedback control, and presented the sufficient conditions of existence and stability of semi-trivial solution and positive period-1 solution. Jiao [13] considered a stage-structured Holling mass defense predator–prey model with time delay and impulsive transmitting on predators. Akhmet [14] discussed the global stability of the predator–prey system with diffusion affected by impulses. Zhang [15] studied the effect of periodic forcing and impulsive perturbations on predator–prey model with Holling type IV functional response. Wang [16] discussed the stability of predator–prey system with Watt-type functional response and impulsive perturbations on the predator. Lu [17] put forward two impulsive strategies in biological control of pesticide. It is important to point out that the uncertainties happen frequently in predator–prey system due modeling errors, measurement inaccuracy, mutations in the evolutionary processes and so on. It is well known that the uncertainties often lead to instability. Therefore, the stability analysis of uncertain predator–prey system is very important. However, to the best of our
* Corresponding author. Address: College of Mathematics and Physics Science, Chongqing University, Chongqing 400030, PR China. Tel.: +86 23 66700256. E-mail address: [email protected] (D. Li). 0960-0779/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.06.021
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
1573
knowledge, no reports about the stability results for uncertain predator–prey systems and few for uncertain impulsive dynamical systems have been presented in the open literatures. In this paper, we shall investigate the uncertain Lotka–Volterra predator–prey system with impulsive effects and derive the corresponding stability criterion. The organization of this paper is as follows. In the next section, the problems investigated in this paper are formulated, and some preliminaries are presented. We state and prove our main results in Section 3. Then, an illustrative example is given to show the effectiveness of the obtained results in Section 4. 2. Problem formulation and preliminaries Based on the Lotka–Volterra model, we propose the uncertain Lotka–Volterra predator–prey system with impulsive effects as follows:
x_ 1 ¼ x1 ðl1 r12 x2 Þ x_ 2 ¼ x2 ðl2 þ r 21 x1 Þ x1 ðt þ0 Þ ¼ x10 > 0 t–tk
ð2:1Þ
x2 ðt þ0 Þ ¼ x20 > 0
Dxjt¼tk , ¼ xðt þk Þ xðt k Þ ¼ Uðk; xÞ; k ¼ 1; 2; where x1 ðtÞ, x2 ðtÞ ðx1 ðtÞ > 0; x2 ðtÞ > 0Þ denote the species density of the preys and the predators at time t, respectively. The coefficient l1 > 0 denotes the birth rate of the preys, l2 > 0 denotes the birth rate of the predators, the other two coefficients r 12 and r 12 (both positive) describe interactions between the species. Moreover, the coefficients r 12 and r 21 are not constant but lie in some interval characterized by r12 2 N½l1 ; g 1 and r21 2 N½l2 ; g 2 due to the uncertainties. Obviously, x ¼ ðl2 =r 21 ; l1 =r12 Þ is one of the equilibrium points of system (2.1). In the following, in order to discuss the stability of system (2.1), we present some preliminary results with respect to impulsive differential equations initially. We denote by Rn the n-dimensional Euclidean space. Let Rþ ¼ ½0; þ1Þ, N ¼ f1; 2; g. The following sets will be used in the sequel. v ¼ f/ 2 CðRþ ; Rþ Þ, strictly increasing and /ð0Þ ¼ 0g, PC ¼ fw : Rþ ! Rþ is continuous everywhere except at finite number of point tk ðk 2 NÞ at which w is left continuous the þ right limit wðtþ k Þ exists, wðt k Þ–wðt k Þg
Sq ¼ fx 2 Rn : kxk 6 qg: The nominal impulsive system is given by
x_ ¼ f ðt; xÞ t–t k
Dx ¼ Uðk; xÞ t ¼ tk ; k 2 N xðt þ0 Þ ¼ x0 P 0 Definition 2.1 [19]. Let V : Rþ Rn ! Rþ , then V is said to belong to class
ð2:2Þ
t0 (a Lyapunov-like function), if
(i) V is continuous in ðtk1 ; tk Rn and for each x 2 Rn , t 2 ðt k1 ; t k , k 2 N
Vðt; yÞ ¼ Vðtþk1 ; xÞ
lim
ðt;yÞ!ðt þ ;xÞt>t k1
ð2:3Þ
k1
exists; (ii) V is locally Lipschizian in x 2 Sq , and for all t P t0 , Vðt; 0Þ 0.
Definition 2.2 [19]. For ðt; xÞ 2 ðt k1 ; t k Rn , we define
1 Dþ Vðt; xÞ ¼ limþ sup ½Vðt þ h; x þ hf ðt; xÞÞ Vðt; xÞ h h!0
ð2:4Þ
Definition 2.3. The uncertain impulsive predator–prey system is called stable, asymptotically stable, if the trivial solution of the system is stable, asymptotically stable.
3. The stability of the uncertain predator–prey system with impulsive effects The effective trivial solution of system (2.1) is ðl2 =r21 ; l1 =r 12 Þ. By the Definition 2.3, we shall discuss the stability of the trivial solution to study the stability of system (2.1). First, the conversion is applied as follows:
1574
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
x01 ¼ x1
l2 r 21
x02 ¼ x2
;
l1
ð3:1Þ
r 12
Applying (3.1) to system (2.1), and if x01 is replaced by x1 , x02 is replaced by x2 , then the system (2.1) can be rewritten as the following form:
x_ 1 ¼ l2
r12 x2 r 12 x1 x2 r21
r 21 x1 þ r 21 x1 x2 r 12 x1 ðt þ0 Þ ¼ x10 > 0 t–tk
x_ 2 ¼ l1
ð3:2Þ
x2 ðt þ0 Þ ¼ x20 > 0
Dxjt¼tk , ¼ xðt þk Þ xðt k Þ ¼ Uðk; xÞ; k 2 N
Let Uðk; xÞ ¼ C k x, where C k ¼
0 . The system (3.2) can be described as c2k
c1k 0
x_ ¼ A1 A2 A3 x þ Bx1 x2
t–t k Dx ¼ Uðk; xÞ ¼ C 0k x t ¼ t k xðtþ0 Þ ¼ x0
where
A1 ¼
0
l2
l1 0
ð3:3Þ
r21 ; A2 ¼ 0
0 r 12
; A3 ¼
1=r 12
0
0
1=r21
; B¼
r 12 r 21
:
Then we can discuss the stability of origin of system (3.3) to obtain the stability of system (2.1). In order to obtain stability result for system (3.3), we shall establish some lemmas first. Lemma 3.1 [20]. Let X 2 Rnn be a n n positive definite matrix and Y 2 Rnn a symmetric matrix. Then for any x 2 Rn ,t 2 Rþ , the following inequality holds:
xT Yx 6 kmax ðX 1 YÞxT Xx:
ð3:4Þ
where kmax ðÞ is the maximal eigenvalue function. Lemma 3.2. If r12 2 N½l1 ; g 1 ,r 21 2 N½l2 ; g 2 , Let R , ¼ fR 2 R22 : R ¼ diagðe1 ; e2 Þ; jei j 6 1; i ¼ 1; 2g, the following statements are hold. e ¼ g 2 0 , then A2 ¼ r21 0 e ¼ l2 0 and Q can be written as (1) Let P 0 g1 0 r12 0 l1
ee F A2 ¼ A20 þ e ER
ð3:5Þ
where
A20 ¼
1 e e ð P þ Q Þ; 2
_
(2) Let P ¼
1 e e ~ Þ e ¼ ðh H ð Q PÞ; ij 22 ¼ 2
0 1=g 2
1=g 1 0
e ¼ 1=l1 and Q 0
R 2 R ;
e E¼e F¼
1=r 12 0 , then A3 ¼ 0 1=l2
" pffiffiffiffiffi # h1 0 pffiffiffiffiffi : 0 h2 0 1=r 21
can be written as
_ _ _
A3 ¼ A30 þ E R F
ð3:6Þ
where
A30 ¼
1 _ _ ðP þ Q Þ; 2
_
_
H ¼ ðh Þ22 ¼
Remarks. Clearly, for any
ij
P
2
P
1 _ _ ðQ P Þ; R 2 R ; 2
_
_
E¼F¼
" pffiffiffiffiffi # h1 0 pffiffiffiffiffi : 0 h2
, we get __
_
_
eR eTR eT ¼ R e 6 I, R RT ¼ RT R 6 I, where I is the 2 2 identity matrix. (1) RRT ¼ RT R 6 I, R (2) by Lemma 3.2, the system (3.3) can be rewritten as _ _ _
ee F ÞðA30 þ E R F Þx þ Bx1 x2 ER x_ ¼ A1 ðA20 þ e
t–tk
Dx ¼ Uðk; xÞ ¼ C k x t ¼ t k xðtþ0 Þ ¼ x0 _ _ _
e; e where E; R; F; e E; R F ; E ; R; F are defined as in Lemma 3.2.
ð3:7Þ
1575
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
Lemma 3.3 [20]. If
2nT
X
P
2
P
, then for any positive scalar k > 0 and for any n 2 Rn , the following inequality holds:
1 k
g 6 nT n þ kgT g
ð3:8Þ
especially, the inequality holds for k ¼ 1. Lemma 3.4 [21]. Given any real matrices R1 ; R2 ; R3 of appropriate dimensions and a scalar e > 0 such that 0 < R3 ¼ RT3 . Then, the following inequality holds:
RT1 R2 þ RT2 R1 6 eRT1 R3 R1 þ e1 RT2 R1 3 R2 :
ð3:9Þ
especially, the inequality holds for e ¼ 1; R3 ¼ E0 (identity matrix). Theorem 3.1. Assume that there exists a positive definite symmetric matrix C. Let M ¼ 2g 2 x10 and bk denote the maximum eigenvalue of C1 ðI þ C k ÞT CðI þ C k Þ.
0
1 __ _ _ C1 fAT30 AT20 AT1 C þ CA1 A20 A30 þ ðCA1 A20 E E T AT20 AT1 CÞ þ F T F þðCA1 eEÞðCA1 eEÞT A @ kðaÞ ¼ kmax _ _ _ _ FTe F A30 þ CA1 e Ee E T AT1 C þ kmax ðE T e FTe F E ÞF T F g þAT30 e
ð3:10Þ
if ðkðaÞ þ MÞ ðtkþ1 t k Þ 6 lnðcbk Þ; for k 2 N; c > 1; k P 0: Then the system (2.1) is robust asymptotically stable. Proof. Let V ¼ xT Cx, where C is a positive definite matrix. Clearly V belongs to T
T
t0 . When t ¼ tk , by Lemma 3.1, we have
T
1
Vðt þk ; xÞ ¼ ðx þ C k xÞ Cðx þ C k xÞ ¼ xT ðI þ C k Þ CðI þ C k Þxkmax ðC ðI þ C k Þ CðI þ C k ÞÞxT Cx ¼ bk Vðt; xÞ
ð3:11Þ
When t–tk , we have
r 12 x1 x2 BT x1 x2 Cx þ xT CBx1 x2 ¼ ½ r12 x1 x2 r21 x1 x2 Cx þ xT C r21 x1 x2 r x 0 r 0 12 2 12 x2 Cx þ xT C ¼ xT x 6 2r 21 x1 ðt þ0 ÞxT Cx 6 2g 2 x10 xT Cx ¼ MVðt; xÞ 0 r 21 x1 0 r21 x1 ð3:12Þ By the Lemma 3.2, then
Dþ Vðt; xÞ ¼ x_ T Cx þ xT Cx_ ¼ ðA1 A2 A3 x þ Bx1 x2 ÞT Cx þ xT CðA1 A2 A3 x þ Bx1 x2 Þ _ _ _
_ _ _
ee ee F ÞT AT1 þ BT x1 x2 ÞCx þ xT CðA1 ðA20 þ e F ÞðA30 þ E R F Þx þ Bx1 x2 Þ ER ER ¼ ðxT ðA30 þ E R F ÞT ðA20 þ e _ _ _
_ _ _
ee ee F ÞT AT1 C þ CA1 e ER F A30 ER ¼ xT fAT30 AT20 AT1 C þ CA1 A20 A30 þ ðE R F ÞT AT20 AT1 C þ CA1 A20 E R F þAT30 ð e _ _ _
_ _ _
ee ee F ÞT AT1 C þ CA1 e ER F E R F gx þ BT x1 x2 Cx þ xT CBx1 x2 þ ðE R F ÞT ð e ER
ð3:13Þ
By the assumption, Lemmas 3.3, 3.4 and (3.12), (3.13). __
_ _
Dþ Vðt; xÞ 6 xT fAT30 AT20 AT1 C þ CA1 A20 A30 þ ðCA1 A20 E E T AT20 AT1 CÞ þ F T F þðCA1 e EÞðCA1 e EÞT þ AT30 e FTe F A30 þ CA1 e Ee E T AT1 C _ _ _
_ _ _
F E R F gx þ MVðt; xÞ þ ðe F E R F ÞT e By the Lemma 3.1 _ _ _
_ _ _
_ _
_
_ _ _
_
_
_ _
F E R F x ¼ ðR F xÞT E T e FTe F E ðR F xÞ 6 kmax ðE T e FTe F E ÞxT F T F x xT ð e F E R F ÞT e
ð3:14Þ
Then, we have __
_ _
EÞðCA1 e EÞT Dþ Vðt; xÞ 6 xT fAT30 AT20 AT1 C þ CA1 A20 A30 þ ðCA1 A20 E E T AT20 AT1 CÞ þ F T F þðCA1 e _
_ _ _
FTe F A30 þ CA1 e Ee E T AT1 C þ kmax ðE T e FTe F E ÞF T F gx þ MVðt; xÞ þ AT30 e 6 ðkðaÞ þ MÞVðt; xÞ Let k ¼ 1, for all t 2 ðt 0 ; t 1 , by (3.15), we have
Vðt; xÞ 6 Vðt 0 ; xÞ expððkðaÞ þ MÞðt t 0 ÞÞ: Then
Vðt 1 ; xÞ 6 Vðt0 ; xÞ expððkðaÞ þ MÞðt 1 t 0 ÞÞ:
ð3:15Þ
1576
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
By (3.11), we have
Vðtþ1 ; xÞ 6 b1 Vðt 1 ; xÞ 6 b1 Vðt0 ; xÞ expððkðaÞ þ MÞðt 1 t0 ÞÞ: For all t 2 ðt 1 ; t 2 , we have
Vðt; xÞ 6 Vðt þ1 ; xÞ expððkðaÞ þ MÞðt t1 ÞÞ 6 b1 Vðt 0 ; xÞ expððkðaÞ þ MÞðt t 0 ÞÞ: Similarly, for all k 2 N and t 2 ðtk ; tkþ1 , we have
Vðt; xÞ 6 bk b2 b1 Vðt0 ; xÞ expððkðaÞ þ MÞðt t 0 ÞÞ:
ð3:16Þ
By (3.10), we have
bk expððkðaÞ þ MÞðtkþ1 tk ÞÞ 6
1
C
;
k2N
ð3:17Þ
Hence, for all t 2 ðt k ; t kþ1 ðk 2 NÞ, we have
Vðt; xÞ 6 bk b2 b1 Vðt0 ; xÞ expððkðaÞ þ MÞðt t 0 ÞÞ ¼ Vðt 0 ; xÞ½bk expððkðaÞ þ MÞðt k t k1 ÞÞ ½b1 expððkðaÞ þ MÞðt 1 t0 ÞÞ expððkðaÞ þ MÞðt tk ÞÞ 1 6 Vðt 0 ; xÞ k expððkðaÞ þ MÞðt t k ÞÞ
c
ð3:18Þ
Thus, if t ! 1, then k ! 1 and Vðt; xÞ ! 0. The proof is complete. h Corollary 3.1. Suppose that all the impulsive spacing are uniform, i.e. Dt k ,tkþ1 t k d for all k 2 N. If d 6 mink ðkða1ÞþMÞ lnðcbk Þ , the system (2.1) is robust asymptotically stable, where kðaÞ, M, c and bk are defined as in Theorem 3.1. Corollary 3.2. Based on the conditions of Corollary 3.1 and the applying the uniform impulsive control at all the time t k , i.e. for all 1 lnðcbÞ, the system (2.1) is robust asymptotically stable, where b ¼ kmax ðC1 ðI þ CÞT CðI þ CÞÞ. k, C k ¼ C. If d 6 ðkðaÞþMÞ 4. Illustrative example In this section, we give an example to demonstrate the effectiveness of our results.
Fig. 4.1. The state program of the system with impulsive control.
1577
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
Example 4.1. Consider the following uncertain predator–prey impulsive system
x_ 1 ¼ x1 ðl1 r12 x2 Þ x_ 2 ¼ x2 ðl2 þ r 21 x1 Þ xðt10 Þ ¼ x10 > 0 t–t k
ð4:1Þ
xðt20 Þ ¼ x20 > 0
Dxjt¼tk , ¼ xðt þk Þ xðt k Þ ¼ Uðk; xÞ ¼ Cx; k 2 N The parameters of system (2.1) are as follows:
l1 ¼ 0:2; l2 ¼ 0:16; r12 2 N½0:05; 0:13; r21 2 N½0:25; 0:36; C ¼
2=3
0
0
2=3
:
Set x10 ¼ x20 ¼ 0:1. Let Vðt; xÞ ¼ xT x. Suppose r 12 and r 21 be random of the above interval, respectively. It is easy to obtain kðaÞ ¼ 21:3709, b ¼ 0:1111 by computing them with Matlab 7.0. By Theorem 3.1, when c ¼ 2 and Dtk 6 0:0701, ðkðaÞ þ MÞ d 6 lnðcbÞ, then the system is asymptotically stable. The stable system state program of the system with impulsive control is showed as Fig. 4.1.
5. Conclusions In this paper, we have investigated impulsive control of Lotka–Volterra predator–prey system. The uncertainties in the system are considered. The sufficient conditions of the robust asymptotic stability are established by employing the method of Lyapunov functions in Theorem 3.1. In Section 4, the validity of the results is demonstrated by a numerical example. The illustrated example shows that the system is robust asymptotic stable under the conditions of Theorem 3.1. Acknowledgement The work described in this paper was supported by the National Natural Science Foundation of China (Grant No. 60604007). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Lakshmikantham V, Bainov DD, Simeonov PS. Theory of impulsive differential equations. Singapore: World Scientific; 1989. Samoilenko AM, Perestyuk NA. Impulsive differential equations. Singapore: World Scientific; 1995. Chen F, Wen X. Asymptotic stability for impulsive functional differential equation. J Math Anal Appl 2007;336:1149–60. Sun J, Zhang Y. Impulsive control of a nuclear spin generator. J Comput Appl Math 2003;157:235–42. Yang T, Yang L, Yang C. Impulsive control of Lorenz system. Physica D 1997;110:18–24. Yoshikawa Shoji, Yamada Katsuhiko. Impulsive control for angular momentum management of tumbling spacecraft. Acta Astron 2007;60:810–9. Chen D, Sun J, Wu Q. Impulsive control and its application to Lü’s chaotic system. Chaos, Solitons & Fractals 2004;21:1135–42. Ballinger G, Liu X. Permanence of population growth models with impulsive effects. Math Comput Model 1997;26:59–72. Zhang Y, Xiu Z, Chen L. Dynamic complexity of a two-prey one-predator system with impulsive effect. Chaos, Solitons & Fractals. 2005;26:131–9. Song X, Li Y. Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect. Chaos, Solitons & Fractals 2007;33:463–78. Liu B, Teng Z, Liu W. Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations. Chaos, Solitons & Fractals 2007;31:356–70. Jiang G, Lu Q. Impulsive state feedback control of a predator-prey model. J Comput Appl Math 2007;200:193–207. Jiao J, Meng X, Chen L. A stage-structured Holling mass defence predator–prey model with impulsive perturbations on predators. Appl Math Comput 2007;189:1448–58. Akhmet M, Beklioglu M, Ergenc T, Tkachenko V. An impulsive ratio-dependent predator–prey system with diffusion. Nonlinear Anal Real World Appl 2006;7:1255–67. Zhang S, Tan D, Chen L. Chaos in periodically forced Holling type IV predator–prey system with impulsive perturbations. Chaos, Solitons & Fractals 2006;27:980–90. Wang X, Wang W, Lin X. Chaotic behavior of a Watt-type predator–prey system with impulsive control strategy. Chaos, Solitons & Fractals 2008;37:706–18. Lu Z, Chi X, Chen L. Impulsive control strategies in biological control of pesticide. J Theor Popul Biol 2003;64:39–47. Liu Z, Yuan R. Stability and bifurcation in a delayed predator–prey system with Beddington–DeAngelis functional response. J Math Anal Appl 2004;296:521–37. Yan J, Shen J. Impulsive stabilization of functional differential equations by Lyapunov–Razumikhin functions. Nonlinear Anal 1999;37:245–55. Liu B, Liu X, Liao X. Robust stability of uncertain impulsive dynamical systems. J Math Anal Appl 2004;290:519–33. Sanchez E, Perez J. Input-to-state stability analysis for dynamic NN. IEEE Trans Circuits Syst 1999;46:1395–8.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Impulsive control of uncertain Lotka–Volterra predator–prey system Dong Li a,b,*, Shilong Wang b, Xiaohong Zhang c, Dan Yang c a b c
College of Mathematics and Physics Science, Chongqing University, Chongqing 400030, PR China College of Mechanical Engineering, Chongqing University, Chongqing 400030, PR China College of Software Engineering, Chongqing University, Chongqing 400030, PR China
a r t i c l e
i n f o
Article history: Accepted 23 June 2008
Communicated by Prof. Ji-Huan He
a b s t r a c t In this letter, we investigate the impulsive control of Lotka–Volterra predator–prey system. The uncertainties in the system are considered. The model parameters which character the uncertainties are formulated through matrix analysis. The sufficient conditions of the asymptotic stability are established by employing the method of Lyapunov functions. Finally, the validity of the results is demonstrated by a numerical example. Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved.
1. Introduction The predator–prey system is an important model in population dynamics. The stability of predator–prey system has been studied extensively since the theoretical work of Lotka (1926), Volterra (1931), Nicholson and Bailey (1935) and the experimental work of Gause (1934). On the other hand, impulsive effects exist in a variety of evolutionary processes where states are changed abruptly at certain moments. This complex dynamical behavior can be modeled by impulsive differential equations. The theory of impulsive differential systems has been developed by numerous mathematicians [1–3,18]. As for the application of impulsive differential equations, systems with impulsive effects are used for practical problems in most cases [4–17]. Especially, there are many results about the application of impulsive differential equations to ecology. For examples, Ballinger [8] established some criteria for permanence of populations which undergo impulsive effects at fixed times. Zhang [9] investigated the dynamic complexity of a two-prey one-predator system with impulsive perturbation at fixed moments. Song [10] discussed the dynamic behaviors of a Holling II two-prey and one-predator system with impulsive effects. Liu [11] investigated the dynamic behaviors of a classical periodic Lotka–Volterra competing system with impulsive perturbations, and gives the conditions for global stability of trivial and semi-trivial periodic solutions and the conditions for persistence. Jiang [12] investigated the dynamics of a predator–prey model with impulsive state feedback control, and presented the sufficient conditions of existence and stability of semi-trivial solution and positive period-1 solution. Jiao [13] considered a stage-structured Holling mass defense predator–prey model with time delay and impulsive transmitting on predators. Akhmet [14] discussed the global stability of the predator–prey system with diffusion affected by impulses. Zhang [15] studied the effect of periodic forcing and impulsive perturbations on predator–prey model with Holling type IV functional response. Wang [16] discussed the stability of predator–prey system with Watt-type functional response and impulsive perturbations on the predator. Lu [17] put forward two impulsive strategies in biological control of pesticide. It is important to point out that the uncertainties happen frequently in predator–prey system due modeling errors, measurement inaccuracy, mutations in the evolutionary processes and so on. It is well known that the uncertainties often lead to instability. Therefore, the stability analysis of uncertain predator–prey system is very important. However, to the best of our
* Corresponding author. Address: College of Mathematics and Physics Science, Chongqing University, Chongqing 400030, PR China. Tel.: +86 23 66700256. E-mail address: [email protected] (D. Li). 0960-0779/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.06.021
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
1573
knowledge, no reports about the stability results for uncertain predator–prey systems and few for uncertain impulsive dynamical systems have been presented in the open literatures. In this paper, we shall investigate the uncertain Lotka–Volterra predator–prey system with impulsive effects and derive the corresponding stability criterion. The organization of this paper is as follows. In the next section, the problems investigated in this paper are formulated, and some preliminaries are presented. We state and prove our main results in Section 3. Then, an illustrative example is given to show the effectiveness of the obtained results in Section 4. 2. Problem formulation and preliminaries Based on the Lotka–Volterra model, we propose the uncertain Lotka–Volterra predator–prey system with impulsive effects as follows:
x_ 1 ¼ x1 ðl1 r12 x2 Þ x_ 2 ¼ x2 ðl2 þ r 21 x1 Þ x1 ðt þ0 Þ ¼ x10 > 0 t–tk
ð2:1Þ
x2 ðt þ0 Þ ¼ x20 > 0
Dxjt¼tk , ¼ xðt þk Þ xðt k Þ ¼ Uðk; xÞ; k ¼ 1; 2; where x1 ðtÞ, x2 ðtÞ ðx1 ðtÞ > 0; x2 ðtÞ > 0Þ denote the species density of the preys and the predators at time t, respectively. The coefficient l1 > 0 denotes the birth rate of the preys, l2 > 0 denotes the birth rate of the predators, the other two coefficients r 12 and r 12 (both positive) describe interactions between the species. Moreover, the coefficients r 12 and r 21 are not constant but lie in some interval characterized by r12 2 N½l1 ; g 1 and r21 2 N½l2 ; g 2 due to the uncertainties. Obviously, x ¼ ðl2 =r 21 ; l1 =r12 Þ is one of the equilibrium points of system (2.1). In the following, in order to discuss the stability of system (2.1), we present some preliminary results with respect to impulsive differential equations initially. We denote by Rn the n-dimensional Euclidean space. Let Rþ ¼ ½0; þ1Þ, N ¼ f1; 2; g. The following sets will be used in the sequel. v ¼ f/ 2 CðRþ ; Rþ Þ, strictly increasing and /ð0Þ ¼ 0g, PC ¼ fw : Rþ ! Rþ is continuous everywhere except at finite number of point tk ðk 2 NÞ at which w is left continuous the þ right limit wðtþ k Þ exists, wðt k Þ–wðt k Þg
Sq ¼ fx 2 Rn : kxk 6 qg: The nominal impulsive system is given by
x_ ¼ f ðt; xÞ t–t k
Dx ¼ Uðk; xÞ t ¼ tk ; k 2 N xðt þ0 Þ ¼ x0 P 0 Definition 2.1 [19]. Let V : Rþ Rn ! Rþ , then V is said to belong to class
ð2:2Þ
t0 (a Lyapunov-like function), if
(i) V is continuous in ðtk1 ; tk Rn and for each x 2 Rn , t 2 ðt k1 ; t k , k 2 N
Vðt; yÞ ¼ Vðtþk1 ; xÞ
lim
ðt;yÞ!ðt þ ;xÞt>t k1
ð2:3Þ
k1
exists; (ii) V is locally Lipschizian in x 2 Sq , and for all t P t0 , Vðt; 0Þ 0.
Definition 2.2 [19]. For ðt; xÞ 2 ðt k1 ; t k Rn , we define
1 Dþ Vðt; xÞ ¼ limþ sup ½Vðt þ h; x þ hf ðt; xÞÞ Vðt; xÞ h h!0
ð2:4Þ
Definition 2.3. The uncertain impulsive predator–prey system is called stable, asymptotically stable, if the trivial solution of the system is stable, asymptotically stable.
3. The stability of the uncertain predator–prey system with impulsive effects The effective trivial solution of system (2.1) is ðl2 =r21 ; l1 =r 12 Þ. By the Definition 2.3, we shall discuss the stability of the trivial solution to study the stability of system (2.1). First, the conversion is applied as follows:
1574
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
x01 ¼ x1
l2 r 21
x02 ¼ x2
;
l1
ð3:1Þ
r 12
Applying (3.1) to system (2.1), and if x01 is replaced by x1 , x02 is replaced by x2 , then the system (2.1) can be rewritten as the following form:
x_ 1 ¼ l2
r12 x2 r 12 x1 x2 r21
r 21 x1 þ r 21 x1 x2 r 12 x1 ðt þ0 Þ ¼ x10 > 0 t–tk
x_ 2 ¼ l1
ð3:2Þ
x2 ðt þ0 Þ ¼ x20 > 0
Dxjt¼tk , ¼ xðt þk Þ xðt k Þ ¼ Uðk; xÞ; k 2 N
Let Uðk; xÞ ¼ C k x, where C k ¼
0 . The system (3.2) can be described as c2k
c1k 0
x_ ¼ A1 A2 A3 x þ Bx1 x2
t–t k Dx ¼ Uðk; xÞ ¼ C 0k x t ¼ t k xðtþ0 Þ ¼ x0
where
A1 ¼
0
l2
l1 0
ð3:3Þ
r21 ; A2 ¼ 0
0 r 12
; A3 ¼
1=r 12
0
0
1=r21
; B¼
r 12 r 21
:
Then we can discuss the stability of origin of system (3.3) to obtain the stability of system (2.1). In order to obtain stability result for system (3.3), we shall establish some lemmas first. Lemma 3.1 [20]. Let X 2 Rnn be a n n positive definite matrix and Y 2 Rnn a symmetric matrix. Then for any x 2 Rn ,t 2 Rþ , the following inequality holds:
xT Yx 6 kmax ðX 1 YÞxT Xx:
ð3:4Þ
where kmax ðÞ is the maximal eigenvalue function. Lemma 3.2. If r12 2 N½l1 ; g 1 ,r 21 2 N½l2 ; g 2 , Let R , ¼ fR 2 R22 : R ¼ diagðe1 ; e2 Þ; jei j 6 1; i ¼ 1; 2g, the following statements are hold. e ¼ g 2 0 , then A2 ¼ r21 0 e ¼ l2 0 and Q can be written as (1) Let P 0 g1 0 r12 0 l1
ee F A2 ¼ A20 þ e ER
ð3:5Þ
where
A20 ¼
1 e e ð P þ Q Þ; 2
_
(2) Let P ¼
1 e e ~ Þ e ¼ ðh H ð Q PÞ; ij 22 ¼ 2
0 1=g 2
1=g 1 0
e ¼ 1=l1 and Q 0
R 2 R ;
e E¼e F¼
1=r 12 0 , then A3 ¼ 0 1=l2
" pffiffiffiffiffi # h1 0 pffiffiffiffiffi : 0 h2 0 1=r 21
can be written as
_ _ _
A3 ¼ A30 þ E R F
ð3:6Þ
where
A30 ¼
1 _ _ ðP þ Q Þ; 2
_
_
H ¼ ðh Þ22 ¼
Remarks. Clearly, for any
ij
P
2
P
1 _ _ ðQ P Þ; R 2 R ; 2
_
_
E¼F¼
" pffiffiffiffiffi # h1 0 pffiffiffiffiffi : 0 h2
, we get __
_
_
eR eTR eT ¼ R e 6 I, R RT ¼ RT R 6 I, where I is the 2 2 identity matrix. (1) RRT ¼ RT R 6 I, R (2) by Lemma 3.2, the system (3.3) can be rewritten as _ _ _
ee F ÞðA30 þ E R F Þx þ Bx1 x2 ER x_ ¼ A1 ðA20 þ e
t–tk
Dx ¼ Uðk; xÞ ¼ C k x t ¼ t k xðtþ0 Þ ¼ x0 _ _ _
e; e where E; R; F; e E; R F ; E ; R; F are defined as in Lemma 3.2.
ð3:7Þ
1575
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
Lemma 3.3 [20]. If
2nT
X
P
2
P
, then for any positive scalar k > 0 and for any n 2 Rn , the following inequality holds:
1 k
g 6 nT n þ kgT g
ð3:8Þ
especially, the inequality holds for k ¼ 1. Lemma 3.4 [21]. Given any real matrices R1 ; R2 ; R3 of appropriate dimensions and a scalar e > 0 such that 0 < R3 ¼ RT3 . Then, the following inequality holds:
RT1 R2 þ RT2 R1 6 eRT1 R3 R1 þ e1 RT2 R1 3 R2 :
ð3:9Þ
especially, the inequality holds for e ¼ 1; R3 ¼ E0 (identity matrix). Theorem 3.1. Assume that there exists a positive definite symmetric matrix C. Let M ¼ 2g 2 x10 and bk denote the maximum eigenvalue of C1 ðI þ C k ÞT CðI þ C k Þ.
0
1 __ _ _ C1 fAT30 AT20 AT1 C þ CA1 A20 A30 þ ðCA1 A20 E E T AT20 AT1 CÞ þ F T F þðCA1 eEÞðCA1 eEÞT A @ kðaÞ ¼ kmax _ _ _ _ FTe F A30 þ CA1 e Ee E T AT1 C þ kmax ðE T e FTe F E ÞF T F g þAT30 e
ð3:10Þ
if ðkðaÞ þ MÞ ðtkþ1 t k Þ 6 lnðcbk Þ; for k 2 N; c > 1; k P 0: Then the system (2.1) is robust asymptotically stable. Proof. Let V ¼ xT Cx, where C is a positive definite matrix. Clearly V belongs to T
T
t0 . When t ¼ tk , by Lemma 3.1, we have
T
1
Vðt þk ; xÞ ¼ ðx þ C k xÞ Cðx þ C k xÞ ¼ xT ðI þ C k Þ CðI þ C k Þxkmax ðC ðI þ C k Þ CðI þ C k ÞÞxT Cx ¼ bk Vðt; xÞ
ð3:11Þ
When t–tk , we have
r 12 x1 x2 BT x1 x2 Cx þ xT CBx1 x2 ¼ ½ r12 x1 x2 r21 x1 x2 Cx þ xT C r21 x1 x2 r x 0 r 0 12 2 12 x2 Cx þ xT C ¼ xT x 6 2r 21 x1 ðt þ0 ÞxT Cx 6 2g 2 x10 xT Cx ¼ MVðt; xÞ 0 r 21 x1 0 r21 x1 ð3:12Þ By the Lemma 3.2, then
Dþ Vðt; xÞ ¼ x_ T Cx þ xT Cx_ ¼ ðA1 A2 A3 x þ Bx1 x2 ÞT Cx þ xT CðA1 A2 A3 x þ Bx1 x2 Þ _ _ _
_ _ _
ee ee F ÞT AT1 þ BT x1 x2 ÞCx þ xT CðA1 ðA20 þ e F ÞðA30 þ E R F Þx þ Bx1 x2 Þ ER ER ¼ ðxT ðA30 þ E R F ÞT ðA20 þ e _ _ _
_ _ _
ee ee F ÞT AT1 C þ CA1 e ER F A30 ER ¼ xT fAT30 AT20 AT1 C þ CA1 A20 A30 þ ðE R F ÞT AT20 AT1 C þ CA1 A20 E R F þAT30 ð e _ _ _
_ _ _
ee ee F ÞT AT1 C þ CA1 e ER F E R F gx þ BT x1 x2 Cx þ xT CBx1 x2 þ ðE R F ÞT ð e ER
ð3:13Þ
By the assumption, Lemmas 3.3, 3.4 and (3.12), (3.13). __
_ _
Dþ Vðt; xÞ 6 xT fAT30 AT20 AT1 C þ CA1 A20 A30 þ ðCA1 A20 E E T AT20 AT1 CÞ þ F T F þðCA1 e EÞðCA1 e EÞT þ AT30 e FTe F A30 þ CA1 e Ee E T AT1 C _ _ _
_ _ _
F E R F gx þ MVðt; xÞ þ ðe F E R F ÞT e By the Lemma 3.1 _ _ _
_ _ _
_ _
_
_ _ _
_
_
_ _
F E R F x ¼ ðR F xÞT E T e FTe F E ðR F xÞ 6 kmax ðE T e FTe F E ÞxT F T F x xT ð e F E R F ÞT e
ð3:14Þ
Then, we have __
_ _
EÞðCA1 e EÞT Dþ Vðt; xÞ 6 xT fAT30 AT20 AT1 C þ CA1 A20 A30 þ ðCA1 A20 E E T AT20 AT1 CÞ þ F T F þðCA1 e _
_ _ _
FTe F A30 þ CA1 e Ee E T AT1 C þ kmax ðE T e FTe F E ÞF T F gx þ MVðt; xÞ þ AT30 e 6 ðkðaÞ þ MÞVðt; xÞ Let k ¼ 1, for all t 2 ðt 0 ; t 1 , by (3.15), we have
Vðt; xÞ 6 Vðt 0 ; xÞ expððkðaÞ þ MÞðt t 0 ÞÞ: Then
Vðt 1 ; xÞ 6 Vðt0 ; xÞ expððkðaÞ þ MÞðt 1 t 0 ÞÞ:
ð3:15Þ
1576
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
By (3.11), we have
Vðtþ1 ; xÞ 6 b1 Vðt 1 ; xÞ 6 b1 Vðt0 ; xÞ expððkðaÞ þ MÞðt 1 t0 ÞÞ: For all t 2 ðt 1 ; t 2 , we have
Vðt; xÞ 6 Vðt þ1 ; xÞ expððkðaÞ þ MÞðt t1 ÞÞ 6 b1 Vðt 0 ; xÞ expððkðaÞ þ MÞðt t 0 ÞÞ: Similarly, for all k 2 N and t 2 ðtk ; tkþ1 , we have
Vðt; xÞ 6 bk b2 b1 Vðt0 ; xÞ expððkðaÞ þ MÞðt t 0 ÞÞ:
ð3:16Þ
By (3.10), we have
bk expððkðaÞ þ MÞðtkþ1 tk ÞÞ 6
1
C
;
k2N
ð3:17Þ
Hence, for all t 2 ðt k ; t kþ1 ðk 2 NÞ, we have
Vðt; xÞ 6 bk b2 b1 Vðt0 ; xÞ expððkðaÞ þ MÞðt t 0 ÞÞ ¼ Vðt 0 ; xÞ½bk expððkðaÞ þ MÞðt k t k1 ÞÞ ½b1 expððkðaÞ þ MÞðt 1 t0 ÞÞ expððkðaÞ þ MÞðt tk ÞÞ 1 6 Vðt 0 ; xÞ k expððkðaÞ þ MÞðt t k ÞÞ
c
ð3:18Þ
Thus, if t ! 1, then k ! 1 and Vðt; xÞ ! 0. The proof is complete. h Corollary 3.1. Suppose that all the impulsive spacing are uniform, i.e. Dt k ,tkþ1 t k d for all k 2 N. If d 6 mink ðkða1ÞþMÞ lnðcbk Þ , the system (2.1) is robust asymptotically stable, where kðaÞ, M, c and bk are defined as in Theorem 3.1. Corollary 3.2. Based on the conditions of Corollary 3.1 and the applying the uniform impulsive control at all the time t k , i.e. for all 1 lnðcbÞ, the system (2.1) is robust asymptotically stable, where b ¼ kmax ðC1 ðI þ CÞT CðI þ CÞÞ. k, C k ¼ C. If d 6 ðkðaÞþMÞ 4. Illustrative example In this section, we give an example to demonstrate the effectiveness of our results.
Fig. 4.1. The state program of the system with impulsive control.
1577
D. Li et al. / Chaos, Solitons and Fractals 41 (2009) 1572–1577
Example 4.1. Consider the following uncertain predator–prey impulsive system
x_ 1 ¼ x1 ðl1 r12 x2 Þ x_ 2 ¼ x2 ðl2 þ r 21 x1 Þ xðt10 Þ ¼ x10 > 0 t–t k
ð4:1Þ
xðt20 Þ ¼ x20 > 0
Dxjt¼tk , ¼ xðt þk Þ xðt k Þ ¼ Uðk; xÞ ¼ Cx; k 2 N The parameters of system (2.1) are as follows:
l1 ¼ 0:2; l2 ¼ 0:16; r12 2 N½0:05; 0:13; r21 2 N½0:25; 0:36; C ¼
2=3
0
0
2=3
:
Set x10 ¼ x20 ¼ 0:1. Let Vðt; xÞ ¼ xT x. Suppose r 12 and r 21 be random of the above interval, respectively. It is easy to obtain kðaÞ ¼ 21:3709, b ¼ 0:1111 by computing them with Matlab 7.0. By Theorem 3.1, when c ¼ 2 and Dtk 6 0:0701, ðkðaÞ þ MÞ d 6 lnðcbÞ, then the system is asymptotically stable. The stable system state program of the system with impulsive control is showed as Fig. 4.1.
5. Conclusions In this paper, we have investigated impulsive control of Lotka–Volterra predator–prey system. The uncertainties in the system are considered. The sufficient conditions of the robust asymptotic stability are established by employing the method of Lyapunov functions in Theorem 3.1. In Section 4, the validity of the results is demonstrated by a numerical example. The illustrated example shows that the system is robust asymptotic stable under the conditions of Theorem 3.1. Acknowledgement The work described in this paper was supported by the National Natural Science Foundation of China (Grant No. 60604007). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Lakshmikantham V, Bainov DD, Simeonov PS. Theory of impulsive differential equations. Singapore: World Scientific; 1989. Samoilenko AM, Perestyuk NA. Impulsive differential equations. Singapore: World Scientific; 1995. Chen F, Wen X. Asymptotic stability for impulsive functional differential equation. J Math Anal Appl 2007;336:1149–60. Sun J, Zhang Y. Impulsive control of a nuclear spin generator. J Comput Appl Math 2003;157:235–42. Yang T, Yang L, Yang C. Impulsive control of Lorenz system. Physica D 1997;110:18–24. Yoshikawa Shoji, Yamada Katsuhiko. Impulsive control for angular momentum management of tumbling spacecraft. Acta Astron 2007;60:810–9. Chen D, Sun J, Wu Q. Impulsive control and its application to Lü’s chaotic system. Chaos, Solitons & Fractals 2004;21:1135–42. Ballinger G, Liu X. Permanence of population growth models with impulsive effects. Math Comput Model 1997;26:59–72. Zhang Y, Xiu Z, Chen L. Dynamic complexity of a two-prey one-predator system with impulsive effect. Chaos, Solitons & Fractals. 2005;26:131–9. Song X, Li Y. Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect. Chaos, Solitons & Fractals 2007;33:463–78. Liu B, Teng Z, Liu W. Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations. Chaos, Solitons & Fractals 2007;31:356–70. Jiang G, Lu Q. Impulsive state feedback control of a predator-prey model. J Comput Appl Math 2007;200:193–207. Jiao J, Meng X, Chen L. A stage-structured Holling mass defence predator–prey model with impulsive perturbations on predators. Appl Math Comput 2007;189:1448–58. Akhmet M, Beklioglu M, Ergenc T, Tkachenko V. An impulsive ratio-dependent predator–prey system with diffusion. Nonlinear Anal Real World Appl 2006;7:1255–67. Zhang S, Tan D, Chen L. Chaos in periodically forced Holling type IV predator–prey system with impulsive perturbations. Chaos, Solitons & Fractals 2006;27:980–90. Wang X, Wang W, Lin X. Chaotic behavior of a Watt-type predator–prey system with impulsive control strategy. Chaos, Solitons & Fractals 2008;37:706–18. Lu Z, Chi X, Chen L. Impulsive control strategies in biological control of pesticide. J Theor Popul Biol 2003;64:39–47. Liu Z, Yuan R. Stability and bifurcation in a delayed predator–prey system with Beddington–DeAngelis functional response. J Math Anal Appl 2004;296:521–37. Yan J, Shen J. Impulsive stabilization of functional differential equations by Lyapunov–Razumikhin functions. Nonlinear Anal 1999;37:245–55. Liu B, Liu X, Liao X. Robust stability of uncertain impulsive dynamical systems. J Math Anal Appl 2004;290:519–33. Sanchez E, Perez J. Input-to-state stability analysis for dynamic NN. IEEE Trans Circuits Syst 1999;46:1395–8.