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Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453
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Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier .com/locate /cnsns
Permanence and global attractivity of a periodic predator–prey systemwith mutual interference and impulses
Zhong Li ⇑, Fengde Chen, Mengxin HeCollege of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China
a r t i c l e i n f o a b s t r a c t
Article history:Received 8 September 2010Received in revised form 14 May 2011Accepted 16 May 2011Available online 23 May 2011
Keywords:Predator–preyImpulseMutual interferencePermanenceGlobal attractivity
1007-5704/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.cnsns.2011.05.026
⇑ Corresponding author.E-mail addresses: [email protected] (Z. Li)
In this paper, we consider a periodic predator–prey system with mutual interference andimpulses. By constructing a suitable Lyapunov function and using the comparison theoremof impulsive differential equation, sufficient conditions which ensure the permanence andglobal attractivity of the system are obtained. We also present some examples to verify ourmain results.
� 2011 Elsevier B.V. All rights reserved.
1. Introduction
The traditional periodic predator–prey system with periodic coefficients has been studied extensively (see [1–5]). Butthere are few papers considering the mutual interference between the predators and preys, which was introduced by Hassellin 1971. From the experiments, Hassell introduced the concept of mutual interference constant m(0 < m 6 1) into a Volterramodel as follows (see [6–8]):
x0 ¼ xgðxÞ �uðxÞym;
y0 ¼ yð�dþ buðxÞym�1 � qðyÞÞ:ð1:1Þ
Recently, Wang [9] investigated a special non-autonomous system corresponding to system (1.1) as follows
x0ðtÞ ¼ xðtÞðr1ðtÞ � b1ðtÞxðtÞ � cðtÞymðtÞÞ;y0ðtÞ ¼ yðtÞð�r2ðtÞ þ bcðtÞxðtÞym�1 � b2ðtÞyðtÞÞ;
ð1:2Þ
where k > 0, and ri, bi, i = 1, 2, c are T-periodic continuous functions. By utilizing Mawhin’s coincidence degree theorem andconstructing a suitable Lyapunov functional, Wang [9] investigated the existence and global stability of positive periodicsolution of system (1.2). In [9], mutual interference m needs to be a rational number in the interval (0,1). Furthermore, whenthe coefficient of system (1.2) are continuous functions, Wang [10] investigated the permanence and global asymptoticalstability of system (1.2). In [10], mutual interference m is allowed to be any real-valued number in the interval (0,1), there-fore, the results improve those of [9]. For more details in this direction, please see [11–14].
. All rights reserved.
, [email protected], [email protected] (F. Chen), [email protected] (M. He).
Z. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453 445
However, ecological systems are often deeply perturbed by human exploit activities such as planting and harvesting etc.,which are not suitable to be considered continually. To accurately describe the system, one needs to use the impulsivedifferential equations. The theory of impulsive differential equations is now being recognized much richer content thanthe corresponding theory of differential equations without impulses, also it represents a more natural framework for math-ematical modeling of many real world phenomena [15]. For more details in this direction, please see [16–19].
The main purpose of this paper is to study the permanence and global attractivity of the following T-periodic predator–prey system with impulses:
x0ðtÞ ¼ xðtÞðr1ðtÞ � b1ðtÞxðtÞ � cðtÞymðtÞÞ;y0ðtÞ ¼ yðtÞð�r2ðtÞ þ bcðtÞxðtÞym�1ðtÞ � b2ðtÞyðtÞÞ; t – sk;
x sþk� �
¼ ð1þ hkÞxðskÞ;y sþk� �
¼ ð1þ gkÞyðskÞ; t ¼ sk; k ¼ 1;2; . . . ;
ð1:3Þ
where x(t) denotes the density of the prey species at time t; y(t) denotes the density of the predator species at time t; b > 0,ri(t), bi(t), i = 1, 2 and c(t) are continuous periodic functions defined on [0,+1) with a common period T > 0; bi(t), i = 1, 2, andc(t) are strictly positive; sk ? +1(t ? +1) and 0 = s0 < s1 < s2 < s3 < � � � < sk < sk+1 < � � � Assume that hk, gk, k = 1, 2, . . ., areconstants and there exists an integer q > 0 such that
hkþq ¼ hk; gkþq ¼ gk; skþq ¼ sk þ T:
In this paper, the growth rate ri(t), i = 1, 2 of species are not necessarily positive, and this is realistic since the environmentfluctuates randomly (e.g. seasonal effect of weather condition, temperature, mating habits and food supplies) and in somebad condition, ri(t), i = 1, 2 may be negative. So we assume that:
m½ri� ¼1T
Z T
0riðtÞdt > 0; i ¼ 1;2
and natural for biological meanings:
1þ hk > 0; 1þ gk > 0; k ¼ 1;2; . . . :
Obviously, the models in [18] are the special cases of system (1.3) with m = 1, that is, there is no mutual interferencebetween the predator and prey. So we only study system (1.3) in the case of 0 < m < 1.
Definition 1.1. For any ðt; FðtÞÞ 2 ½sk�1; skÞ �R2þ, the right-hand derivative D+V(t,F(t)) along the solution F(t,F0) of system
(1.3) is defined by
DþVðt; FðtÞÞ ¼ lim infh!0þ
1h½Vðt þ h; Fðt þ hÞÞ � Vðt; FðtÞÞ�:
By the basic theories of impulsive differential equations in [15], system (1.3) has a unique solutionF(t) = F(t,F0) 2 PC([0,+1), R2) and PC([0,+1), R2) = {/:[0,+1) ? R2 is continuous for t – sk, also / s�k
� �and / sþk
� �exist with
/ s�k� �
¼ /ðskÞ; k ¼ 1;2; . . .g for each initial value F(0) = F0 2 R2.
The organization of this paper is as follows. In Section 2, we introduce some useful lemmas. In Section 3, we study thepermanence and global attractivity of system (1.3). In Section 4, some numerical simulations are presented to illustratethe feasibility of our main results. In the last section, we give a brief discussion of our results.
2. Preliminaries
In this section, we present some preliminary results required in the sequel. Firstly, we introduce an important comparisontheorem of impulsive differential equation [15].
Lemma 2.1. Assume that m 2 PC[R+,R] with points of discontinuity at t = tk and is left continuous at t = tk, k = 1, 2, . . ., and
D mðtÞ 6 gðt;mðtÞÞ; t–tk; k ¼ 1;2; . . . ;
m tþk� �
6 /kðmðtkÞÞ; t ¼ tk; k ¼ 1;2; . . . ;ð2:1Þ
where g 2 C[R+ � R+,R],/k 2 C[R,R] and /k(u) is nondecreasing in u for each k = 1, 2, . . .. Let r(t) be the maximal solution of thescalar impulsive differential equation
_u ¼ gðt;uÞ; t–tk; k ¼ 1;2; . . . ;
u tþk� �
¼ /kðuðtkÞÞP 0; t ¼ tk; tk > t0; k ¼ 1;2; . . . ;
u tþ0� �
¼ u0
ð2:2Þ
existing on [t0,1), then m tþ0� �
6 u0 implies m(t) 6 r(t), t P t0.
446 Z. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453
Remark 2.1. In Lemma 2.1, assume that inequality (2.1) reversed. Let p(t) be the minimal solution of (2.2) existing on(t0,+1). Then, m tþ0
� �P u0 implies m(t) P p(t), t P t0.
Lemma 2.2. Let (x(t),y(t))T be any solution of system (1.3) such that x(0+) > 0 and y(0+) > 0, then x(t) > 0 and y(t) > 0 for all t P 0.
Proof. From the first equation of (1.3), one has
x0ðtÞ ¼ PðtÞxðtÞ; t – sk;
where
PðtÞ ¼ r1ðtÞ � b1ðtÞxðtÞ � cðtÞymðtÞ:
Thus we have
xðtÞ ¼Y
0<sk<t
ð1þ hkÞxð0þÞ expZ t
0PðsÞds
� �> 0;
because of x(0+) > 0. Similarly, it follows from the second equation of (1.3) that y(t) > 0. This completes the proof of Lemma2.2. Consider a periodic logistic equation with impulses
x0ðtÞ ¼ xðtÞðbðtÞ � aðtÞðxðtÞÞhÞ; t – sk;
x sþk� �
¼ 1þ h0k� �
xðskÞ; t ¼ sk; k ¼ 1;2; . . . ;ð2:3Þ
where h is a positive constants; a(t) and b(t) are continuous T-periodic functions with a(t) > 0 andm½b� > 0; h0kþq ¼ h0k; skþq ¼ sk þ T. Let x(t) be any solution of system (2.3) with the initial value x(0+) > 0. h
Lemma 2.3 [19].
(1): If
Xq
k¼1
ln 1þ h0k� �
þ Tm½b� > 0;
then system (2.3) has a unique T-periodic solution x⁄(t), which is globally asymptotically stable in the sense thatlimt?+1jx(t) � x⁄(t)j = 0.
(2): If
Xq
k¼1
ln 1þ h0k� �
þ Tm½b� < 0;
then limt?1x(t) = 0.
Lemma 2.4 [19]. If x⁄(t) is the unique T-periodic positive solution of system (2.3), and x�eðtÞ is the unique T-periodic positive solu-tion of the following system
x0ðtÞ ¼ xðtÞðbeðtÞ � aðtÞxðtÞÞ; t – sk;
x sþk� �
¼ 1þ h0k� �
xðskÞ; t ¼ sk; k ¼ 1;2; . . . ;
where be(t) is the continuous T-periodic function with m[be] > 0 and lime?0be(t) = b(t). Then
lime!0jx�eðtÞ � x�ðtÞj ¼ 0:
(I) Consider the following periodic Logistic equation with impulses
x0ðtÞ ¼ xðtÞðr1ðtÞ � b1ðtÞxðtÞÞ; t – sk;
x sþk� �
¼ ð1þ hkÞxðskÞ; t ¼ sk; k ¼ 1;2; . . . :ð2:4Þ
If
Xq
k¼1
lnð1þ hkÞ þ Tm½r1� > 0 ðA1Þ
Z. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453 447
holds, then it follows from Lemma 2.3 that system (2.4) has a unique T-periodic solution x⁄(t), which is globally asymptot-ically stable in the sense that
limt!þ1
jxðtÞ � x�ðtÞj ¼ 0;
where x(t) is any solution of system (2.4) with the initial value x(0+) > 0.(II) Consider the following periodic Logistic equation with impulses
v 0ðtÞ ¼ vðtÞðr2ðtÞ � bcðtÞx�ðtÞv1�mðtÞÞ; t–sk;
v sþk� �
¼ 1ð1þ gkÞ
vðskÞ; t ¼ sk; k ¼ 1;2; . . . :ð2:5Þ
If
�Xq
k¼1
lnð1þ gkÞ þ Tm½r2� > 0 ðH1Þ
holds, then it follows from Lemma 2.3 that system (2.5) has a unique T-periodic solution v⁄(t), which is globally asymptot-ically stable in the sense that
limt!þ1
jvðtÞ � v�ðtÞj ¼ 0;
where v(t) is any solution of system (2.5) with the initial value v(0+) > 0. For convenience, we denote y�ðtÞ ¼ 1v�ðtÞ.
(III) Consider the following periodic Logistic equation with impulses
x0ðtÞ ¼ xðtÞðr1ðtÞ � cðtÞðy�ðtÞÞm � b1ðtÞxðtÞÞ; t – sk;
x sþk� �
¼ ð1þ hkÞxðskÞ; t ¼ sk; k ¼ 1;2; . . . :ð2:6Þ
If
Xqk¼1
lnð1þ hkÞ þ Tm½r1ðtÞ � cðtÞðy�ðtÞÞm� > 0 ðH2Þ
holds, then it follows from Lemma 2.3 that system (2.6) has a unique T-periodic solution x⁄(t), which is globally asymptot-ically stable in the sense that
limt!þ1
jxðtÞ � x�ðtÞj ¼ 0;
where x(t) is any solution of system (2.6) with the initial value x(0+) > 0.
3. Main results
In this section, we present out our main results for system (1.3).
Lemma 3.1. Assume that (A1) and (H1) hold, and (x(t),y(t))T is any solution of system (1.3) with x(0+) > 0 and y(0+) > 0, then thereexist constants Mi > 0, i = 1, 2 such that
lim supt!þ1
xðtÞ 6 M1; lim supt!þ1
yðtÞ 6 M2:
Proof. From system (1.3), we have
x0ðtÞ 6 xðtÞðr1ðtÞ � b1ðtÞxðtÞÞ; t – sk;
x sþk� �
¼ ð1þ hkÞxðskÞ; t ¼ sk; k ¼ 1;2; . . . :
Consider the following system
z0ðtÞ ¼ zðtÞðr1ðtÞ � b1ðtÞwðtÞÞ; t – sk;
z sþk� �
¼ ð1þ hkÞzðskÞ; t ¼ sk; k ¼ 1;2; . . . :ð3:1Þ
By Lemma 2.1, we have x(t) 6 z(t), where z(t) is the solution of (3.1) with z(0+) = x(0+). By (A1), it follows from Lemma 2.3 thatsystem (3.1) has a unique T-periodic solution x⁄(t), which is globally asymptotically stable. For any positive constant e > 0,there exists a T1 > 0 such that for t > T1,
jzðtÞ � x�ðtÞj < e:
Let M1 = sup{x⁄(t)jt 2 [0,T]}, then it follows that
xðtÞ 6 zðtÞ < x�ðtÞ þ e 6 M1 þ e; for t > T1:
448 Z. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453
Setting e ? 0, we obtain
lim supt!þ1
xðtÞ 6 M1:
Define yðtÞ ¼ 1uðtÞ. it follows from the second equation of system (1.3) that
u0ðtÞ ¼ uðtÞðr2ðtÞ þ b2ðtÞ1
uðtÞ � bcðtÞxðtÞu1�mðtÞÞ; t – sk;
u sþk� �
¼ 1ð1þ gkÞ
uðskÞ; t ¼ sk; k ¼ 1;2; . . . :
For t > T1 and t – sk (k = 1,2, . . .), we obtain
u0ðtÞP uðtÞðr2ðtÞ � bcðtÞðx�ðtÞ þ eÞu1�mðtÞÞ:
Consider the following system
z0ðtÞ ¼ zðtÞðr2ðtÞ � bcðtÞðx�ðtÞ þ eÞz1�mðtÞÞ; t – sk;
z sþk� �
¼ 1ð1þ gkÞ
zðskÞ; t ¼ sk; k ¼ 1;2; . . . :ð3:2Þ
By Lemma 2.1, for all t > T1, we have u(t) P z(t), where z(t) is the solution of (3.2) with z Tþ1� �
¼ u Tþ1� �
. By (H1) and Lemma 2.3,system (3.2) admits a unique T-periodic solution v�eðtÞ, which is globally asymptotically stable. Setting e ? 0, from Lemma2.4, we have
v�eðtÞ ! v�ðtÞ:
Let M�2 ¼ inffv�ðtÞjt 2 ½0; T�g, and for any positive constant e1 > 0, there exists a T2 > T1 such that
uðtÞ ¼ 1yðtÞP zðtÞ > v�ðtÞ � e1 P M�
2 � e1; for t > T2:
Setting e1 ? 0, we obtain
lim supt!þ1
yðtÞ 6 1M�
2def¼
M2:
This completes the proof of Lemma 3.1. h
Theorem 3.1. Assume that (H1), (H2) hold, and further assume that
Xq
k¼1
lnð1þ gkÞ þ Tm½�r2ðtÞ þ bcðtÞx�ðtÞðy�ðtÞÞm�1� > 0; ðH3Þ
where x⁄(t) and y⁄(t) are the unique T-periodic solutions of systems (2.5) and (2.6), respectively, then the species x and y arepermanence, that is, for any positive solution (x(t),y(t))T of system (1.3), there exist positive constants Mi and mi, i = 1, 2 such that
m1 6 lim inft!þ1
xðtÞ 6 lim supt!þ1
xðtÞ 6 M1;
m2 6 lim inft!þ1
yðtÞ 6 lim supt!þ1
yðtÞ 6 M2:
Proof. It is not difficult to show that (H2) implies (A1). Therefore, from Lemma 3.1, there exist constants Mi, i = 1, 2 such that
lim supt!þ1
xðtÞ 6 M1; lim supt!þ1
yðtÞ 6 M2:
From the proof of Lemma 3.1, for e > 0 small enough, there exists a T0 > 0 such that for t > T0,
yðtÞ 6 1v�ðtÞ � e
def¼
y�eðtÞ: ð3:3Þ
By the boundedness of b and c(t), and conditions (H2) and (H3), we have
Xq
k¼1
lnð1þ hkÞ þ Tm r1ðtÞ � cðtÞ y�eðtÞ� �m
h i> 0; ð3:4Þ
Xq
k¼1
lnð1þ gkÞ þ Tm �r2ðtÞ þ bcðtÞðx�ðtÞ � eÞ y�eðtÞ� �m�1
h i> 0: ð3:5Þ
Z. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453 449
From system (1.3) and inequality (3.3), for t > T0 and t – sk, we obtain
x0ðtÞP xðtÞ r1ðtÞ � cðtÞ y�eðtÞ� �m � b1ðtÞxðtÞ
� �:
Consider the following system
z0ðtÞ ¼ zðtÞðr1ðtÞ � cðtÞðy�eðtÞÞm � b1ðtÞzðtÞÞ; t – sk;
z sþk� �
¼ ð1þ hkÞzðskÞ; t ¼ sk; k ¼ 1;2; . . . :ð3:6Þ
By Lemma 2.1, for all t > T0, we have x(t) P z(t), where z(t) is the solution of (3.6) with z(T0+) = x(T
0+). From Lemma 2.3 andinequality (3.4), system (3.6) admits a unique T-periodic solution x⁄e(t), which is globally asymptotically stable. Settinge ? 0, from Lemma 2.4, we have x⁄e(t) ? x⁄(t). Let m1 = inf{x⁄(t)jt 2 [0,T]} and for a positive constant e small enough, thereexists a T 01 > T 0 such that
xðtÞP zðtÞ > x�ðtÞ � e P m1 � e; for t > T 01:
Setting e ? 0, we obtain
lim inft!þ1
xðtÞP m1:
From system (1.3), for t > T 01 and t – sk, we obtain
y0ðtÞP y �r2ðtÞ þ bcðtÞðx�ðtÞ � eÞ y�eðtÞ� �m�1 � b2ðtÞyðtÞ
� �:
Consider the following system
w0ðtÞ ¼ wðtÞð�r2ðtÞ þ bcðtÞðx�ðtÞ � eÞ y�eðtÞ� �m�1 � b2ðtÞwðtÞÞ; t – sk;
w sþk� �
¼ ð1þ gkÞwðskÞ; t ¼ sk; k ¼ 1;2; . . . :ð3:7Þ
By Lemma 2.1, for all t > T 01, we have y(t) P w(t), where w(t) is the solution of (3.7) with w T 0þ1� �
¼ y T 0þ1� �
. From Lemma 2.3and inequality (3.5), system (3.7) admits a unique T-periodic solution y⁄e(t), which is globally asymptotically stable. Settinge ? 0, from Lemma 2.4, we have y⁄e(t) ? y⁄(t), where y⁄(t) is the unique T-periodic globally asymptotically stable solution ofthe following system
p0ðtÞ ¼ pðtÞð�r2ðtÞ þ bcðtÞx�ðtÞðy�ðtÞÞm�1 � b2ðtÞpðtÞÞ; t – sk;
p sþk� �
¼ ð1þ gkÞpðskÞ; t ¼ sk; k ¼ 1;2; . . . :
Let m2 = inf{y⁄(t)jt 2 [0,T]} and for a positive constant e small enough, there exists a T 02 > T 01 such that
yðtÞP wðtÞ > y�ðtÞ � e P m2 � e; for t > T 02:
Setting e ? 0, we obtain
lim inft!þ1
yðtÞP m2:
This completes the proof of Theorem 3.1. h
Theorem 3.2. Assume that (H1)–(H3) hold. Assume further that there exist positive constants a and d such that
mint2½0;T�
ab1ðtÞ � dbcðtÞmm�12
� > 0; ðH4Þ
mint2½0;T�
db2ðtÞ þ dbcðtÞð1�mÞm1Mm�22 � acðtÞmmm�1
2
n o> 0; ðH5Þ
then the species x and y are globally attractive, that is, for any positive solutions (x1(t), y1(t))T and (x2(t),y2(t))T of system (1.3), onehas
limt!þ1
jx1ðtÞ � x2ðtÞj ¼ 0; limt!þ1
jy1ðtÞ � y2ðtÞj ¼ 0:
Proof. For any small enough positive e, from Theorem 3.1, it immediately follows that there exists a large enough T0 > 0 suchthat for all t > T0, one has
m1 � e 6 xiðtÞ 6 M1 þ e; m2 � e 6 yiðtÞ 6 M2 þ e: ð3:8Þ
Consider a Lyapunov function as follows
VðtÞ ¼ V1ðtÞ þ V2ðtÞ;
450 Z. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453
where
V1ðtÞ ¼ aj ln x1ðtÞ � ln x2ðtÞj;V2ðtÞ ¼ dj ln y1ðtÞ � ln y2ðtÞj:
For t > T0, and t – sk, k = 1, 2, . . ., calculating the upper right derivatives of V1(t) and V2(t), respectively, we have
DþV1ðtÞ ¼ asgnðx1ðtÞ � x2ðtÞÞ �b1ðtÞðx1ðtÞ � x2ðtÞÞ � cðtÞ ym1 ðtÞ � ym
2 ðtÞ� �� �
6 �ab1ðtÞjx1ðtÞ � x2ðtÞj þ acðtÞjym1 ðtÞ � ym
2 ðtÞj ¼ �ab1ðtÞjx1ðtÞ � x2ðtÞj þ acðtÞmðn1ðtÞÞm�1jy1ðtÞ � y2ðtÞj
and
DþV2ðtÞ ¼ dsgnðy1ðtÞ � y2ðtÞÞ bcðtÞ x1ðtÞym�11 ðtÞ � x2ðtÞym�1
2 ðtÞ� �
� b2ðtÞðy1ðtÞ � y2ðtÞÞ� �
6 �dbcðtÞx1ðtÞjym�11 ðtÞ � ym�1
2 ðtÞj þ dbcðtÞym�12 jx1ðtÞ � x2ðtÞj � db2ðtÞjy1ðtÞ � y2ðtÞj
6 �dbcðtÞx1ðtÞð1�mÞðn2ðtÞÞm�2jy1ðtÞ � y2ðtÞj þ dbcðtÞym�12 jx1ðtÞ � x2ðtÞj � db2ðtÞjy1ðtÞ � y2ðtÞj;
where ni(t), i = 1, 2, lie between y1(t) and y2(t).By the mean value theorem and (3.8), for any closed interval contained in t 2 (sk, sk+1], k = p, p + 1, . . . and sp > T0, it follows
that
1M1 þ e
jx1ðtÞ � x2ðtÞj 6 j ln x1ðtÞ � ln x2ðtÞj 61
m1 � ejx1ðtÞ � x2ðtÞj;
1M2 þ e
jy1ðtÞ � y2ðtÞj 6 j ln y1ðtÞ � ln y2ðtÞj 61
m2 � ejy1ðtÞ � y2ðtÞj:
ð3:9Þ
For the above small enough e, it follows from conditions (H4) and (H5) that there exist positive constants g and h suchthat
mint2½0;T�fab1ðtÞ � dbcðtÞðm2 � eÞm�1g > g;
mint2½0;T�fdb2ðtÞ þ dbcðtÞð1�mÞðm1 � eÞðM2 þ eÞm�2 � acðtÞmðm2 � eÞm�1g > g;
g minm1 � e
a;m2 � e
d
n o> h:
ð3:10Þ
Therefore, for t 2 (sk,sk+1], k = p, p + 1, . . . and sp > T0, from (3.8)–(3.10), we obatin
DþVðtÞ6� ab1ðtÞ�dbcðtÞym�12
� �jx1ðtÞ�x2ðtÞj�ðdb2ðtÞþdbcðtÞx1ðtÞð1�mÞðn2ðtÞÞm�2�acðtÞmðn1ðtÞÞm�1Þjy1ðtÞ�y2ðtÞj
6�ðab1ðtÞ�dbcðtÞðm2�eÞm�1Þjx1ðtÞ�x2ðtÞj�ðdb2ðtÞþdbcðtÞð1�mÞðm1�eÞðM2þeÞm�2
�acðtÞmðm2�eÞm�1Þ�jy1ðtÞ�y2ðtÞj6�gðjx1ðtÞ�x2ðtÞjþjy1ðtÞ�y2ðtÞjÞ
6�gm1�e
aajlnx1ðtÞ� lnx2ðtÞjþ
m2�ed
djlny1ðtÞ� lny2ðtÞj� �
6�hVðtÞ: ð3:11Þ
For t = sk, k = 1, 2, . . ., we have
V sþk� �
¼ aj ln x1 sþk� �
� ln x2 sþk� �j þ dj ln y1 sþk
� �� ln y2 sþk
� �j
¼ aj lnðð1þ hkÞx1ðskÞÞ � lnðð1þ hkÞx2ðskÞÞj þ dj lnðð1þ gkÞy1ðskÞÞ � lnðð1þ gkÞy2ðskÞÞj ¼ VðskÞ: ð3:12Þ
The above analysis shows that, for all t > sp > T0,
DþVðtÞ 6 �hVðtÞ: ð3:13Þ
Applying the differential inequality theorem and the variation of constants formula of solutions of first-order linear differ-ential equation, we have
VðtÞ 6 VðspÞ expð�hðt � spÞÞ:
It is obvious that V(t) ? 0 as t ? +1, that is
limt!þ1
j ln x1ðtÞ � ln x2ðtÞj ¼ 0; limt!þ1
j ln y1ðtÞ � ln y2ðtÞj ¼ 0:
It follows from (3.9) that
limt!þ1
jx1ðtÞ � x2ðtÞj ¼ 0; limt!þ1
jy1ðtÞ � y2ðtÞj ¼ 0:
This completes the proof of Theorem 3.2. If hk = gk = 0, then system (1.3) is transformed into system (1.2). Similar to the anal-ysis of Theorems 3.1 and 3.2, we have the following corollaries. h
Z. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453 451
Corollary 3.1. Assume that
m½r1ðtÞ � cðtÞðy�ðtÞÞm� > 0; ðH01Þ
m �r2ðtÞ þ bcðtÞx�ðtÞðy�ðtÞÞm�1h i
> 0; ðH02Þ
where x⁄(t) and y⁄(t) are the unique T-periodic solutions of systems (2.5) and (2.6) with hk = gk = 0, respectively, then the spe-cies x and y are permanent, that is, for any positive solution (x(t),y(t))T of system (1.2), there exist positive constants Mi andmi, i = 1, 2 such that
m1 6 lim inft!þ1
xðtÞ 6 lim supt!þ1
xðtÞ 6 M1;
m2 6 lim inft!þ1
yðtÞ 6 lim supt!þ1
yðtÞ 6 M2:
Corollary 3.2. Assume that H01� �
; H02� �
; ðH4Þ and (H5) hold. Then the species x and y are globally attractive, that is, for any positivesolutions (x1(t),y1(t))T and (x2(t),y2(t))T of system (1.2), one has
limt!þ1
jx1ðtÞ � x2ðtÞj ¼ 0; limt!þ1
jy1ðtÞ � y2ðtÞj ¼ 0:
Remark 3.1. Form Corollary 3.1, system (1.2) is permanent, therefore, by Theorem 2 in [2], system (1.2) admits at least oneposiitve T-periodic solution. Furthermore, it follows from Corollary 3.2 that system (1.2) has only one positive T-periodicsolution which is globally asymptotically stable.
4. Examples
By using Matlab software, the following examples show the feasibility of our main result. Consider the following system
x0ðtÞ ¼ xðtÞ 5� ð4þ 0:5 cosðptÞÞxðtÞ � y23ðtÞ
� �;
y0ðtÞ ¼ yðtÞ �0:5þ 1:2xðtÞy�13ðtÞ � ð4:5þ 0:5 sinðptÞÞyðtÞ
� �; t – sk;
x sþk� �
¼ ð1þ hkÞxðskÞ;y sþk� �
¼ ð1þ gkÞyðskÞ; t ¼ sk ¼ kT; k ¼ 1;2; . . . ;
ð4:1Þ
where T = 2.
Example 4.1. Take hk = h = 0.5 and gk = g = 0.8. Conditions (H1)–(H3) are respectively equivalent to inequalities
� lnð1þ gÞ þ Tm½r2� � 0:4122 > 0;
lnð1þ hÞ þ Tm½r1ðtÞ � cðtÞðy�ðtÞÞm�P 6:8422 > 0;
lnð1þ gÞ þ Tm �r2ðtÞ þ bcðtÞx�ðtÞðy�ðtÞÞm�1h i
P 0:8742 > 0:
Taking a = 1.5, d = 1, one could verify
mint2½0;T�fab1ðtÞ � dbcðtÞmm�1
2 gP 1:7979 > 0;
mint2½0;T�fdb2ðtÞ þ dbcðtÞð1�mÞm1Mm�2
2 � acðtÞmmm�12 gP 1:2134 > 0:
The above inequalities show that conditions (H1)–(H5) hold. Form Theorems 3.1 and 3.2, system (4.1) is permanent and glo-bal attractivity. Fig. 1 shows the dynamics behavior of system (4.1).
Example 4.2. Take hk = gk = 0, that is, we consider system (4.1) without impulses. Conditions H01� �
and H02� �
are respectivelyequivalent to inequalities
m½r1ðtÞ � cðtÞðy�ðtÞÞm�P 7:3701 > 0;
m �r2ðtÞ þ bcðtÞx�ðtÞðy�ðtÞÞm�1h i
P 0:7139 > 0:
Taking a = 1.5, d = 1, one could easily verify
mint2½0;T�
ab1ðtÞ � dbcðtÞmm�12
� P 2:3573 > 0;
mint2½0;T�
db2ðtÞ þ dbcðtÞð1�mÞm1Mm�22 � acðtÞmmm�1
2
n oP 1:7788 > 0:
0 5 10 15 200.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
time t
xx
0 5 10 15 200.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
time t
y
y
Fig. 1. Dynamic behavior of the solution (x(t),y(t))T of system (4.1) with hk = gk = 0 and the initial conditions (1.2,0.3)T, (0.8,0.4)T and (1.4,0.2)T, respectively.
0 5 10 15 200.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time t
x an
d y
xy
Fig. 2. Dynamic behavior of the solution (x(t),y(t))T of system (4.1) with hk = h = 0.5 and gk = g = 0.8 and the initial conditions (1.2,0.4)T, (1,0.6)T and(0.8,0.2)T, respectively.
452 Z. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 444–453
The above inequalities show that conditions H01� �
; H02� �
; ðH4Þ and (H5) hold. Form Corollaries 3.1 and 3.2 and Remark 3.1, sys-tem (4.1) is permanent and has only one positive 2-periodic solution which is globally asymptotically stable. Fig. 2 shows thedynamics behavior of system (4.1).
5. Conclusion
In this paper, we consider a periodic predator–prey system with mutual interference and impulses. By constructing a suit-able Lyapunov function and using the comparison theorem of impulsive differential equation, the permanence and globalattractivity of the system are investigated. There are still many interesting and challenging questions that need to be studied.For example, whether our results can be applied to study some more generalized systems? Such questions are clearly worthyof further investigations.
Acknowledgements
This work was supported by the foundation of Fujian Education Bureau (JB09004), the Technology Innovation PlatformProject of Fujian Province (2009J1007) and the program for Science and Technology Development Foundation of FuzhouUniversity (2009-XY-18) and (2010-XY-19).
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