Positive periodic solutions for impulsive predator–prey model with dispersion and time delays

Applied Mathematics and Computation 217 (2010) 661–676

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Positive periodic solutions for impulsive predator–prey model with dispersion and time delays Ruixi Liang a,*, Jianhua Shen b a b

School of Mathematical Sciences and Computing Technologies, Central South University, Changsha, Hunan 410075, China Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

a r t i c l e

i n f o

a b s t r a c t In this paper, we study the existence and global attractivity of positive periodic solutions for impulsive predator–prey systems with dispersion and time delays. By using coincidence degree theorem, a set of easily verifiable sufficient conditions are obtained for the existence of at least one strictly positive periodic solutions, and by means of a suitable Lyapunov functional, the uniqueness and global attractivity of positive periodic solution is presented. Some known results subject to the underlying systems without impulses are improved and generalized. Ó 2010 Elsevier Inc. All rights reserved.

Keywords: Periodic solution Global attractivity Time delay Dispersion Impulses

1. Introduction The effect of environment change in the growth and diffusion of a species in a heterogenous habitat is a very interesting subject in the ecological literature (see [2–8,16,19]). Because of human activities, the location of manufacturing industries, the pollution of the atmosphere, of soil, of rivers, etc., more and more habitats are broken into patches and some of them are polluted. In some of these patches, without the contribution from other patches, the species will go to extinction. As a consequence, the species will occupy a week patchy environment. Since the pioneering theoretical work by Skellam [19], many works have focused on the effect of spatial factors which play a crucial rule in the stability and persistence of a population ([3,4,6–9,12,13] and reference cited therein). Xu et al. [4] investigated the following two-species non-autonomous path-system with time delays:

8 0 > < x1 ðtÞ ¼ x1 ðtÞ½r1 ðtÞ  a11 ðtÞx1 ðtÞ  a13 ðtÞyðtÞ þ D1 ðtÞ½x2 ðtÞ  x1 ðtÞ; x02 ðtÞ ¼ x2 ðtÞ½r2 ðtÞ  a22 ðtÞx2 ðtÞ  a23 ðtÞyðtÞ þ D2 ðtÞ½x1 ðtÞ  x2 ðtÞ; > : 0 y ðtÞ ¼ yðtÞ½r3 ðtÞ þ a31 ðtÞx1 ðt  s1 Þ þ a32 ðtÞx2 ðt  s1 Þ  a33 ðtÞyðt  s2 Þ;

ð1:1Þ

with initial conditions

xi ðhÞ ¼ /i ðhÞ; /i ð0Þ > 0;

yðhÞ ¼ wðhÞ;

wð0Þ > 0;

/i ;

h 2 ½s; 0; w 2 Cð½s; 0Þ; Rþ Þ;

i ¼ 1; 2;

where xi(t) denote the density of species x in patch i, i = 1, 2, at time t, and y(t) denotes the total predator population for both patches. ri(t) is the intrinsic growth rate of the prey at patch i, i = 1, 2; aii(t) (i = 1, 2) are the density-dependent coefficients of the prey at patch i; a13(t) and a23(t) are the capturing rates of the predator in patch 1 and patch 2, respectively, a31(t)/a13(t) and a32(t)/a23(t) are the conversion rates of nutrients into the reproduction of the predator, r3(t) is the death rate of the * Corresponding author. E-mail address: [email protected] (R. Liang). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.06.003

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predator; Di(t) is dispersion rate of prey species x, i = 1, 2. s = max{s1, s2}. s1 is the delay due to gestation, that is, mature adult predators can only contribute to the production of predator biomass. In addition, the model have included the term a33(t)y(t  s2) in the dynamics of the predator y to incorporate the negative feedback of predator crowding. In [4], the authors determined sufficient conditions on parameters of the model that ensure the existence, uniqueness and global stability of positive periodic solution of the system. However, in population dynamics, many evolutionary processes experience short-time rapid changes after undergoing relatively long smooth variation. For examples, the harvesting and stocking occur at fixed moments, and some species usually immigrate at the same time every year, etc. If we still thought of the population dynamical systems with these phenomena as continuous systems, it would be unreasonable or incorrect. We should establish systems with impulsive effects. Recently, theories for impulsive differential equations have been introduced into population dynamics [1,5,10,11]. To the authors knowledge, the population dynamical systems with diffusion and impulsive effects are seldom discussed. In this paper, we mainly study the following impulsive system:

8 0 x ðtÞ ¼ x1 ðtÞ½r 1 ðtÞ  a11 ðtÞx1 ðtÞ  a13 ðtÞyðtÞ þ D1 ðtÞ½x2 ðtÞ  x1 ðtÞ; > > > 10 > > > < x2 ðtÞ ¼ x2 ðtÞ½r 2 ðtÞ  a22 ðtÞx2 ðtÞ  a23 ðtÞyðtÞ þ D2 ðtÞ½x1 ðtÞ  x2 ðtÞ; y0 ðtÞ ¼ yðtÞ½r 3 ðtÞ þ a31 ðtÞx1 ðt  s1 Þ þ a32 ðtÞx2 ðt  s1 Þ  a33 ðtÞyðt  s2 Þ; > > > > Dxi ðt k Þ ¼ bik xi ðt k Þ; > > : Dyðt k Þ ¼ b3k yðtk Þ;

t – tk ; t – tk ; t – tk ;

ð1:2Þ

i ¼ 1; 2; k ¼ 1; 2; . . . ; k ¼ 1; 2; . . . ;

where bikxi(tk) (i = 1, 2) and b3ky(tk) represent the population xi(t) and y(t) at tk regular harvest pulse. Throughout this paper, for system (1.2) the following conditions are assumed. (C1) ri(t), aij(t)(i, j = 1, 2, 3), D1(t) and D2(t) are continuously positive periodic functions with periodic x, s1 and s2 are nonnegative constants. (C2) 1 < bik 6 0, i = 1, 2, 3 for all k 2 N and there exists a positive integer q such that tk+q = tk + x, bi(k+q) = bik, i = 1, 2, 3 and tk  s1, tk  s2 – tm. In the following, we shall use the notation

f ¼ 1

x

Z 0

x

f ðsÞds;

f L ¼ min f ðtÞ; t2½0;x

f M ¼ max f ðtÞ: t2½0;x

Without loss of generality, we shall assume tk – 0, x and [0, x] \ {tk} = {t1,t2, . . . , tq}. It is clear that without the impulses system (1.2) reduces to system (1.1). The existence of positive periodic solutions of (1.1) is investigated in [4], and the following result was obtained. Theorem 1.1. In addition to (C1), assume further that the following hold: M M (H2) aL33 ðr1  D1 Þ  aM 13 Aða31 þ a32 Þ > 0; L M M (H3) a33 ðr2  D2 Þ  a23 Aða31 þ aM 32 Þ > 0; L M  M M (H4) aL31 aM 22 ðr 1  D1 Þ þ a32 a11 ðr 2  D2 Þ  r 3 a11 a22 > 0;where

( ) M M ðr 1  D1 ÞM þ DM 1 ðr 2  D2 Þ þ D2 A ¼ max ; : aL11 aL22

Then system (1.1) has at least one positive x-periodic solution. The Example 2 in [4] shows that Theorem 1.1 has room for improvement. The organization of this paper is as follows. In the next section, we establish some simple criteria for the existence of positive periodic solutions of system (1.2). We also note that our results improve Theorem 1.1 as bik  0, because our results do not need the condition (H2) or (H3). In Section 3, the uniqueness and global attactivity of periodic solutions are presented. Finally, we give an example to show our results.

2. Existence of periodic solutions In this section, by using continuation theorem which was proposed in [14] by Gaines and Mawhin, we will establish the existence conditions of at least one positive periodic solution to system (1.2). To do so, we need to make some preparations. Let X, Z be real Banach spaces, L: Dom L  X ? Z be a Fredholm mapping of index zero (index L = dim Ker L  codim Im L), and let P: X ? X, Q: Z ? Z be continuous projectors such that Im P = Ker L, Ker Q = Im L and X = Ker L  Ker P, Z = Im L  Im Q. Denote by LP the restriction of L to Dom L \ Ker P, KP: Im L ? Ker P \ Dom L the inverse (to LP), and J: Im Q ? Ker L an isomorphism of Im Q onto Ker L. For convenience, we first introduce Mawhin’s continuation theorem [14] as follows:

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R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

Lemma 2.1. Let X  X be an open bounded set. Let L be a Fredholm mapping of index zero and N be L-compact on X. Assume (a) Lx – kNx for each k 2 (0,1), x 2 oX \ Dom L, (b) for each x 2 Ker L \ oX, QNx – 0,  \ DomL. (c) deg {JQN,X \ Ker L,0} – 0. Then Lx = Nx has at least one solution in X To prove the main conclusion by means of the continuation theorem, we need introduce some function spaces.   Let PC(R, R3) = {x:R ? R3jx be continuous at t – tk ; xðt þ k Þ; xðt k Þ exist and xðt k Þ ¼ xðt k Þ; k ¼ 1; 2; . . . ; g, let X = {(u1(t), u2(t), u3(t))T 2 PC(R, R3):ui(t + x) = ui(t), i = 1, 2, 3} with the norm

kðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞkT ¼

3 X

sup jui ðtÞj;

i¼1 t2½0;x

and

Y ¼ X  R3q with the norm kukY ¼ kxk þ kyk; for u 2 Y; x 2 X; y 2 R3q ; here j  j denotes the Euclidean norm. Then X and Y are Banach spaces. Theorem 2.1. In addition to (C1) and (C2), assume further that the following hold: M M L (C3) aL33 ðr1  D1 Þx  aM 13 Aða31 þ a32 Þx þ a33 B1 > 0;

(C4)

aL31 aM ½ðr 1 D1 ÞxþB1 þaL32 aM ½ðr 2 D2 ÞxþB2  22 11 r 3 xB3

Bi ¼

q X

lnð1 þ bik Þ;

M > aM 11 a22 ; where A is defined in Theorem 1.1, and

i ¼ 1; 2; 3:

k¼1

Then system (1.2) has at least one positive x-periodic solution. Proof. Let

u1 ðtÞ ¼ ln½x1 ðtÞ;

u2 ðtÞ ¼ ln½x2 ðtÞ;

u3 ðtÞ ¼ ln½yðtÞ;

ð2:1Þ

then system (1.2) can be translated to

u01 ðtÞ ¼ r 1 ðtÞ  D1 ðtÞ  a11 ðtÞeu1 ðtÞ  a13 ðtÞeu3 ðtÞ þ D1 ðtÞeu2 ðtÞu1 ðtÞ ; u02 ðtÞ ¼ r 2 ðtÞ  D2 ðtÞ  a22 ðtÞeu2 ðtÞ  a23 ðtÞeu3 ðtÞ þ D2 ðtÞeu1 ðtÞu2 ðtÞ ; u03 ðtÞ ¼ r 3 ðtÞ þ a31 ðtÞeu1 ðts1 Þ þ a32 ðtÞeu2 ðts1 Þ  a33 ðtÞeu3 ðts2 Þ ;

Dui ðtk Þ ¼ lnð1 þ bik Þ;

i ¼ 1; 2; 3; k ¼ 1; 2; . . .

ð2:2Þ



It is easy to see that if system (2.2) has one x-periodic solution ðu 1 ðtÞ; u 2 ðtÞ; u 3 ðtÞÞT , then ðx 1 ðtÞ; x 2 ðtÞ; y ðtÞÞT ¼ ðexp½u 1 ðtÞ; exp½u 2 ðtÞ; exp½u 3 ðtÞÞT is a positive x-periodic solution of (1.2). Therefore, to complete the proof, we need only to prove that (2.2) has one x-periodic solution. Let L: Dom L  X ? Y, u ? (u0 , Du(t1), . . . , Du(tq)),

32 3 2 31 3 2 lnð1 þ b1q Þ lnð1 þ b12 Þ lnð1 þ b11 Þ r 1 ðtÞ  D1 ðtÞ  a11 ðtÞeu1 ðtÞ  a13 ðtÞeu3 ðtÞ þ D1 ðtÞeu2 ðtÞu1 ðtÞ 6 7 6 7 6 7C B6 7 Nu ¼ @4 r 2 ðtÞ  D2 ðtÞ  a22 ðtÞeu2 ðtÞ  a23 ðtÞeu3 ðtÞ þ D2 ðtÞeu1 ðtÞu2 ðtÞ 5; 4 lnð1 þ b21 Þ 5; 4 lnð1 þ b22 Þ 5;. .. ; 4 lnð1 þ b2q Þ 5A: lnð1 þ b3q Þ lnð1 þ b31 Þ lnð1 þ b32 Þ r3 ðtÞ þ a31 ðtÞeu1 ðts1 Þ þ a32 ðtÞeu2 ðts1 Þ  a33 ðtÞeu3 ðts2 Þ 02

Evidently

Ker L ¼ fu : uðtÞ ¼ c 2 R3 ; t 2 ½0; xg; ( Z x

Im L ¼

z ¼ ðf ; a1 ; . . . ; aq Þ 2 Y :

f ðsÞds þ

0

q X

) ak ¼ 0 ;

k¼1

and

dimKerL ¼ 3 ¼ codim Im L: So Im L is closed in Y, L is a Fredholm mapping of index zero. Define

Px ¼

1

x

Z

x

xðtÞdt;

0

Qz ¼ Qðf ; a1 ; a2 ; . . . ; aq Þ ¼

1

x

"Z 0

x

f ðsÞds þ

q X k¼1

#

!

ak ; 0; . . . ; 0 :

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R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

It is easy to show that P and Q are continuous projectors satisfying

Im P ¼ Ker L; Im L ¼ Ker Q ¼ ImðI  QÞ: Furthermore, by an easy computation, we can find that the inverse KP: Im L ? Ker P \ Dom L has the form

K P ðzÞ ¼

Z

t

f ðsÞds þ

0

X

ak 

tk
1

x

Z 0

x

Z

t

f ðsÞdsdt 

0

q X

ak :

k¼1

Thus

00 1 R x

1 1 u1 ðtÞ  a13 ðtÞeu3 ðtÞ x 0 ½r 1 ðtÞ  D1 ðtÞ  a11 ðtÞe C BB C C BB C q P C BB C u2 ðtÞu1 ðtÞ 1 þD1 ðtÞe dt þ x lnð1 þ b1k Þ; C C BB C BB C k¼1 C BB C C BB 1 R x C u ðtÞ u ðtÞ C BB x 0 ½r 2 ðtÞ  D2 ðtÞ  a22 ðtÞe 2  a23 ðtÞe 3 C C BB C C BB C q QNu ¼ BB C; 0; . . . ; 0C; P u1 ðtÞu2 ðtÞ 1 C BB C þD2 ðtÞe dt þ x lnð1 þ b2k Þ; C C BB C BB C k¼1 C BB R C C BB 1 x u1 ðts1 Þ u2 ðts1 Þ C C BB x 0 ½r 3 ðtÞ þ a31 ðtÞe C þ a32 ðtÞe C BB C C BB C q A @@ A P a33 ðtÞeu3 ðts2 Þ dt þ x1 lnð1 þ b3k Þ; k¼1

and

2Rt 0

½r1 ðsÞ  D1 ðsÞ  a11 ðsÞeu1 ðsÞ  a13 ðsÞeu3 ðsÞ þ D1 ðsÞeu2 ðsÞu1 ðsÞ ds þ

P

lnð1 þ b1k Þ

3

7 6 t>t k 7 6Rt P 7 6 ½r2 ðsÞ  D2 ðsÞ  a22 ðsÞeu2 ðsÞ  a23 ðsÞeu3 ðsÞ þ D2 ðsÞeu1 ðsÞu2 ðsÞ ds þ lnð1 þ b Þ 2k K P ðI  Q ÞNu ¼ 6 0 7 t>t k 7 6R P 5 4 t u1 ðss1 Þ u2 ðss1 Þ u3 ðss2 Þ ½r 3 ðsÞ þ a31 ðsÞe þ a32 ðsÞe  a33 ðsÞe ds þ lnð1 þ b3k Þ 0 t>t k

3 lnð1 þ b Þ 1k 7 6x 0 7 6 k¼1 7 6 R R q P x t 61 u2 ðsÞ u3 ðsÞ u1 ðsÞu2 ðsÞ  a23 ðsÞe þ D2 ðsÞe ds þ lnð1 þ b2k Þ 7  6 x 0 0 ½r2 ðsÞ  D2 ðsÞ  a22 ðsÞe 7 7 6 k¼1 7 6 q R R P 5 4 1 x t u1 ðss1 Þ u2 ðss1 Þ u3 ðss2 Þ ½r ðsÞ þ a ðsÞe þ a ðsÞe  a ðsÞe ds þ lnð1 þ b Þ 3 31 32 33 3k 0 x 0 2

1

Rx Rt

½r ðsÞ  D1 ðsÞ  a11 ðsÞeu1 ðsÞ  a13 ðsÞeu3 ðsÞ þ D1 ðsÞeu2 ðsÞu1 ðsÞ ds þ 0 1

q P

k¼1

3 q P u1 ðsÞ u3 ðsÞ u2 ðsÞu1 ðsÞ  ½r ðsÞ  D ðsÞ  a ðsÞe  a ðsÞe þ D ðsÞe ds þ lnð1 þ b Þ 1 11 13 1 1k 7 6 x 2 0 1 7 6 k¼1 7 6 q  R P 6 t  1 t ½r ðsÞ  D ðsÞ  a ðsÞeu2 ðsÞ  a ðsÞeu3 ðsÞ þ D ðsÞeu1 ðsÞu2 ðsÞ ds þ lnð1 þ b2k Þ 7 6 x 2 0 2 7: 2 22 23 2 7 6 k¼1 7 6  q P 5 4 t 1 R t u1 ðss1 Þ u2 ðss1 Þ u3 ðss2 Þ þ a32 ðsÞe  a33 ðsÞe ds þ lnð1 þ b3k Þ x  2 0 ½r 3 ðsÞ þ a31 ðsÞe 2

t

Rt 1

k¼1

Clearly, QN and KP(I  Q)N are continuous. Using Lemma 2.4 in [1], it is not difficult to show that QNðXÞ; K p ðI  Q ÞNðXÞ are relatively compact for any open bounded set X  X. Hence N is L-compact on X for any open bounded set X  X. Now we reach the position to search for an appropriate open, bounded subset X for the application of the continuation theorem. Corresponding to equation Lu = kNu, k 2 (0,1), we have

u01 ðtÞ ¼ k½r 1 ðtÞ  D1 ðtÞ  a11 ðtÞeu1 ðtÞ  a13 ðtÞeu3 ðtÞ þ D1 ðtÞeu2 ðtÞu1 ðtÞ ; u02 ðtÞ ¼ k½r 2 ðtÞ  D2 ðtÞ  a22 ðtÞeu2 ðtÞ  a23 ðtÞeu3 ðtÞ þ D2 ðtÞeu1 ðtÞu2 ðtÞ ;

ð2:3Þ

u03 ðtÞ ¼ k½r 3 ðtÞ þ a31 ðtÞeu1 ðts1 Þ þ a32 ðtÞeu2 ðts1 Þ  a33 ðtÞeu3 ðts2 Þ ;

Dui ðtk Þ ¼ k lnð1 þ bik Þ;

i ¼ 1; 2; 3; k ¼ 1; 2; . . . :

Since ui(t) (i = 1, 2, 3) are x-periodic functions, we need only to prove the result in the interval [0, x]. Integrating (2.3) over the interval [0, x] leads to

Z

x

a11 ðtÞeu1 ðtÞ dt þ

Z

0

Z 0

x

a13 ðtÞeu3 ðtÞ dt ¼ 0

x

a22 ðtÞeu2 ðtÞ dt þ

Z

x

ðr 1 ðtÞ  D1 ðtÞÞdt þ

0

x

a23 ðtÞeu3 ðtÞ dt ¼ 0

Z

Z 0

Z

x

D1 ðtÞeu2 ðtÞu1 ðtÞ dt þ

0

x

ðr 2 ðtÞ  D2 ðtÞÞdt þ

Z 0

q X

lnð1 þ b1k Þ;

ð2:4Þ

lnð1 þ b2k Þ;

ð2:5Þ

k¼1

x

D2 ðtÞeu1 ðtÞu2 ðtÞ dt þ

q X k¼1

665

R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

Z

x

a31 ðtÞeu1 ðts1 Þ dt þ

Z

0

x

a32 ðtÞeu2 ðts1 Þ dt ¼

Z

0

x

r 3 ðtÞdt þ

0

Z

x

a33 ðtÞeu3 ðts2 Þ dt 

0

q X

lnð1 þ b3k Þ:

ð2:6Þ

k¼1

It follows from (2.3)–(2.6) that

Z

x

ju1 ðtÞjdt <

0

¼2

Z

a11 ðtÞeu1 ðtÞ dt þ 2

ju02 ðtÞjdt < ¼2

Z

ju03 ðtÞjdt <

0

¼2

Z

x

x

D1 ðtÞeu2 ðtÞu1 ðtÞ dt þ q X

a13 ðtÞeu3 ðtÞ dt  Z

k¼1

x

D2 ðtÞeu1 ðtÞu2 ðtÞ dt þ

x

r 3 ðtÞdt þ

q X

x

a23 ðtÞeu3 ðtÞ dt 

k¼1

Z

0

x

a31 ðtÞeu1 ðts1 Þ dt þ

a22 ðtÞeu2 ðtÞ dt þ

x

x

a32 ðtÞeu2 ðts1 Þ dt þ

0

x

a23 ðtÞeu3 ðtÞ dt

x

D2 ðtÞdt;

a32 ðtÞeu2 ðts1 Þ dt þ

q X

Z 0

0

Z

a13 ðtÞeu3 ðtÞ dt

D1 ðtÞdt;

x

Z

x

x

0

Z

0

Z

Z 0

0

Z

lnð1 þ b2k Þ þ 2

0

a31 ðtÞeu1 ðts1 Þ dt þ 2

a11 ðtÞeu1 ðtÞ dt þ

0

0

Z

x

lnð1 þ b1k Þ þ 2

0

Z

Z 0

x

ðr 2 ðtÞ þ D2 ðtÞÞdt þ

a22 ðtÞeu2 ðtÞ dt þ 2

x

Z

x

0

Z 0

0

Z

x

0

Z

ðr 1 ðtÞ þ D1 ðtÞÞdt þ

x

x

0

x

0

0

Z

Z

lnð1 þ b3k Þ 6 2

Z

Z

x

a33 ðtÞeu3 ðts2 Þ dt

0

x

a31 ðtÞeu1 ðts1 Þ dt þ 2

0

k¼1

Z

x

a32 ðtÞeu2 ðts1 Þ dt:

0

ð2:7Þ Multiplying the first equation of (2.3) by eu1 ðtÞ and integrating over [0, x], we obtain



p X

b1k eu1 ðtk Þ þ

Z

x

a11 e2u1 ðtÞ dt 6 ðr 1  D1 ÞM

Z

0

k¼1

x

0

eu1 ðtÞ dt þ DM 1

Z

x

eu2 ðtÞ dt:

0

Since 1 < b1k 6 0, so we have

Z

x

a11 ðtÞe2u1 ðtÞ dt 6 ðr 1  D1 ÞM

Z

0

x

eu1 ðtÞ dt þ DM 1

0

Z

x

eu2 ðtÞ dt; 0

which yields

aL11

Z

x

e2u1 ðtÞ dt 6 ðr 1  D1 ÞM

Z

0

x

0

eu1 ðtÞ dt þ DM 1

Z

x

eu2 ðtÞ dt:

ð2:8Þ

0

Similarly, multiplying the second equation in (2.3) by eu2 ðtÞ and integrating over [0, x] gives

aL22

Z

x

e2u2 ðtÞ dt < ðr 2  D2 ÞM

0

Z

x 0

eu2 ðtÞ dt þ DM 2

Z

x

eu1 ðtÞ dt:

ð2:9Þ

0

By using the inequalities

Z

x

eui ðtÞ dt

2

6x

0

Z

x

e2ui ðtÞ dt;

i ¼ 1; 2;

0

and (2.8) and (2.9), we have

aL11 aL22 If

Rx 0

Z

x

eu1 ðtÞ dt

0

Z

x

eu2 ðtÞ dt

2

< xðr 1  D1 ÞM

Z

Z

x

0

< xðr 2  D2 ÞM

0

Z

x

0

eu2 ðtÞ dt 6

aL11

2

Rx 0

eu1 ðtÞ dt þ DM 1 x eu2 ðtÞ dt þ DM 2 x

Z

x

eu2 ðtÞ dt;

ð2:10Þ

eu1 ðtÞ dt:

ð2:11Þ

0

Z

x

0

eu1 ðtÞ dt, then we derive from (2.10) that

x

eu1 ðtÞ dt

2

< xðr 1  D1 ÞM

0

Z 0

x

eu1 ðtÞ dt þ DM 1 x

Z

x

eu1 ðtÞ dt;

0

which implies

Z

x

eu2 ðtÞ dt 6

0

If

Rx 0

0

x

eu1 ðtÞ dt <

0

eu1 ðtÞ dt 6

Z

Z

Rx 0

xðr1  D1 ÞM þ xDM1 aL11

:

ð2:12Þ

:

ð2:13Þ

eu2 ðtÞ dt, then we can conclude

x

eu1 ðtÞ dt 6

Z 0

x

eu2 ðtÞ dt <

xðr2  D2 ÞM þ xDM2 aL22

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R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

Set

A ¼ max

( ) M M ðr 1  D1 ÞM þ DM 1 ðr 2  D2 Þ þ D2 ; ; aL11 aL22

ð2:14Þ

so, from (2.12)–(2.14) we get

Z

x

eui ðtÞ dt < xA;

i ¼ 1; 2:

ð2:15Þ

0

Since u(t) 2 X, there exist ni, gi 2 [0, x] such that

ui ðgi Þ ¼ max ui ðtÞ;

ui ðni Þ ¼ min ui ðtÞ; t2½0;x

i ¼ 1; 2; 3:

t2½0;x

ð2:16Þ

From (2.15) and (2.16), we see that

ui ðni Þ < ln A; Noticing that

aL33

Z

Rx 0

i ¼ 1; 2:

eu1 ðts1 Þ dt ¼

x 0

eu3 ðtÞ dt ¼ aL33 ¼

Z

Z

x

ð2:17Þ

Rx 0

x

eu1 ðtÞ dt and

eu3 ðts2 Þ dt 6

Rx 0

Z

0

x

Rx 0

eu3 ðtÞ dt, from (2.6) and (2.15), one obtains

a33 ðtÞeu3 ðts2 Þ dt

0

a31 ðtÞeu1 ðts1 Þ dt þ

Z

0

Z

eu3 ðts2 Þ dt ¼

x

a32 ðtÞeu2 ðts1 Þ dt 

Z

0

x

6

a31 ðtÞeu1 ðts1 Þ dt þ

Z

0

x

r3 ðtÞdt þ

0

x

a32 ðtÞeu2 ðts1 Þ dt ¼

Z

0

q X

lnð1 þ b3k Þ

k¼1

x

a31 ðtÞeu1 ðtÞ dt þ

0

Z 0

x M a32 ðtÞeu2 ðtÞ dt 6 ðaM 31 þ a32 ÞxA;

which implies

Z

x

eu3 ðtÞ dt 6

0

M ðaM 31 þ a32 ÞxA aL33

ð2:18Þ

and

u3 ðn3 Þ 6 ln

M ðaM 31 þ a32 ÞA : aL33

ð2:19Þ

It follows from (2.7), (2.15) and (2.18) that

Z 0

Z

0

M X ðaM 31 þ a32 ÞxA þ 2xD1  lnð1 þ b1k Þ :¼ c1 aL33 k¼1

M ju02 ðtÞjdt 6 2aM 22 xA þ 2a23 

M X ðaM 31 þ a32 ÞxA þ 2xD2  lnð1 þ b2k Þ :¼ c2 L a33 k¼1

q

q

x

0

Z

M ju01 ðtÞjdt 6 2aM 11 xA þ 2a13 

x

ð2:20Þ

x M ju03 ðtÞjdt 6 2ðaM 31 þ a32 ÞxA :¼ c 3 :

Thus, from (2.17), (2.19) and (2.20), we have

8 Rt 0 P > > < u1 ðn1 Þ þ n1 u1 ðsÞds þ n lnð1 þ b1k Þ; t 2 ½0; n1  > : u1 ðn1 Þ þ n1 u1 ðsÞds  6 u1 ðn1 Þ þ

t6t k 6n 1

Z

x

ju01 ðtÞjdt 

0

< ln A þ c1 

q X

lnð1 þ b1k Þ

k¼1 q X

lnð1 þ b1k Þ

ð2:21Þ

k¼1

u2 ðtÞ 6 u2 ðn2 Þ þ

Z

x 0

ju02 ðtÞjdt 

q X k¼1

lnð1 þ b2k Þ < ln A þ c2 

q X k¼1

lnð1 þ b2k Þ

ð2:22Þ

R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

u3 ðtÞ 6 u3 ðn3 Þ þ

Z

x

ju03 ðtÞjdt 

0

X

M X ðaM 31 þ a32 ÞA þ c3  lnð1 þ b3k Þ: L a33 k¼1

667

q

lnð1 þ b3k Þ < ln

k¼1

ð2:23Þ

From (2.4) and (2.18) we obtain

Z

x

a11 ðtÞeu1 ðtÞ dt >

Z

0

x

ðr 1 ðtÞ  D1 ðtÞÞdt 

Z

0

q X

x

a13 ðtÞeu3 ðtÞ dt þ

0

P xðr 1  D1 Þ  aM 13 

lnð1 þ b1k Þ

k¼1

xAðaM31 þ aM32 Þ aL33

þ

q X

lnð1 þ b1k Þ > 0

k¼1

which implies

u1 ðg1 Þ > ln

xðr1  D1 Þ  aM13

xAðaM þaM Þ 31 32 aL33

þ

Pq

k¼1

lnð1 þ b1k Þ :¼ d1 :

31 a

This, together with (2.20), leads to

u1 ðtÞ P u1 ðg1 Þ 

Z

x

ju01 ðtÞjdt þ

0

q X

lnð1 þ b1k Þ > d1  c1 þ

q X

k¼1

lnð1 þ b1k Þ:

ð2:24Þ

k¼1

Let

( R1 ¼ max j ln Aj þ c1 

q X

lnð1 þ b1k Þ; jd1 j þ c1 

k¼1

q X

) lnð1 þ b1k Þ ;

k¼1

it follows from (2.21) and (2.24) that

max ju1 ðtÞj < R1 :

ð2:25Þ

t2½0;x

From (2.5) we have

22 eu2 ðg2 Þ P ðr2  D2 Þ þ D2 eu1 ðn1 Þu2 ðg2 Þ  a

Z

x

a23 ðtÞeu3 ðtÞ dt þ

0

lnð1 þ b2k Þ

k¼1

> ðr 2  D2 Þ þ D2 eu1 ðn1 Þ  eu2 ðg2 Þ  aM 23  P ðr2  D2 Þ þ D2 eR1  eu2 ðg2 Þ  aM 23 

q X

xAðaM31 þ aM32 Þ aL33

q X

þ

lnð1 þ b2k Þ

k¼1

xAðaM31 þ aM32 Þ aL33

þ

q X

lnð1 þ b2k Þ;

k¼1

which implies

u2 ðg2 Þ

e

P

dþ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 22 D2 eR1 d þ 4a ;  2a22

where d ¼ ðr2  D2 Þ  aM 23

xAðaM þaM Þ 31 32 aL33

þ

Pq

k¼1

lnð1 þ b2k Þ, so

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 22 D2 eR1 d þ d þ 4a u2 ðg2 Þ P ln :¼ d2 : 22 2a

ð2:26Þ

It follows from (2.20) and (2.26) that

u2 ðtÞ P u2 ðg2 Þ 

Z

x 0

ju02 ðtÞjdt þ

q X

lnð1 þ b2k Þ P d2  c2 þ

k¼1

q X

lnð1 þ b2k Þ:

k¼1

This, together with (2.22), leads to

max ju2 ðtÞj < maxfj ln Aj þ c2 

t2½0;x

From (2.4) and (2.5), we have

q X k¼1

lnð1 þ b2k Þ; jd2 j þ c2 

q X k¼1

lnð1 þ b2k Þg :¼ R2 :

ð2:27Þ

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R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

aM 11 aM 22

Z

x

eu1 ðtÞ dt þ aM 13

0

Z 0

x

eu2 ðtÞ dt þ aM 23

Z

x

eu3 ðtÞ dt P ðr1  D1 Þx þ 0

Z

x

eu3 ðtÞ dt P ðr2  D2 Þx þ 0

q X k¼1 q X

lnð1 þ b1k Þ; ð2:28Þ lnð1 þ b2k Þ;

k¼1

We derive from (2.6) that

Z

x

a33 ðtÞeu3 ðts2 Þ dt ¼

Z

0

x

Z

a31 ðtÞeu1 ðts1 Þ dt þ

0

a32 ðtÞeu2 ðts1 Þ dt 

0

P aL31 ¼ aL31

x

Z

x 0

Z

Z

x

0

eu1 ðtÞ dt þ aL32

x

eu2 ðts1 Þ dt  r 3 x þ

0

r 3 ðtÞdt þ

x

eu2 ðtÞ dt  r 3 x þ

Rx 0

q X

q X



0

lnð1 þ b3k Þ

lnð1 þ b3k Þ

eu3 ðtÞ dt þ

q P

lnð1 þ b1k Þ

k¼1

þ aL32

aM 11 Rx

lnð1 þ b3k Þ

k¼1

k¼1

P aL31 ðr2  D2 Þx  aM 23

q X

k¼1

0

ðr 1  D1 Þx  aM 13

x

0

Z

eu1 ðts1 Þ dt þ aL32

Z

eu3 ðtÞ dt þ

q P

lnð1 þ b2k Þ

k¼1

 r3 x þ

aM 22

q X

lnð1 þ b3k Þ

ð2:29Þ

k¼1

From (2.28), (2.29) and condition (C4), we have

u3 ðg3 Þ P ln 

aL31 aM 22 ½ðr 1  D1 Þx þ

Pq

L M k¼1 lnð1 þ b1k Þ þ a32 a11 ½ðr 2  D2 Þ L M M M L M L M a11 a22 a33 þ a11 a32 a23 þ aM 22 a31 a13

xþ

Pq

k¼1

Pq  M  aM 11 a22 ½r 3 x  k¼1 lnð1 þ b3k Þ :¼ d3 : M M L M M L M aL11 aM 22 a33 þ a11 a32 a23 þ a22 a31 a13

lnð1 þ b2k Þ

ð2:30Þ

So, from (2.20) and (2.30), we obtain

u3 ðtÞ P u3 ðg3 Þ 

Z 0

x

ju03 ðtÞjdt þ

q X

lnð1 þ b3k Þ P d3  c3 þ

k¼1

q X

lnð1 þ b3k Þ:

ð2:31Þ

k¼1

This, together with (2.23), leads to

max ju3 ðtÞj < R3 ;

t2½0;x

here

( )  q q M X X  ðaM  31 þ a32 ÞA  R3 ¼ max ln lnð1 þ b3k Þ; jd3 j þ c3  lnð1 þ b3k Þ :  þ c3  aL 33

k¼1

k¼1

Clearly, R1, R2 and R3 are independent of k. Similar to the proof of Theorem 2.1 of [4], we can find a sufficiently large M > 0, denote the set

X ¼ fuðtÞ ¼ ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞT 2 X : kuk < M; uðtþk Þ 2 X;

k ¼ 1; 2; . . . ; qg;

it follows that for each u 2 Ker L \ oX, QNu – 0 and

degfJQNu; X \ KerL; 0g ¼ 1 – 0: By now we have proved that X verifies all the requirements in Lemma 2.1. Hence (2.2) has at least one x-periodic solution. Accordingly, system (1.2) has at least one positive x-periodic solution. The proof is complete. h Remark 1. If bik  0, (i = 1, 2, 3), k = 1, 2 . . . , then (1.2) is translated to (1.1). In this case, the conditions (C3) and (C4) are the same as (H2) and (H4) of Theorem 1.1, but we see that (H3) is not need in here. Hence our result improve and generalized the corresponding result of [4]. Similarly, we can easily get the following theorem. Theorem 2.2. The conclusion of Theorem (2.1) is still valid if the condition C3 in Theorem 2.1 is replaced by the following M M L (C 3 ) aL33 ðr2  D2 Þx  aM 23 Aða31 þ a32 Þx þ a33 B2 > 0.

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R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

3. Uniqueness and global stability of periodic solutions Under the hypotheses (C1)  (C2), we consider the nonimpulsive delay differential equation

Y

z01 ðtÞ ¼ z1 ðtÞ½r1 ðtÞ  A11 ðtÞz1 ðtÞ  A13 ðtÞz3 ðtÞ þ D1 ðtÞ½z2 ðtÞ

ð1 þ b2k Þð1 þ b1k Þ1  z1 ðtÞ

0
Y

z02 ðtÞ ¼ z2 ðtÞ½r2 ðtÞ  A22 ðtÞz2 ðtÞ  A23 ðtÞz3 ðtÞ þ D2 ðtÞ½z1 ðtÞ

ð1 þ b1k Þð1 þ b2k Þ1  z2 ðtÞ

ð3:1Þ

0
z03 ðtÞ ¼ z3 ðtÞ½r 3 ðtÞ þ A31 ðtÞz1 ðt  s1 Þ þ A32 ðtÞz2 ðt  s1 Þ  A33 ðtÞz3 ðt  s2 Þ with initial conditions

zi ðhÞ ¼ /i ðhÞ; z3 ðhÞ ¼ wðhÞ; h 2 ½s; 0 /i ð0Þ > 0; wð0Þ > 0; /i ; w 2 Cð½s; 0Þ; Rþ Þ; where

Q

Aii ðtÞ ¼ aii ðtÞ

ð1 þ bik Þ;

ð3:2Þ

i ¼ 1; 2;

ði ¼ 1; 2; 3Þ;

0
Q

A13 ðtÞ ¼ a13 ðtÞ

ð1 þ b3k Þ; A23 ðtÞ ¼ a23 ðtÞ

0
Q

A31 ðtÞ ¼ a31 ðtÞ

Q

ð1 þ b3k Þ

0
ð1 þ b1k Þ;

0
Q

A32 ðtÞ ¼ a32 ðtÞ

ð1 þ b2k Þ;

t P 0:

0
The following lemmas will be used in the proofs of our results. The proof of the first lemma is similar to that of Theorem 1 in [15], and it will be omitted. Lemma 3.1. Assume that (C1)  (C2) hold. Then Q Q (i) if z(t) = (z1(t),z2(t),z3(t))T is a solution of (3.1) on [  s, + 1), then xi ðtÞ ¼ 0 0 such that

0 < zi ðtÞ 6 M i ;

ði ¼ 1; 2; 3Þ;

where

M1 ¼ M2 >

M 1 ;

e 1 ðtÞ ¼ D1 ðtÞ D

M 1

Y

for t P T 2 ;

( ) eM M eM rM 1 þ D 1 r2 þ D 2 ¼ max ; ; AL11 AL22

ð3:3Þ

M3 ¼

M AM 31 M 1 þ A32 M 2

AL33

M

M

eðA31 M1 þA32 M2 Þs2 ;

ð1 þ b2k Þð1 þ b1k Þ1 ;

ð3:4Þ

0
e 2 ðtÞ ¼ D2 ðtÞ D

Y

ð1 þ b1k Þð1 þ b2k Þ1 :

0
Proof. We define

VðtÞ ¼ maxfz1 ðtÞ; z2 ðtÞg: Calculating the upper-right derivative of V(t) along the positive solution of system (3.1), we have the following (P1) If z1(t) > z2(t) or z1(t) = z2(t), then

"

Dþ VðtÞ ¼ z01 ðtÞ ¼ z1 ðtÞ½r1 ðtÞ  A11 ðtÞz1 ðtÞ  A13 ðtÞz3 ðtÞ þ D1 ðtÞ z2 ðtÞ

Y 0
L L M eM eM 6 z1 ðtÞ½r M 1  A11 z1 ðtÞ þ D 1 z2 ðtÞ 6 z1 ðtÞ½r 1 þ D 1  A11 z1 ðtÞ:

(P2) If z1(t) < z2(t) or z1(t) = z2(t), then L eM Dþ VðtÞ ¼ z02 ðtÞ 6 z2 ðtÞ½r M 2 þ D 2  A22 z2 ðtÞ:

# ð1 þ b2k Þð1 þ b1k Þ1  z1 ðtÞ

670

R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

It follows from (P1) and (P2) that L eM Dþ VðtÞ 6 zi ðtÞ½r M i þ D i  Aii zi ðtÞ; ði ¼ 1 or 2Þ:

ð3:5Þ

By (3.5) we can derive (A) If max{z1(0), z2(0)} 6 M1, then max{z1(t), z2(t)} 6 M1, t P 0. L eM (B) If max{z1(0), z2(0)} > M1. Let a ¼ maxfM 1 ðr M i þ D i  Aii M 1 Þ; i ¼ 1; 2g, (a > 0). We consider the following two possibilities: (a) V(0) = z1(0) > M1. (z1(0) P z2(0)). (b) V(0) = z2(0) > M1. (z1(0) < z2(0)). If (a) holds, then there exists e > 0 such that if t 2 [0,e), then V(t) = z1(t) > M1, and we have

Dþ VðtÞ ¼ z01 ðtÞ < a < 0: If (b) holds, then there exists e > 0 such that if t 2 [0,e), then V(t) = z2(t) > M1, and also we have

Dþ VðtÞ ¼ z02 ðtÞ < a < 0: From what has been discussed above, we can conclude that if V(0) > M1, then V(t) is strictly monotone decreasing with speed at least a. Therefore there exists a T1 > 0 such that if t P T1, then

VðtÞ ¼ maxfz1 ðt; z2 ðtÞg 6 M1 :

ð3:6Þ

In addition, from the third equation of system (3.1) and (3.6) we derive that for t > T1 + s1, M L z03 ðtÞ 6 z3 ðtÞ½AM 31 M 1 þ A32 M 2  A33 z3 ðt  s2 Þ:

A similar argument in proof of Lemma 2.1 in [16] shows that there exists a T2 P T1 + s1 such that

z3 ðtÞ 6

M AM 31 M 1 þ A32 M 2

AL33

M

M

eðA31 M1 þA32 M2 Þs2 :¼ M3 ; for t P T 2 ;

which completes the proof.

h

Lemma 3.3. Let (C1)  (C2) hold. Assume further that ðC 5 ÞðAL31 þ AL32 Þ min Then there exist positive constants T > T2 and mi (i = 1, 2, 3) such that

nrL DM AM M 1

1

AM 11

13

3

;

M r L2 DM 2 A23 M 3

AM 22

mi < zi ðtÞ; ði ¼ 1; 2; 3Þ for t P T;

o

> rM 3 .

ð3:7Þ

in which

( ) M M M L r L1  DM 1  A13 M 3 r 2  D2  A23 M 3 ; m1 ¼ m2 < min ; AM AM 11 22 m3 <

m 3

¼

AL31 m1 þ AL32 m2  r M 3 AM 33

ð3:8Þ

;

and M3 are defined in (3.4). Proof. Define

VðtÞ ¼ minfz1 ðtÞ; z2 ðtÞg: Then calculating to lower-right derivative of V(t) along the positive solution of system (3.1). Similar to the discussion for inequality (3.5), it is easy to obtain by (3.3)

( Dþ VðtÞ ¼ z0i ðtÞði ¼ 1 or 2Þ P

M M z1 ðtÞ½rL1  AM 11 z1 ðtÞ  A13 M 3  D1 ði ¼ 1Þ M M for t P T 2 ; z2 ðtÞ½rL2  AM 22 z1 ðtÞ  A23 M 3  D2  ði ¼ 2Þ

where T2 be defined in Lemma 3.2. (I) If V(T2) = min{z1(T2),z2(T2)} P m1, then min{z1(t),z2(t)} P m1, t P T2. (II) If V(T2) = min{z1(T2),z2(T2)} < m1, and let

l ¼ minfz1 ðT 2 ÞðrL1  AM11 m1  AM13 M3  DM1 Þ; z2 ðT 2 ÞðrL2  AM22 m1  DM2 Þg:

671

R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

There are three cases: (i) V(T2) = z1(T2) < m1,(z1(T2) < z2(T2)). (ii) V(T2) = z2(T2) < m1,(z2(T2) < z1(T2)). (iii) V(T2) = z1(T2) = z2(T2) < m1. If (i) holds, then there exists e > 0 such that if t 2 [T2,T2 + e) we have V(t) = z1(t) and

Dþ VðtÞ ¼ z01 ðtÞ > l > 0: If (ii) holds, similar to (i), there exists [T2, T2 + e) such that if t 2 [T2,T2 + e), we have V(t) = z2(t) and

Dþ VðtÞ ¼ z02 ðtÞ > l > 0: If (iii) holds, in the same way also there exists [T2, T2 + e) such that if t 2 [T2, T2 + e), we have V(t) = zi(t) (i = 1 or 2), and

Dþ VðtÞ ¼ z_ i ðtÞ > l > 0 ði ¼ 1 or 2Þ: From (i)-(iii), we know that if V(T2) < m1, V(t) will strictly monotonically in crease with speed l. So there exists T3 > T2 such that if t P T3, we have

VðtÞ ¼ minfz1 ðtÞ; z2 ðtÞg P m1 : From the third equation of system (3.1), we know that L L M z03 ðtÞ P z3 ðtÞ½r M 3 þ A31 m1 þ A32 m2  A33 z3 ðt  s2 Þ; for t P T 3 þ s1 :

AL31 m1

AL32 m2

AL31 m1

AL32 m2

ð3:9Þ

AM 33 m3 ;

By (C5), þ  rM þ  rM ðk > 0Þ, then if z3(t) 6 m3 for all t P T3 + s1 + s2, (3.9) 3 > 0. Let k ¼ 3  implies that z03 ðtÞ > kz3 ðtÞ. This will lead to a contradiction. Hence there must exist a T4 P T3 + s1 + s2 such that z3(T4) > m3. If z3(t) > m3 for all t P T4, then the conclusion holds. If not, suppose z3 ðtÞ 6 m3 , where t > T 4 . Then from the above discussion there exists t* and t** such that

z3 ðt Þ ¼ z3 ðt Þ ¼ m3 and z3 ðtÞ < m3 for all t < t < t ; where T 4 6 t < t < t . Now suppose z3(t) with t* 6 t 6 t** attains its maximum at tˇ, t* < tˇ < t**. Then since z03 ðtÞ ¼ 0, (3.9) implies L L M  r M 3 þ A31 m1 þ A32 m2  A33 z3 ðt  s2 Þ 6 0;

this leads to

z3 ðt  s2 Þ P

AL31 m1 þ AL32 m2  r M 3 AM 33

> m3 :

From (3.9) we have L L z03 ðtÞ=z3 ðtÞ P rM 3 þ A31 m1 þ A32 m2 :

Integrating the above inequality from t  s2 to tˇ, we have L L L L M M z3 ðtÞ > z3 ðt  s2 ÞeðA31 m1 þA32 m2 r3 Þs2 > m3 eðA31 m1 þA32 m2 r3 Þs2 > m3 ;

this contradicts to z3(t) < m3 for all t* < t < t**. So we have

z3 ðtÞ > m3 for all t P T 4 : Let T > T4, then for t P T, (3.7) holds. The proof is complete. h Now we formulate the uniqueness and global stability of the x-periodic solution x*(t) in Theorem 2.1. It is immediate that if x*(t) is globally asymptotically stable then x*(t) is in fact unique. Theorem 3.1. In addition to (C15), assume further that (C6) limt?1 inf Ai(t) > 0, where

Z tþs1 þs2 eM D 2  A31 ðt þ s1 ÞM3 A33 ðsÞds; m2 tþs1 Z tþs1 þs2 eM D A2 ðtÞ ¼ A22 ðtÞ  A32 ðt þ s1 Þ  1  A32 ðt þ s1 ÞM3 A33 ðsÞds; ; m2 tþs1

A1 ðtÞ ¼ A11 ðtÞ  A31 ðt þ s1 Þ 

A3 ðtÞ ¼ A33 ðtÞ  A13 ðtÞ  A23 ðtÞ  ðr3 ðtÞ þ A31 ðtÞM 1 þ A32 ðtÞM 2 þ A33 ðtÞM 3 Þ  A33 ðt þ s2 ÞM 3

Z

Z

tþs2

A33 ðsÞds

t tþ2s2

A33 ðsÞds

ð3:10Þ

tþs2

Then system (1.2) has a unique positive x-periodic solution x ðtÞ ¼ ðx 1 ðtÞ; x 2 ðtÞ; y ðtÞÞT which is globally asymptotically stable.

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Proof. Due to the conclusion of Theorem 2.1, we only need to show the global asymptotic stability of the positive periodic solution of (1.2). Let x ðtÞ ¼ ðx 1 ðtÞ; x 2 ðtÞ; y ðtÞÞT be a positive x-periodic solution of system (1.2), x(t) = (x1(t), x2(t), y(t))T be Q Q any positive solution of (1.2). Thus z ðtÞ ¼ ðz 1 ðtÞ; z 2 ðtÞ; z 3 ðtÞÞT ; ðz i ðtÞ ¼ 0 0, M1 and mi (defined by (3.4) and (3.8), respectively) such that for all t P T,

mi < z i ðtÞ 6 M i ;

mi < zi ðtÞ 6 Mi ; i ¼ 1; 2; 3:

ð3:11Þ

We define

V 1 ðtÞ ¼ j ln z 1 ðtÞ  ln z1 ðtÞj þ j ln z 2 ðtÞ  ln z2 ðtÞj:

ð3:12Þ

Calculating the upper-right derivative of V1(t) along the solution of (3.1), it follows for t P T that

Dþ V 1 ðtÞ ¼

2  0 X z ðtÞ i¼1

i z i ðtÞ



 z i ðtÞ sgnðz i ðtÞ  zi ðtÞÞ zi ðtÞ

   e 1 ðtÞ z2 ðtÞ  z2 ðtÞ þ sgnðz 2 ðtÞ ¼ sgnðz 1 ðtÞ  z1 ðtÞÞ A11 ðtÞðz 1 ðtÞ  z1 ðtÞÞ  A13 ðtÞðz 3 ðtÞ  z3 ðtÞÞþ D z1 ðtÞ z1 ðtÞ

  e 2 ðtÞ z1 ðtÞ  z1 ðtÞ  z2 ðtÞÞ  A22 ðtÞðz 2 ðtÞ  z2 ðtÞÞ  A23 ðtÞðz 3 ðtÞ  z3 ðtÞÞ þ D z 2 ðtÞ z2 ðtÞ 6 A11 ðtÞjz 1 ðtÞ  z1 ðtÞj þ ðA13 ðtÞ þ A23 ðtÞÞjz 3 ðtÞ  z3 ðtÞj  A22 ðtÞjz 2 ðtÞ  z2 ðtÞj þ D11 ðtÞ þ D22 ðtÞ;

ð3:13Þ

where

D11 ðtÞ ¼

 8 z2 ðtÞ z2 ðtÞ e > > < D 1 ðtÞ z 1 ðtÞ  z1 ðtÞ ; z1 ðtÞ > z1 ðtÞ;  > > :D e 1 ðtÞ z2 ðtÞ  z2 ðtÞ ; z ðtÞ < z1 ðtÞ: 1 z1 ðtÞ z ðtÞ 1

 8 e 2 ðtÞ z1 ðtÞ  z1 ðtÞ ; z ðtÞ > z2 ðtÞ; > D > 2 ðtÞ ðtÞ z z < 2 2

D22 ðtÞ ¼

 > > :D e 2 ðtÞ z1 ðtÞ  z1 ðtÞ ; z ðtÞ < z2 ðtÞ: 2 z ðtÞ z2 ðtÞ 2

We estimate D11(t) under the following two cases: (i) If z 1 ðtÞ P z1 ðtÞ, then

D11 ðtÞ 6

e 1 ðtÞ eM D D jz ðtÞ  z2 ðtÞj: ðz ðtÞ  z2 ðtÞÞ 6 z 1 ðtÞ 2 m1 2

(ii) If z 1 ðtÞ < z1 ðtÞ, then

D11 ðtÞ 6

e 1 ðtÞ eM D D ðz2 ðtÞ  z 2 ðtÞÞ 6 1 jz 2 ðtÞ  z2 ðtÞj: z1 ðtÞ m1

Combining the conclusions in (i)–(ii), we obtain

D11 ðtÞ 6

eM D 1 jz ðtÞ  z2 ðtÞj: m1 2

ð3:14Þ

A similar argument in the discussion above shows that

D22 ðtÞ 6

eM D 2 jz ðtÞ  z1 ðtÞj: m2 1

ð3:15Þ

It follows from (3.14) and (3.15) that

Dþ V 1 ðtÞ 6 A11 ðtÞjz 1 ðtÞ  z1 ðtÞj  A22 ðtÞjz 2 ðtÞ  z2 ðtÞj þ ðA13 þ A23 ðtÞÞðtÞjz 3 ðtÞ  z3 ðtÞj þ þ

eM D 2 jz ðtÞ  z1 ðtÞj: m2 1

eM D 1 jz ðtÞ  z2 ðtÞj m1 2 ð3:16Þ

R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

673

Define

V 21 ðtÞ ¼ j ln z 3 ðtÞ  ln z3 ðtÞj:

ð3:17Þ

Calculating the upper-right derivative of V21(t) along the solution of (3.1), we derive for t P T that

  z 03 ðtÞ z03 ðtÞ sgnðz 3 ðtÞ  z3 ðtÞÞ ¼ sgnðz 3 ðtÞ  z3 ðtÞÞ A33 ðtÞðz 3 ðt  s2 Þ  z3 ðt  s2 ÞÞþA31 ðtÞðz 1 ðt  s1 Þ  z3 ðtÞ z3 ðtÞ

 z1 ðt  s1 ÞÞ þ A32 ðtÞðz 2 ðt  s1 Þ  z2 ðt  s1 ÞÞ ¼ sgnðz 3 ðtÞ  z3 ðtÞÞ A33 ðtÞðz 3 ðtÞ  z3 ðtÞÞ þ A31 ðtÞðz 1 ðt  s1 Þ Z t z1 ðt  s1 ÞÞþA32 ðtÞðz 2 ðt  s1 Þ  z2 ðt  s1 ÞÞ þ A33 ðtÞ ðz 03 ðuÞ  z03 ðuÞÞdu ð3:18Þ

Dþ V 21 ðtÞ ¼



ts2

By substituting (3.1) into (3.18), we obtain  Dþ V 21 ðtÞ ¼ sgnðz 3 ðtÞ  z3 ðtÞÞ A33 ðtÞðz 3 ðtÞ  z3 ðtÞÞ þ A31 ðz 1 ðt  s1 Þ  z1 ðt  s1 ÞÞ þ A32 ðz 2 ðt  s1 Þ Z t ½ðr3 ðuÞ þ A31 ðuÞz1 ðu  s1 ÞþA32 ðuÞz2 ðu  s1 Þ  A33 ðuÞz3 ðu  s2 ÞÞðz 3 ðuÞ  z3 ðuÞÞ  z2 ðt  s1 ÞÞþA33 ðtÞ ts

2

þA31 ðuÞz 3 ðuÞðz 1 ðu  s1 Þ  z1 ðu  s1 ÞÞ þ A32 ðuÞz 3 ðuÞðz 2 ðu  s1 Þ  z2 ðu  s1 ÞÞA33 ðuÞz 3 ðuÞðz 3 ðu  s2 Þ  z3 ðu  s2 ÞÞdu :

ð3:19Þ

It follows from (3.7) and (3.19) that for t P T + s

Dþ V 21 ðtÞ 6 A33 jz 3 ðtÞ  z3 ðtÞj þ A31 ðtÞjz 1 ðt  s1 Þ  z1 ðt  s1 Þj þ A32 ðtÞjz 2 ðt  s1 Þ  z2 ðt  s1 Þj Z t ½ðr 3 ðuÞ þ A31 ðuÞM 1 þ A32 ðuÞM 2 þ A33 ðuÞM3 Þjz 3 ðuÞ  z3 ðuÞj þ A31 ðuÞM3 jz 1 ðu  s1 Þ þ A33 ðtÞ ts2

 z1 ðu  s1 Þj þ A32 ðuÞM 3 jz 2 ðu  s1 Þ  z2 ðu  s1 Þj þ A33 ðuÞM 3 jz 3 ðu  s2 Þ  z3 ðu  s2 Þjdu

ð3:20Þ

Define

V 22 ðtÞ ¼

Z

t

A31 ðs þ s1 Þjz 1 ðsÞ  z1 ðsÞjds þ

ts1

Z

tþs2

Z

Z

t

ts1

A32 ðs þ s1 Þjz 2 ðsÞ  z2 ðsÞjds

t

A33 fðr 3 ðuÞ þ A31 ðuÞM 1 þ A32 ðuÞM 2 þA33 ðuÞM 3 Þjz 3 ðuÞ  z3 ðuÞj þ A31 ðuÞM 3 jz 1 ðu  s1 Þ  z1 ðu  s1 Þj

ð3:21Þ þ A32 ðuÞM3 jz 2 ðu  s1 Þ  z2 ðu  s1 ÞjþA33 ðuÞM3 jz 3 ðu  s2 Þ  z3 ðu  s2 Þj du ds:

þ

ss2

t

It follows from (3.20) and (3.21) that for t P T + s

Dþ V 21 ðtÞ þ V 022 ðtÞ 6 A33 ðtÞjz 3 ðtÞ  z3 ðtÞj þ A31 ðt þ s1 Þjz 1 ðtÞ  z1 ðtÞj þ A32 ðt þ s1 Þjz 2 ðtÞ  z2 ðtÞj Z tþs2  þ A33 ðsÞds ðr 3 ðtÞ þ A31 ðtÞM1 þ A32 ðtÞM2 þ A33 ðtÞM 3 Þjz 3 ðtÞ  z3 ðtÞjþA31 ðtÞM 3 jz 1 ðt  s1 Þ  z1 ðt  s1 Þj t

þA32 ðtÞM 3 jz 2 ðt  s1 Þ  z2 ðt  s1 ÞjþA33 ðtÞM 3 jz 3 ðt  s2 Þ  z3 ðt  s2 Þj

ð3:22Þ

We now define

V 2 ðtÞ ¼ V 21 ðtÞ þ V 22 ðtÞ þ V 23 ðtÞ;

ð3:23Þ

in which

V 23 ðtÞ ¼ M 3

Z

Z

t

ts1

uþs1 þs2

A33 ðsÞA31 ðu þ s1 Þjz 1 ðuÞ  z1 ðuÞjds du þ M3

uþs1

Z

 z2 ðuÞjds du þ M3

t

ts2

Z

uþ2s2

uþs2

Z

t

Z

ts1

uþs1 þs2

uþs1

A33 ðsÞA32 ðu þ s1 Þjz 2 ðuÞ

A33 ðsÞA33 ðu þ s2 Þjz 3 ðuÞ  z3 ðuÞjds du:

ð3:24Þ

It follows from (3.22)–(3.24) that for t P T + s

Dþ V 2 ðtÞ 6 A33 ðtÞjz 3 ðtÞ  z3 ðtÞj þ A31 ðt þ s1 Þjz 1 ðtÞ  z1 ðtÞj þ A32 ðt þ s1 Þjz 2 ðtÞ  z2 ðtÞj þ

Z t

tþs2

 A33 ðsÞds ðr 3 ðtÞ þ A31 ðtÞM1 þ A32 ðtÞM2 þ A33 ðtÞM 3 Þjz 3 ðtÞ  z3 ðtÞj

þ A31 ðt þ s1 ÞM 3

Z

tþs1 þs2

tþs1

þ A33 ðt þ s2 ÞM3

Z

tþ2s2

tþs2

A33 ðsÞdsjz 1 ðtÞ  z1 ðtÞj þ A32 ðt þ s1 ÞM 3

A33 ðsÞdsjz 3 ðtÞ  z3 ðtÞj :

Z

tþs1 þs2

tþs1

A33 ðsÞdsjz 2 ðtÞ  z2 ðtÞj ð3:25Þ

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Now, define a Lyapunove functional V(t) as

VðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ:

ð3:26Þ

Then it follows from (3.16), (3.25) and (3.26) that for t P T + s

Dþ VðtÞ 6 

3 X

Ai ðtÞjz i ðtÞ  zi ðtÞj;

ð3:27Þ

i¼1

where Ai(t) (i = 1, 2, 3) are defined in (3.10). By hypothesis (C6), there exist constants ai > 0 (i = 1, 2, 3) and T* P T + s such that

Ai ðtÞ P ai > 0 for t P T :

ð3:28Þ

Integrating both sides of (3.27) on interval [T*,t]

VðtÞ þ

3 Z X i¼1

T

t

Ai ðsÞjz i ðsÞ  zi ðsÞjds 6 VðT Þ:

ð3:29Þ

It follows from (3.28) and (3.29) that 3 X

ai

i¼1

Z T

t

jz i ðsÞ  zi ðsÞjds 6 VðT Þ < 1; for t P T :

ð3:30Þ

* * 0 Since z 0 i ðtÞ and zi ðtÞ ði ¼ 1; 2; 3Þ are bounded for t P T , so jzi ðtÞ  zi ðtÞj ði ¼ 1; 2; 3Þ are uniformly continuous on [T , 1). By Barbalat’s lemma [17], we have

"

Y

lim jz i ðtÞ  zi ðtÞj ¼ lim

t!1

t!1

0
" lim jz ðtÞ t!1 3

 z3 ðtÞj ¼ lim

t!1

# ð1 þ bik Þ1 ðx i ðtÞ  xi ðtÞÞ ¼ 0;

Y

ði ¼ 1; 2Þ

# ð1 þ b3k Þ

1

ðx 3 ðtÞ

 x3 ðtÞÞ ¼ 0:

0
Hence

lim jx i ðtÞ  xi ðtÞj ¼ 0; ði ¼ 1; 2Þ;

t!1

lim jy ðtÞ  yðtÞj ¼ 0:

t!1

By Theorems 7.4 and 8.2 in [18], we know that the positive periodic solution x ðtÞ ¼ ðx 1 ðtÞ; x 2 ðtÞ; y ðtÞÞ of Eq. (1.2) is uniformly asymptotically stable. The proof of Theorem 3.1 is complete. h

4. An example In this section, we give an example to illustrate the feasibility of our main results. Example 4.1. We consider the following impulsive Lotka–Volterra predator–prey model with prey dispersal and time delays:

x01 ðtÞ ¼ x1 ðtÞ½8 þ sin t  5x1 ðtÞ  6yðtÞ þ 0:5½x2 ðtÞ  x1 ðtÞ;

t – tk ;

x02 ðtÞ

¼ x2 ðtÞ½2  sin t  5x2 ðtÞ  5yðtÞ þ 3½x1 ðtÞ  x2 ðtÞ; t – t k ;        1 1 1 þ x2 t   8y t  ; y ðtÞ ¼ yðtÞ 1 þ ð3 þ sin tÞx1 t  10 10 1000 0

t – tk ;

ð4:1Þ

Dxi ðt k Þ ¼ bik xi ðt k Þ; i ¼ 1; 2; k ¼ 1; 2; . . . ; Dyðt k Þ ¼ b3k yðt k Þ; k ¼ 1; 2; . . . ; We fix the parameters bik ¼  16 ; i ¼ 1; 2; b3k ¼  18 ; t kþ2 ¼ t k þ 2p; ½0; 2p \ ftk g ¼ ft 1 ; t 2 g. Obviously, ðr2  D2 Þ ¼ 1. Thus, the result in Xu et al. [4] cannot be applied to this case. However, it is easy to verified that our result is feasible. In fact, simple computation shows that

A ¼ max and

( ) M M ðr 1  D1 ÞM þ DM 9 1 ðr 2  D2 Þ þ D2 ¼ ; ; 5 aL11 aL22

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R. Liang, J. Shen / Applied Mathematics and Computation 217 (2010) 661–676

x1 x2 y

1.2 1 0.8 0.6 0.4 0.2 0

0

10

20

30

40

50

60

Fig. 1. The existence of positive 2p-periodic solution of system (4.1).

2 X

lnð1 þ bik Þ ¼ 2 ln 5  2 ln 6; i ¼ 1; 2;

k¼1

2 X

lnð1 þ b3k Þ ¼ 2 ln 7  2 ln 8:

k¼1

Therefore we have M M L aL33 ðr1  D1 Þx  aM 13 Aða31 þ a32 Þx þ a33 B1 ¼ 12p  16ðln 6  ln 5Þ > 0;

and L M aL31 aM 140p  30ðln 6  ln 5Þ M 22 ½ðr 1  D1 Þx þ B1  þ a32 a11 ½ðr 2  D2 Þx þ B2  ¼ > 25 ¼ aM 11 a22 : r 3 x  B3 2p þ 2 ln 8  2 ln 7

The above inequalities show that system (4.1) satisfies all the assumptions in Theorem 2.1. Thus, by Theorem 2.1, system (4.1) has at least one positive 2p-periodic solution.Numerical simulation shows that system (4.1) has at least one positive 2p-periodic solution (see Fig. 1). Therefore, in some sense, we generalize the results in [4]. 5. Conclusion In this paper, a model which describes the non-autonomous periodic predator–prey system with prey dispersal and impulse is proposed. By using coincidence degree theorem, a set of easily sufficient conditions are obtained for the existence of at least one positive periodic solution. We also present the uniqueness and global attractivity of positive periodic solution by means of Lyapunov functional method. From the view point of biology, the mathematical results are full of biological meanings and can be used to provide reliable foundation for making control strategy. The conditions of Theorem 2.1 show that human activities may save the extinct species living in a week environment (r2  D2 < 0, see Example 4.1). Hence, the results obtained imply impulsive perturbation is one of important control factors in determining the existence of periodic solutions of system (2.1). Numerical simulation is given to illustrate the feasibility of our main result. Acknowledgements The authors would like to thank the anonymous referees for their careful reading of the manuscript and excellent comments, which improved the presentation of this paper. Also, this work is supported by the National Science Foundation of China (10871062). References [1] D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific, Technical, New Yoke, 1993. [2] L.L. Wang, W.T. Li, Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator–prey model with Holling type functional response, J. Comput. Appl. Math. 162 (2004) 341–357.

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[3] S. Chen, J. Zhang, T. Yong, Existence of positive periodic solution for nonautonomous predator–prey system with diffusion and time delay, J. Comput. Appl. Math. 159 (2003) 375–386. [4] R. Xu, M.A.J. Chaplain, F.A. Davidson, Periodic solution for a delayed predator–prey model of prey dispersal in two-patch environments, Nonlinear Anal. RWA 5 (2004) 183–206. [5] J. Zhen, Zh. Ma, M. Han, The existence of periodic solutions of the n-species Lotka–Volterra competition systems with impulsive, Chaos, Solitons & Fractals 22 (2004) 181–188. [6] J. Zhang, L. Chen, X. Chen, Persistence and global stability for two-species nonautonomous competition Lotka–Volterra patch-system with time delay, Nonlinear Anal. TMA 37 (1999) 1019–1028. [7] R. Xu, L. Chen, Persistence and stability for a two-species ratio-dependent predator–prey system with time delay in a two-patch environment, Comput. Math. Appl. 40 (2000) 577–588. [8] X.Y. Song, L.S. Chen, Persistence and global stability for nonautonomous predator–prey systems with diffusion and time delay, Comput. Math. Appl. 35 (1998) 33–40. [9] M. Kouche, N. Tatar, Existence and global attractivity of a periodic solution to a nonautonomous dispersal system with delay, Appl. Math. Model. 31 (2007) 780–793. [10] S. Lenci, G. Rega, Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos, Solitons & Fractals 11 (2000) 2453–2472. [11] X. Liu, L. Chen, Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator, Chaos, Solitons & Fractals 16 (2003) 311–320. [12] W. Sun et al, On positive periodic solution of periodic competition Lotka–Valterra system with time delay and diffusion, Chaos, Solitons & Fractals 33 (2007) 971–978. [13] Z. Teng, Z. Lu, The effect of dispersal on single-species nonautonomous dispersal models with delays, J. Math. Biol. 42 (2001) 439–454. [14] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. [15] J. Yan, A. Zhao, oscillation and stability of linear impulsive delay differential equation, J. Math. Anal. Appl. 227 (1998) 187–194. [16] W. Wang, Z. Ma, Harmless delays for uniform persistence, J. Math. Anal. Appl. 158 (1991) 256–268. [17] L. Barbalat, Systemes d’equations differentielles d’oscillations nonlineaires, Rev. Roumaine Math. Pures Appl. 4 (1959) 267–270. [18] T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc. Japan, Tokyo, 1966. [19] J.G. Skellam, Random dispersal in theoretical populations, Biometrika 38 (1951) 16–218.