Periodicity and stability of a nonlinear periodic integro-differential prey-competition model with infinite delays

Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885 www.elsevier.com/locate/cnsns

Periodicity and stability of a nonlinear periodic integro-differential prey-competition model with infinite delays Fengde Chen a

a,* ,

Xiangdong Xie

b

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350002, China b Department of Mathematics, Ningde Teachers College, Ningde 352100, China

Received 15 September 2004; received in revised form 12 November 2005; accepted 17 November 2005 Available online 18 January 2006

Abstract A nonlinear integro-differential prey-competition model with infinite delays is proposed. By using the method of coincidence degree and constructing a suitable Lyapunov functional, some sufficient conditions are obtained for the existence of an unique globally attractive strictly positive (componentwise) periodic solution.  2005 Elsevier B.V. All rights reserved. PACS: 02.30.Hq Keywords: Periodic solution; Prey-competition; Lyapunov functional; Globally asymptotic stable; Integro-differential; Infinite delay

1. Introduction Recently, Fan and Wang [1] and Chen [2] studied the following integro-differential competition system ( ) Z t n n X X y_ i ðtÞ ¼ y i ðtÞ ri ðtÞ  aij ðtÞy j ðtÞ  bij ðtÞ K ij ðt  uÞy j ðuÞ du ; i ¼ 1; 2; . . . ; n; ð1:1Þ j¼1

j¼1

1

Rx where aij, bij, ri, j = 1, 2, . . . , n, are continuous x-periodic functions with aij(t) P 0, bij(t) P 0, 0 ri ðsÞ ds > 0, and the delay R þ1kernels Kij : [0, +1) R þ1 ! [0, +1), i, j = 1, 2, . . . , n, are measurable x-periodic normalized functions such that 0 K ij ðsÞ ds ¼ 1; 0 sK ij ðsÞ ds < þ1. In mathematical ecology, (1.1) denotes a model of the dynamics of an n-species system in which each individual competes with all others of the system for common resources and the intra-species and inter-species competition involve a time delay extending over the entire past as denoted by Kij in (1.1). In [1,2], by using the method of coincidence degree and constructing a suitable

*

Corresponding author. E-mail addresses: [email protected] (F. Chen), [email protected] (X. Xie).

1007-5704/$ - see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.11.004

F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885

877

Lyapunov functional, some excellent results which guarantee the existence of globally attractive strictly positive (componentwise) periodic solution of the system are obtained. For more works on system (1.1), one could refer to [1–4] and the references therein. Already, as was pointed out by Berryman [5], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Li and Lu [6] introduced the following more complicated preycompetition model ! n X aij x_ i ðtÞ ¼ xi ðtÞ bi ðtÞ  aij ðtÞxj ðtÞ ; 1 6 i 6 m; j¼1

x_ i ðtÞ ¼ xi ðtÞ bi ðtÞ þ

m X j¼1

a aij ðtÞxj ij ðtÞ



n X

! a aij ðtÞxj ij ðtÞ

ð1:2Þ ;

m þ 1 6 i 6 n;

j¼mþ1

where xi (1 6 i 6 m) are the density the prey species, and xi (m + 1 6 i 6 n) are the density of predator species, aij > 0 (i, j = 1, 2, . . . , n). In [6], they gave sufficient conditions for the existence of a unique globally attractive periodic solution of system (1.2). Recently, for general nonautonomous case of system (1.2), Zhao and Chen [7] obtained a set of sufficient conditions which guarantee the permanence and the global attractivity of the system (1.2). For more works on this direction, one could refer to [6–13] and the reference cited therein. For single species or multispecies Lotka–Volterra type competition model, concerned with the existence of positive periodic solution of the system, many excellent results have been obtained (see [14–19] and the references therein). It is of interesting to consider the more complicated prey-competition case. However, to the best of the authors’ knowledge, seldom did scholars consider the multispecies prey-competition system with delays, in particular, the nonlinear delayed competition-prey system. Stimulated by the works of [1–4,6–19], we introduce the following more complicated competition-prey model in this paper:   8 n n Rt P P bik aik > > x_ i ðtÞ ¼ xi ðtÞ ri ðtÞ  aik ðtÞxk ðtÞ  bik ðtÞ 1 K ij ðt  uÞxk ðuÞ du ; > > > k¼1 k¼1 > > > > i ¼ 1; 2; . . . ; m; > >  > < m m Rt P P b x_ i ðtÞ ¼ xi ðtÞ ri ðtÞ þ aik ðtÞxak ik ðtÞ þ bik ðtÞ 1 K ij ðt  uÞxk ik ðuÞ du ð1:3Þ > k¼1 k¼1 > >  > n n > Rt P P > bik aik > >  a ðtÞx ðtÞ  b ðtÞ K ðt  uÞx ðuÞ du ; ik ik ij > k k 1 > > k¼mþ1 k¼mþ1 > : i ¼ m þ 1; . . . ; n; where xi(t) denote the density of prey (i = 1, 2, . . . , m)/predator (i = m + 1, . . . , n) species Xi at time t; aik, bik, i, k = 1, . . . , n, are positive constants; ri(t), aik(t), bik(t), (i, k = 1, . . . , n) are continuous periodic functions defined on [0, +1) with a common periodic x > 0; ri(t) (i = 1, . . . , m), aii(t) (i = 1, 2, . . . , n) are strictly positive. ij : [0, +1) ! [0, +1), RKþ1 R þ1 i, j = 1, 2, . . . , n, are measurable x-periodic normalized functions such that K ij ðsÞ ds ¼ 1, 0 sK ij ðsÞ ds < þ1, which denotes the delay kernels. In addition to the predator-prey rela0 tionship, the competition among prey species and the competition among predator species are simultaneously considered. We consider (1.3) together with the following initial conditions: xi ðhÞ ¼ /i ðhÞ P 0;

h 2 ð1; 0; /i ð0Þ > 0;

sup /i ðhÞ < þ1;

ð1:4Þ

h2ð1;0

where /i are continuous on (1, 0], respectively. The aim of this paper is, by further developing the analysis technique of [1,2,7,14], to give sufficient conditions which guarantee the existence of unique global attractive positive periodic solution of system (1.3). This paper is organized as follows. In Section 2, we study the existence of at least one positive periodic solution with strictly positive components of (1.3)–(1.4) by using the continuation theorem of coincidence degree theory proposed in [20]. In Section 3, by constructing a suitable Lyapunov functional, sufficient conditions on the existence of unique globally attractive positive periodic solution of system (1.3) are obtained.

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F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885

Throughout this paper, we shall use the following notations: • We always use i = 1, 2, . . . , n; j = 1, . . . , m unless otherwise stated. • Let f(t) be a nonnegative continuous x-periodic function defined on (1, +1), we set Z 1 x l u  f ¼ min f ðtÞ; f ¼ max f ðtÞ; f ¼ f ðtÞ dt. t2½0;x t2½0;x x 0 • (Æ)n·m denotes n · m matrix.

2. Existence of positive periodic solution In order to obtain the existence of positive periodic solutions of (1.3), for the reader’s convenience, we shall summarize in the following a few concepts and results from [20] that will be basic for this paper. Let X, Z be normalized vector spaces, L : Dom L  X ! Z be a linear mapping, N : X ! Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L = Codim Im L < +1 and Im L is closed in Z. If L is a Fredholm mapping of index zero there exist continuous projectors P : X ! X and Q : Z ! Z such that Im P = Ker L, Im L = Ker Q = Im (I  Q). It follows that LjDom L \ Ker P : (I  P)X ! Im L is invertible. We denote the inverse of that map by KP. Let X be an open bounded subset of X. The mapping N will be called L-compact on X if QN ðXÞ is bounded and K P ðI  QÞN : X ! X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q ! Ker L. In the proof of our existence theorem below, we will use the continuation theorem of Gaines and Mawhin([20, p40]). Lemma 2.1 (Continuation Theorem). Let L be a Fredholm mapping of index zero and let N be L-compact on X. Suppose (a) For each k 2 (0, 1), every solution x of Lx = kNx is such that x 62 oX; (b) QNx 5 0 for each x 2 oX \ Ker L and

degfJQN ; X \ Ker L; 0g 6¼ 0.  Then the equation Lx = Nx has at least one solution lying in DomL \ X. Lemma 2.2. The domain Rnþ ¼ fðx1 ; . . . ; xn Þjxi > 0; i ¼ 1; 2; . . . ; ng is invariant with respect to (1.3). Proof. Since xi ðtÞ ¼ xi ð0Þ exp

(Z

t

ri ðsÞ 

0

xi ðtÞ ¼ xi ð0Þ exp

(Z



k¼mþ1

bik ðsÞ

aik ðsÞxak ik ðsÞ



k¼1 t

ri ðsÞ þ

0 n X

n X

m X

bik ðsÞ

aik ðsÞxak ik ðsÞ þ

m X

Z

)

s

K ij ðs 

b uÞxk ik ðuÞ du

;

i ¼ 1; 2; . . . ; m;

1

k¼1

k¼1

Z

n X

bik ðsÞ

k¼1

Z

s b

K ij ðs  uÞxk ik ðuÞ du 

1

n X

aik ðsÞxak ik ðsÞ

k¼mþ1

)

s b

K ij ðs  uÞxk ik ðuÞ du ;

i ¼ m þ 1; . . . ; n;

1

the assertion of the Lemma follows immediately for all t 2 [0, +1).

h

Theorem 2.1. Assume that the system of algebraic equations gðuÞ ¼ ðgi ðuÞÞn1 ¼ 0;

ð2:1Þ

F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885

879

where def

n X

def

k¼1 m X

gi ðuÞ ¼ ri  gi ðuÞ ¼ ri 

aik uak ik  aik uak ik 

k¼1

n X k¼1 m X

b

bik uk ik ; b

bik uk ik þ

k¼1

i ¼ 1; 2; . . . ; m; n X k¼mþ1

aik uak ik þ

n X

T

has finite solutions u ¼ ðu1 ; . . . ; un Þ 2 Rnþ with ui > 0 and ðiÞ

rli >

n X

auik eaik H k þ

k¼1;k6¼i

n X

buik ebik H k ;

P

u

sgn J g ðu Þ 6¼ 0. In addition, if

i ¼ 1; 2; . . . ; m;

k¼1

m X ðalik eaik Lk þ blik ebik Lk Þ > rui þ

ðiiÞ

b

bik uk ik ; i ¼ m þ 1; . . . ; n;

k¼mþ1

n X k¼mþ1;k6¼i

k¼1

auik eaik H k þ

n X

buik ebik H k ;

i ¼ m þ 1; . . . ; n;

k¼mþ1

where def

Hi ¼

1 ru ln il ; aii aii

i ¼ 1; 2; . . . ; m;

Pm 1 rl þ k¼1 ðauik eaik H k þ buik ebik H k Þ ln i ; i ¼ m þ 1; . . . ; n; aii alii Pn Pn rli  k¼1;k6¼i auik eaik H k  k¼1 buik ebik H k def 1 ln ; i ¼ 1; 2; . . . ; m. Li ¼ aii alii def

Hi ¼

Then (1.3)–(1.4) has at least one x-periodic solution with strictly positive components, say, x ðtÞ ¼ ðx1 ðtÞ; . . . ; T xn ðtÞÞ , and there exist positive constants ki, li such that ki 6 xi ðtÞ 6 li . Proof. Since the solution of Eqs. (1.3)–(1.4) remain positive for t P 0, we can let xi ðtÞ ¼ ey i ðtÞ .

ð2:2Þ

System (1.3) is then transformed into 8 n n Rt P P > > aik ðtÞeaik y k ðtÞ  bik ðtÞ 1 K ij ðt  uÞebik y k ðuÞ du; > y_ i ðtÞ ¼ ri ðtÞ  > > k¼1 k¼1 > > > > > i ¼ 1; 2; . . . ; m; > > > < m m Rt P P y_ i ðtÞ ¼ ri ðtÞ þ aik ðtÞeaik y k ðtÞ þ bik ðtÞ 1 K ij ðt  uÞebik y k ðuÞ du > k¼1 k¼1 > > > > n n Rt P P > > > aik ðtÞeaik y k ðtÞ  bik ðtÞ 1 K ij ðt  uÞebik y k ðuÞ du;  > > > k¼mþ1 k¼mþ1 > > : i ¼ m þ 1; . . . ; n.

ð2:3Þ

Take X ¼ Z ¼ fyðtÞ ¼ ðy 1 ðtÞ; . . . ; y n ðtÞÞ 2 CðR; Rn Þjyðt þ xÞ ¼ yðtÞg and define kyk ¼

n  X i¼1

2 !1=2 max jy i ðtÞj ;

t2½0;x

for any y 2 X ðor ZÞ.

Then X and Z are both Banach spaces when they are endowed with the norm kÆk. For any y 2 X and i = 1, 2, . . . , n. Let Ki,y(t) : R ! R be defined by

880

F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885 n X

Ki;y ¼ ri ðtÞ 

aik ðtÞe

aik y k ðtÞ

n X



k¼1

Z

bik ðtÞ

t

K ij ðt  uÞebik y k ðuÞ du;

1

k¼1

i ¼ 1; 2; . . . ; m. Ki;y ¼ ri ðtÞ þ

m X

aik ðtÞeaik y k ðtÞ þ

k¼1 n X



m X

bik ðtÞ

Z

aik ðtÞeaik y k ðtÞ 

k¼mþ1

K ij ðt  uÞebik y k ðuÞ du 1

k¼1 n X

t

bik ðtÞ

Z

t

K ij ðt  uÞebik y k ðuÞ du;

1

k¼mþ1

i ¼ m þ 1; . . . ; n. Then Ki,y 2 C(R, R). Moreover, because of the periodicity of x(t) we can easily check that Ki,y is x-periodic. Let  T Ny ¼ K1;y ; K2;y ; . . . ; Kn;y ; x 2 X ; Z Z dyðtÞ 1 x 1 x ; Py ¼ yðtÞ dt; y 2 X ; Qz ¼ zðtÞ dt; z 2 Z. Ly ¼ y_ ¼ dt x 0 x 0 Then it follows that



Ker L ¼ fy 2 X : y ¼ h 2 Rn g;

Im L ¼

z2Z:

Z



x

zðtÞ dt ¼ 0

is closed in Z;

0

dim Ker L ¼ n ¼ Codim Im L < þ1; and P, Q are continuous projectors such that Im P ¼ Ker L;

Ker Q ¼ Im L ¼ ImðI  QÞ.

Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) KP : Im L ! Ker P \ Dom L reads Z

K P ðzÞ ¼

t

zðsÞ ds 

0

1 x

Z

x

Z

t

zðsÞ ds dt. 0

0

Thus  QNy ¼

1 x

Z

x

K1;y ðsÞ ds; . . . ;

0

Z

1 x

T

x

Kn;y ðsÞ ds

;

0

T

K P ðI  QÞNy ¼ ðn1 ; . . . nn Þ ; where ni ¼

1 x

Z

x

Ki;y ðsÞ ds 

0

1 x

Z 0

x

Z



t

Ki;y ðsÞ ds dt 

0

t 1  x 2

Z

x

Ki;y ðsÞ ds. 0

 is compact for Obviously, QN and KP(I  Q)N are continuous. It is not difficult to show that K P ðI  QÞN ðXÞ  is clearly bounded. Thus, N any open bounded set X  X by using Arzela-Ascoli theorem. Moreover, QN ðXÞ  with any open bounded set X  X. The isomorphism J of Im Q onto Ker L can be the idenis L-compact on X tity mapping, since Im Q = Ker L. Now we reach the position to search for an appropriate open bounded subset X for the application of the continuation theorem (Lemma 2.1). Corresponding to the operator equation Lx = kNx, k 2 (0, 1), we have " # Z t n n X X aik y k ðtÞ bik y k ðuÞ y_ i ðtÞ ¼ k ri ðtÞ  aik ðtÞe  bik ðtÞ K ij ðt  uÞe du ; k¼1

i ¼ 1; 2; . . . ; m;

k¼1

1

F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885

" y_ i ðtÞ ¼ k ri ðtÞ þ

m X

aik ðtÞeaik y k ðtÞ þ

k¼1



n X

m X

Z

bik ðtÞ

aik ðtÞeaik y k ðtÞ 

k¼mþ1

t

K ij ðt  uÞebik y k ðuÞ du

1

k¼1 n X

bik ðtÞ

Z

881

#

t

K ij ðt  uÞebik y k ðuÞ du ;

1

k¼mþ1

i ¼ m þ 1; . . . ; n.

ð2:4Þ

Assume that y = y(t) 2 X is a solution of (2.4) for some k 2 (0, 1). Since y 2 X, there exist hi 2 [0, x] such that y i ðhi Þ ¼ max y i ðtÞ; t2½0;x

i ¼ 1; 2; . . . ; n.

Then, for i = 1, 2, . . . , n, "

n X

0 ¼ y_ i ðhi Þ ¼ k ri ðhi Þ 

aik ðhi Þe

ð2:5Þ

aik y k ðhi Þ



k¼1

n X

bik ðhi Þ

Z

K ij ðhi  uÞe

i ¼ 1; 2; . . . ; m; " Z m m X X aik y k ðhi Þ aik ðhi Þe þ bik ðhi Þ 0 ¼ y_ i ðhi Þ ¼ k ri ðhi Þ þ 

n X

aik ðhi Þe

n X



k¼mþ1

du ;

bik ðhi Þ

hi

K ij ðhi  uÞebik y k ðuÞ du

1

k¼1 aik y k ðhi Þ

bik y k ðuÞ

1

k¼1

k¼1

#

hi

Z

ð2:6Þ

#

hi

K ij ðhi  uÞe

bik y k ðuÞ

du ;

1

k¼mþ1

i ¼ m þ 1; . . . ; n. The first equation of (2.6) implies that y i ðhi Þ 6

1 ru ln il :¼ H i ; aii aii

i ¼ 1; 2; . . . ; m.

ð2:7Þ

The second equation of (2.6) together with (2.7) imply that m P rli þ ðauik eaik H k þ buik ebik H k Þ 1 k¼1 y i ðhi Þ 6 ln :¼ H i ; i ¼ m þ 1; . . . ; n. aii alii

ð2:8Þ

Again, since y 2 X, there exist fi 2 [0, x] such that y i ðfi Þ ¼ min y i ðtÞ; i ¼ 1; 2; . . . ; n.

ð2:9Þ

t2½0;x

Then, for i = 1, 2, . . . , n, " 0 ¼ y_ i ðfi Þ ¼ k ri ðfi Þ 

n X

aik ðfi Þe

aik y k ðfi Þ

k¼1



n X

bik ðfi Þ

Z

K ij ðfi  uÞe

i ¼ 1; 2; . . . ; m; " Z m m X X 0 ¼ y_ i ðfi Þ ¼ k ri ðfi Þ þ aik ðfi Þeaik y k ðfi Þ þ bik ðfi Þ 

n X

aik ðfi Þe

k¼mþ1



n X k¼mþ1

bik ðfi Þ

fi

du ;

K ij ðfi  uÞebik y k ðuÞ du

1

k¼1 aik y k ðfi Þ

bik y k ðuÞ

1

k¼1

k¼1

#

fi

Z

fi

K ij ðfi  uÞe

ð2:10Þ

# bik y k ðuÞ

du ;

1

i ¼ m þ 1; . . . ; n. The first equation of (2.10) together with (2.7) and (2.8) imply that Pn Pn rli  k¼1;k6¼i auik eaik H k  k¼1 buik ebik H k 1 ln :¼ Li ; i ¼ 1; 2; . . . ; m. y i ðfi Þ P aii alii

ð2:11Þ

882

F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885

The second equation of (2.10) together with (2.7), (2.8) and (2.11) imply that y i ðfi Þ P

1 i ln :¼ Li ; aii alii

i ¼ m þ 1; . . . ; n;

ð2:12Þ

where  i ¼ rui þ

m X ðalik eaik Lk þ blik ebik Lk Þ 

n X

auik eaik H k 

k¼mþ1;k6¼i

k¼1

n X

buik ebik H k .

k¼mþ1

It follows from (2.7), (2.8) and (2.11), (2.12) that, for i = 1, 2, . . . , n, max jxi ðtÞj 6 max fjLi j; jH i jg :¼ K i .

ð2:13Þ

t2½0;x

Pn 1=2 Obviously, Ki’s are independent of k. Set H ¼ C þ ð i¼1 K 2i Þ ; where C is a positive number sufficiently large T  such that the solutions of (2.1) satisfies kðlnfu1 g; . . . ; lnfun gÞ k < C; then kxk < H. Let X :¼ {x = T (x1, . . . , xn) 2 Xkxk < H}. It is clear that X verifies requirement (a) in Lemma 2.1. Also we know from (2.4) and (2.13) that Lx 5 kNx for x 2 oX and k 2 (0, 1), that is X verifies the requirement (a) of Lemma 2.1. When x 2 oX \ Ker L = oX \ Rn, x is a constant vector in Rn with kxk = H, then QNx ¼ ðGi Þn1 6¼ 0; where Gi ¼ ri 

n X

aik eaik y k 

k¼1 m X

Gi ¼ ri þ

k¼1

n X

bik ebik y k ;

k¼1 m X

aik eaik y k þ

i ¼ 1; 2; . . . ; m;

bik ebik y k 

k¼1

n X k¼mþ1

aik eaik y k 

n X

bik ebik y k ;

k¼mþ1

i ¼ m þ 1; . . . ; n. Furthermore, in view of the assumption in Theorem 2.1, it is easy to see that degfJQN ; X \ Ker L; 0g 6¼ 0; where the isomorphism J of Im Q onto Ker L can be the identity mapping, since Im Q = Ker L. By now we have proved that X verifies all the requirements in Lemma 2.1. Hence, Eq. (2.3) has at least one x-periodic solution y*(t) in X. Set xi ðtÞ ¼ expfy i ðtÞg, then by the medium of (2.2) we know that x ðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞT is a positive x-periodic solution of Eq. (1.3). The boundedness of x*(t) implies the existence of positive constant ki, li in Theorem 2.1. The proof of Theorem 2.1 is completed. h 3. Global attractivity From Theorem 2.1 we know that system (1.3) with initial condition (1.4) has at least one positive periodic T solution x ðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞ and there exist positive constants ki, li such that ki 6 xi ðtÞ 6 li ; i ¼ 1; 2; . . . ; n. Now let c be positive constant such that 0 < c 6 mini{ki} and making the change of variable yi(t) = xi(t)/c, i = 1, 2, . . . , n. Then system (1.3) is transformed into   8 n n Rt P P bik aik > aik bik > > > y_ i ðtÞ ¼ y i ri ðtÞ  k¼1 c aik ðtÞy k ðtÞ  k¼1 c bik ðtÞ 1 K ij ðt  uÞy k ðuÞ du ; > > > > > i ¼ 1; 2; . . . ; m; > >  > < m m Rt P P b caik aik ðtÞy ak ik ðtÞ þ cbik bik ðtÞ 1 K ij ðt  uÞy k ik ðuÞ du y_ i ðtÞ ¼ y i ri ðtÞ þ ð3:1Þ > k¼1 k¼1 > >  > n n > Rt P P > b > >  caik aik ðtÞy ak ik ðtÞ  cbik bik ðtÞ 1 K ij ðt  uÞy k ik ðuÞ du ; > > > k¼mþ1 k¼mþ1 > : i ¼ m þ 1; . . . ; n.

F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885 T

883

T

Obviously, y  ðtÞ ¼ ðy 1 ðtÞ; . . . ; y n ðtÞÞ ¼ ðx1 ðtÞ=c; . . . ; xn ðtÞ=cÞ is the periodic solution of system (3.1). And if the periodic solution of system (3.1) is globally attractive, so is the periodic solution of system (1.3). Theorem 3.1. In addition to the assumption of Theorem 2.1, assume further that ðiiiÞ aii P maxfaki ; bki g;

1 6 k; i 6 n;

k

ðivÞ

n X

aii

qi c aii ðtÞ >

aki

qk c aki ðtÞ þ

k¼1;k6¼i

n X

qk c

bki

Z

þ1

K ki ðsÞbki ðt þ sÞ ds;

i ¼ 1; 2; . . . ; n;

0

k¼1

where qi, i = 1, 2, . . . , n are positive constants. Then system (1.3)–(1.4) has a unique positive periodic solution x ðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞT which is globally asymptotic stable. Proof. From condition (iv) and the periodicity of coefficients of system (3.1) there exists a positive constant A such that Z þ1 n n X X aii aki bki qk c aki ðtÞ  qk c K ki ðsÞbki ðt þ sÞ ds > A; i ¼ 1; 2; . . . ; n. qi c aii ðtÞ  k¼1;k6¼i

0

k¼1

Let y(t) = (y1(t), . . . , yn(t))T be any positive solution of system (3.1) and y  ðtÞ ¼ ðy 1 ðtÞ; . . . ; y n ðtÞÞT be the positive periodic solution of system (3.1) (the existence of such a positive periodic solution is guarantee by Theorem 2.1). Now constructing a Lyapunov functional V(t) as follow V ðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ;

ð3:2Þ

where V 1 ðtÞ ¼

n X

qi j ln y i ðtÞ  ln y i ðtÞj;

ð3:3Þ

t P 0;

i¼1

and V 2 ðtÞ ¼

n X n X i¼1

qi c

bik

Z

þ1

K ik ðsÞ

Z

0

k¼1

t

bik ðs þ uÞjðy k ðuÞÞ

bik



ðy k ðuÞÞbik j du

þ

i¼1

qi c

k¼1

ds;

t P 0.

ð3:4Þ

ts

From definition of V(t) one has Xn q j ln y i ð0Þ  ln y i ð0Þj V ð0Þ 6 i¼1 i n n X X



bik

bik

max bik ðtÞ sup jðUi ðtÞÞ

t2½0;x



b ðy i ðtÞÞ ik j

t2ð1;0

Z

!

þ1

sK ij ðsÞ ds

0

< þ1

ð3:5Þ

and V ðtÞ P V 1 ðtÞ ¼

n X

qi j ln y i ðtÞ  ln y i ðtÞj;

ð3:6Þ

t P 0.

i¼1

By direct computation, we have n n n X X X a a a a qi caii aii ðtÞjðy i ðtÞÞ ii  ðy i ðtÞÞ ii j þ qi caik aik ðtÞjðy k ðtÞÞ ik  ðy k ðtÞÞ ik j Dþ V ðtÞ 6  i¼1 k¼1;k6¼i

i¼1

þ

n X n X i¼1

þ

k¼1

t

1

k¼1

n X n X i¼1

qi cbik bik ðtÞ

Z

qi cbik

Z 0

K ik ðt  uÞjðy k ðuÞÞbik  ðy k ðuÞÞbik j du

þ1

K ik ðsÞbik ðt þ sÞ dsjðy k ðtÞÞ

bik

b

 ðy k ðtÞÞ ik j

884

F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885

Z

þ1

K ik ðsÞbik ðtÞjðy k ðt  sÞÞbik  ðy k ðt  sÞÞbik j ds 0 i¼1 k¼1 n n n X X X  qi caii aii ðtÞjðy i ðtÞÞaii  ðy i ðtÞÞaii j þ qk caki aki ðtÞjðy i ðtÞÞaki i¼1 i¼1 k¼1;k6¼i

 6

n X n X

þ

n X n X i¼1

qi c

bik

qk c

bki

Z

þ1

bki

 ðy i ðtÞÞaki j

b

 ðy i ðtÞÞ ki j.

K ki ðsÞbki ðt þ sÞ dsjðy i ðtÞÞ 0

k¼1

¼ ¼ 1; 2; . . . ; n we know y*(t) P 1. Observe that y = jax  bxj is an increasing function for From a P 1, and x > 0. For aii P maxk{aki, bki}, we get y i ðtÞ

xi ðtÞ=c; i

jðy i ðtÞÞ

aki

 ðy i ðtÞÞ ki j 6 jðy i ðtÞÞ ii  ðy i ðtÞÞ ii j;

a

a

a

jðy i ðtÞÞ

bki

 ðy i ðtÞÞ ki j 6 jðy i ðtÞÞ ii  ðy i ðtÞÞ ii j;

b

a

a

i; k ¼ 1; 2; . . . ; n.

Therefore þ

D V ðtÞ 6 

n X

aii

qi c aii ðtÞ 

aki

qk c aki ðtÞ

k¼1;k6¼i

i¼1

6 A

n X

n X

a

n X

qk c

k¼1

bki

Z

!

þ1

a

a

K ki ðsÞbki ðt þ sÞ ds jðy i ðtÞÞ ii  ðy i ðtÞÞ ii j

0

a

jðy i ðtÞÞ ii  ðy i ðtÞÞ ii j.

i¼1

Integrating both sides of above inequality with respect to t, we have Z tX n a a jðy i ðsÞÞ ii  ðy i ðsÞÞ ii j ds 6 V ð0Þ < þ1; t P 0. V ðtÞ þ A 0

ð3:7Þ

i¼1

(3.7) shows that 0 6 V ðtÞ 6 V ð0Þ;

ð3:8Þ

for all t P 0;

and Z

t

0

n X

jðy i ðsÞÞaii  ðy i ðsÞÞaii j ds 6

i¼1

V ð0Þ < þ1; t P 0. A

(3.9) implies that n X a a jðy i ðtÞÞ ii  ðy i ðtÞÞ ii j 2 L1 ½0; þ1Þ.

ð3:9Þ

ð3:10Þ

i¼1

Obviously y i ðtÞ; i ¼ 1; 2; . . . ; n are uniformly bounded from below and above, and so ln y i ðtÞ is bounded. From (3.6), one has j ln y i ðtÞ  ln y i ðtÞj 6

1 1 V ðtÞ 6 V ð0Þ; qi qi

which is equivalent to y i ðtÞ exp



V ð0Þ qi

6 y i ðtÞ 6

y i ðtÞ exp



V ð0Þ . qi

ð3:11Þ

(3.11) show that yi(t), i = 1, 2, . . . , n are uniformly bounded. ThisPfact together with (3.1) lead to n a a y_ i ðtÞ; y_ i ðtÞ; i ¼ 1; 2; . . . ; n are uniformly bounded on [0, +1). Therefore i¼1 jðy i ðtÞÞ ii  ðy i ðtÞÞ ii j is uniformly Pn a a continuous on [0, +1). From (3.10) we know that i¼1 jðy i ðtÞÞ ii  ðy i ðtÞÞ ii j is integrable on [0, +1). By Barbalat’s Lemma (Lemma 1.2.2 and Lemma 1.2.3, Gopalsamy [21]), we can conclude that a

a

lim jðy i ðtÞÞ ii  ðy i ðtÞÞ ii j ¼ 0.

t!þ1

F. Chen, X. Xie / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 876–885

885

Therefore, lim jy i ðtÞ  y i ðtÞj ¼ 0.

t!þ1

The proof of Theorem 3.1 is completed.

h

Acknowledgements This work was supported by the National Natural Science Foundation of China (10501007), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002) and the Foundation of Fujian Education Bureau (JA04156). References [1] Fan M, Wang K. Positive periodic solutions of a periodic integro-differential competition system with infinite delays. ZAMM Z Angew Math Mech 2001;81(3):197–203. [2] Chen Y. New results on positive periodic solutions of a periodic integro-differential competition system. Appl Math Comput 2004;153(2):557–65. [3] Teng ZD. Permanence and stability of Lotka–Volterra type N-species competitive systems. Acta Math Sinica 2002;45(5):905–18 [in Chinese]. [4] Murakami S. Almost periodic solutions of a system of integro-differential equations. Tohoku Math J 1987;39:71–9. [5] Berryman AA. The origins and evolution of predator-prey theory. Ecology 1982;75:1530–5. [6] Li CR, Lu SJ. The qualitative analysis of N-species periodic coefficient, nonlinear relation, prey-competition systems. Appl Math-JCU 1997;12(2):147–56 [in Chinese]. [7] Zhao JD, Chen WC. The qualitative analysis of N-species nonlinear prey-competition systems. Appl Math Comput 2004;149(2):567–76. [8] Fan M, Wang K. Global periodic solutions of a generalized N-species Gilpin-Ayala competition model. Comput Math Appl 2000;40:1141–51. [9] Yang P, Xu R. Global attractivity of the periodic Lotka–Volterra system. J Math Anal Appl 1999;233(1):221–32. [10] Chen FD. On a nonlinear non-autonomous predator–prey model with diffusion and distributed delay. J Comput Appl Math 2005;180(1):33–49. [11] Zhao JD, Chen WC. Global asymptotic stability of a periodic ecological model. Appl Math Comput 2004;147(3):881–92. [12] Huo HF, Li WT. Periodic solutions of a periodic Lotka–Volterra system with delay. Appl Math Comput 2004;156(3):787–803. [13] Yan J, Feng Q. Global existence and oscillation in a nonlinear delay equation. Nonlinear Anal 2001;43:101–8. [14] Chen FD, Shi JL. Periodicity in a Logistic type system with several delays. Comput Math Appl 2004;48(1–2):35–44. [15] Chen FD et al. Periodicity in a food-limited population model with toxicants and state dependent delays. J Math Anal Appl 2003;288(1):132–42. [16] Chen FD, Lin FX, Chen XX. Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control. Appl Math Comput 2004;158(1):45–68. [17] Chen FD. Positive periodic solutions of neutral Lotka–Volterra system with feedback control. Appl Math Comput 2005;162(3):1279–302. [18] Li YK. Periodic solutions for delay Lotka–Volterra competition systems. J Math Anal Appl 2000;246(1):230–44. [19] Yang ZH, Cao JD. Positive periodic solutions of neutral Lotka–Volterra system with periodic delays. Appl Math Comput 2004;149(3):661–87. [20] Gaines RE, Mawhin JL. Coincidence degree and nonlinear differential equations. Berlin: Springer-Verlag; 1977. [21] Gopalsamy K. Stability and oscillations in delay differential equations of population dynamics. Dordrecht/Norwell, MA: Kluwer Academic Press; 1992.