Periodic solutions for a predator–prey model with Holling-type functional response and time delays

Periodic solutions for a predator–prey model with Holling-type functional response and time delays

Applied Mathematics and Computation 161 (2005) 637–654 www.elsevier.com/locate/amc Periodic solutions for a predator–prey model with Holling-type fun...

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Applied Mathematics and Computation 161 (2005) 637–654 www.elsevier.com/locate/amc

Periodic solutions for a predator–prey model with Holling-type functional response and time delays Rui Xu *, M.A.J. Chaplain, F.A. Davidson Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK

Abstract A delayed periodic Holling-type predator–prey model without instantaneous negative feedback is investigated. By using the continuation theorem of coincidence degree theory and by constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for the existence, uniqueness and global stability of positive periodic solutions to the model. Numerical simulation is carried out to illustrate the feasibility of our main results. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Predator–prey system; Time delay; Periodic solution; Lyapunov functional; Global stability

1. Introduction In [1], Bazykin proposed the following predator–prey system: 

 bv u_ ¼ u a  eu  ; 1 þ au   du  gv ; v_ ¼ v  c þ 1 þ au

ð1:1Þ

* Corresponding author. Permanent address: Department of Mathematics, Institute of Shijiazhuang Mechanical Engineering, Shijiazhuang 050003, Hebei Province, PR China. E-mail address: [email protected] (R. Xu).

0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.054

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where uðtÞ, vðtÞ represent the densities of prey and predator population, respectively. a, b, c, d, a, e, g are positive parameters. System (1.1) is called Holling-type-II predator–prey model in the literature. This system is an extension of the familiar Lotka–Volterra system, in which the divisor 1 þ au is missing, i.e., a ¼ 0. a is interpreted as a constant handling time for each prey captured. For low prey biomass (au  1) this response approximates the classical Lotka–Volterra one. d=b is the ratio of conversion of prey biomass into predator biomass. It is investigated in [1] for the stability of equilibrium, Hopf bifurcation, global existence of limit cycles, global attractivity of equilibria, and codimension two bifurcations. The global behavior of system (1.1) has been discussed by many authors (see, for example, [1–5] and the references cited therein). We note that any biological or environmental parameters are naturally subject to fluctuation in time. As Cushing [6] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc.). Thus, the assumption of periodicity of the parameters is a way of incorporating the periodicity of the environment. On the other hand, it is generally recognized that some kinds of time delays are inevitable in population interactions(see, for example, [4,5,7,8] and the references cited therein). Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. The effect of these kinds of delays on the asymptotic behavior of populations has been studied by a number of authors (see, for example, [9–11]). In this paper we are concerned with the effects of the periodicity of ecological and environmental parameters and time delays due to gestation and negative feedbacks on the global dynamics of predator–prey systems with Holling-type-II functional response. To do so, we consider the following delayed differential system:   a12 ðtÞx2 ðtÞ x_1 ðtÞ ¼ x1 ðtÞ r1 ðtÞ  a11 ðtÞx1 ðt  s1 ðtÞÞ  ; 1 þ mx1 ðtÞ   ð1:2Þ a21 ðtÞx1 ðt  s2 ðtÞÞ  a22 ðtÞx2 ðt  s3 ðtÞÞ x_2 ðtÞ ¼ x2 ðtÞ  r2 ðtÞ þ 1 þ mx1 ðt  s2 ðtÞÞ with initial conditions xi ðhÞ ¼ /i ðhÞ;

h 2 ½s; 0 ; /i ð0Þ > 0;

/i 2 Cð½s; 0Þ; Rþ Þ;

i ¼ 1; 2;

ð1:3Þ

where x1 ðtÞ, x2 ðtÞ denote the densities of prey and predator population, respectively; m is a positive constant, s ¼ maxfsi ðtÞ; t 2 ½0; x ; i ¼ 1; 2; 3g.

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s1 ðtÞ P 0 and s3 ðtÞ P 0 denote the time delays due to negative feedbacks of the prey and the predator population, respectively. s2 ðtÞ is a time delay due to gestation, that is, mature adult predators can only contribute to the reproduction of predator biomass. a11 ðtÞ and a22 ðtÞ are the intra-specific competition rates of the prey and the predator, respectively. a12 ðtÞ is the capturing rate of the predator, a21 ðtÞ=a12 ðtÞ is the conversion rate of nutrients into the reproduction of the predator. In this paper, for system (1.2) we always assume that for all i; j ¼ 1; 2: (A1) ri ðtÞ, aij ðtÞ are continuously positive periodic functions with period x; (A2) si ðtÞ is nonnegative and continuously differentiable periodic function with period x, and mint2½0;x ð1  s_ i ðtÞÞ > 0, where s_ i ðtÞ ¼ dsi ðtÞ=dt, i ¼ 1; 2; 3. It is well known by the fundamental theory of functional differential equations [12] that system (1.2) has a unique solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ satisfying initial conditions (1.3). It is easy to verify that solutions of system (1.2) corresponding to initial conditions (1.3) are defined on ½0; þ1Þ and remain positive for all t P 0. In this paper, the solution of system (1.2) satisfying initial conditions (1.3) is said to be positive. The organization of this paper is as follows. In the next section, by using the continuation theorem of coincidence degree theory, we discuss the existence of positive x-periodic solutions of system (1.2) with initial conditions (1.3). In Section 3, by means of suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for the uniqueness and global stability of the positive periodic solutions of system (1.2). Numerical simulation is presented to illustrate the validity of our main results.

2. Existence of positive periodic solutions In this section, by using Gaines and MawhinÕs continuation theorem, we show the existence of positive periodic solutions of (1.2). To this end, we first introduce the following notations. Let X ; Y be real Banach spaces, L : Dom L X ! Y a linear mapping, and N : X ! Y a continuous mapping. The mapping L is called a Fredholm mapping of index zero if dimKer L ¼ codim Im L < þ1 and Im L is closed in Y . If L is a Fredholm mapping of index zero and there exist continuous projectors P : X ! X , and Q : Y ! Y such that Im P ¼ Ker L, Ker Q ¼ Im L ¼ ImðI  QÞ, then the restriction LP of L to Dom L \ Ker P : ðI  P ÞX ! Im L is invertible. Denote the inverse of LP by KP . If X is an open bounded subset of X , the mapping N will be called L-compact on X if QN ðXÞ is bounded and KP ðI  QÞN : X ! X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q ! Ker L.

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For convenience of use, we introduce the continuation theorem (see [13, p. 40]) as follows. Lemma 2.1. Let X X be an open bounded set. Let L be a Fredholm mapping of  Assume index zero and N be L-compact on X. (a) for each k 2 ð0; 1Þ, x 2 oX \ Dom L, Lx 6¼ kNx; (b) for each x 2 oX \ Ker L, QNx 6¼ 0; (c) degfJQN ; X \ Ker L; 0g 6¼ 0.  \ Dom L. Then Lx ¼ Nx has at least one solution in X In what follows we shall use the following notations: 1 f ¼ x

Z

x

f ðtÞ dt; 0

f L ¼ min f ðtÞ; t2½0;x

f M ¼ max f ðtÞ; ½0;x

where f is a continuous x-periodic function. We are now in a position to state and prove our result on the existence of positive periodic solutions to system (1.2). Theorem 2.1. System (1.2) has at least one positive x-periodic solution provided that (H1) r1 ð a21  mr2 Þ   a11r2 e2r1 x > 0. Proof. Since solutions of (1.2) and (1.3) remain positive for all t P 0, we let u1 ðtÞ ¼ ln½x1 ðtÞ ;

u2 ðtÞ ¼ ln½x2 ðtÞ :

ð2:1Þ

On substituting (2.1) into system (1.2), we obtain du1 ðtÞ a12 ðtÞ eu2 ðtÞ ¼ r1 ðtÞ  a11 ðtÞ eu1 ðts1 ðtÞÞ  ; dt 1 þ m eu1 ðtÞ du2 ðtÞ a21 ðtÞ eu1 ðts2 ðtÞÞ ¼ r2 ðtÞ þ  a22 ðtÞ eu2 ðts3 ðtÞÞ : dt 1 þ m eu1 ðts2 ðtÞÞ

ð2:2Þ

It is easy to see that if system (2.2) has one x-periodic solution ðu1 ðtÞ; u2 ðtÞÞT , then x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞT ¼ ðexp½u1 ðtÞ ; exp½u2 ðtÞ ÞT is a positive x-periodic solution of system (1.2). Therefore, to complete the proof, it suffices to show that system (2.2) has at least one x-periodic solution. Take X ¼ Y ¼ fðu1 ðtÞ; u2 ðtÞÞT 2 CðR; R2 Þ : u1 ðt þ xÞ ¼ u1 ðtÞ; u2 ðt þ xÞ ¼ u2 ðtÞg

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and T

kðu1 ðtÞ; u2 ðtÞÞ k ¼ max ju1 ðtÞj þ max ju2 ðtÞj; t2½0;x

t2½0;x

here j  j denotes the Euclidean norm. Then X and Y are Banach spaces with the norm k  k. Set  T du1 ðtÞ du2 ðtÞ T ; L : DomL \ X ! X ; Lðu1 ðtÞ; u2 ðtÞÞ ¼ ; dt dt where DomL ¼ fðu1 ðtÞ; u2 ðtÞÞT 2 C 1 ðR; R2 Þg and N : X ! X , 2 3 u ðtÞ   12 ðtÞ e 2 r1 ðtÞ  a11 ðtÞ eu1 ðts1 ðtÞÞ  a1þm eu1 ðtÞ u1 5: N ¼4 u2 a21 ðtÞ eu1 ðts2 ðtÞÞ u2 ðts3 ðtÞÞ r2 ðtÞ þ 1þm eu1 ðts2 ðtÞÞ  a22 ðtÞ e Define two projectors P and Q as       1 Rx u u ðtÞ dt u P 1 ¼ Q 1 ¼ x1 R0x 1 ; u2 u2 u2 ðtÞ dt x 0



u1 u2

 2 X ¼ Y:

It is clear that Ker L ¼ fx j x 2 X ; x ¼ h; h 2 R2 g; Z x Im L ¼ y j y 2 Y ; yðtÞ dt ¼ 0 is closed in Y ; 0

and dimKer L ¼ codim Im L ¼ 2: Therefore, L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that Im P ¼ Ker L;

Ker Q ¼ Im L ¼ Im ðI  QÞ:

Furthermore, the inverse KP of LP exists KP : Im L ! Dom L \ Ker P , Z t Z Z 1 x t yðsÞ ds  yðsÞ ds dt: KP ðyÞ ¼ x 0 0 0

and

is

given

Then QN : X ! Y and KP ðI  QÞN : X ! X are given respectively by i 2 3 Rx h u ðtÞ 1 12 ðtÞ e 2 r1 ðtÞ  a11 ðtÞ eu1 ðts1 ðtÞÞ  a1þm dt x 0 eu1 ðtÞ h i 7 6 u1 ðts2 ðtÞÞ QNx ¼ 4 1 R x ;  r ðtÞ þ a21 ðtÞ e  a ðtÞ eu2 ðts3 ðtÞÞ dt 5 x

0

2

1þm eu1 ðts2 ðtÞÞ

22

by

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KP ðI  QÞNx ¼

Z

t

0

1 NxðsÞ ds  x

Z

x

0

Z



t

NxðsÞ ds dt 

0

t 1  x 2

Z

x

NxðsÞ ds: 0

Clearly, QN and KP ðI  QÞN are continuous. In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset X. Corresponding to the operator equation Lx ¼ kNx, k 2 ð0; 1Þ, we derive   du1 ðtÞ a12 ðtÞ eu2 ðtÞ u1 ðts1 ðtÞÞ ¼ k r1 ðtÞ  a11 ðtÞ e  ; dt 1 þ m eu1 ðtÞ ð2:3Þ   du2 ðtÞ a21 ðtÞ eu1 ðts2 ðtÞÞ u2 ðts3 ðtÞÞ ¼ k  r2 ðtÞ þ  a22 ðtÞ e : dt 1 þ m eu1 ðts2 ðtÞÞ Suppose that ðu1 ðtÞ; u2 ðtÞÞT 2 X is a solution of (2.3) for a certain k 2 ð0; 1Þ. Integrating (2.3) over the interval ½0; x gives Z x Z x Z x a12 ðtÞ eu2 ðtÞ a11 ðtÞ eu1 ðts1 ðtÞÞ dt þ dt ¼ r1 ðtÞ dt; ð2:4Þ 1 þ m eu1 ðtÞ 0 0 0 Z x Z x Z x a21 ðtÞ eu1 ðts2 ðtÞÞ u2 ðts3 ðtÞÞ a22 ðtÞ e dt þ r2 ðtÞ dt ¼ dt: ð2:5Þ 1 þ m eu1 ðts2 ðtÞÞ 0 0 0 It follows from (2.3)–(2.5) that  Z x Z x a12 ðtÞ eu2 ðtÞ 0 u1 ðts1 ðtÞÞ ju1 ðtÞj dt < r1 ðtÞ þ a11 ðtÞ e þ dt ¼ 2r1 x; 1 þ m eu1 ðtÞ 0 0   Z x Z x a21 ðtÞ eu1 ðts2 ðtÞÞ 2a21 x 0 u2 ðts3 ðtÞÞ : ju2 ðtÞj dt < r2 ðtÞ þ þ a22 ðtÞ e dt < u ðts ðtÞÞ 1 2 m 1 þ me 0 0 ð2:6Þ T

Since ðu1 ðtÞ; u2 ðtÞÞ 2 X , there exist ni ; gi 2 ½0; x such that ui ðni Þ ¼ min ui ðtÞ; t2½0;x

ui ðgi Þ ¼ max ui ðtÞ; t2½0;x

i ¼ 1; 2:

ð2:7Þ

It follows from (2.4) and (2.7) that Z

x

a11 ðtÞ eu1 ðts1 ðtÞÞ dt < r1 x;

0

which yields u1 ðn1 Þ < ln

r1 : 11 a

We derive from (2.6) and (2.8) that

ð2:8Þ

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u1 ðtÞ 6 u1 ðn1 Þ þ

Z

x

ju01 ðtÞj dt < ln 0

r1 þ 2r1 x: 11 a

643

ð2:9Þ

It follows from (2.5), (2.7) and (2.9) that Z x Z x a21 ðtÞ eu1 ðts2 ðtÞÞ a21 x ; a22 ðtÞ eu2 ðts3 ðtÞÞ dt 6 dt 6 u ðts ðtÞÞ 1 2 m 1 þ me 0 0 which implies u2 ðn2 Þ 6 ln

 a21 : m a22

ð2:10Þ

Thus we derive from (2.6) and (2.10) that Z x  2a21 x a21 : ju02 ðtÞj dt < ln þ u2 ðtÞ 6 u2 ðn2 Þ þ m m a 22 0

ð2:11Þ

On the other hand, it follows from (2.5) that Z

x

a21 ðtÞ eu1 ðts2 ðtÞÞ dt P

0

Z

x

r2 ðtÞ dt;

0

which yields u1 ðg1 Þ P ln

r2 : 21 a

ð2:12Þ

We derive from (2.6) and (2.12) that u1 ðtÞ P u1 ðg1 Þ 

Z

x

ju01 ðtÞj dt > ln 0

r2  2r1 x;  a21

which, together with (2.9), yields    ( )  r   r    1  2  max ju1 ðtÞj < max  ln  þ 2r1 x;  ln  þ 2r1 x :¼ R1 :    a21  t2½0;x a11 

ð2:13Þ

ð2:14Þ

Similarly, it follows from (2.5) that Z x Z x Z x a21 ðtÞ eu1 ðts2 ðtÞÞ a22 ðtÞ eu2 ðts3 ðtÞÞ dt ¼ dt  r2 ðtÞ dt 1 þ m eu1 ðts2 ðtÞÞ 0 0 0  a21 x eu1 ðn1 Þ P  r2 x; 1 þ m eu1 ðn1 Þ which implies  a22 ð1 þ m eu1 ðn1 Þ Þ eu2 ðg2 Þ P ð a21  mr2 Þ eu1 ðn1 Þ  r2 :

ð2:15Þ

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Noting that u1 ðg1 Þ 6 u1 ðn1 Þ þ

Z

x

ju01 ðtÞj dt < u1 ðn1 Þ þ 2r1 x; 0

r1 6  a11 eu1 ðg1 Þ þ  a12 eu2 ðg2 Þ ; we derive from (2.15) that  a22 ð1 þ m eu1 ðn1 Þ Þ eu2 ðg2 Þ P

ð a21  mr2 Þðr1  a12 eu2 ðg2 Þ Þ  r2 ;  a11 e2r1 x

which, together with (2.9), implies u2 ðg2 Þ P ln

r1 ð a21  mr2 Þ   a11r2 e2r1 x :  a11 Þ þ a12 ða21  mr2 Þ a11  a22 e2r1 x ð1 þ mr1 e2r1 x =

ð2:16Þ

It follows from (2.6) and (2.16) that u2 ðtÞ P u2 ðg2 Þ 

Z

x

ju02 ðtÞj dt 0

> ln

11r2 e2r1 x r1 ð a21  mr2 Þ  a 2a21 x ;  2 r x 2 r x 1 1  m a11 Þ þ a12 ða21  mr2 Þ a11  a22 e ð1 þ mr1 e =

which, together with (2.11), leads to  (   a21 x a21  2  ; max ju2 ðtÞj < max  ln þ  m t2j0;xj m a22    )  2a x  r1 ð a21  mr2 Þ   a11r2 e2r1 x   21 :¼ R2 : þ  ln   m a11 Þ þ  a12 ða21  mr2 Þ  a11 a22 e2r1 x ð1 þ mr1 e2r1 x = ð2:17Þ Clearly, R1 and R2 in (2.14) and (2.17) are independent of k. Denote M ¼ R1 þ R2 þ R0 , where R0 is taken sufficiently large such that each solution T ða ; b Þ of the following algebraic equations:  a12 eb ¼ 0; 1 þ m ea a  a21 e  r2 þ  a22 eb ¼ 0 1 þ m ea

r1   a11 ea 

satisfies kða ; b ÞT k ¼ ja j þ jb j < M (if it exists) and the following:

ð2:18Þ

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  )  ( (  r   r    a21    1   2  max  ln ;  ln  þ max  ln ;    ma22  a11   a21   )   r1 ð a21  mr2 Þ   a11r2    ln  < M:   a11 Þ þ  a12 ð a21  mr2 Þ  a11 a22 ð1 þ mr1 = T

645

ð2:19Þ

T

We now take X ¼ fðu1 ðtÞ; u2 ðtÞÞ 2 X : kðu1 ; u2 Þ k < Mg. This satisfies the T condition (a) in Lemma 2.1. When ðu1 ðtÞ; u2 ðtÞÞ 2 oX \ Ker L ¼ oX \ R2 , T 2 ðu1 ; u2 Þ is a constant vector in R with ju1 j þ ju2 j ¼ M. If system (2.18) has at least one solution, then we have #     "  a12 eu2 r1   a11 eu1  1þm u1 0 eu1 QN ¼ 6¼ :  a21 eu1 u2 u2 0 r2 þ 1þm eu1   a22 e If system (2.18) does not have a solution, we can directly derive     u 0 QN 1 6¼ : u2 0 This proves that condition (b) in Lemma 2.1 is satisfied. Finally, we will prove that condition ðcÞ in Lemma 2.1 holds. To this end, we define / : Dom L  ½0; 1 ! X by " /ðu1 ; u2 ; lÞ ¼

r1   a11 eu1 a21 eu1   a22 eu2 1þm eu1

#

"

#  a12 eu2  1þm u 1 e þl ; r2

where l 2 ½0; 1 is a parameter. When ðu1 ðtÞ; u2 ðtÞÞT 2 oX \ Ker L ¼ oX \ R2 , ðu1 ; u2 ÞT is a constant vector in R2 with ju1 j þ ju2 j ¼ M. We will show T that when ðu1 ; u2 Þ 2 oX \ Ker L, /ðu1 ; u2 ; lÞ 6¼ 0. Otherwise, there is a conT stant vector ðu1 ; u2 Þ with ju1 j þ ju2 j ¼ M satisfying /ðu1 ; u2 ; lÞ ¼ 0, that is r1   a11 eu1  l

 a12 eu2 ¼ 0; 1 þ m eu1

 a21 eu1  a22 eu2  lr2 ¼ 0: 1 þ m eu1 A similar argument in (2.14) and (2.17) shows that   ) (  r   r   1   2  ju1 j < max  ln ;  ln  ;   a11    a21  )   (    r1 ð a21   a21  mr2 Þ  a11r2   ju2 j < max  ln  : ;  ln  m a22    a11  a22 ð1 þ mr1 =a11 Þ þ a12 ða21  mr2 Þ 

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It follows from (2.19) that ju1 j þ ju2 j < M; which leads to a contradiction. Using the property of topological degree and taking J ¼ I : Im Q ! Ker L; ðu1 ; u2 ÞT ! ðu1 ; u2 ÞT , we have T

T

degðJQN ðu1 ; u2 Þ ; X \ Ker L; ð0; 0Þ Þ T

¼ degð/ðu1 ; u2 ; 1Þ; X \ Ker L; ð0; 0Þ Þ T

¼ degð/ðu1 ; u2 ; 0Þ; X \ Ker L; ð0; 0Þ Þ 0 1 !T u1  e a 21 a11 eu1 ; ¼ deg @ r1    a22 eu2 ; X \ Ker L; ð0; 0ÞT A: 1 þ m eu1 The system of algebraic equations r1   a11 eu1 ¼ 0;  a21 eu1  a22 eu2 ¼ 0 1 þ m eu1 has a unique solution ðu1 ; u2 Þ which satisfies u1 ¼ ln

r1 ;  a11

u2 ¼ ln

 a21r1 :  a11 þ mr1 Þ a22 ð

Thus, a direct calculation shows that    u2  a22 e 

  a11 eu1  degðJQN ðu1 ; u2 Þ ; X \ Ker L; ð0; 0Þ Þ ¼ sign a21 eu1  u 2 1 T

0

T

ð1þm e Þ





¼ signfa11 a22 eu1 þu2 g ¼ 1:  is equicontinuous Finally, it is easy to show that the set fKP ðI  QÞNx j x 2 Xg and uniformly bounded. By using the Arzela–Ascoli theorem, we see that  ! X is compact. Consequently, N is L-compact. KP ðI  QÞN : X By now we have proved that X satisfies all the requirements in Lemma 2.1. Hence, (2.2) has at least one x-periodic solution. As a consequence, system (1.2) has at least one positive x-periodic solution. The proof is complete. h

3. Uniqueness and global stability We now proceed to the discussion on the uniqueness and global stability of the x-periodic solution x ðtÞ in Theorem 2.1. It is immediate that if x ðtÞ is

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647

globally asymptotically stable then x ðtÞ is in fact unique. We first derive certain upper bound estimates for solutions of system (1.2). T

Theorem 3.1. Let xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ denote any positive solution of system (1.2) with initial conditions (1.3). Then there exists a T > 0 such that if t > T , 0 < x1 ðtÞ 6 M1 ;

0 < x2 ðtÞ 6 M2 ;

ð3:1Þ

where M1 ¼

r1M rM s e1 ; aL11

M2 ¼

aM 21 M1 aM e 21 M1 s : aL22

ð3:2Þ

The proof of Theorem 3.1 is similar to that of Lemma 2.1 in [10], we therefore omit it here. We now formulate the uniqueness and global stability of the positive xperiodic solutions of system (1.2). Theorem 3.2. In addition to (H1), assume further that ðH2Þ

lim inf Ai ðtÞ > 0; t!1

where Z r1 ðr1 ðtÞÞ 1 1 a11 ðr1 1 ðtÞÞM1 A1 ðtÞ ¼ a11 ðtÞ  ma12 ðtÞM2  a11 ðsÞ ds 1 1  s_ 1 ðr1 ðtÞÞ r1 ðtÞ 1 Z r1 ðtÞ 1 a21 ðr1 2 ðtÞÞ a11 ðsÞ ds   ½r1 ðtÞ þ a11 ðtÞM1 þ a12 ðtÞM2 1 _ ðr 1  s 2 t 2 ðtÞÞ Z r1 ðr1 ðtÞÞ 1 3 2 a21 ðr2 ðtÞÞ  M2 a22 ðsÞ ds; 1 ðtÞÞ 1  s_ 2 ðr1 r2 ðtÞ 2 ! Z 1 1 r1 ðtÞ a11 ðsÞ ds  ðr2 ðtÞ þ a21 ðtÞM1 A2 ðtÞ ¼ a22 ðtÞ  a12 ðtÞ 1 þ m t Z r1 ðtÞ Z r1 ðr1 ðtÞÞ 3 3 3 a22 ðr1 3 ðtÞÞ þ a22 ðtÞM2 Þ a22 ðsÞ ds  M2 a22 ðsÞ ds  1 1 _ ðr ðtÞÞ 1  s 3 t r3 ðtÞ 3 ð3:3Þ

in which r1 i ðtÞ is inverse function of ri ðtÞ ¼ t  si ðtÞ (i ¼ 1; 2; 3). Then system (1.2) and (1.3) has a unique positive x-periodic solution x ðtÞ ¼ ðx1 ðtÞ; x2 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) and (1.3). Let x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞT be a positive x-periodic solution of (1.2) and (1.3), and yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞÞT be any positive solution of system (1.2). It follows from Theorem 3.1 that there exist positive constants T and Mi (defined by (3.2)), such that for all t P T , 0 < x1 ðtÞ 6 Mi ;

0 < yi ðtÞ 6 Mi ;

i ¼ 1; 2:

ð3:4Þ

We define V11 ðtÞ ¼ j ln x1 ðtÞ  ln y1 ðtÞj:

ð3:5Þ

Calculating the upper right derivative of V11 ðtÞ along solutions of (1.2), it follows that ! x_ 1 ðtÞ y_ 1 ðtÞ D V11 ðtÞ ¼  sgnðx1 ðtÞ  y1 ðtÞÞ x1 ðtÞ y1 ðtÞ ¼ sgnðx1 ðtÞ  y1 ðtÞÞ  a11 ðtÞ½x1 ðt  s1 ðtÞÞ  y1 ðt  s1 ðtÞÞ a12 ðtÞx2 ðtÞ a12 ðtÞy2 ðtÞ þ  1 þ mx1 ðtÞ 1 þ my1 ðtÞ  ¼ sgnðx1 ðtÞ  y1 ðtÞÞ  a11 ðtÞ½x1 ðt  s1 ðtÞÞ  y1 ðt  s1 ðtÞÞ a12 ðtÞðx2 ðtÞ  y2 ðtÞÞ ma12 ðtÞy2 ðtÞðx1 ðtÞ  y1 ðtÞÞ þ  1 þ mx1 ðtÞ ð1 þ mx1 ðtÞÞð1 þ my1 ðtÞÞ ( a12 ðtÞðx2 ðtÞ  y2 ðtÞÞ ¼ sgnðx1 ðtÞ  y1 ðtÞÞ  a11 ðtÞðx1 ðtÞ  y1 ðtÞÞ  1 þ mx1 ðtÞ ) Z t ma12 ðtÞy2 ðtÞðx1 ðtÞ  y1 ðtÞÞ  þ a11 ðtÞ þ ð_x1 ðuÞ  y_ 1 ðuÞÞ du ð1 þ mx1 ðtÞÞð1 þ my1 ðtÞÞ ts1 ðtÞ þ

6  a11 ðtÞjx1 ðtÞ  y1 ðtÞj þ a12 ðtÞjx2 ðtÞ  y2 ðtÞj Z   t      ð_x1 ðuÞ  y_ 1 ðuÞÞ du: þ ma12 ðtÞy2 ðtÞjx1 ðtÞ  y1 ðtÞj þ a11 ðtÞ  ts1 ðtÞ  ð3:6Þ On substituting (1.2) into (3.6), we derive that

R. Xu et al. / Appl. Math. Comput. 161 (2005) 637–654

649

Dþ V11 ðtÞ 6  a11 ðtÞjx1 ðtÞ  y1 ðtÞj þ a12 ðtÞjx2 ðtÞ  y2 ðtÞj Z   t    x ðuÞ r1 ðuÞ þ ma12 ðtÞy2 ðtÞjx1 ðtÞ  y1 ðtÞj þ a11 ðtÞ  ts1 ðtÞ 1  a12 ðuÞx2 ðuÞ  a11 ðuÞx1 ðu  s1 ðuÞÞ  1 þ mx1 ðuÞ    a12 ðuÞy2 ðuÞ   y1 ðuÞ r1 ðuÞ  a11 ðuÞy1 ðu  s1 ðuÞÞ  du  1 þ my1 ðuÞ ¼ a11 ðtÞjx1 ðtÞ  y1 ðtÞj þ a12 ðtÞjx2 ðtÞ  y2 ðtÞj þ ma12 ðtÞy2 ðtÞjx1 ðtÞ Z   t  r1 ðuÞ  a11 ðuÞy1 ðu  s1 ðuÞÞ  y1 ðtÞj þ a11 ðtÞ  ts1 ðtÞ  a12 ðuÞy2 ðuÞ  ðx1 ðuÞ  y1 ðuÞÞ ð1 þ mx1 ðuÞÞð1 þ my1 ðuÞÞ  a11 ðuÞx1 ðuÞðx1 ðu  s1 ðuÞÞ  y1 ðu  s1 ðuÞÞÞ  a12 ðuÞx1 ðuÞ   ðx ðuÞ  y2 ðuÞÞ du 6  a11 ðtÞjx1 ðtÞ  y1 ðtÞj   1 þ mx1 ðuÞ 2 þ a12 ðtÞjx2 ðtÞ  y2 ðtÞj þ ma12 ðtÞy2 ðtÞjx1 ðtÞ  y1 ðtÞj Z t þ a11 ðtÞ ½r1 ðuÞ þ a11 ðuÞy1 ðu  s1 ðuÞÞ þ a12 ðuÞy2 ðuÞ jx1 ðuÞ ts1 ðtÞ

 y1 ðuÞj þ a11 ðuÞx1 ðuÞjx1 ðu  s1 ðuÞÞ  y1 ðu  s1 ðuÞÞj a12 ðuÞ  jx2 ðuÞ  y2 ðuÞj du: þ m

ð3:7Þ

It follows from (3.4) and (3.7) that for t > T þ s, Dþ V11 ðtÞ 6  a11 ðtÞjx1 ðtÞ  y1 ðtÞj þ a12 ðtÞjx2 ðtÞ  y2 ðtÞj Z t  þ ma12 ðtÞM2 jx1 ðtÞ  y1 ðtÞj þ a11 ðtÞ ½r1 ðuÞ þ a11 ðuÞM1 ts1 ðtÞ

þ a12 ðuÞM2 jx1 ðuÞ  y1 ðuÞj þ a11 ðuÞM1 jx1 ðu  s1 ðuÞÞ a12 ðuÞ  jx2 ðuÞ  y2 ðuÞj du:  y1 ðu  s1 ðuÞÞj þ m

ð3:8Þ

Define V12 ðtÞ ¼

Z t

r1 ðtÞ 1

Z

t

r1 ðsÞ

a11 ðsÞ ½r1 ðuÞ þ a11 ðuÞM1 þ a12 ðuÞM2 jx1 ðuÞ  y1 ðuÞj

þ a11 ðuÞM1 jx1 ðu  s1 ðuÞÞ  y1 ðu  s1 ðuÞÞj þ

a12 ðuÞ  jx2 ðuÞ  y2 ðuÞj du ds: m

ð3:9Þ

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R. Xu et al. / Appl. Math. Comput. 161 (2005) 637–654

We obtain from (3.8) and (3.9) that for t P T þ s Dþ V11 ðtÞ þ V_12 ðtÞ 6  ða11 ðtÞ  ma12 ðtÞM2 Þjx1 ðtÞ  y1 ðtÞj þ a12 ðtÞjx2 ðtÞ  y2 ðtÞj Z r1 ðtÞ 1 a11 ðsÞ ds ½r1 ðtÞ þ a11 ðtÞM1 þ a12 ðtÞM2 jx1 ðtÞ þ t

 y1 ðtÞj þ a11 ðtÞM1 jx1 ðt  s1 ðtÞÞ  y1 ðt  s1 ðtÞÞj a12 ðtÞ  þ jx2 ðtÞ  y2 ðtÞj : m

ð3:10Þ

We now define V1 ðtÞ ¼ V11 ðtÞ þ Vl2 ðtÞ þ V13 ðtÞ;

ð3:11Þ

where V13 ðtÞ ¼ M1

Z

t ts1 ðtÞ

Z

r1 ðr1 ðlÞÞ 1 1

ðlÞ r1 1

a11 ðsÞa11 ðr1 1 ðlÞÞ  jx ðlÞ  y1 ðlÞj ds dl: 1 1  s_ 1 ðr1 ðlÞÞ 1 ð3:12Þ

It then follows from (3.10)–(3.12) that for t P T þ s, Dþ V1 ðtÞ 6  ða11 ðtÞ  ma12 ðtÞM2 Þjx1 ðtÞ  y1 ðtÞj ! Z 1 1 r1 ðtÞ a11 ðsÞ ds jx2 ðtÞ  y2 ðtÞj þ a12 ðtÞ 1 þ m t Z r1 ðtÞ 1 þ ½r1 ðtÞ þ a11 ðtÞM1 þ a12 ðtÞM2 a11 ðsÞ dsjx1 ðtÞ  y1 ðtÞj þ

a11 ðr1 1 ðtÞÞM1 1  s_ 1 ðr1 1 ðtÞÞ

Z

t r1 ðr1 ðtÞÞ 1 1

ðtÞ r1 1

a11 ðsÞ dsjx1 ðtÞ  y1 ðtÞj:

ð3:13Þ

Similarly, we define V2 ðtÞ ¼ V21 ðtÞ þ V22 ðtÞ þ V23 ðtÞ;

ð3:14Þ

where ð3:15Þ V21 ðtÞ ¼ j ln x2 ðtÞ  ln y2 ðtÞj; Z t a21 ðr1 2 ðsÞÞ jx1 ðsÞ  y1 ðsÞj ds V22 ðtÞ ¼ 1 _ 1  s ðr ðsÞÞ 2 ts2 ðtÞ 2 Z r1 ðtÞ Z t 3 a22 ðsÞ½ðr2 ðuÞ þ a21 ðuÞM1 þ a22 ðuÞM2 Þjx2 ðuÞ  y2 ðuÞj þ t

r3 ðsÞ

þ a21 ðuÞM2 jx1 ðu  s2 ðuÞÞ  y1 ðu  s2 ðuÞÞj þ a22 ðuÞM2 jx2 ðu  s3 ðuÞÞ  y2 ðu  s3 ðuÞÞj du ds;

ð3:16Þ

R. Xu et al. / Appl. Math. Comput. 161 (2005) 637–654

V23 ðtÞ ¼ M2

Z

Z

t ts2 ðtÞ

þ M2

Z

r1 ðr1 ðlÞÞ 3 2

ðlÞ r1 2

t

ts3 ðtÞ

Z

651

a22 ðsÞa21 ðr1 2 ðlÞÞ  jx1 ðlÞ  y1 ðlÞj ds dl 1  s_ 2 ðr1 2 ðlÞÞ

r1 ðr1 ðlÞÞ 3 3

a22 ðsÞa22 ðr1 3 ðlÞÞ  jx2 ðlÞ  y2 ðlÞj ds dl: 1  s_ 3 ðr1 3 ðlÞÞ

ðlÞ r1 3

ð3:17Þ Calculating the upper right derivative of V2 ðtÞ along solutions of (1.2), we derive for t P T þ s that a21 ðr1 2 ðtÞÞ jx1 ðtÞ  y1 ðtÞj 1  s_ 2 ðr1 ðtÞÞ 2 Z r1 ðtÞ 3 þ ðr2 ðtÞ þ a21 ðtÞM1 þ a22 ðtÞM2 Þ a22 ðsÞ dsjx2 ðtÞ  y2 ðtÞj

Dþ V2 ðtÞ 6  a22 ðtÞjx2 ðtÞ  y2 ðtÞj þ

þ

a21 ðr1 2 ðtÞÞ M2 1  s_ 2 ðr1 2 ðtÞÞ

þ M2

a22 ðr1 3 ðtÞÞ 1  s_ 3 ðr1 3 ðtÞÞ

Z

t r1 ðr1 ðtÞÞ 3 2

ðtÞ r1 2

Z

r1 ðr1 ðtÞÞ 3 3

ðtÞ r1 3

a22 ðsÞ dsjx1 ðtÞ  y1 ðtÞj a22 ðsÞ dsjx2 ðtÞ  y2 ðtÞj:

ð3:18Þ

We now define a Lyapunov functional V ðtÞ as V ðtÞ ¼ V1 ðtÞ þ V2 ðtÞ:

ð3:19Þ

It then follows from (3.13), (3.18) and (3.19) that for t P T þ s Dþ V ðtÞ 6  A1 ðtÞjx1 ðtÞ  y1 ðtÞj  A2 ðtÞjx2 ðtÞ  y2 ðtÞj;

ð3:20Þ

where A1 ðtÞ and A2 ðtÞ are defined in (3.3). By hypothesis (H2), there exist positive constants a1 , a2 and T  P T þ s such that if t P T  , Ai ðtÞ P ai > 0:

ð3:21Þ

Integrating both sides of (3.20) on interval ½T  ; t , 2 Z t X Ai ðsÞjxi ðsÞ  yi ðsÞj ds 6 V ðT  Þ: V ðtÞ þ i¼1

T

It follows from (3.21) and (3.22) that Z t 2 X ai jxi ðsÞ  yi ðsÞj ds 6 V ðT  Þ V ðtÞ þ i¼1

T

Therefore, V ðtÞ is bounded on ½T  ; 1Þ and also

for t P T  :

ð3:22Þ

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R. Xu et al. / Appl. Math. Comput. 161 (2005) 637–654

Z

1

T

jxi ðsÞ  yi ðsÞj ds < 1;

i ¼ 1; 2:

By Theorem 3.1, jxi ðtÞ  yi ðtÞj (i ¼ 1; 2) are bounded on ½T  ; 1Þ. On the other hand, it is easy to see that x_ i ðtÞ and y_ i ðtÞ (i ¼ 1; 2) are bounded for t P T  . Therefore, jxi ðtÞ  yi ðtÞj (i ¼ 1; 2) are uniformly continuous on ½T  ; 1Þ. By BarbalatÕs lemma [4, Lemmas 1.2.2 and 1.2.3], we conclude that lim jxi ðtÞ  yi ðtÞj ¼ 0:

t!1

h

The proof is complete.

Remark. If si ðtÞ  si , where si (i ¼ 1; 2; 3) are nonnegative constants, then assumption (H2) can be simplified. In this case, s ¼ maxfs1 ; s2 ; s3 g. We therefore have the following result. Corollary 3.1. Let si ðtÞ  si , where si (i ¼ 1; 2; 3) are nonnegative constants. In addition to (H1) assume further that Z tþ2s1 ðH3Þ lim inf a11 ðtÞ  ma12 ðtÞM2  a11 ðt þ s1 ÞM1 a11 ðsÞ ds t!1

 ½r1 ðtÞ þ a11 ðtÞM1 þ a12 ðtÞM2  M2 a21 ðt þ s2 Þ

Z

Z t

tþs2 þs3

a22 ðsÞ ds

tþs1 tþs1

a11 ðsÞ ds  a21 ðt þ s2 Þ > 0;

tþs2



  Z 1 ðtþs1 Þ ðH4Þ lim inf a22 ðtÞ  a12 ðtÞ 1 þ a11 ðsÞ ds  ðr2 ðtÞ þ a21 ðtÞM1 t!1 m t Z tþs3 Z tþ2s3 þ a22 ðtÞM2 Þ a22 ðsÞ ds  M2 a22 ðt þ s3 Þ a22 ðsÞ ds > 0: t

tþs3

Then (1.2) and (1.3) has a unique positive x-periodic solution which is globally asymptotically stable. We now give an example to show the feasibility of our main result on the existence of positive periodic solutions of system (1.2). Example 1. As an example, we consider the following delayed system:   x2 ðtÞ x_ 1 ðtÞ ¼ x1 ðtÞ 1 þ 0:1 sin t  0:1x1 ðt  0:1Þ  ; 1 þ x1 ðtÞ   ð3:23Þ 1 9x1 ðt  0:3Þ  x2 ðt  0:1Þ : x_ 2 ðtÞ ¼ x2 ðtÞ  5 ð2 þ sin tÞ þ 10 1 þ x1 ðt  0:3Þ

R. Xu et al. / Appl. Math. Comput. 161 (2005) 637–654

653

3.5 x1 x2 3

solution

2.5

2

1.5

1

0.5

0 0

5

10

15

20

25 time t

30

35

40

45

50

Fig. 1. The positive 2p-periodic solution found by numerical integration of system (3.23) with initial condition ð/1 ðhÞ; /2 ðhÞÞ  ð2; 2Þ.

It is easy to verify that the coefficients of system (3.23) satisfy (H1). By Theorem 2.1, we see that system (3.23) has at least one positive periodic solutions. Numerical integration can be carried out by using MATLAB. As shown in Fig. 1, numerical simulation also confirms that system (3.23) admits at least one positive periodic solution.

References [1] A.D. Bazykin, Structural and dynamic stability of model predator–prey systems, Int. Inst. Appl. Syst. Analysis, Laxenburg, 1976. [2] J. Hainzl, Stability and Hopf bifurcation in a predator–prey system with several parameters, SIAM Appl. Math. 48 (1988) 170–190. [3] N.D. Kazarinoff, P. Van Den Driessche, A model predator–prey system with functional response, Math. Biosci. 39 (1978) 125–134. [4] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht/Norwell, MA, 1992. [5] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [6] J.M. Cushing, Periodic time-dependent predator–prey system, SIAM J. Appl. Math. 32 (1977) 82–95.

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[7] J.M. Cushing, Integro-differential equations and delay models in population dynamics, in: Lecture Notes in Biomathematics, vol. 20, Springer-Verlag, Berlin/Heidelberg/New York, 1977. [8] H.I. Freedman, V.S.H. Rao, The tradeoff between mutual interference and time lag in predator–prey models, Bull. Math. Biol. 45 (1983) 991–1004. [9] H.I. Freedman, J. So, P. Waltman, Coexistence in a model of competition in the chemostat incorporating discrete time delays, SIAM J. Appl. Math. 49 (1989) 859–870. [10] W. Wang, Z. Ma, Harmless delays for uniform persistence, J. Math. Anal. Appl. 158 (1991) 256–268. [11] E. Beretta, Y. Kuang, Global analysis in some delayed ratio-dependent predator–prey systems, Nonlinear Anal., TMA 32 (1998) 381–408. [12] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, Heidelberg, 1977. [13] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.