Permanence and extinction of a periodic predator–prey delay system with functional response and stage structure for prey

Applied Mathematics and Computation 184 (2007) 931–944 www.elsevier.com/locate/amc

Permanence and extinction of a periodic predator–prey delay system with functional response and stage structure for prey Hong Zhang

a,b,*

, Lansun Chen b, Rongping Zhu

a

a

b

Department of Mathematics, Jiangsu University, ZhenJiang, JiangSu 212013, PR China Department of Applied Mathematics, Dalian University of Technology, DaLian, LiaoNing 116024, PR China

Abstract In this paper, we consider a periodic coefficients predator–prey system with functional response and infinite delay, in which the prey has a history that takes them through two stages, immature and mature. Sufficient conditions which guarantee the permanence and extinction of the system are obtained. Finally, we give an example. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Predator–prey system; Infinite delay; Stage structure; Permanence; Extinction

1. Introduction It is well known that past history as well as current conditions can influence population dynamics and such interactions has motivated the introduction of delays in population growth. There are several books [1– 4]devoted to investigations of the dynamic behavior of delay differential equations. Recently, Stage structure models have received much attraction [6–11,19]. This is not only because they are much more simple than the models governed by partial differential equations but also they can exhibit phenomena similar to those of partial differential models [6], and many important physiological parameters can be incorporated. The single species model with stage structure was studied by Aiello and Freedman [7]. Two species models with stage structure were investigated by Wang and Chen [8], Xiao and Chen [9] and Magnusson [10]. Zhang, Chen and Neumann [12] proposed the following autonomous stage structure predator–prey system: 8 0 2 > < x1 ¼ ax2  r1 x1  bx1  gx1  b1 x1 x3 ; ð1Þ x02 ¼ bx1  r2 x2 ; > : 0 x3 ¼ x3 ðr þ kb1 x1  g1 x3 Þ;

*

Corresponding author. E-mail address: [email protected] (H. Zhang).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.195

932

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

where a, b, b1, g, g1, r, r1, r2 and k are all positive constants, k is a digesting constant. Sufficient conditions which ensure the permanence of two species and extinction of one or two species are obtained. On the other hand, since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment. More realistic and interesting models should take into account both the seasonality of the changing environment [13,14]. This motivated Cui and Song [15] to consider the following periodic nonautonomous predator–prey model with stage structure for prey: 8 2 > < x_1 ¼ aðtÞx2  bðtÞx1  dðtÞx1  pðtÞx1 y; ð2Þ x_2 ¼ cðtÞx1  f ðtÞx22 ; > : y_ ¼ y½gðtÞ þ hðtÞx1  qðtÞy; where a(t), b(t), c(t), d(t), f(t), g(t), h(t), p(t) and q(t) are all continuous positive x-periodic functions. x1 and x2 denote the density of immature and mature population (prey) respectively, and y is the density of the predator that only prey on x1 (immature prey). They obtained a set of sufficient and necessary condition which guarantee the permanence of the above system. Recently, maybe stimulated by the works of Teng and Chen [16], Cui and Sun [18] further incorporated infinite delay to system (2) and investigated the following model: 8 R0 > x_ ¼ aðtÞx2  bðtÞx1  dðtÞx21  pðtÞx1 1 k 12 ðsÞyðt þ sÞ ds; > < 1 x_2 ¼ cðtÞx1  f ðtÞx22 ; ð3Þ h i > > : y_ ¼ y gðtÞ þ hðtÞ R 0 k ðsÞx ðt þ sÞ ds  qðtÞ R 0 k ðsÞyðt þ sÞ ds : 1

21

1

1

22

Under the assumption that the coefficients in (3) are all x-periodic R xand continuous R x for t P 0, a(t), b(t), c(t), d(t) and f(t) are all positive, p(t), h(t) and q(t) are nonnegative, and 0 qðtÞ dt >R 0, 0 gðtÞ dt P 0. The functions 0 kij(s) (i, j = 1, 2) defined on R ¼ ð1; 0 are nonnegative and integrable, 1 k ij ðsÞ ¼ 1. By using analysis technique, they obtained a set of sufficient and necessary conditions which guarantee the permanence of the system. Noticing that (1)–(3) are modified from the classical Lotka–Volterra predator–prey system:  0 x ¼ xða  bx  cyÞ; ð4Þ y 0 ¼ yðd þ ex  fyÞ: A predators functional response is its per capita feeding rate on prey. Holling [20,21] suggested that the predator should not be able to consume an unlimited number of prey as the prey population increases. That is, in the Lotka–Volterra equations, the number of prey consumed per predator is unlimited as the prey population increases. The number of prey removed is cxy, so that the number of prey eaten per predator is unlimited as x increases to infinity. Holling proposed three models of the rate of prey capture per predator as a function of prey population density: Types I, II, and III. In 2001, Skalski and Gilliam reviewed the literature on functional response curves and presented statistical evidence from 19 predator–prey systems that three predatordependent functional responses [22–25], i.e., models that are functions of both prey and predator abundance because of predator interference, can provide better descriptions of predator feeding over a range of predator–prey abundances. No single functional response best described all of the data sets. To our knowledge, seldom did scholars consider the stage structure predator–prey system with functional response and infinite delay. This paper is largely motivated by the above-mentioned fact. We consider the following system: 8 x2 ð1ecx1 Þ R 0 > x_1 ¼ aðtÞx2  bðtÞx1  dðtÞx21  pðtÞ 1 P þx2 k ðsÞyðt þ sÞ ds; > 1 12 > 1 > < x_2 ¼ cðtÞx1  f ðtÞx22 ; ð5Þ  > R 2  > 2 ðtþsÞð1ecx1 ðtþsÞ Þ R > x 0 0 > y_ ¼ y gðtÞ þ hðtÞ : k ðsÞ 1 P þx2 ðtþsÞ ds  y  qðtÞ 1 k 22 ðsÞyðt þ sÞ ds ; 1 21 1

where x1 and x2 denote the density of immature and mature population (prey) respectively. y is the density of the predator that only prey on x1. The coefficients in (5) are all continuous positive x-periodic for

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

933

t P 0. P, c are The functions kij(s) (i, j = 1, 2) defined on R ¼ ð1; 0 are nonnegative and R 0 positive constants. x2 integrable, 1 k ij ðsÞ ¼ 1. P þx1 2 ðHolling typeÞ and 1  ecx1 ðIvlev typeÞ represent the functional response of 1

predator to prey. The biological background for (5) can be found in [3,15,16]. Let C þ ¼ f/ ¼ ð/1 ; /2 ; /3 Þ : /i ðtÞ is continuous and nonnegative on R and/i ð0Þ > 0; i ¼ 1; 2; 3g. In this paper, we always assume that solutions of (5) satisfy the initial conditions: xi ðsÞ ¼ ui ðsÞ;

yðsÞ ¼ wðsÞ;

ði ¼ 1; 2Þ; ðu1 ; u2 ; wÞ 2 C þ ; s 2 ð1; 0:

ð6Þ

The main purpose of this paper is to find a set of easily verifiable sufficient conditions for the permanence and extinction of the system (5). The present paper is organized as follows. In Section 2, we introduce some notations and definitions, give some preliminary results needed in later sections, and then state the main results of this paper. We then prove, in Section 3, the main results of (5) by using analysis technique. Finally, in Section 4, we work out an example. 2. Main results Let f(t) be a continuous x-periodic function defined on [0, +1), we set Z x Ax ðf Þ ¼ x1 f ðtÞ dt; f U ¼ max f ðtÞ; f L ¼ min f ðtÞ: t2½0;x

t2½0;x

0

Definition 2.1. The system x_ ¼ F ðt; xÞ, x 2 Rn is said to be permanent if there are constants M P m > 0 such that every positive solution xðtÞ ¼ ðx1 ðtÞ; . . . ; xn ðtÞÞ 2 Rnþ ¼ fðx1 ; . . . ; xn Þ : xi > 0; i ¼ 1; . . . ; ng of this system, satisfies m 6 lim inf xi ðtÞ 6 lim sup xi ðtÞ 6 M: t!1

t!1

Lemma 2.2. [19] The system (

x_1 ¼ aðtÞx2  bðtÞx1  dðtÞx21 ; x_2 ¼ cðtÞx1  f ðtÞx22 ;

ð7Þ

has a positive x-periodic solution ðx1 ðtÞ; x2 ðtÞÞ which is globally asymptotically stable with respect to R2þ0 ¼ fðx1 ; x2 Þ : x1 > 0; x2 > 0g. Lemma 2.3. For the following nonautonomous differential equation u_ ¼ u½aðtÞ  bðtÞu  cðtÞu2 ;

ð8Þ L

L

where a(t), b(t) and c(t) are x–periodic continuous functions, c , b P 0 and Ax ðbÞ > 0, there is a constant M > 0 such that every positive solution u(t) of (8) satisfies limsupt!1u(t) 6 M. Proof. The proof is obvious, in fact, u_ ¼ u½aðtÞ  bðtÞu  cðtÞu2  6 u½aðtÞ  bðtÞu. From [17], we have there exists a constant M such that the solution x(t) of the Logistic equation x_ ¼ x½aðtÞ  bðtÞx satisfies limsupt!1x(t) 6 M. Using comparison theorem, the proof is completed. h Theorem 2.4. Assume Z

Ax

0

2



x ðt þ sÞð1  ecx1 ðtþsÞ Þ gðtÞ þ hðtÞ k 21 ðsÞ 1 ds 2 P þ x1 ðt þ sÞ 1

! > 0;

where ðx1 ðtÞ; x2 ðtÞÞ is the positive x-periodic solution of (7). Then system (5) is permanent.

ð9Þ

934

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

Let  ( 1) be some positive constant and Z 0  2 ðx ðt þ sÞ þ Þ ð1  ecðx1 ðtþsÞþÞ Þ k 21 ðsÞ 1 ds þ hðtÞ: kðtÞ ¼ gðtÞ þ hðtÞ 2 P þ ðx1 ðt þ sÞ þ Þ 1

Theorem 2.5. Assume Z

Ax and l¼

2

0



x ðt þ sÞð1  ecx1 ðtþsÞ Þ gðtÞ þ hðtÞ k 21 ðsÞ 1 ds 2 P þ x1 ðt þ sÞ 1 Z

! 60

ð10Þ

0

k 22 ðsÞ expfkU sg ds < 1;

ð11Þ

1

where ðx1 ðtÞ; x2 ðtÞÞ is the positive x-periodic solution of (7). Then for any solution (x1, x2, y) of (5), y(t) ! 0 as t ! 1. 3. Proof of main results Lemma 3.1. There exist positive constants Mx and My such that lim sup xi ðtÞ 6 M x ;

lim sup yðtÞ 6 M y :

t!1

t!1

Proof. Obviously, R3þ is a positively invariant set of system (5). Given any positive solution (x1(t), x2(t), y(t)) of (5) with initial conditions (6), we have ( x_1 6 aðtÞx2  bðtÞx1  dðtÞx21 ; x_2 ¼ cðtÞx1  f ðtÞx22 : Next, we consider the following auxiliary equations ( u_1 ¼ aðtÞu2  bðtÞu1  dðtÞu21 ; u_2 ¼ cðtÞu1  f ðtÞu22 ;

ð12Þ

by Lemma 2.2, it follows that (12) has a globally asymptotically stable positive x-periodic solution ðx1 ðtÞ; x2 ðtÞÞ. Let (u1(t), u2(t)) be the solution of (12) with u1(0) = x1(0), u2(0) = x1(0). By the vector comparison theorem [5], we obtain xi ðtÞ 6 ui ðtÞ

ði ¼ 1; 2Þ

for all t P 0. From the global asymptotic stability of ðx1 ðtÞ; x2 ðtÞÞ, for any positive constant e, there exists a T0 > 0 such that for all t P T0, jui ðtÞ  xi ðtÞj < e;

i ¼ 1; 2:

Hence, for all t P T0, we derive xi ðtÞ 6 xi ðtÞ þ e; i ¼ 1; 2: pffiffiffi  Let M x ¼ maxt2½0;x xi ðtÞ þ e; P , we get xi ðtÞ 6 M x ;

i ¼ 1; 2:

Consequently, lim sup xi ðtÞ 6 M x ; t!1

i ¼ 1; 2:

ð13Þ

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

935

Let a(t) = g(t) + h(t) and let the constant s > 0 be such that Z

0

k 22 ðsÞ expðaU sÞ ds > 0:

ð14Þ

s

From

R0 1

k 21 ðsÞ ds ¼ 1, for any t P T0 + s we have

 Z y_ 6 y gðtÞ þ hðtÞ

0

k 21 ðsÞ

1

 x21 ðt þ sÞð1  ecx1 ðtþsÞ Þ ds 6 y½gðtÞ þ hðtÞ ¼ yaðtÞ: P þ x21 ðt þ sÞ

Hence, for any t P t + s P T0 + s (s 6 0) we obtain yðt þ sÞ P yðtÞ exp

Z

tþs

aðnÞ dn P yðtÞ expðaU sÞ: t

It follows from the above inequality that for any t P T0 + 2s, we have " y_ 6 y aðtÞ  yðtÞ  qðtÞ

Z

2 #

0

k 22 ðsÞyðt þ sÞ ds

" 6 y aðtÞ  yðtÞ  qðtÞ

1

" 6 y aðtÞ  yðtÞ  qðtÞ

Z

0

k 22 ðsÞ expðaU sÞ ds

2

Z

#

2 #

0

k 22 ðsÞyðt þ sÞ ds s

y 2 ðtÞ :

s

Let u(t) be the solution of the auxiliary equation " u_ ¼ u aðtÞ  uðtÞ  qðtÞ

Z

0 U

k 22 ðsÞ expða sÞ ds

2

# 2

u ðtÞ

s

with the initial condition u(T0 + 2s) = y(T0 + 2s). Then we derive yðtÞ 6 uðtÞ

for all t P T 0 þ 2s:

ð15Þ

From Lemma 2.3, we know that there is a constant My > 0 such that lim sup uðtÞ 6 M y : t!1

Consequently, by (15), we have ð16Þ

lim sup yðtÞ 6 M y : t!1

The proof is completed.

h

Lemma 3.2. There is a positive constant qx (qx < Mx) such that lim inf xi ðtÞ P qx : t!1

Proof. By Lemma 3.1, there exists a positive constant T1 > T0 + 2s such that 0 < xi ðtÞ 6 M x ði ¼ 1; 2Þ;

0 < yðtÞ 6 M y ; t P T 1 :

Obviously, there exists a constant r > 0 such that Z r H0 k 12 ðsÞ ds < M y ; 1

ð17Þ

936

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

where H0 = sup{y(t + s)jt P 0, s 6 0}. Hence, From (13) and (16), for every t P T1 + r, we have Z x2 ð1  ecx1 Þ r x_1 ¼ aðtÞx2  bðtÞx1  dðtÞx21  pðtÞ 1 k 12 ðsÞyðt þ sÞ ds P þ x21 1 Z x2 ð1  ecx1 Þ 0 k 12 ðsÞyðt þ sÞ ds  pðtÞ 1 P þ x21 r Z x21 ð1  ecx1 Þ r 2 k 12 ðsÞ ds P aðtÞx2  bðtÞx1  dðtÞx1  pðtÞH 0 P þ x21 1 Z x21 ð1  ecx1 Þ 0 k 12 ðsÞ ds  pðtÞM y P þ x21 r P aðtÞx2  bðtÞx1  dðtÞx21  2M y pðtÞ

ð18Þ

ð19Þ

M x ð1  ecM x Þ x1 ; P þ M 2x

x_2 ¼ cðtÞx1  f ðtÞx22 : Consider the following auxiliary system: 8 h i < u_ ¼ aðtÞu  bðtÞ þ 2M x M y pðtÞð1ecM x Þ u  dðtÞu2 ; 1 2 1 1 P þM 2x : u_ ¼ cðtÞu  f ðtÞu2 : 2

1

ð20Þ

2

Let (u1(t), u2(t)) is the solution of system (20) with the initial condition (u1(T1 + r), u2(T1 + r)) = (x1(T1 + r), x2(T1 + r)), then for all t P T1 + r, xi ðtÞ P ui ðtÞ: By Lemma 2.2, (20) has a positive x-periodic solution ð ub1  ðtÞ; ub2  ðtÞÞ, which is globally asymptotically stable. By the global asymptotic stability of ubi  ðtÞ ði ¼ 1; 2Þ, for any a sufficiently small e* (>0), there exists T2 > T1 + r such that for all t P T2, ui ðtÞ P ubi  ðtÞ  e :

: Hence, for all t P T2, xi ðtÞ P qx ¼ max06t6x f ubi  ðtÞ  e g. So we have lim inf xi ðtÞ P qx :



t!1

Lemma 3.3. Suppose that (9) holds, then there is a positive constant .y (.y < My) such that ð21Þ

lim sup yðtÞ > .y : t!1



Proof. By (9), we can choose constant e0 < 12 mint2½0;x fx1 ðtÞg, where x1 ðtÞ; x2 ðtÞ is the unique positive solution of system (12) such that Ax ðwe0 ðtÞÞ > 0;

ð22Þ

where we0 ðtÞ ¼ gðtÞ þ hðtÞ

Z

2

0

k 21 ðsÞ 1



ðx1 ðt þ sÞ  e0 Þ ð1  ecðx1 ðtþsÞe0 Þ Þ P þ ðx1 ðt þ sÞ  e0 Þ

2

Consider the following equations with a positive parameter l: 8 h i < x_ ¼ aðtÞx  bðtÞ þ 2lM x pðtÞð1ecM x Þ x  dðtÞx2 ; 1 2 1 1 P þM 2x : x_ ¼ cðtÞx  f ðtÞx2 : 2

1

2

ds  4qðtÞe20  e0 :

ð23Þ

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

937



 By Lemma 2.2, (23) has a positive x-periodic solution x1l ðtÞ; x2l ðtÞ , which is globally asymptotically stable. Let (x1l(t), x2l(t)) be the solution of (23) with initial condition xil ð0Þ ¼ xi ð0Þ, where x ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ is the positive periodic solution of (7). Hence, for the above e0, there exists T3 > T2 such that jx1l ðtÞ  x1l ðtÞj < e0 =4

ð24Þ

for t P T 3 :

By the continuity of the solution in the parameter, we have x1l ðtÞ ! x1 ðtÞ uniformly in [T3, T3 + x] as l ! 0. Hence for e0 > 0, there exists l0 = l0(e0) (0 < l0 < e0) such that jx1l ðtÞ  x1 ðtÞj < e0 =4;

0 6 l 6 l0 ; t 2 ½T 3 ; T 3 þ x:

ð25Þ

Thus from (24) and (25), we get jx1l ðtÞ  x1 ðtÞj < e0 =2;

0 6 l 6 l0 ; t 2 ½T 3 ; T 3 þ x:

Since x1l ðtÞ and x1 ðtÞ are all x-periodic, we have jx1l ðtÞ  x1 ðtÞj < e0 =2;

ð26Þ

0 6 l 6 l0 ; t P 0:

Choose a constant l1 (0 < l1 < l0, l1 < e0), from (26), we derive x1l1 ðtÞ P x1 ðtÞ  e0 =2;

ð27Þ

t P 0:

Suppose that (21) is not true, then for the above e0, there exists / 2 C+ such that lim sup yðt; /Þ < l1 ; t!1

where (x1 (t, /), x2(t, /), y(t, /)) is the solution of (5) with the initial condition (x1(h), x2(h), x3(h)) = (/(h), /(h), /(h)). So, there exists T4 (>T3) such that yðt; /Þ < l1 ;

ð28Þ

t P T 4:

On the other hand, Lemma 3.1 shows that there exist an enough large constant T5 (>T4) such that x1 ðt; /Þ < M x ; t P T 5 : R0 Also, from 1 k ij ðsÞ ds ¼ 1 ði; j ¼ 1; 2Þ, we choose a positive constant s0 such that Z s0 H0 kðsÞ ds < l1 ;

ð29Þ

ð30Þ

1

where k(t) = k12(t) + k21(t) + k22(t) and H0 is defined in the proof of Lemma 3.2. For any t P T5 + s0, we have x_1 ðt; /Þ ¼ aðtÞx2 ðt; /Þ  bðtÞx1 ðt; /Þ  dðtÞx21 ðt; /Þ  pðtÞ  pðtÞ

x21 ðt; /Þð1  ecx1 ðt;/Þ Þ P þ x21 ðt; /Þ

P aðtÞx2 ðt; /Þ  bðtÞx1 ðt; /Þ   pðtÞl1

Z

x21 ðt; /Þð1  ecx1 ðt;/Þ Þ P þ x21 ðt; /Þ

Z

r0

0

k 12 ðsÞyðt þ sÞ ds

r0

dðtÞx21 ðt; /Þ

x21 ðt; /Þð1  ecx1 ðt;/Þ Þ P þ x21 ðt; /Þ

k 12 ðsÞyðt þ sÞ ds

1

Z

x2 ðt; /Þð1  ecx1 ðt;/Þ Þ  pðtÞH 0 1 P þ x21 ðt; /Þ

Z

r0

k 12 ðsÞ ds 1

0

k 12 ðsÞ ds

r0

P aðtÞx2 ðt; /Þ  bðtÞx1 ðt; /Þ  dðtÞx21 ðt; /Þ  2l1 pðtÞ

M x ð1  ecM x Þ x1 ðt; /Þ; P þ M 2x

x_2 ðt; /Þ ¼ cðtÞx1 ðt; /Þ  f ðtÞx22 ðt; /Þ: Let ðu1l1 ; u2l1 Þ be the solution of (23) with l = l1 and ðu1l1 ðT 5 þ s0 Þ; u2l1 ðT 5 þ s0 ÞÞ ¼ ðx1 ðT 5 þ s0 Þ; x2 ðT 5 þ s0 ÞÞ, then by the vector comparison theorem, we obtain xi ðt; /Þ P uil1 ðtÞ;

i ¼ 1; 2; t P T 5 þ s0 :

ð31Þ

938

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

  By the global asymptotic stability of x1l1 ðtÞ; x2l1 ðtÞ , for the given e0 > 0 there exists T6 > T5 + s0 such that u1l1 ðtÞ > x1l1 ðtÞ  e0 =2;

t P T6

and hence, by (27), we derive x1 ðt; /Þ > x1 ðtÞ  e0 ;

ð32Þ

t P T 6:

Therefore, for t P T6 + s0, we have " _ /Þ ¼ yðt; /Þ gðtÞ þ hðtÞ yðt;

Z

0

k 21 ðsÞ

ðx1 ðt þ s; /ÞÞ2 ð1  ecx1 ðtþs;/Þ Þ P þ ðx1 ðt þ s;/ÞÞ2

1

" P yðt;/Þ gðtÞ þ hðtÞ " P yðt;/Þ gðtÞ þ hðtÞ

Z

k 21 ðsÞ

k 22 ðsÞyðt þ s;/Þds

P þ ðx1 ðt þ s;/Þ  e0 Þ2

0

ðx1 ðt þ s; /Þ  e0 Þ2 ð1  ecðx1 ðtþs;/Þe0 Þ Þ

ds  l1  4qðtÞl21 #



k 21 ðsÞ

P þ ðx1 ðt þ s;/Þ  e0 Þ2

1

#



ðx1 ðt þ s; /Þ  e0 Þ ð1  ecðx1 ðtþs;/Þe0 Þ Þ

1

Z

2 #

0

1

2

0

ds  yðt;/Þ  qðtÞ

Z

ds  e0  4qðtÞe20

P yðt;/Þwe0 ðtÞ:

Integrating the above inequality from T6 + s0 to t yields yðt; /Þ P yðT 6 þ s0 Þ exp

Z

t T 6 þs0

 we0 ðsÞ ds :

By (22) we know that y(t, /) ! 1 as t ! 1, which is a contradiction. The proof is completed.

h

Lemma 3.4. Assume that (9) holds, there exists a positive constant dy (dy < My) such that any solution (x1, x2, y) of system (5) with initial conditions satisfies ð33Þ

lim inf yðtÞ P dy : t!1

Proof. Suppose that (33) is not true, there must exist a sequence {/k}  C+ such that lim inf yðt; /k Þ < t!1

.y ðk þ 1Þ

2

;

k ¼ 1; 2; . . .

and by Lemma 3.3, we have limsupt!1y(t, uk) > .y, k = 1, 2, . . . Hence, for each k, we choose two time seðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ ðkÞ ðkÞ ðkÞ quences fsðkÞ q g and ft q g, satisfying 0 < s1 < t 1 < s2 < t 2 <    < sq < t q <    and sq ! 1 as q ! 1, and

yðsðkÞ q ; /k Þ ¼ .y ðk þ 1Þ

2

.y ; kþ1

< yðt; /k Þ <

yðtðkÞ q ; /k Þ ¼ .y ; kþ1

.y ðk þ 1Þ

2

;

ðkÞ t 2 ðsðkÞ q ; t q Þ:

ð34Þ ð35Þ

By Lemma 3.1, for a given positive integer k, there exists Te ðkÞ > 0 such that xi(t, /k) 6 Mx (i = 1, 2) and y(t, /k) 6 My for all t P Te ðkÞ . Further, there is a constant r(k) > 0 such that ðkÞ H1

Z

rðkÞ

kðsÞ ds < M y ; 1

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944 ðkÞ

939 ðkÞ

where H 1 ¼ supfyðt þ s; /k Þ : t P 0; s 6 0g. Because of sðkÞ q ! 1 as q ! 1, there is a positive integer K 1 ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ e e such that sq > T þ r as q P K 1 . For any t P T þ rðkÞ , we have " y_ ðt; /k Þ P yðt; /k Þ gðtÞ  yðt; /k Þ  qðtÞ

Z

0

2 # k 22 ðsÞyðt þ s; /k Þ ds

1

2 P yðt; /k Þ4gðtÞ  yðt; /k Þ  qðtÞ

Z

rðkÞ

k 22 ðsÞyðt þ s; /k Þ ds þ

1

Z

0 rðkÞ

!2 3 k 22 ðsÞyðt þ s; /k Þ ds 5

P yðt; /k Þ½gðtÞ  M y  4M 2y qðtÞ: ðkÞ

ðkÞ Integrating the above inequality from sðkÞ q to t q , for any q P K 1 , we get

ðkÞ yðtðkÞ q ; /k Þ P yðsq ; /k Þ exp

Z

!

ðkÞ

tq ðkÞ

½gðtÞ  M y  4M 2y qðtÞ dt :

sq

Obviously, we derive Z

ðkÞ

tq

ðkÞ

½gðtÞ þ M y þ 4M 2y qðtÞ dt P lnðk þ 1Þ for q P K 1 :

ðkÞ

sq

Hence, in view of the periodicity of g(t) and q(t), we get ðkÞ

ðkÞ tðkÞ q  sq ! 1;

as k ! 1; q P K 1 :

ð36Þ

By (22), (34) and (36), there are positive constants T and N0 such that

and

.y < e0 ; kþ1 > 2T ;

yðsðkÞ q ; /k Þ ¼

ð37Þ

ðkÞ tðkÞ q  sq

ð38Þ

Z

j

0

we0 ðtÞ dt > 0;

ð39Þ

ðkÞ

for k P N0, q P K 1 , and j > T. (37) implies that ðkÞ t 2 ½sðkÞ q ; t q ;

yðt; /k Þ < e0 ;

ð40Þ

ðkÞ

for k P N0, q P K 1 . Noticing that sðkÞ q ! 1 as q ! 1 and ðkÞ ðkÞ ðkÞ K 2 > K 1 such that for all q > K 2 , we obtain ðkÞ H1

Z eT ðkÞ sðkÞ 0 q r 1

R0 1

k ij ðsÞ ds ¼ 1 ði; j ¼ 1; 2Þ, for any k there exists

1 kðsÞ ds < e0 2

ð41Þ

and Z

r0

My 1

1 kðsÞ ds < e0 ; 2

ð42Þ

where r0 > 0 and k(t) = k12 + k21(t) + k22(t). By (36), there exists a positive integer N1 such that ðkÞ 0 tðkÞ q  sq > r

ðkÞ

for k > N 1 ; q P K 2 :

940

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944 ðkÞ

0 ðkÞ For k > N1, q P K 2 and sðkÞ q þ r 6 t 6 t q , from (40)–(42), we have

dx1 ðt; /k Þ ¼ aðtÞx2 ðt; /k Þ  bðtÞx1 ðt; /k Þ  dðtÞx21 ðt; /k Þ dt Z ðkÞ pðtÞx21 ðt; /k Þð1  ecx1 ðt;/k Þ Þ eT k 12 ðu  tÞyðu; /k Þ du  P þ x21 ðt; /k Þ 1 Z ðkÞ pðtÞx21 ðt; /k Þð1  ecx1 ðt;/k Þ Þ sq  k 12 ðu  tÞyðu; /k Þ du P þ x21 ðt; /k Þ eT ðkÞ Z pðtÞx21 ðt; /k Þð1  ecx1 ðt;/k Þ Þ t  k 12 ðu  tÞyðu; /k Þ du ðkÞ P þ x21 ðt; /k Þ sq P aðtÞx2 ðt; /k Þ  bðtÞx1 ðt; /k Þ  dðtÞx21 ðt; /k Þ Z ðkÞ pðtÞx21 ðt; /k Þð1  ecx1 ðt;/k Þ Þ ðkÞ eT t  H k 12 ðsÞ ds 1 P þ x21 ðt; /k Þ 1 Z sðkÞ q t pðtÞx21 ðt; /k Þð1  ecx1 ðt;/k Þ Þ M  k 12 ðsÞ ds y 2 P þ x1 ðt; /k Þ 1 Z 0 pðtÞx21 ðt; /k Þð1  ecx1 ðt;/k Þ Þ e0 k 12 ðsÞ ds  P þ x21 ðt; /k Þ 1 P aðtÞx2 ðt; /k Þ  bðtÞx1 ðt; /k Þ  dðtÞx21 ðt; /k Þ Z ðkÞ pðtÞM x x1 ðt; /k Þð1  ecM x Þ ðkÞ eT t  H k 12 ðsÞ ds 1 P þ M 2x 1 Z sðkÞ q t pðtÞM x x1 ðt; /k Þð1  ecM x Þ M k 12 ðsÞ ds  y P þ M 2x 1 Z 0 pðtÞM x x1 ðt; /k Þð1  ecM x Þ  e0 k 12 ðsÞ ds P þ M 2x 1   2e0 pðtÞM x ð1  ecM x Þ ¼ aðtÞx2 ðt; /k Þ  bðtÞ þ x1 ðt; /k Þ  dðtÞx21 ðt; /k Þ; P þ M 2x dx2 ðt; /k Þ ¼ cðtÞx1 ðt; /k Þ  f ðtÞx22 ðt; /k Þ: dt 0 ðkÞ 0 ðkÞ 0 Let ðu1e0 ; u2e0 Þ be the solution of (23) with l = e0 and ðu1l1 ðsðkÞ q þ s Þ; u2l1 ðsq þ s ÞÞ ¼ ðx1 ðsq þ s Þ; ðkÞ 0 x2 ðsq þ s ÞÞ, then by the vector comparison theorem, we obtain

xi ðt; /k Þ P uie0 ðtÞ;

0 ðkÞ i ¼ 1; 2; t 2 ½sðkÞ q þ s ; t q :

ð43Þ ðkÞ

ðkÞ

From limq!1 sðkÞ q ¼ 1 and Lemmas 3.1 and 3.2, we obtain that for any k there is a K 3 > K 2 such that for any ðkÞ q P K3 , 0 qx 6 xi ðsðkÞ q þ r ; /k Þ 6 M x ;

i ¼ 1; 2:

For l = e0, Eq. (23) has a globally asymptotically stable positive x-periodic solution ðx1l ðtÞ; x2l ðtÞÞ. From the periodicity of (23) we know that the periodic solution ðx1l ðtÞ; x2l ðtÞÞ also is globally uniformly asymptotically stable. Hence, there is a T7 > T, and T7 is independent of any k and q, such that e0 u1e0 ðtÞ > x1l ðtÞ  2 ðkÞ

0 for all t P T 7 þ sðkÞ q þ r and q P K 3 . Consequently, by (27),

u1e0 ðtÞ > x1 ðtÞ  e0

ð44Þ

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

941

ðkÞ

0 ðkÞ ðkÞ for all t P T 7 þ sðkÞ q þ r and q P K 3 . By (36), there is a N2 P N1 such that t q  sq P 2T for all k P N2 and ðkÞ 0 q P K 3 , where T P T7 + r . Hence, from (43) and (44) we obtain

ð45Þ x1 ðt; /k Þ P x1 ðtÞ  e0 ; h i ðkÞ ðkÞ ðkÞ ðkÞ for all t 2 T þ sðkÞ , k P N2 and q P K 3 . Since, for any t 2 ½T þ sðkÞ q ; tq q þ r0 ; t q , k P N2 and q P K 3 , by (5), (41) and (42), we have 2 Z 0 dyðt; /k Þ x2 ðt þ s; /k Þð1  ecx1 ðtþs;/k Þ Þ P yðt; /k Þ4gðtÞ þ hðtÞ ds  yðt; /k Þ  qðtÞ k 21 ðsÞ 1 dt P þ x21 ðt þ s; /k Þ r0 

Z eT ðkÞ 1

k 22 ðu  tÞyðu; /k Þ du þ

2

ðkÞ

sq

eT ðkÞ

k 22 ðu  tÞyðu; /k Þ du þ

Z

t ðkÞ

!2 3 k 22 ðu  tÞyðu; /k Þ du 5

sq

0

x21 ðt þ s; /k Þð1  ecx1 ðtþs;/k Þ Þ ds  yðt; /k Þ  qðtÞ P þ x21 ðt þ s; /k Þ r0 !2 3 Z eT ðkÞ t Z sðkÞ Z 0 q t ðkÞ k 22 ðsÞ ds þ M y k 22 ðsÞ ds þ e0 k 22 ðsÞ ds 5 H1

P yðt; /k Þ4gðtÞ þ hðtÞ



Z

Z

"

k 21 ðsÞ

1

P yðt; /k Þ gðtÞ þ hðtÞ

1

Z

1

2

0

k 21 ðsÞ

r0

#



ðx1 ðt þ s; /k Þ  e0 Þ ð1  ecðx1 ðtþs;/k Þe0 Þ Þ P þ ðx1 ðt þ s; /k Þ  e0 Þ

2

ds  e0 

4qðtÞe20

¼ yðt; /k Þwe0 ðtÞ: ðkÞ

0 ðkÞ Integrating from T þ sðkÞ q þ r to t q for any k P N2 and q P K 3 we obtain Z tðkÞ q ðkÞ 0 yðtðkÞ ; / Þ P yðT þ s þ r ; / Þ exp we0 ðtÞ dt: k k q q ðkÞ

T þsq þr0

Hence, by (34) and (35) we finally have Z tðkÞ q .y .y .y P exp we0 ðtÞ dt > : 2 2 2 ðkÞ ðk þ 1Þ ðk þ 1Þ ðk þ 1Þ T þsq þr0 This leads to a contradiction. The proof is completed.

h

Proof of Theorem 2.4. This theorem now follows from Lemmas 3.1–3.4. h Proof of Theorem 2.5. Assume # Z 0 Z x"  2 x1 ðt þ sÞð1  ecx1 ðtþsÞ Þ ds dt 6 0: gðtÞ þ hðtÞ k 21 ðsÞ P þ x1 2 ðt þ sÞ 0 1 We will show that limt!1y(t) = 0. In fact, we know that for any given 0 < e < 1, there exist  < e and 0 > 0 such that # Z x" Z 0  2 ðx1 ðt þ sÞ þ Þ ð1  ecðx1 ðtþsÞþÞ Þ 1 gðtÞ þ hðtÞ k 21 ðsÞ ds þ hðtÞ  qðtÞl2 dt 2 P þ ðx1 ðt þ sÞ þ Þ2 0 1 Z x Z x 1 hðtÞ dt  l2 qðtÞ dt < 0 ; ð46Þ 6 2 0 0 R0 where l ¼ 1 k 22 ðsÞ expðkU sÞ ds. Since ( x_ 1 6 aðtÞx2  bðtÞx1  dðtÞx21 ; x_ 2 ¼ cðtÞx1  f ðtÞx22

942

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

for all t P 0. Let ðx1 ðtÞ; x2 ðtÞÞ be the solution of (7) with initial condition xi ð0Þ ¼ xi ð0Þ ði ¼ 1; 2Þ. By the vector comparison theorem we obtain xi ðtÞ 6 xi ðtÞ (i = 1, 2), t P 0. Obviously, by the global asymptotic stability of x*(t), there is a T , for all t P T , we have xi ðtÞ 6 xi ðtÞ þ  ði ¼ 1; 2Þ:

ð47Þ

Choose a constant s1 > 0 such that Z s1 kðsÞ ds < ; 1 rffiffiffi Z 0 l U : k 22 ðsÞ expðk sÞ ds > 2 s1

ð48Þ ð49Þ

For any t P T þ s1 , by (47) and (48) we have " # Z 0  2 ðx1 ðt þ sÞÞ ð1  ecx1 ðtþsÞ Þ k 21 ðsÞ ds þ hðtÞ y_ 6 y gðtÞ þ hðtÞ 2 P þ ðx1 ðt þ sÞÞ 1 " # Z 0  2 ðx1 ðt þ sÞ þ Þ ð1  ecðx1 ðtþsÞþÞ Þ 6 y gðtÞ þ hðtÞ k 21 ðsÞ ds þ hðtÞ 6 ykðtÞ: 2 P þ ðx1 ðt þ sÞ þ Þ s1

ð50Þ

Hence, by (49), for any t P t þ s P T þ s1 , we obtain "

Z y_ 6 y kðtÞ  qðtÞ 

2 #

0

k 22 ðsÞyðt þ sÞ ds

" 6 y kðtÞ  qðtÞ

s1

Z

0

2 # k 22 ðsÞ expðk sÞ ds y 2 U

s1



1 6 y kðtÞ  lqðtÞy 2 : 2 If y(t) P  for all t P T þ 2s1 , then we have 

 1 2 y_ 6 y kðtÞ  lqðtÞ : 2

ð51Þ

Consequently, by (46) we obtain yðtÞ 6 yðT þ 2s1 Þ exp

Z

t T þ2s1



 1 kðsÞ  lqðsÞ2 ds ! 0 2

as t ! 1, which leadsto a contradiction. Hence, there is a t1 P T þ 2s1 such that y(t1) < . Let MðÞ ¼ maxtP0 jkðtÞj þ 12 lqðtÞ2 . We know that M() is bounded for  2 [0, 1]. We then show that yðtÞ 6  expðMðÞxÞ;

ð52Þ

t P t1 :

Otherwise, there are t3 > t2 > t1 such that y(t3) >  exp(M()x), y(t2) =  and y(t) >  for all t 2 (t2, t3]. Let h P 0 be an integer such that t3 2 (t2 + hx, t2 + (h + 1)x]. Then from (51) we have 

Z t2 þhx Z t3   1 1 kðtÞ  lqðtÞ2 dt ¼  exp þ kðtÞ  lqðtÞ2 dt 2 2 t2 t2 þhx t2

Z t3    1 <  exp kðtÞ  lqðtÞ2 dt 6  expðMðÞxÞ: 2 t2 þhx

 expðMðÞxÞ < yðt3 Þ 6 yðt2 Þ exp

Z

t3



This leads to a contradiction. Hence, inequality (52) holds. Further, in view of the arbitrariness of , we have y(t) ! 0 as t ! 1. The proof is completed. h

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

943

4. Example Example 4.1 8 x2 ð1e2x1 Þ R 0 2 > > x2  x1  x21  12 1 2þx2 k ðsÞyðt þ sÞ ds; x_1 ¼ 2þsin > 1 12 t > 1 > < 2cos t x2 ; x_2 ¼ 2x1  ð2þsin tÞ2 2  > R 2  > > R0 x21 ðtþsÞð1e2x1 ðtþsÞ Þ 0 > 1 > ds  y  1 k 22 ðsÞyðt þ sÞ ds : : y_ ¼ y ð1  eÞ þ 3ð2 þ sin tÞ 1 k 21 ðsÞ 2þx2 ðtþsÞ

ð53Þ

1

Example 4.2 8 x2 ð1e2x1 Þ R 0 2 > > x_1 ¼ 2þsin x2  x1  x21  12 1 2þx2 k ðsÞyðt þ sÞ ds; > 1 12 t > 1 > > > 2cos t > x2 ; x_2 ¼ 2x1  ð2þsin > > tÞ2 2 <  R 2 x2 ðtþsÞð1e2x1 ðtþsÞ Þ tÞ 0 _ y ¼ y ð3  cos2 tÞ þ ðe 1Þð2þsin k 21 ðsÞ 1 2þx2 ðtþsÞ ds > 2 > 1 e > 1 > > > 2  >

 R 0 > > sin t > y  5 þ k ðsÞyðt þ sÞ ds : : 1 22 3 We consider the subsystem of (53) and (54): ( 2 x_1 ¼ 2þsin x  x1  x21 ; t 2 2cos t x_2 ¼ 2x1  ð2þsin x2 : tÞ2 2

ð54Þ

ð55Þ

Obviously, (55) admits an unique positive 2p-periodic solution (1, 2 + sin t). By simple computation, we derive system (53) is permanent, and y(t) ! 0 as t ! 1 in system (54). References [1] J.K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977. [2] V.B. Kolmanovskii, V.R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986. [3] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Boston, 1991. [4] Kuang Yang, Delay Differential Equations with Applications in Population Dynamics, Springer, New York, 1993. [5] V. Lakshmikantham, V.M. Matrosov, S. Sivasundaram, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Kluwer Academic Publishers, Dordrecht, Boston, London, 1991. [6] J.R. Bence, R.M. Nisbet, Space limited recruitment in open systems: The importance of time delays, Ecology 70 (1989) 1434–1441. [7] W.G. Aiello, H.I. Freedman, A time delay model of single species growth with stage structure, Math. Biosci. 101 (1990) 139–156. [8] W. Wang, L. Chen, A predator–prey system with stage-structure for predator, Comput. Math. Appl. 33 (8) (1997) 83–91. [9] Y. Xiao, L. Chen, Global stability of a predator–prey system with stage structure for the predator, Acta Math. Sinica, English Series 19 (2) (2003) 1–11. [10] K.G. Magnusson, Destabilizing effect of cannibalism on a structured predator–prey system, Math. Biosci. 155 (1999) 61–75. [11] H.L. Smith, Systems of ordinary differential equations which generate an order preserving flow, SIAM Rev. 30 (1988) 87–98. [12] X. Zhang, L. Chen, A.U. Neuman, The stage-structure predator–prey model and optimal harvesting policy, Math. Biosci. 101 (2000) 139–153. [13] J.M. Cushing, Periodic time dependent predator–prey system, SIAM J. Math. 32 (1977) 82–95. [14] G. Krukonis, W.M. Schaffer, Population cycles in mammals and birds: does periodicity scale with body size? J. Theor. Biol. 148 (1991) 469–493. [15] J.A. Cui, X.Y. Song, Permanence of a predator–prey system with stage structure, Discrete Continuous Dynam. Syst., Series B 4 (3) (2004) 547–554. [16] Z. Teng, L. Chen, Permanence and extinction of periodic predator–prey systems in a patchy environment with delay, Nonlinear Anal. 4 (2003) 335–364. [17] Z. Teng, The almost periodic Kolmogorov competitive systems, Nonlinear Anal. 42 (2000) 1221–1230. [18] J.A. Cui, Y. Sun, Permanence of predator–prey system with infinite delay, Electron. J. Differ. Equat. 81 (2004) 1–12. [19] J. Cui, L. Chen, W. Wang, The effect of dispersal on population growth with stage-structure, Comput. Math. Appl. 39 (2000) 91–102.

944

H. Zhang et al. / Applied Mathematics and Computation 184 (2007) 931–944

[20] C.S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memorand. Entomol. Soc. Canada 45 (1965). [21] C.S. Holling, The functional response of invertebrate prey to prey density, Memorand. Entomol. Soc. Canada 48 (1966) 1–48. [22] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol. 44 (1975) 331–340. [23] D.L. DeAngelis, R.A. Goldstein, R.V. O’Neill, A model for tropic interaction, Ecology 56 (1975) 881–892. [24] P.H. Crowley, E.K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. North Am. Benthol. Soc. 8 (1989) 211–221. [25] M.P. Hassell, G.C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature 223 (1969) 1133–1136.