Permanence and extinction analysis for a delayed periodic predator–prey system with Holling type II response function and diffusion

Applied Mathematics and Computation 216 (2010) 3002–3015

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

Permanence and extinction analysis for a delayed periodic predator–prey system with Holling type II response function and diffusion Zijian Liu a,*, Shouming Zhong a,b a b

School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, PR China Key Laboratory for NeuroInformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu 610054, PR China

a r t i c l e

i n f o

Keywords: Lotka–Volterra system Dispersal Delay Permanence Periodic solution

a b s t r a c t In this paper, it is studied that two species predator–prey Lotka–Volterra type dispersal system with delay and Holling type II response function, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of the patches and cannot disperse. Sufficient conditions of integrable form for the boundedness, permanence, extinction and the existence of positive periodic solution are established, respectively. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction The effect of environment change in the growth and diffusion of species in a heterogeneous habitat is a subject of considerable interest in the ecological literature. Since the pioneering theoretical work by Skellam [20], many works have focused on the effect of spatial factors which play a crucial rule in the persistence and stability of a population [4,5,9,16,27,29,32]. And two-species predator–prey discrete dispersal systems have been extensively studied in many articles (see [8,11,14,23]). Most of the existing models deal with autonomous population systems and indicate that a dispersal process in an ecological system is often considered to have a stabilizing influence on the system [6], but it is also probably destabilizing the system [18]. In [16], Kuang and Takeuchi showed that discrete diffusions are capable of both stabilizing and destabilizing a given ecosystem. For time-dependent predator–prey systems in patchy environments, existing results have largely been restricted to permanence and extinction analysis due to the increased complexity of global analysis. Owing to many natural and man-made factors such as low birth rate, high death rate, hunting, decreasing habitats, aggravating living environment, etc., some predator animals become very rare and even liable to extinction. Hence, in [8], Cui and Chen studied a time-dependent predator–prey system where the predator and prey disperse among patches, and showed that dispersal can make the prey and predator permanent even if the prey live in some poor patches. And in many other articles ([7,21,24]), authors usually assumed that the predator’s density is regulated only by predation and is density-independent, which is much identical with the real biological background. So, in this paper, we will consider the instance that the predator’s density is regulated only by predation and is density-independent. On the other hand, based on experiments, in [12], Holling suggested three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka–Volterra system more realistic. For example, he proposed the form

UðxÞ ¼

mx ; aþx

* Corresponding author. E-mail addresses: [email protected] (Z. Liu), [email protected] (S. Zhong). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.012

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

3003

as a Holling type II response function, the study of this can be seen in many articles [1,19,28,30]. He also proposed the Holling type III response function in the following form:

UðxÞ ¼

mx2 ; a þ x2

which also have been discussed in many articles no matter autonomous and nonautonomous, such as [13,17]. However, the effects of a periodically varying environment and time delay play an important role in the permanence and extinction of population dynamic systems (e.g., [2,3,7,10,15]). Thus, the assumptions of periodicity of the parameters and time delay of species during the course of dispersion and conversion of nutrients into the reproduction are effective way to characterize and investigate dispersal population systems. Moreover, in [31], the authors discussed two species time-delayed periodic predator–prey Lotka–Volterra type systems with dispersal but without response function, the authors get some sufficient conditions on the boundedness, permanence and existence of positive periodic solution for the systems, which enlighten us to study this type nonautonomous periodic predator–prey systems with dispersal and Holling type II response function, and to see wether can we get some good conditions or not. Motivated by the arguments above, in this paper, we consider the following two species periodic predator–prey Lotka– Volterra type system with Holling type II response function:

  X Z t n dx1 ðtÞ yðsÞ k1 ðt  sÞ d1j ðtÞ½xj ðtÞ  x1 ðtÞ; ¼ x1 ðtÞ a1 ðtÞ  b1 ðtÞx1 ðtÞ  cðtÞ ds þ dt 1 þ mx1 ðsÞ tr j¼1 n X dxi ðtÞ dij ðtÞ½xj ðtÞ  xi ðtÞ; ¼ xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ þ dt j¼1   Z t dyðtÞ x1 ðsÞ k2 ðt  sÞ ¼ yðtÞ eðtÞ þ f ðtÞ ds ; dt 1 þ mx1 ðsÞ tp

i ¼ 2; 3; . . . ; n; ð1:1Þ

where t 2 R+0 = [0, 1), xi(i 2 I = {1, 2, . . . , n}) denote the population density of the prey species in the ith patch, y presents the population density of the predator species, ai(t), bi(t), the intrinsic growth rate and density-dependent coefficient of the prey in the ith patch, respectively, c(t) the capturing rate of the predator, e(t) the death rate of the predator, f(t) the rate of conversion of nutrients into the reproduction of the predator and dij(t)(i, j 2 I, i – j) the dispersal rate of the prey species from the ith patch to the jth patch, m is a nonnegative constant, the term x1/(1 + mx1) denotes the functional response of the predator. And the function ki(s)(i = 1, 2), defined on [0, r] and [0, p] respectively, are both nonnegative and integrable. That is, Rp Rt Rr k1 ðsÞds ¼ 1 and 0 k2 ðsÞds ¼ 1. Then we know that the term tr k1 ðt  sÞyðsÞ=ð1 þ mx1 ðsÞÞds represents the negative effect 0 to the growing rate of the prey population at time t due to the intervention of predator during the time t  r to t, and the Rt term tp k2 ðt  sÞx1 ðsÞ=ð1 þ mx1 ðsÞÞds represents the positive effect to the growing rate of the predator population at time t due to the predation during the time t  p to t. In this paper, we always assume that the functions ai(t), bi(t), dij(t)(i, j 2 I, i – j),c(t),e(t) and f(t) are continuous and periodic defined on R+0 with common period x > 0 and dii = 0(i 2 I) for all t 2 R+0. Our main purpose is to establish a series of criteria on the ultimate boundedness, permanence, extinction and the existence of the periodic solution of the prey and predator species for system (1.1). The method used in this paper is motivated by the works given by Teng and Chen in [26], and Zhang and Teng in [31]. The organization of this paper is as follows. Section 2 presents some basic assumptions and useful lemmas. In Section 3, we state and prove the main results for two species periodic predator–prey Lotka–Volterra type system with Holling type II response function and diffusion. Finally, a conclusion is given in Section 4. 2. Preliminaries In this section, we will give some preliminary knowledge that will be used in the following sections. Let set C+ = {/ = (/1, /2, . . . , /n+1) 2 C: /i(i = 1, 2, . . . , n + 1) is nonnegative on [s, 0] and /i(0) > 0}, where s = max{p, r}. For ecological reasons, we always assume that solutions of system (1.1) satisfy the following initial conditions:

xi ðsÞ ¼ /i ðsÞ;

yðsÞ ¼ /nþ1 ðsÞ for all s 2 ½s; 0; i 2 I;

ð2:1Þ

where / = (/1, /2, . . . , /n+1) 2 C+. It is easy to prove that the functional of the right side system (1.1) is continuous and satisfy a local Lipschitz condition with respect to / in the space R  C. Therefore, by the fundamental theory of functional differential equations, for any / 2 C+ system (1.1) has a unique solution (x(t,/), y(t,/)) = (x1(t,/), x2(t,/), . . . , ,xn(t,/), y(t,/)) satisfying the initial condition (2.1). It is also easy to prove that the solution (x(t,/), y(t,/)) is positive, that is xi(t,/) > 0(i 2 I) and y(t,/) > 0 in its maximal interval of the existence. In this paper, such a solution of system (1.1) is called a positive solution. We define that system (1.1) to be permanent, if there are constants M P m > 0 such that

m 6 lim inf xi ðtÞ 6 lim sup xi ðtÞ 6 M; t!1

and

t!1

i2I

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Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

m 6 lim inf yðtÞ 6 lim sup yðtÞ; 6 M t!1

t!1

for any positive solution (x(t), y(t)) = (x1(t), x2(t), . . . , xn(t), y(t)) of system (1.1). Let h(t) be a x-periodic continuous function defined on R. We define

½h ¼

1

x

Z

x

m

hðtÞdt;

h ¼ max hðtÞ; t2R

0

l

h ¼ min hðtÞ: t2R

Firstly, we consider the following single species logistic equation with dispersal in n-patches n X dxi ðtÞ dij ðtÞ½xj ðtÞ  xi ðtÞ; ¼ xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ þ dt j¼1

i 2 I;

ð2:2Þ

where x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 Rnþ0 ¼ fðx1 ; x2 ; . . . ; xn Þ 2 Rn ; xi P 0; i 2 Ig. xi is the population density of species x in the ith patch, ai(t), bi(t) and dij(t)(i,j 2 I, i – j) are continuous and x-periodic functions defined on R and dii  0 for all t 2 R. Further, we asl l sume that bi > 0 and dij > 0 for all i,j 2 I and i – j. We introduce the following assumptions. (H1) There is a nonempty subset I1  I such that [A] > 0, where

( AðtÞ ¼ min ai ðtÞ  i2I1

n X

dij ðtÞ þ

j¼1

X

) dki ðtÞ :

k2I1

(H2) There is a nonempty subset I1  I such that ½A > 0, where

( AðtÞ ¼ min ai ðtÞ  i2I1

X

) dij ðtÞ

and I2 ¼ I  I1 :

j2I2

We have the following result. Lemma 2.1 [26]. If assumption (H1) or (H2) holds, then Eq. (2.2) has a unique globally asymptotically stable positive x-periodic solution. Further, for convenience, we need the following vector comparison theorem. Firstly, we say the system of differential equations

_ xðtÞ ¼ Fðt; xðtÞÞ;

x 2 Rn

is cooperative if the off-diagonal elements of DxF(t,x(t)) are nonnegative and competitive if the off-diagonal elements are nonpositive, where DxF(t,x(t)) is the n  n matrix derivative of F with respect to x. Lemma 2.2 [22]. Let x(t) and y(t) be solutions of

_ xðtÞ ¼ Fðt; xðtÞÞ;

x 2 Rn

_ yðtÞ ¼ Gðt; yðtÞÞ;

y 2 Rn ;

and

respectively, where both systems are assumed to have a uniqueness property for initial value problems. Assume both x(t) and y(t) belong to a domain D # Rn for [t0, t1], in which one of the two systems is cooperative and

Fðt; zÞ 6 Gðt; zÞ;

ðt; zÞ 2 ½t 0 ; t 1   D:

If x (t0) 6 y (t0), then x (t1) 6 y (t1). 3. Main results On system (1.1), in addition to the assumptions (H1) and (H2), we further need the following assumption: (H3) Functions bi(t) > 0,dij(t) > 0(i, j 2 I,i – j), c(t) > 0, f(t) > 0 and [e] > 0 for all t 2 R.   Let x ðtÞ ¼ x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ be the unique positive x-periodic solution of Eq. (2.2). With respect to the permanence of system (1.1) we have the following result. Theorem 3.1. Suppose that assumptions (H1) or (H2) and ( H3) hold. If

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

x

Z

eðtÞ þ f ðtÞ

Z

t

k2 ðt  sÞ

tp

0

 x1 ðsÞ ds dt > 0; 1 þ mx1 ðsÞ

3005

ð3:1Þ

then system (1.1) is permanent. Proof. Here we prove Theorem 3.1 only under the assumptions (H1) and (H3). The proof under the assumptions (H2) and (H3) is similar. We will use the following several propositions to complete the proof of Theorem 3.1. h Proposition 1. There is a constant M > 0 such that limsupt?1xi(t) < M (i 2 I) and limsupt?1y (t) < M for any positive solution (x (t), y (t)) = (x1(t), . . . , ,xn(t), y(t)) of system (1.1). Proof. Choose a constant M1 > maxt2Rjx*(t)j, where jx ðtÞj ¼

dxi ðtÞ 6 xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ þ dt

n X

Pn

dij ðtÞ½xj ðtÞ  xi ðtÞ;

 i¼1 xi ðtÞ.

Since

i 2 I;

j¼1

by Lemma 2.2 (the vector comparison theorem) we obtain

xi ðtÞ 6 xi ðtÞ for all t P 0; i 2 I;

ð3:2Þ

where  xðtÞ ¼ ð x1 ðtÞ;  x2 ðtÞ; . . . ;  xn ðtÞÞ is the solution of comparison Eq. (2.2) with the initial condition  xð0Þ ¼ xð0Þ. By the global asymptotic stability of x*(t), there is a T1 > 0 such that

xi ðtÞ 6 M1

for all t P T 1 ; i 2 I:

Hence, by (3.2) we have

xi ðtÞ 6 M1

for all t P T 1 ; i 2 I:

Consequently,

lim sup xi ðtÞ 6 M1 ;

i 2 I:

ð3:3Þ

t!1

Next, we will proof that there is a constant M2 > 0 such that

lim sup yðtÞ 6 M2 :

ð3:4Þ

t!1

Choose 0 < e < M1, then from (H3) we can choose the constant M0 > M1 such that

eða1 ðtÞ  c0 ðtÞM0 Þ þ

n X

d1j ðtÞM1 < e;

ð3:5Þ

j¼1

for all t P 0 and

½eðtÞ þ f ðtÞe < e;

ð3:6Þ

where c0(t) = c(t)/(1 + mM1). We first prove

lim inf yðtÞ 6 M 0 :

ð3:7Þ

t!1

Otherwise, there is a constant T2 P T1 such that y(t) > M0 for all t P T2. If x1(t) P e for all t P T3 P T2 + s, then directly from system (1.1) we have

Z t" Z x1 ðtÞ 6 x1 ðT 3 Þ exp a1 ðuÞ  cðuÞ T3

6 x1 ðT 3 Þ exp

Z

t

T3

# n X k1 ðu  sÞyðsÞ xj ðuÞ ds þ du d1j ðuÞ x1 ðuÞ ur 1 þ mx1 ðsÞ j¼1 " # n X 1 eða1 ðuÞ  c0 ðuÞM0 Þ þ d1j ðuÞM1 du < x1 ðT 3 Þ exp½ðt  T 3 Þ:

e

u

j¼1

Hence, we obtain x1(t) ? 0 as t ? 1. This leads to a contradiction with x1(t) P e for all t P T3. Therefore, there is a t1 > T3 such that x1(t1) < e. Now, we prove x1(t) < e for all t P t1. Otherwise, there is a t2 > t1 such that x1 (t2) = e and x1(t) < e for all t 2 (t1, t2). Hence, we have dx1dtðt2 Þ P 0. On the other hand, directly from system (1.1) we obtain

  X Z t2 n dx1 ðt 2 Þ yðsÞ k1 ðu  sÞ d1j ðt 2 Þxj ðt 2 Þ 6 eða1 ðt 2 Þ  c0 ðt 2 ÞM0 Þ 6 x1 ðt 2 Þ a1 ðt 2 Þ  cðt 2 Þ ds þ dt 1 þ mx1 ðsÞ t 2 r j¼1 þ

n X j¼1

d1j ðt 2 ÞM 1 < e:

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Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

This leads to a contradiction. Therefore, x1(t) < e for all t P t1. For any t P t1, we can choose an integer pt P 0 such that t 2 [t1 + ptx, t1 + (pt + 1)x). Obviously, we have pt ? 1 as t ? 1. For all t P t1 + s, directly from system(1.1) we obtain

yðtÞ ¼ yðt 1 þ sÞ exp

Z

 Z eðsÞ þ f ðsÞ

t t 1 þs

6 yðt1 þ sÞ exp

(Z

s

k2 ðs  hÞ

sp

t 1 þsþpt x

þ

Z

t 1 þs

)

t

t 1 þsþpt x

 x1 ðhÞ dh ds 1 þ mx1 ðhÞ

½eðsÞ þ f ðsÞeds 6 yðt 1 þ sÞ expðr xÞ expðpt xeÞ;

where r* = max06t6x{je(t)j + f(t)e}. Hence, we obtain y(t) ? 0 as t ? 1. This leads a contradiction with y(t) > M0 for all t P T2. Therefore, (3.7) holds. nþ1 such that Now, we prove that (3.4) is true. Otherwise, there is a sequence of initial functions f/k g  Rþ

lim sup yðt; /k Þ > ð2M 0 þ 1Þk for all k ¼ 1; 2; . . . t!1

n o n o ðkÞ In view of (3.7), we can obtain that, for each k, there are two time sequences sq and t ðkÞ , satisfying q ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ 0 < s1 < t1 < s2 < t2 <    < sðkÞ q < tq <    ðkÞ

and sq ! 1 as q ? 1, such that

yðsðkÞ q ; /k Þ ¼ 2M 0 ;

yðtðkÞ q ; /k Þ ¼ ð2M 0 þ 1Þk

ð3:8Þ

and

  ðkÞ 2M 0 < yðt; /k Þ < ð2M 0 þ 1Þk for all t 2 sðkÞ : q ; tq

ð3:9Þ

By the ultimate boundedness of (x1(t,/k), x2(t,/k), . . . , ,xn(t,/k)), for each k there is a constant T(k) > 0 such that xi(t,/k) < M1 for ðkÞ all t P T(k). Further, for each k there is a K(k) > 0 such that sq > T ðkÞ þ s for all q P K(k). Hence, for any q P K(k) directly from system (1.1) we obtain

Z     ðkÞ ¼ y s exp y t ðkÞ ; / ; / k k q q

ðkÞ

tq



ðkÞ

eðtÞ þ f ðtÞ

Z

t

k2 ðt  sÞ

tp

sq

  h  i ðkÞ ðkÞ ; 6 y sðkÞ q ; /k exp r 1 t q  sq

 Z tðkÞ   q x1 ðsÞ exp ; / ½eðtÞ þ f ðtÞM 1 dt ds dt 6 y sðkÞ k q ðkÞ 1 þ mx1 ðsÞ sq

where r1 = maxt2[0, x]{je(t)j + f(t)M1}. Consequently, by (3.8) we have ðkÞ tðkÞ q  sq P

ln k r1

for all q P K ðkÞ :

ð3:10Þ

R tþL From inequality (3.6), we can choose L > s satisfies exp tþs ½eðtÞ þ f ðtÞedt < 1 and M1exp[  (L  s)] < e. For the above L, by ðkÞ (k) (k) ðkÞ inequality (3.10) there h is NL > 0 such i that t q > sq þ 2L for all k P NL and q P K . Fixed k P hNL and q P K i, we prove that ðkÞ ðkÞ ðkÞ ðkÞ   there must be a t 1 2 sq þ s; sq þ L such that x1 ðt1 ; /k Þ < e. Otherwise, x1(t,/k) P e for all t 2 sq þ s; sq þ L . Directly from system (1.1) and (3.9) we have

Z     ðkÞ 6 x exp x1 sðkÞ þ L; / s þ s ; / 1 k k q q

ðkÞ

sq þL ðkÞ

" a1 ðtÞ  cðtÞ

sq þs

6 M 1 exp

Z

ðkÞ sq þL ðkÞ

sq þs

1

"

e

eða1 ðtÞ  c0 ðtÞM0 Þ þ

Z

t

k1 ðt  sÞ

tr n X

#

# n X yðs; /k Þ xj ðt; /k Þ dt d1j ðtÞ ds þ 1 þ mx1 ðs; /k Þ x1 ðt; /k Þ j¼1

d1j ðtÞM1 dt 6 M 1 exp½ðL  sÞ < e;

ð3:11Þ

j¼1

   which is a contradiction. Next, we prove x1(t) < e for all t 2 ðt1 ; tðkÞ q . Otherwise, there is t 2 > t 1 such that x1 ðt 2 ; /k Þ ¼ e and t 2 ;/k Þ P 0. On the other hand, directly from system (1.1) we have x1(t,/k) < e for all t 2 ðt1 ; t2 Þ. Hence, we obtain dx1 ðdt

" # Z t2 n X dx1 ðt2 ; /k Þ yðs; / Þ k k1 ðt 2  sÞ d1j ðt2 Þxj ðt2 ; /k Þ 6 x1 ðt 2 ; /k Þ a1 ðt 2 Þ  cðt 2 Þ ds þ dt 1 þ mx ðs; / Þ t2 r 1 k j¼1 6 eða1 ðt 2 Þ  c0 ðt2 ÞM 0 Þ þ

n X

d1j ðt2 ÞM 1 < e:

j¼1 ðkÞ

This leads to a contradiction. Therefore, x1(t,/k) < e for all t 2 ½sq þ L; tðkÞ q . Further, from (3.8) and (3.9) we have

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Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

Z   ðkÞ ð2M 0 þ 1Þk ¼ yðt ðkÞ q ; /k Þ 6 y sq þ L þ s; /k exp < ð2M 0 þ 1Þk exp

Z

ðkÞ

tq

½eðtÞ þ f ðtÞedt

ðkÞ

sq þLþs

ðkÞ

tq

½eðtÞ þ f ðtÞedt < ð2M 0 þ 1Þk exp

ðkÞ

sq þLþs

Z

ðkÞ

sq þL ðkÞ

½eðtÞ þ f ðtÞedt < ð2M 0 þ 1Þk;

ð3:12Þ

sq þs

which leads a contradiction. Therefore, (3.4) holds. Choose the constant M > max{M1, M2}. Then we see the conclusion of Proposition 1 is true. h Remark 3.1. Considering the biological meaning of Proposition 1 one can know that it is reasoning. In fact, if the predator species is not ultimately bounded, then the population density of predator species will expand unlimitedly. Since the predation rate of predator species for prey species is strictly positive (i.e., c(t) > 0 in assumption (H3)) in patch 1, the prey species will be extinct in the 1th patch because of the massive preying by the predator species. Since the survival of predator species is absolutely dependent on the prey species in the patch 1, as an opposite result the predator species will be extinct too. Proposition 2. There is a constant g > 0 such that limsupt?1q(t) > g for any positive solution (x (t), y (t)) = (x1(t), x2(t), . . . , ,xn(t), y P (t)) of system (1.1), where qðtÞ ¼ i2I1 xi ðtÞ. Proof. By assumptions (H1) and (H3) we can choose positive constants e0 > 0 and g0 > 0 such that

Z

x

½eðtÞ þ f ðtÞe0 dt < e0

ð3:13Þ

/0 ðtÞdt > g0 ;

ð3:14Þ

0

and

Z

x

0

where

( 0

/ ðtÞ ¼ min ai ðtÞ  i2I1

n X j¼1

dij ðtÞ þ

X

) dki ðtÞ  cðtÞg0  bi ðtÞe0 :

k2I1

Suppose that Proposition 2 is not true. Then there is a positive solution (x(t), y(t)) = (x1(t), x2(t), . . . , ,xn(t), y(t)) of system (1.1) such that

lim sup qðtÞ < e0 :

ð3:15Þ

t!1

Hence, there is a T0 > 0 such that q(t) < e0 for all t P T0. So, xi(t) < e0(i 2 I1) for all t P T0. When 1 2 I1, From system (1.1) we have for all t P T1 P T0 + s,

  Z t dyðtÞ x1 ðsÞ k2 ðt  sÞ ¼ yðtÞ eðtÞ þ f ðtÞ ds 6 yðtÞ½eðtÞ þ f ðtÞe0 : dt 1 þ mx1 ðsÞ tp

ð3:16Þ

Choose an integer pt P 0 such that for any t 2 [T1 + ptx, T1 + (pt + 1)x), clearly pt ? 1 as t ? 1. From (3.16) we have,

yðtÞ 6 yðT 1 Þ exp

Z

t

½eðsÞ þ f ðsÞe0 ds 6 yðT 1 Þ exp

(Z

T 1 þpt x

T1

T1

þ

Z

t

)

½eðsÞ þ f ðsÞe0 ds

T 1 þpt x

6 yðT 1 Þ expðr 1 xÞ expðpt xe0 Þ; where r1 = max06t6x{je(t)j + f(t)e0}. Consequently, we have y(t) ? 0 as t ? 1, then there is T2 P T1 such that y(t) < g0 for all t P T2. From system (1.1) we obtain for all t P T2 + s,

  X Z t n dx1 ðtÞ yðsÞ k1 ðt  sÞ d1j ðtÞ½xj ðtÞ  x1 ðtÞ ¼ x1 ðtÞ a1 ðtÞ  b1 ðtÞx1 ðtÞ  cðtÞ ds þ dt 1 þ mx1 ðsÞ tr j¼1 P x1 ðtÞ½a1 ðtÞ  b1 ðtÞx1 ðtÞ  cðtÞg0  þ

n X

d1j ðtÞ½xj ðtÞ  x1 ðtÞ;

ð3:17Þ

j¼1

for any i – 1, n n X X dxi ðtÞ dij ðtÞ½xj ðtÞ  xi ðtÞ P xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ  cðtÞg0  þ dij ðtÞ½xj ðtÞ  xi ðtÞ: ¼ xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ þ dt j¼1 j¼1

ð3:18Þ

3008

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

Hence, we have

" # n X X X dqðtÞ X dxi ðtÞ xi ðtÞðai ðtÞ  bi ðtÞxi ðtÞ  cðtÞg0 Þ þ dik ðtÞxk ðtÞ  dij ðtÞxi ðtÞ ¼ P dt dt i2I1 i2I1 j¼1 k2I1 " # n X X X ¼ ai ðtÞ  dij ðtÞ þ dki ðtÞ  cðtÞg0  bi ðtÞxi ðtÞ xi ðtÞ P /0 ðtÞqðtÞ; i2I1

j¼1

ð3:19Þ

k2I1

when 1 R I1, by (3.18) we can immediately obtain that

" # n X X X dqðtÞ X dxi ðtÞ ai ðtÞ  dij ðtÞ þ dki ðtÞ  cðtÞg0  bi ðtÞxi ðtÞ xi ðtÞ P /0 ðtÞqðtÞ: ¼ P dt dt i2I i2I j¼1 k2I 1

1

ð3:20Þ

1

By this and (3.19) and (3.14) we finally have q(t) ? 1 as t ? 1, which leads to a contradiction with (3.15). This completes the proof of Proposition 2. h Proposition 3. There is a constant r0 > 0 such that liminft?1q(t) > r0 for any positive solution (x(t), y(t)) = (x1(t), x2(t), . . . , xn(t), y (t)) of system (1.1). Proof. By (3.13) and (3.14) we can choose a sufficiently large constant P such that for any a P P

M exp

Z 0

a

½eðtÞ þ f ðtÞe0 dt < g0

ð3:21Þ

and

Z

a

/0 ðtÞdt > 0:

ð3:22Þ

0

Assume that Proposition 3 is not true. Then there is a sequence {/k}  C+ such that for the solution (x(t,/k), y(t,/k)) = (x1(t,/k), x2(t,/k), . . . , ,xn(t,/k), y(t,/k)) of system (1.1),

lim inf qðt; /k Þ < t!1

g 2k

2

for all k ¼ 1; 2; . . .

P where qðt; /k Þ ¼ i2I1 xi ðt; /k Þ. From Proposition 2 we have limsupt?1q(t,/k) > g. Hence, for each k there are time sequences ðkÞ ðkÞ fsq g and ft q g, satisfying ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ 0 < s1 < t1 < s2 < t2 <    < sðkÞ q < tq <    ðkÞ

and sq ! 1 as q ? 1, such that

  g   g q sðkÞ ; q t ðkÞ q ; /k ¼ q ; /k ¼ 2 k

ð3:23Þ

k

and

g 2

k

< qðt; /k Þ <

g k

  ðkÞ : for all t 2 sðkÞ q ; tq

ð3:24Þ

By Proposition 1, for each k, there is a T(k) > 0 such that

xi ðt; /k Þ < Mði 2 IÞ;

yðt; /k Þ < M

Further, for each k, there is a K we have

(k)

for all t P T ðkÞ :

> 0 such that

ðkÞ sq

>T

ðkÞ

(k)

þ s for all q P K . Hence, for any t 2

h

ðkÞ sq ; t ðkÞ q

i

ð3:25Þ (k)

and q P K , by (3.25)

n X dx1 ðt; /k Þ d1j ðtÞ½xj ðt; /k Þ  x1 ðt; /k Þ; P x1 ðt; /k Þ½a1 ðtÞ  b1 ðtÞx1 ðt; /k Þ  cðtÞM þ dt j¼1

consequently, we have

( ) n X X dqðt; /k Þ xi ðt; /k Þ½ai ðtÞ  bi ðtÞxi ðt; /k Þ  cðtÞM þ dij ðtÞ½xj ðt; /k Þ  xi ðt; /k Þ P dt i2I1 j¼1 ( ) n X X X xi ðt; /k Þ½ai ðtÞ  bi ðtÞM  cðtÞM þ dip ðtÞxp ðt; /k Þ  dij ðtÞxi ðt; /k Þ P wðtÞqðt; /k Þ; P i2I1

p2I1

j¼1

ð3:26Þ

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

3009

where

( wðtÞ ¼ max jai ðtÞj þ bi ðtÞM þ cðtÞM þ i2I1

dij ðtÞ þ

ðkÞ sq

ln k P m w

to

t ðkÞ q ,

X

) dpi ðtÞ :

p2I1

j¼1

Integrating (3.26) from ðkÞ tðkÞ q  sq

n X

we obtain

for all q P K ðkÞ :

So, there is a constant K0 > 0 such that

g k

< e0 ;

ðkÞ t ðkÞ q  sq > 2Q ;

ð3:27Þ

(k) for all k P K0 and q P K(k), where the constant h i Q > P + s. Let k P K0 and q P K in the following discussion. ðkÞ ðkÞ When 1 2 I1, for any t 2 sq þ s; sq þ Q , by (3.24) and (3.27) we have

dyðt; /k Þ 6 yðt; /k Þ½eðtÞ þ f ðtÞe0 : dt ðkÞ

Integrating this from sq þ s to t one can obtain

Z   yðt; /k Þ 6 y sðkÞ þ s ; / exp k q

t

ðkÞ sq þ

½eðsÞ þ f ðsÞe0 ds 6 M exp

Z

s

h i ðkÞ From (3.21) we have for all t 2 sq þ Q ; tðkÞ q

yðt; /k Þ 6 M exp

Z

t

ðkÞ sq þ

½eðsÞ þ f ðsÞe0 ds: s

t ðkÞ

sq þ s

½eðsÞ þ f ðsÞe0 ds < g0 :

h i ðkÞ Then by system (1.1) we have for all t 2 sq þ Q ; tðkÞ q n X dx1 ðt; /k Þ d1j ðtÞ½xj ðt; /k Þ  x1 ðt; /k Þ: P x1 ðt; /k Þ½a1 ðtÞ  b1 ðtÞx1 ðt; /k Þ  cðtÞg0  þ dt j¼1

For any i – 1, n X dxi ðt; /k Þ dij ðtÞ½xj ðt; /k Þ  xi ðt; /k Þ ¼ xi ðt; /k Þ½ai ðtÞ  bi ðtÞxi ðt; /k Þ þ dt j¼1

P xi ðt; /k Þ½ai ðtÞ  bi ðtÞxi ðt; /k Þ  cðtÞg0  þ

n X

dij ðtÞ½xj ðt; /k Þ  xi ðt; /k Þ;

ð3:28Þ

j¼1

then the following can be obtained quickly from the above two inequalities

( ) n X X dqðt; /k Þ X dxi ðt; /k Þ xi ðt; /k Þ½ai ðtÞ  bi ðtÞxi ðt; /k Þ  cðtÞg0  þ dij ðtÞ½xj ðt; /k Þ  xi ðt; /k Þ ¼ P dt dt i2I1 i2I1 j¼1 ( ) n X X X xi ðt; /k Þ½ai ðtÞ  bi ðtÞe0  cðtÞg0  þ dil ðtÞxl ðt; /k Þ  dij ðtÞxi ðt; /k Þ P i2I1

P

X

l2I1

" ai ðtÞ  bi ðtÞe0  cðtÞg0 þ

i2I1

X

dli ðtÞ 

n X

j¼1

#

dij ðtÞ xi ðt; /k Þ P /0 ðtÞqðt; /k Þ;

ð3:29Þ

j¼1

l2I1

when 1 R I1, by (3.28) we directly have

( ) n X X dqðt; /k Þ X dxi ðt; /k Þ xi ðt; /k Þ½ai ðtÞ  bi ðtÞxi ðt; /k Þ  cðtÞg0  þ dij ðtÞ½xj ðt; /k Þ  xi ðt; /k Þ ¼ P dt dt i2I i2 j¼1 1

1

P /0 ðtÞqðt; /k Þ:

ð3:30Þ ðkÞ

Finally, integrating (3.29) or (3.30) from sq þ Q þ s to t ðkÞ q , by (3.22)–(3.24) one can obtain

g 2

k

ðtðkÞ q ; /k Þ

¼q



Pq

sðkÞ q



þ Q þ s; /k exp

Z

ðkÞ

tq ðkÞ

0

/ ðtÞdt P

sq þQþs

which leads to a contradiction. This completes the proof. h

g k

2

exp

Z

ðkÞ

tq ðkÞ

sq þQ þs

/0 ðtÞdt >

g 2

k

;

3010

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

Proposition 4. There is a constant b > 0 such that liminft?1xi(t) > b(i 2 I) for any positive solution (x (t), y (t)) = (x1(t), x2(t), . . . , xn (t), y (t)) of system (1.1). Proof. By Propositions 1 and 3, there is a T0 > 0 such that

X

xi ðtÞ > r0 ;

yðtÞ < M

for all t P T 0 :

i2I1

Then, for all t P T0 + s and any i 2 I we have n X dxi ðtÞ dij ðtÞ½xj ðtÞ  xi ðtÞ P xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ  cðtÞM þ dt j¼1 " # n X dij ðtÞ þ d0 r0 ; P xi ðtÞ ai ðtÞ  bi ðtÞxi ðtÞ  cðtÞM 

ð3:31Þ

j¼1

n o l where d0 ¼ min dij : i; j 2 I; i – j . Obviously, we can choose a constant b > 0, and b is independent of any positive solution of system (1.1), such that for any i 2 I, 0 6 xi 6 b and t 2 R

"

xi ai ðtÞ  bi ðtÞxi  cðtÞM 

n X

#

dij ðtÞ þ d0 r0 > b:

j¼1

Hence, by (3.31) one can obtain

dxi ðtÞ > b for all 0 6 xi ðtÞ 6 b; t P T 0 þ s: dt From this, we finally have

lim inf xi ðtÞ P b;

i 2 I:

t!1

Proposition 4 is completed. h Remark 3.2. Propositions 1–4 show that if we guarantee the assumption (H1) or (H2) and (H3) hold then the prey species must be permanent. If not, then it may be extinct, as a result the predator species y will be extinct too because its survival is absolutely dependent on x. However, when y became extinct, x will not turn to extinction for assumption (H1) or (H2) shows that x has a total positive average growth rate in n patches. Proposition 5. There is a constant c > 0 such that limsupt?1y (t) > c for any positive solution (x (t), y (t)) = (x1(t), x2(t), . . . , ,xn(t), y (t)) of system (1.1). Proof. By (3.1), we can choose a constant e0 > 0 such that

Z

x

eðtÞ þ f ðtÞ

0

 k2 ðt  sÞx1 ðsÞ e ds  f ðtÞ 0 dt > e0 :  tp 1 þ mx1 ðsÞ

Z

t

ð3:32Þ

For any constant a > 0, we consider the following auxiliary system: n X dx1 ðtÞ ¼ x1 ðtÞ½a1 ðtÞ  cðtÞa  b1 ðtÞx1 ðtÞ þ d1j ðtÞ½xj ðtÞ  x1 ðtÞ; dt j¼1 n X dxi ðtÞ ¼ xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ þ dij ðtÞ½xj ðtÞ  xi ðtÞ; dt j¼1

By assumption (H1), we obtain for sufficiently small a > 0

Z 0

x

/a ðtÞdt > 0;

where

( /a ðtÞ ¼ min ai ðtÞ  cðtÞa  i2I1

n X j¼1

dij ðtÞ þ

X k2I1

) dki ðtÞ :

i ¼ 2; 3; . . . ; n:

ð3:33Þ

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

3011

Hence, by Lemma 2.1 we obtain that Eq. (3.33) has a unique globally asymptotically stable positive x-periodic solution   xa ðtÞ ¼ x1a ðtÞ; x2a ðtÞ; . . . ; xna ðtÞ . Let xa(t) = (x1a(t), x2a(t), . . . , xna(t)) be the solution of Eq. (3.33) with initial condition * xa(0) = x (0). Then, for the above given constant e0 > 0 there is a T0 > 0 such that

e0

jxia ðtÞ  xia ðtÞj <

4

i 2 I;

;

ð3:34Þ

for all t P T0. On the other hand, by the continuity of solutions with respect to parameters, we can obtain that when a ? 0, xa(t) uniformly for t 2 [T0, T0 + x] converges to x*(t). Hence, there is a 0 < a0 < e0 such that for any 0 6 a 6 a0 we have

jxia ðtÞ  xi ðtÞj <

e0 4

i 2 I;

;

ð3:35Þ

for all t 2 [T0, T0 + x]. From (3.34) and (3.35) and the periodicity of xa ðtÞ and x*(t), we finally have

jxia ðtÞ  xi ðtÞj <

e0 2

i 2 I;

;

ð3:36Þ

for all t 2 R and 0 6 a 6 a0. Suppose that Proposition 5 is not true. Then for any constant 0 < a < a0 there is a positive solution (x(t), y(t)) = (x1(t), x2(t), . . . , xn(t), y(t)) of system (1.1) such that limsupt?1y(t) < a. Hence, there is a T1 > 0 such that y(t) < a for all t P T1. Then for all t P T1 + s we obtain n X dx1 ðtÞ P x1 ðtÞ½a1 ðtÞ  cðtÞa  b1 ðtÞx1 ðtÞ þ d1j ðtÞ½xj ðtÞ  x1 ðtÞ; dt j¼1

and n X dxi ðtÞ dij ðtÞ½xj ðtÞ  xi ðtÞ i ¼ 2; 3; . . . ; n: ¼ xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ þ dt j¼1

Hence, by Lemma 2.2 we have

xi ðtÞ P xia ðtÞ i 2 I;

ð3:37Þ

t P T1 + s, where xa(t) = (x1a(t), x2a(t), . . . , xna(t)) is the solution of Eq. (3.33) with initial condition xa(T1 + s) = x(T1 + s). By the asymptotic stability of the positive x-periodic solution xa ðtÞ, we obtain that there is a T2 > T1 + s such that

jxia ðtÞ  xia ðtÞj <

e0 2

i 2 I;

;

ð3:38Þ

for all t P T2. Hence, by (3.36)–(3.38) we finally have

xi ðtÞ P xi ðtÞ  e0 ;

i 2 I;

ð3:39Þ

for all t P T2. Since for all t P T2 + s,

    Z t Z t dyðtÞ x1 ðsÞ x1 ðsÞ  e0 k2 ðt  sÞ k2 ðt  sÞ ¼ yðtÞ eðtÞ þ f ðtÞ ds P yðtÞ eðtÞ þ f ðtÞ ds dt 1 þ mx1 ðsÞ 1 þ mðx1 ðsÞ  e0 Þ tp tp   Z t  k2 ðt  sÞx1 ðsÞ P yðtÞ eðtÞ þ f ðtÞ ds  f ðtÞe0 ;  tp 1 þ mx1 ðsÞ integrating this from T2 + s to t, one can obtain

yðtÞ P yðT 2 þ sÞ exp

Z



t

T 2 þs

eðtÞ þ f ðtÞ

 k2 ðt  sÞx1 ðsÞ e ds  f ðtÞ 0 ds:  tp 1 þ mx1 ðsÞ

Z

t

Hence, by (3.32) we finally obtain y(t) ? 1 as t ? 1, which leads to a contradiction. Then the conclusion of Proposition 5 is true. h Proposition 6. There is a constant l > 0 such that liminft?1y (t) > l for any positive solution (x (t), y (t)) = (x1(t), x2(t), . . . , xn(t), y (t)) of system (1.1). Proof. If Proposition 6 is not true, then there is a sequence {/k}  C+ such that

lim inf yðt; /k Þ < t!1

c kþ1

for all k ¼ 1; 2; . . . ;

where c is given in Proposition 5, c < e0 and the constantne0 o satisfies (3.32). By Proposition 5, we have limn inequality o ðkÞ and tðkÞ , satisfying supt?1y(t) > c. Hence, for each k there are time sequences sq q

3012

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015 ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ 0 < s1 < t1 < s2 < t2 <    < sðkÞ q < tq <    ðkÞ

and sq ! 1 as q ? 1, such that

  y sðkÞ q ; /k ¼ c;

  y t ðkÞ q ; /k ¼

c

ð3:40Þ

kþ1

and

c kþ1

ðkÞ < yðt; /k Þ < c for all t 2 ðsðkÞ q ; t q Þ:

ð3:41Þ

By system (1.1) we have

dyðt; /k Þ P eðtÞyðt; /k Þ; dt ðkÞ

for any k, integrating this inequality from sq to t ðkÞ q we have

Z     ðkÞ P y s exp y t ðkÞ ; / ; / k k q q

ðkÞ

tq ðkÞ

½eðtÞdt:

sq

Consequently ðkÞ tðkÞ q  sq P

lnðk þ 1Þ e0

for any k ¼ 1; 2; . . .

ð3:42Þ

where e0 = max06t6x{je(t)j}. By (3.32), there are constants P > 0 and h > 0 such that for any a P P we have

Z

a

 eðtÞ þ f ðtÞ

 k2 ðt  sÞx1 ðsÞ ds  f ðtÞe0 dt > h:  tp 1 þ mx1 ðsÞ

Z

0

t

ð3:43Þ

h i ðkÞ , Further, from system (1.1) we have for any t 2 sq þ s; tðkÞ q n X dx1 ðt; /k Þ d1j ðtÞ½xj ðt; /k Þ  x1 ðt; /k Þ P x1 ðt; /k Þ½a1 ðtÞ  b1 ðtÞx1 ðt; /k Þ  cðtÞc þ dt j¼1

ð3:44Þ

n X dxi ðt; /k Þ dij ðtÞ½xj ðt; /k Þ  xi ðt; /k Þ i ¼ 2; 3; . . . ; n: P xi ðt; /k Þ½ai ðtÞ  bi ðtÞxi ðt; /k Þ þ dt j¼1

ð3:45Þ

xi ðt; /k Þ P xic ðtÞ;

ð3:46Þ

and

    ðkÞ ðkÞ Let xc(t) = (x1c(t), x2c(t), . . . , xnc(t)) is the solution of Eq. (3.33) with a = c and initial condition xc sq þ s; /k ¼ x sq þ s; /k . Then by (3.44), (3.45) and Lemma 2.2, it follows that

for all t 2

b6

h

ðkÞ sq

þs

xi ðsðkÞ q

; tðkÞ q

i 2 I;

i ðkÞ . By limq!1 sq ¼ 1 and Proposition 1 and 4 we obtain that for any k, there is a K(k) such that

þ s; /k Þ 6 M;

i 2 I;

ð3:47Þ

for all q P K(k). For the parameter a = c, Eq. (3.33) has a globally asymptotically stable positive x-periodic solution  xc ðtÞ ¼ x1c ðtÞ; x2c ðtÞ; . . . ; xnc ðtÞ . From the periodicity of Eq. (3.33) we know that the periodic solution xc ðtÞ also is globally uniformly asymptotically stable. Hence, by (3.46) there is a T0 > P, and T0 is independent of any k and q, such that

xic ðtÞ > xic ðtÞ  for all t P T 0 þ

ðkÞ sq

e0 2

; ðkÞ

þ s and q P K(k). So, by (3.36), for all t P T 0 þ sq þ s and q P K(k) we have

xic ðtÞ > xi ðtÞ  e0 :

ð3:48Þ

By (3.42), there is a N > 0 such that ðkÞ tðkÞ q  sq P 2L;

for all k P N and q P K(k), where L P T0 + s is a constant. Hence, from (3.46) and (3.48) we finally obtain

xi ðt; /k Þ P xi ðt; /k Þ  e0 ; for all t 2

h

ðkÞ sq

þ

L; tðkÞ q

i 2 I;

i h i ðkÞ , k P N and q P K(k). Since, for any t 2 sq þ L þ s; t ðkÞ , k P N and q P K(k), by (3.49) we have q

ð3:49Þ

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

3013

    Z t Z t dyðt; /k Þ x1 ðs; /k Þ k2 ðt  sÞx1 ðsÞ k2 ðt  sÞ e ¼ yðt; /k Þ eðtÞ þ f ðtÞ ds P yðt; /k Þ eðtÞ þ f ðtÞ ds  f ðtÞ ; 0  dt 1 þ mx1 ðs; /k Þ tp tp 1 þ mx1 ðsÞ ðkÞ

integrating above from sq þ L þ s to t ðkÞ q , one can obtain

Z     ðkÞ y tðkÞ q ; /k P y sq þ L þ s; /k exp

ðkÞ

tq ðkÞ



eðtÞ þ f ðtÞ

sq þLþs

Z

 k2 ðt  sÞx1 ðsÞ ds  f ðtÞe0 dt:  tp 1 þ mx1 ðsÞ t

Hence, by (3.40) and (3.43) we finally have

 k2 ðt  sÞx1 ðsÞ ds  f ðtÞe0 dt  ðkÞ kþ1 sq þLþs tp 1 þ mx1 ðsÞ   Z Z tðkÞ t q c k2 ðt  sÞx1 ðsÞ c eðtÞ þ f ðtÞ e P exp ds  f ðtÞ ; dt P 0  ðkÞ kþ1 kþ1 tp 1 þ mx1 ðsÞ sq þLþs Z   P y sðkÞ q þ L þ s; /k exp

c

ðkÞ

tq

 Z eðtÞ þ f ðtÞ

t

which leads to a contradiction. This completes the proof. h Remark 3.3. From Lemma 2.1, one can obtain that the prey species will approach a positive periodic stable state x ðtÞ ¼ x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ when there is not the predator species y. Therefore, Propositions 5 and 6 show that if the positive periodic stable state x*(t) of x can guarantee that y obtain a positive total average growth, that is condition

Z

x

eðtÞ þ f ðtÞ

Z

t

k2 ðt  sÞ

tp

0

 x1 ðsÞ ds dt > 0; 1 þ mx1 ðsÞ

then y will be permanent. Finally, from Propositions 1–6 we see that Theorem 3.1 is proved and this completes the proof. Corollary 3.1. Suppose the assumptions (H1) or (H2) and (H3) hold. If

Z

x

eðtÞ þ f ðtÞ

Z

t

k2 ðt  sÞ

tp

0

 x1 ðsÞ ds dt > 0; 1 þ mx1 ðsÞ

then system(1.1) has a positive x-periodic solution. From the result of Teng and Chen in [25] on the existence of positive periodic solutions for general delayed periodic nspecies Kolmogorov system, the Corollary 3.1 can be obtained easily. There we omit the proof of it. Theorem 3.2. Suppose that assumptions (H1) or (H2) and (H3) hold. If

Z

x

eðtÞ þ f ðtÞ

Z

t

k2 ðt  sÞ

tp

0

 x1 ðsÞ ds dt < 0; 1 þ mx1 ðsÞ

then for any positive solution of system (1.1), x(t) ? x*(t) and y(t) ? 0 as t ? 1. Proof. For any positive solution (x(t), y(t)) = (x1(t), x2(t), . . . , xn(t), y(t)) of system (1.1), directly from system (1.1) we have n X dxi ðtÞ dij ðtÞ½xj ðtÞ  xi ðtÞ; 6 xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ þ dt j¼1

i 2 I;

for all t P 0. Let ~ xðtÞ ¼ ð~ x1 ðtÞ; ~ x2 ðtÞ; . . . ; ~ xn ðtÞÞ be the solution of Eq. (2.2) with initial condition ~ xð0Þ ¼ xð0Þ. By Lemma 2.2 it can xi ðtÞ ði 2 IÞ for all t P 0. Obviously, for any constant e > 0, by the global asymptotic stability of x*(t), there is a be seen xi ðtÞ 6 ~ xi ðtÞ 6 xi ðtÞ þ eði 2 IÞ for all t P T0. Hence, we have T0 > 0 such that ~

xi ðtÞ 6 xi ðtÞ þ e;

i 2 I;

ð3:50Þ

for all t P T0. By the conditions of Theorem 3.2, we can choose a constant e < e0 < 1 such that

Z 0

x

eðtÞ þ f ðtÞ

Z

t

k2 ðt  sÞ

tp

For all t P T0 + s, by (3.50) we have

 x1 ðsÞ dt < e0 ; e ds þ f ðtÞ 0 1 þ mx1 ðsÞ

ð3:51Þ

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Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

" #   Z t Z t dyðtÞ x1 ðsÞ x1 ðsÞ þ e    ds k2 ðt  sÞ k2 ðt  sÞ ¼ yðtÞ eðtÞ þ f ðtÞ ds 6 yðtÞ eðtÞ þ f ðtÞ dt 1 þ mx1 ðsÞ 1 þ m x1 ðsÞ þ e tp tp   Z t  x1 ðsÞ k2 ðt  sÞ ds þ f ðtÞe0 ; 6 yðtÞ eðtÞ þ f ðtÞ 1 þ mx1 ðsÞ tp Integrating above from T0 + s to t, we have

yðtÞ 6 yðT 0 þ sÞ exp

Z

t

 eðtÞ þ f ðtÞ

T 0 þs

Z

t

k2 ðt  sÞ

tp

 x1 ðsÞ ds þ f ðtÞe0 dt:  1 þ mx1 ðsÞ

By (3.51) we finally obtain y(t) ? 0 as t ? 1. Further, by a similar argument as in the proof Proposition 5 we can obtain that

xi ðtÞ P xi ðtÞ  e;

i 2 I;

 for all t P T2. Therefore, from the inequality (3.50) and the arbitrariness of e we obtain ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞ ! x1 ðtÞ; x2 ðtÞ;  . . . ; xn ðtÞÞ as t ? 1. This completes the proof of Theorem 3.2. h Considering the following predator density-dependent system

  X Z t n dx1 ðtÞ yðsÞ k1 ðt  sÞ d1j ðtÞ½xj ðtÞ  x1 ðtÞ; ¼ x1 ðtÞ a1 ðtÞ  b1 ðtÞx1 ðtÞ  cðtÞ ds þ dt 1 þ mx1 ðsÞ tr j¼1 n X dxi ðtÞ dij ðtÞ½xj ðtÞ  xi ðtÞ; ¼ xi ðtÞ½ai ðtÞ  bi ðtÞxi ðtÞ þ dt j¼1

i ¼ 2; 3; . . . ; n;

  Z t dyðtÞ x1 ðsÞ k2 ðt  sÞ ¼ yðtÞ eðtÞ þ f ðtÞ ds  gðtÞyðtÞ ; dt 1 þ mx1 ðsÞ tp

ð3:52Þ

where g(t) > 0 is also a x-periodic function, we can obtain sufficient and necessary conditions for the permanence, extinction of system (3.52). Corollary 3.2. Suppose the assumptions (H1) or (H2) and (H3) hold, then system (1.1) is permanent, if and only if

Z

x

eðtÞ þ f ðtÞ

Z

t

k2 ðt  sÞ

tp

0

 x1 ðsÞ ds dt > 0: 1 þ mx1 ðsÞ

Corollary 3.3. Suppose the assumptions (H1) or (H2) and ( H3) hold, then for any positive solution of system(1.1), x (t) ? x*(t) and y(t) ? 0 as t ? 1, if and only if

Z

x

eðtÞ þ f ðtÞ

Z

t

k2 ðt  sÞ

tp

0

 x1 ðsÞ ds dt 6 0: 1 þ mx1 ðsÞ

4. Conclusion In this paper, two species predator–prey Lotka–Volterra type dispersal system with delay and Holling type II response function is studied. We have established the sufficient conditions of integrable form for the boundedness, permanence, extinction and existence of positive periodic solution for the system. One can notice that the system we investigated in this paper is predator density-independent. However, on the predator density-dependent system (3.52), as a corollary, we have given sufficient and necessary conditions for the permanence, extinction of system. Hence, an important open question is whether the results obtained above are also valid to the predator density-independent system (1.1). The key about this question is to check what results will be obtained with the condition

Z

x

eðtÞ þ f ðtÞ

0

Z

t

tp

k2 ðt  sÞ

 x1 ðsÞ ds dt ¼ 0;  1 þ mx1 ðsÞ

permanent or extinction? Acknowledgement This research was supported by National Basic Research Program of China (2010CB732501).

Z. Liu, S. Zhong / Applied Mathematics and Computation 216 (2010) 3002–3015

3015

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