Permanence for a class of periodic time-dependent predator–prey system with dispersal in a patchy-environment

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Chaos, Solitons and Fractals 38 (2008) 1483–1497 www.elsevier.com/locate/chaos

Permanence for a class of periodic time-dependent predator–prey system with dispersal in a patchy-environment q Long Zhang, Zhidong Teng

*

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, PR China Accepted 2 January 2007

Abstract In this paper, we study two species predator–prey Lotka–Volterra type dispersal system with periodic coefficients in two patches, in which both the prey and predator species can disperse between two patches. By utilizing analytic method, sufficient and realistic conditions on permanence and the existence of periodic solution are established. The theoretical results are confirmed by a special example and numerical simulations. Ó 2007 Published by Elsevier Ltd.

1. Introduction As a very classical question in population biology, predator–prey systems in a homogenous environment have been investigated extensively. Many important qualitative analysis, such as boundedness, persistence, extinction, stability for survival of predator and prey species, far more important results have been obtained and collected in some monographs (see [1,3,8,14–16,18,23,25,28] and the references cited therein). Recently, the problem of understanding the impact of habitat fragmentation on populations and communities has attracted considerable attention because of the possible implications for biological conservation. The literature on the effects of habitat patchiness or fragmentation on populations is large and we make no systematic attempt to review it here. A review of empirical studies is given in [7,12,13,20]. Two species predator–prey dispersal systems have been extensively studied, and many good results have been achieved (see [2,4–6,9–11,21,22,24,26,27,32–34] and the references cited therein). As a pioneering work, Kuang and Takeuchi [11] investigated the following autonomous two species prey–predator dispersal system:

q

This work was supported by The National Natural Science Foundation of PR China (10361004), The Major Project of The Ministry of Education and The Scientific Research Programmes of Colleges in Xinjiang (XJEDU2004I12 and XJEDU2005I08). * Corresponding author. E-mail addresses: [email protected] (L. Zhang), [email protected] (Z. Teng). 0960-0779/$ - see front matter Ó 2007 Published by Elsevier Ltd. doi:10.1016/j.chaos.2007.01.154

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dx1 ¼ x1 g1 ðx1 Þ  yp1 ðx1 Þ þ ðx2  x1 Þ; dt dx2 ¼ x2 g2 ðx2 Þ þ yp2 ðx2 Þ þ ðx1  x2 Þ; dt dy ¼ y½sðyÞ þ c1 p1 ðx1 Þ þ c2 p2 ðx2 Þ: dt

ð1:1Þ

Conditions are established for the existence, uniform persistence, and local and global stability of positive steady states. On the other hand, the effects of a periodically varying environment play an important role in the permanence and extinction of population dynamic systems (see [1,2,14,21,25,29,34]). Xu et al. [33] extend the work of Kuang and Takeuchi [11] and considered the following two species predator–prey dispersal system with periodic coefficients: dx1 ðtÞ ¼ x1 ðtÞ½r1 ðtÞ  a11 ðtÞx1 ðtÞ  a13 ðtÞyðtÞ þ D1 ðtÞðx2 ðtÞ  x1 ðtÞÞ; dt dx2 ðtÞ ¼ x2 ðtÞ½r2 ðtÞ  a22 ðtÞx2 ðtÞ  a23 ðtÞyðtÞ þ D2 ðtÞðx1 ðtÞ  x2 ðtÞÞ; dt dyðtÞ ¼ yðtÞ½r3 ðtÞ þ a31 ðtÞx1 ðt  s1 Þ þ a32 ðtÞx2 ðt  s1 Þ  a33 ðtÞyðt  s2 Þ: dt

ð1:2Þ

Existence, uniqueness and global stability of positive periodic solutions were established by applying Gaines and Mawhins continuation theorem of coincidence degree theory and by means of a suitable Lyapunov-functional. However, no matter in the system (1.1) and (1.2) or all of investigated two species predator–prey models specialized in above literatures (and the references cited therein) with dispersal in a heterogeneous environment considered so far, it had been assumed that the predator population have no barriers between patches, i.e., the predator species can arrive to any patches to prey on without restriction by the environment. Obviously, this assumption is too realistic to be true in many circumstance. In fact, in real life, the prey species will be usually affected by the patchy environment, once the food in their patch is exhausted or there are abundant natural enemies in their patch, they will leave in search for new habitats or refuges, at the same time, the predator species also will be confined in a isolated patch, when there is little food to feed on or the last prey is eaten the predators will depart and become predator dispersers. To our best knowledge, the work on the two species predator–prey system with both the predator and prey dispersing in a patchy environment is very few and far between. Mchich et al. [17] considered the following two species autonomous Lotka–Volterra type predator–prey dispersal system: dn1 ds dn2 ds dp1 ds dp2 ds

¼ ðm0 ðp2 Þn2  mðp1 Þn1 Þ þ ½r1 n1  a1 n1 p1 ; ¼ ðmðp1 Þn1  m0 ðp2 Þn2 Þ þ ½r2 n2  a2 n2 p2 ; ð1:3Þ ¼ ðk 0 p2  kp1 Þ þ ½m1 p1 þ b1 n1 p1 ; ¼ ðkp1  k 0 p2 Þ þ ½m2 p2 þ b2 n2 p2 ;

where ni ; pi ði ¼ 1; 2Þ denote the population density of the prey species and predator species in the ith patch respectively, ri ði ¼ 1; 2Þ represent the intrinsic growth rate of the prey population in the ith patch, ai and bi are predation parameters on the ith patch. The term mi is the natural mortality rate of the predator in patch i. Constant parameters k and k 0 represent the predator migration rates from patch 1 to patch 2 and inversely. The prey migration rates are predator density dependent as follows: mðp1 Þ ¼ ap1 þ a0 ;

m0 ðp2 Þ ¼ bp2 þ a0 :

a0 ; a and b are positive parameters,  is a small dimensionless parameter meaning that biotic processes are assumed to be slow. On the assumption that the more predators found on a patch, the more the prey tend to leave this patch, conditions are founded on the existence of unique strictly positive equilibrium which may be stable or unstable, a Hopf bifurcation may occur. Cui and Chen [4] studied a class two species nonautonomous predator–prey dispersal system in n-patches with both the predator and prey species dispersing, a primary work has been done and some results had been obtained on the permanence. However, we found the conditions they established are not very absorbing and satisfying (details will be given in the text), as a elemental research, there are some improvements can be made.

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Based on the arguments above, in this paper, we consider the following two species periodic time-dependent predator–prey Lotka–Volterra type system with prey–predator dispersal in two patches: dx1 ¼ x1 ½b11 ðtÞ  a11 ðtÞx1  c11 ðtÞy 1 ðtÞ þ d 12 ðtÞðx2  x1 Þ; dt dx2 ¼ x2 ½b12 ðtÞ  a12 ðtÞx2  c12 ðtÞy 2 ðtÞ þ d 21 ðtÞðx1  x2 Þ; dt ð1:4Þ dy 1 ¼ y 1 ½b21 ðtÞ  a21 ðtÞy 1 þ c21 ðtÞx1 ðtÞ þ D12 ðtÞðy 2  y 1 Þ; dt dy 2 ¼ y 2 ½b22 ðtÞ  a22 ðtÞy 2 þ c22 ðtÞx2 ðtÞ þ D21 ðtÞðy 1  y 2 Þ; dt where t 2 Rþ0 ¼ ½0; 1Þ; xi ði ¼ 1; 2Þ denote the population density of the prey species in the ith patch, y i ði ¼ 1; 2Þ the population density of the predator species in the ith patch, b1i ðtÞ; a1i ðtÞ, the intrinsic growth rate and density-dependent coefficient of the prey in the ith patch respectively, b2i ðtÞ; a2i ðtÞ the death rate and density-dependent coefficient of the predator species in the ith patch respectively, c1i ðtÞ the capturing rate of the predator in the ith patch, c2i ðtÞ the rate of conversion of nutrients into the reproduction of the predator in the ith patch and d ij ðtÞ; Dij ðtÞ ði; j ¼ 1; 2; i 6¼ jÞ the dispersal rate of the prey species and the predator species from the ith patch to the jth patch respectively. For system (1.4), we always assume that the functions a1i ðtÞ; a2i ðtÞ; b1i ðtÞ; b2i ðtÞ; c1i ðtÞ; c2i ðtÞ; d ij ðtÞ; Dij ðtÞði; j ¼ 1; 2; i 6¼ jÞ are continuous and periodic defined on Rþ0 . with common period x > 0 and d ii ðtÞ ¼ Dii ðtÞ ¼ 0ði ¼ 1; 2Þ for all t 2 Rþ0 . In this paper, as a primary research, our main purpose is to establish sufficient conditions on the boundedness, uniform persistence and the existence of positive periodic solutions for system (1.4). We will find that the dispersal will affect the persistence both of the prey and predator species, that is if the prey or predator species is persistent in one patch, then due to the effects of dispersion the prey or predator species will persistent in the other patch. The organization of this paper is as follows. In Section 2, some basic assumptions, useful lemmas are presented. In Section 3, we show the main results, i.e., the boundedness, permanence and the existence of positive periodic solution of system (1.4). Finally, a specific example is given to illustrate our results. 2. Preliminaries Before going into details, we draw some notations and assumptions, furthermore, quote some useful Lemmas. Let f ðtÞ be a x-periodic continuous function defined on R. We define Z x ½f  ¼ x1 f ðtÞdt; f m ¼ max f ðtÞ; f l ¼ min f ðtÞ: t2R

t2R

0

In this paper, system (1.4) is said to be permanent, if there are positive constants m and M such that for any positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4), m 6 lim inf xi ðtÞ 6 lim sup xi ðtÞ 6 M; t!þ1

i ¼ 1; 2

t!þ1

and m 6 lim inf y i ðtÞ 6 lim sup y i ðtÞ 6 M; t!þ1

i ¼ 1; 2:

t!þ1

Let R4þ0 ¼ fðx1 ; x2 ; y 1 ; y 2 Þ : xi P 0y i P 0; i ¼ 1; 2g and R4þ ¼ fðx1 ; x2 ; y 1 ; y 2 Þ : xi > 0; y > 0; i ¼ 1; 2g. We firstly have the following result. Lemma 2.1. R4þ0 and R4þ are the positively invariant sets of system (1.4). The proof of Lemma 2.1 is simple, here we omit it. From Lemma 2.1, we obtain that for any point ðx10 ; x20 ; y 10 ; y 20 Þ 2 R4þ0 (or R4þ ) the solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) satisfying the initial condition xi ðt0 Þ ¼ xi0 ; y i ðt0 Þ ¼ y i0 ði ¼ 1; 2Þ and for some t0 2 R is nonnegative (or positive) on the maximal existence interval. We first consider the following well-known periodic logistic equation du ¼ u½aðtÞ  bðtÞu; ð2:1Þ dt where aðtÞ and bðtÞ are x-periodic continuous functions, bl P 0 and Ax ðbðtÞÞ > 0. We have the following well-known result.

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Lemma 2.2. (a) There is a constant M > 0 such that for any positive solution uðtÞ of Eq. (2.1), lim supt!1 uðtÞ 6 M. (b) If Ax ðaðtÞÞ > 0, then Eq. (2.1) has a unique globally asymptotically stable positive x-periodic solution. Lemma 2.2 can be found in many articles, for example, see [26]. Next, we consider the following single species logistic equation with dispersal in two patches n X dxi ¼ xi ½b1i ðtÞ  a1i ðtÞxi  þ d ij ðtÞðxj  xi Þ; dt i¼1

i ¼ 1; 2;

ð2:2Þ

where xi is the population density of species x in the ith patch, functions ai ðtÞ; bi ðtÞ and d ij ðtÞ ði; j ¼ 1; 2; i 6¼ jÞ are given in above system (1.4) and are continuous and x-periodic defined on R, and d ii ðtÞ ¼ 0 ði ¼ 1; 2Þ for all t 2 R. Further, we assume that ali > 0 and d lij > 0 for all i; j ¼ 1; 2 and i 6¼ j. For the system (2.2), we introduce the following assumptions. (H1) There is an i 2 f1; 2; 3; 4g, such that ½Ai  > 0, where A1 ðtÞ ¼ b11 ðtÞ  d 12 ðtÞ; A2 ðtÞ ¼ b12 ðtÞ  d 21 ðtÞ; A3 ðtÞ ¼ min fb11 ðtÞ  d 12 ðtÞ þ d 21 ðtÞ; b12 ðtÞ  d 21 ðtÞ þ d 12 ðtÞg; A4 ðtÞ ¼ min fb11 ðtÞ; b12 ðtÞg: For system (2.2), we have the following result. Lemma 2.3. If (H1) holds, then Eq. (2.2) has a unique positive x-periodic solution ðx1 ðtÞ; x2 ðtÞÞ which is globally asymptotically stable. Lemma 2.3 can be found in article [26, Lemma 2]. We further consider the following ordinary differential equation dx ¼ F ðt; xÞ; dt

ð2:3Þ

where t 2 R; x 2 R2 , function F ðt; xÞ : R  X ! R2 is continuous with respect to ðt; xÞ and satisfies the local Lipschitz condition with respect to x, where X  R2 is a region. Let F ðt; xÞ ¼ ðf1 ðt; xÞ; f2 ðt; xÞÞ and x ¼ ðx1 ; x2 Þ. Eq. (2.3) is said to be cooperative on X, if for any i; j ¼ 1; 2 and i 6¼ j; fi ðt; xÞ is nondecreasing with respect to xj on X. Let uðtÞ ¼ ðu1 ðtÞ; u2 ðtÞÞ : ½t0 ; T  ! X be a continuously differentiable function, and let xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ be a solution of Eq. (2.3) defined on ½t0 ; T . We have the following well-known result. Lemma 2.4. If ui ðt0 Þ < xi ðt0 Þ and dui ðtÞ 6 fi ðt; uðtÞÞ dt

for all t 2 ½t0 ; T ; i ¼ 1; 2;

then ui ðtÞ < xi ðtÞ for all t 2 ½t0 ; T  and i ¼ 1; 2. Lemma 2.4 can be found in many articles, for example, see [19].

3. Main results Let ðx1 ðtÞ; x2 ðtÞÞ be the unique positive x-periodic solution of Eq. (2.2). Throughout the paper, we make the following assumptions. (H2) al1 > 0, and al2 > 0, where a1 ðtÞ ¼ mini¼1;2 fa1i ðtÞg; a2 ðtÞ ¼ mini¼1;2 fa2i ðtÞg. Moreover, d lij > 0 ði; j ¼ 1; 2; i 6¼ jÞ; c1i ðtÞ > 0; c2i ðtÞ P 0; i ¼ 1; 2. (H3) There is an i 2 f1; 2; 3; 4g, such that ½Bi  > 0, where B1 ðtÞ ¼ b21 ðtÞ  D12 ðtÞ þ c21 ðtÞx1 ðtÞ; B2 ðtÞ ¼ b22 ðtÞ  D21 ðtÞ þ c22 ðtÞx2 ðtÞ;   B3 ðtÞ ¼ min b21 ðtÞ  D12 ðtÞ þ D21 ðtÞ þ c21 ðtÞx1 ðtÞ; b22 ðtÞ  D21 ðtÞ þ D12 ðtÞ þ c22 ðtÞx2 ðtÞ ;   B4 ðtÞ ¼ min b21 ðtÞ þ c21 ðtÞx1 ðtÞ; b22 ðtÞ þ c22 ðtÞx2 ðtÞ :

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(H4) ½b2  > 0, where b2 ðtÞ ¼ mini¼1;2 fb2i ðtÞg. (H5) ½g < 0, where gðtÞ ¼ max fb21 ðtÞ  D12 ðtÞ þ D21 ðtÞ; b22 ðtÞ  D21 ðtÞ þ D12 ðtÞg: With respect to the ultimate boundedness of system (1.4), we have the following result. Theorem 3.1. Suppose that assumption (H1) and (H2) hold, then all solution of system (1.4) are ultimate bounded. That is there is a constant M > 0 which is independent with any solution of system (1.4) such that for any positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4), lim sup xi ðtÞ < M; t!1

lim sup y i ðtÞ < M;

i ¼ 1; 2:

t!1

Proof. From (H2) and system (1.4), we can obtain 2 X dxi ðtÞ 6 xi ðtÞ½b1i ðtÞ  a1i ðtÞxi ðtÞ þ d ij ðtÞðxj ðtÞ  xi ðtÞÞ: dt j¼1

ð3:1Þ

Let ðx1 ðtÞ; x2 ðtÞÞ be the solution of (2.2) with the initial value xi ð0Þ ¼ xi ð0Þ ði ¼ 1; 2Þ, by Lemma 2.4, we get xi ðtÞ 6 xi ðtÞ

for all i ¼ 1; 2:

ð3:2Þ ðx1 ðtÞ; x2 ðtÞÞ

Since (H1) and (H2), we see that system (2.2) has a uniquePpositive periodic solution which is globally asymptotically stable, choosing a constant M 1 > maxt2½0;x f 2i¼1 xi ðtÞg, there exists a T 0 > 0 such that xi ðtÞ < M 1 ;

for all t P T 0 and i ¼ 1; 2:

ð3:3Þ

Form (3.2) and (3.3), we can obtain that lim sup xi ðtÞ < M 1

for all i ¼ 1; 2:

ð3:4Þ

t!1

Let rðtÞ ¼ maxi¼1;2 fy i ðtÞg, from (3.2) and (3.3), we have drðtÞ 6 rðtÞ½b2 ðtÞ  a2 ðtÞrðtÞ þ c2 ðtÞM 1  dt

ð3:5Þ

for all t P T 0 , where c2 ðtÞ ¼ maxi¼1;2 fc2i ðtÞg. Therefore, by Lemma 2.4 we have rðtÞ 6 rðtÞ for all t P T 0 ; where rðtÞ is the solution of the following auxiliary system: drðtÞ ¼ rðtÞ½b2 ðtÞ  a2 ðtÞrðtÞ þ c2 ðtÞM 1 ; dt

ð3:6Þ

with initial rð0Þ ¼ rð0Þ. From (H2) and Lemma 2.2 we have that there is a constant M 2 > 0 such that lim sup rðtÞ 6 M 2 , thus, lim sup rðtÞ 6 M 2 . Therefore, we. have lim sup y i ðtÞ 6 M 2

for all i ¼ 1; 2:

Choosing a constant M > maxfM 1 ; M 2 g, we see (3.4) and (3.7) hold. This completes the proof of Theorem 3.1.

ð3:7Þ h

Furthermore, on the permanence of system (1.4), we have the following result. Theorem 3.2. Suppose assumption (H1)–(H3) and (H4) or (H5) hold, then system (1.4) is permanent. Proof. In the following, we will use the following several claims to complete the proof of Theorem 3.2. Claim 1. There is a constant g > 0 such that for any positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) there is an i 2 f1; 2; 3; 4g we have lim sup qi ðtÞ > g; where q1 ðtÞ ¼ x1 ðtÞ; q2 ðtÞ ¼ x2 ðtÞ; q3 ðtÞ ¼ x1 ðtÞ þ x2 ðtÞ; q4 ðtÞ ¼ mini¼1;2 fxi ðtÞg.

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Proof. For convenience of proof, we take i = 1 in the assumption (H1), the proof of i = 2, 3, 4 is similar, we omit it here, and we always prove the following Claim 2 under the situation i = 1. h Case 1. Firstly, we prove Claim 1 with assumptions (H1)–(H4). Let a ¼ maxt2½0;x fjb2 ðtÞj þ a2 ðtÞ þ c2 ðtÞg. By assumption (H1) and (H4), we can choose positive constants 0 ; 1 and k1 such that Z x ½b2 ðtÞ  a2 ðtÞ0 þ c2 ðtÞ1 dt < k 1 ; ð3:8Þ 0

and

Z

x

/1 ðtÞdt > k 1 ;

ð3:9Þ

0

where /1 ðtÞ ¼ b11 ðtÞ  d 12 ðtÞ  a11 ðtÞ1  c11 ðtÞ0 expðaxÞ: Suppose that Claim 1 is not true, then there is a positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) such that for any i 2 f1; 2; 3; 4g lim sup qi ðtÞ 6 1 :

ð3:10Þ

t!1

Therefore, there is T 0 > 0 such that xi ðtÞ 6 1

for all t P T 0 and i ¼ 1; 2:

ð3:11Þ

Let r1 ðtÞ ¼ maxi¼1;2 fy i ðtÞg. Calculating the Dini upper right derivation of r1 ðtÞ at time t and by (3.11) we obtain   þ D r1 ðtÞ 6 r1 ðtÞ b2 ðtÞ  a2 ðtÞr1 ðtÞ þ c2 ðtÞ max xi ðtÞ 6 r1 ðtÞ½b2 ðtÞ  a2 ðtÞr1 ðtÞ þ c2 ðtÞ1 : ð3:12Þ i¼1;2

If r1 ðtÞ P 0 for all t P T 0 , integrating (3.12) from T0 to t, we have Z t r1 ðtÞ 6 r1 ðT 0 Þ exp ½b2 ðsÞ  a2 ðsÞ0 þ c2 ðsÞ1 dt: T0

By (3.8) we can obtain r1 ðtÞ ! 0 as t ! 1 which leads to a contradiction. Hence there is T 1 P T 0 such that r1 ðT 1 Þ < 0 . In the following, we will prove r1 ðtÞ 6 0 expðaxÞ for all t P T 1 :

ð3:13Þ

In fact, if (3.13) is not true, then there is t2 > t1 > T 1 such that r1 ðt2 Þ > 0 expðaxÞ; r1 ðt1 Þ ¼ 0 ; r1 ðtÞ > 0 for all t 2 ðt1 ; t2 . Further, we can choose an integer p P 0 such t2 2 ðt1 þ px; t1 þ ðp þ 1ÞxÞ. Integrating (3.12) from t1 to t2, by (3.8) we obtain Z t2 Z t2 r1 ðt2 Þ 6 r1 ðt1 Þ exp ½b2 ðtÞ  a2 ðtÞr1 ðtÞ þ c2 ðtÞ1 dt 6 0 exp ½b2 ðtÞ  a2 ðtÞ0 þ c2 ðtÞ1 dt ¼ 0 exp

Z t1

t1 t1 þpx

þ

Z

t2

t1



½b2 ðtÞ  a2 ðtÞ0 þ c2 ðtÞ1 dt 6 0 expðaxÞ:

t1 þpx

Thus, this leads to a contradiction. Therefore, (3.13) is true, and we have y i ðtÞ 6 0 expðaxÞ

for all t P T 1 i ¼ 1; 2:

ð3:14Þ

Therefore, from (3.11) and (3.14) we have dq1 ðtÞ dx1 ¼ P x1 ½b11 ðtÞ  a11 ðtÞ1  c11 ðtÞ expðaxÞ  d 12 ðtÞ ¼ q1 ðtÞ/1 ðtÞ: dt dt

ð3:15Þ

Integrating (3.15) from T1 to t, by (3.9) we have q1 ðtÞ ! 1 as t ! 1, which leads to a contradiction with (3.11). Therefore, Claim 1 holds.

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Case 2. Now we prove Claim 1 under the assumptions (H1)–(H3) and (H5). Let r2 ðtÞ ¼ y 1 ðtÞ þ y 2 ðtÞ, from (H5) we can select positive constants 00 ; 01 and k 01 such that Z x g0 ðtÞdt < k 01

ð3:16Þ

0

and

Z

x

/01 ðtÞdt > k 01 ;

ð3:17Þ

0

where

  g0 ðtÞ ¼ max b21 ðtÞ þ c21 ðtÞ01  D12 ðtÞ þ D21 ðtÞ; b22 ðtÞ þ c22 ðtÞ01  D21 ðtÞ þ D12 ðtÞ ;

and /01 ðtÞ ¼ b11 ðtÞ  d 12 ðtÞ  a11 ðtÞ01  c11 c11 ðtÞ00 : Suppose that Claim 2 is not true, then there is a solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) such that for any i 2 f1; 2; 3; 4g lim sup qi ðtÞ 6 01 :

ð3:18Þ

Therefore, there is T 0 > 0 such that xi ðtÞ 6 01

for all t P T 0 and i ¼ 1; 2:

ð3:19Þ

Calculating the Dini upper right derivation of r2 ðtÞ we have   dr2 ðtÞ 6 y 1 ðtÞ b21 ðtÞ þ c21 ðtÞ01  D12 ðtÞ þ D21 ðtÞ þ y 2 ðtÞ b22 ðtÞ þ c22 ðtÞ01  D21 ðtÞ þ D12 ðtÞ 6 r2 ðtÞg0 ðtÞ: dt ð3:20Þ Integrating (3.20) from T0 to t, from (3.16) we have r2 ðtÞ ! 10 as t ! 1. Therefore, there is constant T 1 > T 0 such that y i ðtÞ 6 00

for all t P T 1 i ¼ 1; 2:

ð3:21Þ

Hence, from (3.21) we have  dq1 ðtÞ dx1 ðtÞ ¼ P x1 ðtÞ b11 ðtÞ  a11 ðtÞ01  c11 ðtÞ00  d 12 ðtÞ ¼ q1 ðtÞ/1 ðtÞ: dt dt From (3.17), we have q1 ðtÞ ! 1 as t ! 1, which leads to a contradiction with (3.19). Therefore, we complete the proof of the Claim 1. Claim 2. There is a constant c > 0 such that lim supt!1 xi ðtÞ > c ði ¼ 1; 2Þ for any positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4). Proof. Prom Claim 1, we have there is constant g > 0 for any positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) there is an i 2 f1; 2; 3; 4g such that lim supt!1 qi ðtÞ > g. Without loss generality, here we take i ¼ 1. Therefore, for any positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) we have lim supt!1 q1 ðtÞ ¼ lim supt!1 x1 ðtÞ > g. So there is time sequence {tk} and tk ! 1 as n ! 1, such that x1 ðtk Þ > g k ¼ 1; 2; . . . :

ð3:22Þ

By the ultimate boundedness of solution of system (1.4) (see Theorem 3.1), there is a constant T 1 > 0 such that xi ðtÞ < M;

y i ðtÞ < M

for all t P T 1 ; i ¼ 1; 2;

ð3:23Þ

where M > 0 is a constant and independent of any positive solution of system (1.4). Hence, there is a constant B > 0 and B is also independent of any positive solution of system (1.4) such that

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dx1 ðtÞ



dt 6 B for all t P T 1 :

ð3:24Þ

From this, we can obtain that there is a constant d > 0, and d is independent of any positive solution of system (1.4) and k ¼ 1; 2; . . ., such that 1 x1 ðtÞ P g 2

for all t 2 ½tk  d; tk þ d; k ¼ 1; 2; . . . :

ð3:25Þ

Without loss of generality, we can assume for any k 5 m ½tk  d; tk þ d \ ½tm  d; tm þ d ¼ U:

ð3:26Þ

Choose constant M > M such that   d 21 ðtÞ < 0: b12 ðtÞ  ða12 ðtÞ þ c12 ðtÞÞM

ð3:27Þ

Further, we can choose a small enough 0 < c < 12 d 1 gd such that   d 21 ðtÞ > 1 d 1 g x2 ðtÞ½b12 ðtÞ  ða12 ðtÞ þ c12 ðtÞ þ c12 ðtÞÞM 4

ð3:28Þ

for all x2 ðtÞ 6 c, where d 1 ¼ mint2½0;x fd 21 ðtÞg. Suppose that lim supt!1 x2 ðtÞ < c. Then there is a T 2 > T 1 such that for t P T 2 x2 ðtÞ 6 c:

ð3:29Þ

Choose an integer k 0 > 0 such that tk  d > T 2 for all k P k 0 . Then by (3.27) and (3.28) for any k P k 0 and t 2 ½tk  d; tk þ d, we have dx2 ðtÞ   d 21 ðtÞ þ d 21 ðtÞx1 ðtÞ > 1 d 1 g > 0: P x2 ðtÞ½b12 ðtÞ  ða12 ðtÞ þ c12 ðtÞÞM dt 4

ð3:30Þ

Therefore, we finally have x2 ðtk þ dÞ P 12 d 1 gd þ x2 ðtk  dÞ P 12 d 1 gd, i.e., c P 12 d 1 gd. This lead to a contradiction and completes the proof of Claim 2. h Claim 3. There is a constant n > 0 such that for any positive solution (x1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) there is an i 2 f1; 2; 3; 4g we have lim sup si ðtÞ > n; t!1

where s1 ðtÞ ¼ y 1 ðtÞ; s2 ðtÞ ¼ y 2 ðtÞ ¼ y 2 ðtÞ; s3 ðtÞ ¼ y 1 ðtÞ þ y 2 ðtÞ; s4 ðtÞ ¼ mini¼1;2 fy i ðtÞg. Proof. For convenience of proof, here we take i = 3 in (H3), the proof of i = 2, 3, 4 is similar, here we omit it. At the same time, we only prove Claim 3 in assumption (H1)–(H4), the proof under the assumption (H1)–(H3) and (H5) is similar, we omit it here too. By (H3) we can choose constant 0 > 0 and 1 > 0 such that ½B03  > 0, where    B03 ðtÞ ¼ max b21 ðtÞ þ c12 ðtÞ x1 ðtÞ  1  a21 ðtÞ0  D12 ðtÞ þ D21 ðtÞ; b22 ðtÞ þ c22 ðtÞ x1 ðtÞ  1 a22 ðtÞ0  D21 ðtÞD12 ðtÞg: For constant a > 0, we consider the following auxiliary system: 2 X dxi ðtÞ ¼ xi ðtÞ½b1i ðtÞ  c1i ðtÞa  a1i ðtÞxi ðtÞ þ d ij ðtÞ½xj  xi ; dt j¼1

i ¼ 1; 2:

ð3:31Þ

By assumption (H1), for constant 0 < a0 < 0 and any a 2 ð0; a0 Þ we have ½Aa3  > 0, where Aa3 ðtÞ ¼ min fb11 ðtÞ  d 12 ðtÞ þ d 21 ðtÞ  c11 ðtÞa; b12 ðtÞ  d 21 ðtÞ þ d 12 ðtÞ  c12 ðtÞag: Hence, by Lemma 2.3 we have that system (3.31) has a unique globally asymptotically stable positive x-periodic solution (x1a ; x2a ). By the continuity of solution with respect to parameters, we can obtain that xia ðtÞ ði ¼ 1; 2Þ uniformly for t 2 ½0; x converge to xi ðtÞ as a ! 0. Hence, there is a constant a 2 ð0; a0 Þ such that 1 jxia ðtÞ  xi ðtÞj < ; t 2 ½0; x: ð3:32Þ 2

L. Zhang, Z. Teng / Chaos, Solitons and Fractals 38 (2008) 1483–1497

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Suppose that Claim 3 is not true, then there is a positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) such that for any i 2 f1; 2; 3; 4g lim sup si ðtÞ < a :

ð3:33Þ

t!1

Therefore, there is T 1 > 0 such that y i ðtÞ < a for all t P T 1 . And owing to 0 < a < a0 < 0 , we finally get y i ðtÞ < 0

ð3:34Þ

for all t P T 1 :

For all t P T 1 , we can obtain dxi ðtÞ P xi ðtÞ½b1i ðtÞ  c1i ðtÞa  a1i ðtÞxi ðtÞ þ um2j¼1 d ij ðtÞðxj  xi Þ; dt

i ¼ 1; 2:

ð3:35Þ

Hence, by Lemma 2.4, we have xi ðtÞ P xia ðtÞ for all t P T 1 and i ¼ 1; 2;

ð3:36Þ

where ðx1a ðtÞ; x2a ðtÞÞ is the solution of Eq. (3.31) with a ¼ a and the initial value xia ðT 1 Þ ¼ xi ðT 1 Þ for i = 1, 2. By the asymptotic stability of the positive x-periodic solution (x1a ðtÞ; x2a ðtÞÞ, we can get that there is T 2 > T 1 such that 1 jxia ðtÞ  xia ðtÞj < ; i ¼ 1; 2; t P T 2 : ð3:37Þ 2 Hence, by (3.32), (3.36) and (3.37) we get xi ðtÞ P xi ðtÞ  1 ;

for all i ¼ 1; 2; t P T 2 :

ð3:38Þ

Therefore, for all t P T 2 , by (3.34) and (3.38) we have   ds3 ðtÞ P y 1 ðtÞ b21 ðtÞ  a21 ðtÞ0 þ c21 ðtÞ x1 ðtÞ  1  D12 ðtÞ þ D21 ðtÞ dt   þ y 2 ðtÞ b22 ðtÞ þ a22 ðtÞ0 þ c22 ðtÞ x2 ðtÞ  1  D21 ðtÞ þ D12 ðtÞ P s3 ðtÞB03 ðtÞ: Integrating this inequality from T2 to t, we obtain Z t B03 ðsÞds: s3 ðtÞ P s3 ðT 2 Þ exp T2

½B03 

Since > 0, we finally have s3 ðtÞ ! 1 as t ! 1 which leads to a contradiction with (3.33). This completes the proof of Claim 3. h Claim 4. There is a constant b > 0, such that lim supt!1 y i ðtÞ > bði ¼ 1; 2Þ for any positive periodic solution of system (1.4). Proof. From Claim 3, we have there is constant n > 0 for any positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) there is an i 2 f1; 2; 3; 4g such that lim supt!1 si ðtÞ > n. Without loss generality, here we take i = 1. Therefore, for any positive solution ðx1 ðtÞ; x2 ðtÞ; y 1 ðtÞ; y 2 ðtÞÞ of system (1.4) we have lim supt!1 s1 ðtÞ ¼ lim supt!1 y 1 ðtÞ > n. So there is time sequence {tk} and tk ! 1 as n ! 1, such that y 1 ðtk Þ > n

k ¼ 1; 2; . . .

ð3:39Þ

By the ultimate boundedness of solution of system (1.4) (see Theorem 3.1), there is a constant T 1 > 0 such that xi ðtÞ < M;

y i ðtÞ < M

for all t P T 1 ; i ¼ 1; 2;

ð3:40Þ

where M > 0 is a constant and independent of any positive solution of system (1.4). Hence, there is a constant B 0 > 0 and B 0 is also independent of any positive solution of system (1.4) such that



dy 1 ðtÞ

0



ð3:41Þ

dt 6 B for all t P T 1 : From this, we can obtain that there is a constant d > 0, and d is independent of any positive solution of system (1.4) and k = 1, 2, . . ., such that 1 y 1 ðtÞ P n for all t 2 ½tk  d; tk þ d; k ¼ 1; 2; . . . 2

ð3:42Þ

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Without loss of generality, we can assume for any k 5 m ½tk  d; tk þ d \ ½tm  d; tm þ d ¼ U:

ð3:43Þ

Choose constant M > M such that b22 ðtÞ  a22 ðtÞM  D21 ðtÞ < 0:

ð3:44Þ

Further, we can choose a small enough 0 < b < 12 D1 nd such that  1 y 2 ðtÞ b22 ðtÞ  a22 ðtÞM  D21 ðtÞ >  D1 n 4

ð3:45Þ

for all y 2 ðtÞ 6 b, where D1 ¼ mint2½0;x fD21 ðtÞg. Suppose that lim supt!1 y 2 ðtÞ < b, then there is a T 2 > T 1 such that for t P T2 y 2 ðtÞ 6 b:

ð3:46Þ

Choose an integer k 0 > 0 such that tk  d > T 2 for all k P k 0 . Then by (3.44) and (3.45) for any k P k 0 and t 2 ½tk  d; tk þ d, we have  dy 2 ðtÞ 1 P y 2 ðtÞ b22 ðtÞ  a22 ðtÞM  D21 ðtÞ þ D21 ðtÞy 1 ðtÞ > d 1 n > 0: dt 4

ð3:47Þ

Therefore, we finally have y 2 ðtk þ dÞ P 12 d 1 nd þ y 2 ðtk  dÞ P 12 D1 nd, i.e., b P 12 D1 nd. This lead to a contradiction and completes the proof of Claim 4. h By Claims 2, 4 and Theorem 3.1, using the results in [30,31] that uniform weak persistence can induce uniform strong persistence, we can obtain that there is constant l > 0 and l is independent with any positive solution of system (1.4) such that lim inf xi ðtÞ > l;

lim inf y i ðtÞ > l i ¼ 1; 2:

Based on arguments above, we see that Theorem 3.2 is true and this completes the proof.

ð3:48Þ h

Lastly, from Theorem 3.2 and Theorem 1 given by Teng and Chen in [29] on the existence of positive periodic solutions for general delayed periodic n-species Kolmogorov systems, we have the following result. Corollary 3.1. Suppose the assumptions (H1)–(H3) and (H4) or (H5) hold, then system (1.4) has a positive x-periodic solution. Remark 3.1. In this paper, as a primary work, we study a two species nonautonomous predator–prey type Lotka– Volterra system with periodic coefficients and equal dispersal in two patches, sufficient condition on the permanence has been obtained. There is a very important open question that is whether we can extend these results to the more general two species nonautonomous periodic predator–prey system with both predator and prey dispersal in npatches, i.e., n X dxi ðtÞ ¼ xi ðtÞ½b1i ðtÞ  a1i ðtÞxi ðtÞ  c1i ðtÞy i ðtÞ þ d ij ðtÞðxj ðtÞ  xi ðtÞÞ; dt j¼1 n X dy i ðtÞ ¼ y i ðtÞ½b2i ðtÞ  a2i ðtÞy i ðtÞ  c2i ðtÞxi ðtÞ þ Dij ðtÞðy j ðtÞ  y i ðtÞÞ: dt j¼1

ð3:49Þ

Remark 3.2. For system (1.4), in (H4) we suppose that the predator species is density-dependent on the average value, i.e., ½b2  > 0, however, we know in real life, usually the predator species may be density-independent (such work can be seen in [34]), therefore, there is a very important open question that is wether the same results can be true in the situation that the predator species is density-independent in a patchy environment with both predator and prey species dispersing. Remark 3.3. In [4], the authors considered the following nonautonomous two species predator–prey dispersing system in n-patches

L. Zhang, Z. Teng / Chaos, Solitons and Fractals 38 (2008) 1483–1497 n X  dxi ðtÞ Dij ðtÞ xj ðtÞ  aij ðtÞxi ðtÞ ; ¼ xi ðtÞ½bi ðtÞ  ai ðtÞxi ðtÞ  ci ðtÞy i ðtÞ þ dt j¼1 n X  dy i ðtÞ ¼ y i ðtÞ½d i ðtÞ þ ei ðtÞxi ðtÞ  fi ðtÞy i ðtÞ þ kij ðtÞ y j ðtÞ  bij ðtÞy i ðtÞ ; dt j¼1

1493

i ¼ 1; 2; ð3:50Þ i ¼ 1; 2:

Particularly, when n = 2 we have the following system as a special case of system (3.50): dx1 ðtÞ ¼ x1 ðtÞ½b1 ðtÞ  a1 ðtÞx1 ðtÞ  c1 ðtÞy 1 ðtÞ þ D12 ðtÞðx2 ðtÞ  a12 ðtÞx1 ðtÞÞ; dt dx2 ðtÞ ¼ x2 ðtÞ½b2 ðtÞ  a2 ðtÞx2 ðtÞ  c2 ðtÞy 2 ðtÞ þ D21 ðtÞðx1 ðtÞ  a21 ðtÞx2 ðtÞÞ; dt dy 1 ðtÞ ¼ y 1 ðtÞ½d 1 ðtÞ þ e1 ðtÞx1 ðtÞ  f1 ðtÞy 1 ðtÞ þ k12 ðtÞðy 2 ðtÞ  b12 ðtÞy 1 ðtÞÞ; dt dy 1 ðtÞ ¼ y 2 ðtÞ½d 2 ðtÞ þ e2 ðtÞx2 ðtÞ  f2 ðtÞy 2 ðtÞ þ k21 ðtÞðy 1 ðtÞ  b21 ðtÞy 2 ðtÞÞ: dt

ð3:51Þ

They established the permanence of system (3.51) under the assumptions of (H4.1) or (H4.2), i.e., there is a i0 2 f1; 2g such that ½/1  > 0 or ½c1  > 0 where /1 ðtÞ ¼ bi0 ðtÞ  ci0 ðtÞN y  Di0 j ðtÞai0 j ðtÞ; and

j ¼ 1; 2; j 6¼ i

  c1 ðtÞ ¼ min bi ðtÞ  ci ðtÞN y  Dij ðtÞaij ðtÞ þ Dji ðtÞ ; i¼1;2

m  ei N x þ kmij þ  ; N y ¼ max i¼1;2 fil

j ¼ 1; 2; j 6¼ i;

and N x ¼ max i¼1;2

m  jbi j þ Dmij þ  ; l ai

 > 0 is any constant. Contrasting the ½/1  > 0 and ½c1  > 0 with our assumptions (H1) and (H3) we will find that the conditions in ours is much simple and satisfied easily than that in [4]. Remark 3.4. In Eq. (2.2), if assumption (H1) holds, then by Lemma 2.3, Eq. (2.2) has a unique globally asymptotically stable positive x-periodic solution x ðtÞ  ðx1 ðtÞ; x2 ðtÞÞ. Further, by using the result in [22] we will obtain

 m  b1j ¼ K 1 ; j¼1;2 a1j

max xi ðtÞ 6 max t2R

i ¼ 1; 2;

ð3:52Þ

and min xi ðtÞ t2R

( P min j¼1;2

b1j a1j

l )

¼ k 1 ;

i ¼ 1; 2:

ð3:53Þ

Therefore, we can replace xi ðtÞði ¼ 1; 2Þ in assumption (H3) by (3.52), which will further simplify the condition in (H3), i.e., we can obtain (H 03 ) there is an i 2 f1; 2; 3; 4g, such that [B0i  > 0, where B01 ðtÞ ¼ b21 ðtÞ  D12 ðtÞ þ c21 ðtÞk 1 ; B02 ðtÞ ¼ b22 ðtÞ  D21 ðtÞ þ c22 ðtÞk 1 ;   B03 ðtÞ ¼ min b21 ðtÞ  D12 ðtÞ þ D21 ðtÞ þ c21 ðtÞk 1 ; b22 ðtÞ  D21 ðtÞ þ D12 ðtÞ þ c22 ðtÞk 1 ;   B04 ðtÞ ¼ min b21 ðtÞ þ c21 ðtÞk 1 ; b22 ðtÞ þ c22 ðtÞk 1 : Correspondingly, we have the following result with respect to permanence of system (1.4). Corollary 3.2. Suppose assumption (H1)–ðH03 Þ and (H4) or (H5) hold, then system (1.4) is permanent.

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4. Example, numerical simulation and discussing In this paper, we investigate a class of periodic two species predator–prey Lotka–Volterra type systems with dispersal. By using analytic method we give the criteria for the boundedness, permanence, the existence of positively periodic solution of system (1.4). In order to testify the validity of our results, we consider the following two species predator–prey Lotka–Volterra type systems with dispersal in two patches: dx1 dt dx2 dt dy 1 dt dy 2 dt

¼ x1 ðb11 ðtÞ  a11 x1  c11 ðtÞy 1 Þ þ d 12 ðx2  x1 Þ; ¼ x2 ðb12 ðtÞ  a12 x2  c12 ðtÞy 2 Þ þ d 21 ðx1  x2 Þ; ð4:1Þ ¼ y 1 ðb21 ðtÞ  a21 y 1 þ c21 ðtÞx1 Þ þ D12 ðy 2  y 1 Þ; ¼ y 2 ðb22 ðtÞ  a22 y 1 þ c22 ðtÞx1 Þ þ D21 ðy 1  y 2 Þ:

Corresponding prey dispersal system is dx1 ¼ x1 ðb11 ðtÞ  a11 x1 Þ þ d 12 ðx2  x1 Þ; dt dx2 ¼ x2 ðb12 ðtÞ  a12 x2 Þ þ d 21 ðx1  x2 Þ: dt

ð4:2Þ

In system (4.1) and system (4.2), let b11 ðtÞ ¼ 1 þ sinð2ptÞ; b12 ðtÞ ¼ 12 þ cosð2ptÞ; b21 ðtÞ ¼ 0:6 þ sinð2ptÞ; b22 ðtÞ ¼ 0:5 þ sinð2ptÞ; a11 ðtÞ ¼ 0:4; a12 ðtÞ ¼ 0:3; a21 ðtÞ ¼ 0:3; a22 ðtÞ ¼ 0:2; c11 ðtÞ ¼ 0:3; c12 ðtÞ ¼ 0:5; c21 ðtÞ ¼ 0:5; c22 ðtÞ ¼ 0:6; d 12 ¼ 0:6; d 21 ¼ 0:5; D12 ¼ 0:4; D21 ¼ 0:6. We easily verify that for any i 2 f1; 2; 3; 4g½Ai  > 0 holds in assumption (H1). Therefore, from Lemma 2.3 system (4.2) has a unique globally asymptotically stable positive periodic solution (x1 ðtÞ; x2 ðtÞÞ. By numerical simulation, we get 2:0 6 x1 ðtÞ 6 2:65; 1:7 6 x2 ðtÞ 6 2:26 (see Fig. 1). At the same time, it is easy to verify that (H2) and (H4) also hold, and for any i 2 f1; 2; 3; 4g½bi  > 0 holds in assumption (H3). Therefore, from Theorems 3.1 and 3.2, and Corollary 3.1 we obtain that system (4.1) is permanent and have a positively periodic solution. Numerical simulation of these results can be seen in Fig. 2. However, if in system (4.1), b1i ðtÞ; aij ; cij ðtÞ ði; j ¼ 1; 2Þ and d ij ; Dij ði; j ¼ 1; 2; i 6¼ jÞ are given as in above and b21 ðtÞ ¼ 1:3 þ j sinð2ptÞj and b22 ðtÞ ¼ 1:2 þ cosð2ptÞ. It is easy to verify assumption (H1)–(H2) also hold, however, for any i 2 f1; 2; 3; 4g, (H3) does not hold. Therefore, we can not guarantee the permanence of both the prey and predator

Fig. 1. Dynamical behavior of system (4.2). Here, we take the initial value x0 ¼ ðx10 ; x20 Þ ¼ ð1:4; 1:5Þ.

L. Zhang, Z. Teng / Chaos, Solitons and Fractals 38 (2008) 1483–1497

1495

Fig. 2. Dynamical behavior of system (4.1). Here, we take the initial value ðx0 ; y 0 Þ ¼ ðx10 ; x20 ; y 10 ; y 20 Þ ¼ ð1:0; 1:1; 0:8; 0:9Þ.

species (see Fig. 3). From Fig. 3 we can see that the predator species y turn to extinction and prey species x turn to its periodic stable state (x1 ðtÞ; x2 ðtÞ). In Figs. 1 and 2, for the above given numbers, we see for any i 2 f1; 2; 3; 4g; ½Ai  > 0 and ½Bi  > 0 hold in (H1) and (H3) respectively. However, if in system (4.1) and system (4.2), let b11 ðtÞ ¼ 45 þ 15 sinð2ptÞ; b12 ðtÞ ¼ sinð2ptÞ; b21 ðtÞ ¼ 0:3 þ j sinð2ptÞj; b22 ðtÞ ¼ 0:35 þ j sinð27ptÞj; a11 ðtÞ ¼ 0:4; a12 ðtÞ ¼ 0:3; a21 ðtÞ ¼ 0:3; a22 ðtÞ ¼ 0:2; c11 ðtÞ ¼ 0:4; c12 ðtÞ ¼ 0:5; c21 ðtÞ ¼ 0:4  0:4 sinð2ptÞ; c22 ðtÞ ¼ 0:3; d 12 ¼ 0:3 þ 0:7j sinð2ptÞj; d 21 ¼ 0:6 þ 0:4j sinð2ptÞj; D12 ¼ 0:2 þ 0:8 j sinð2ptÞj; D21 ¼ 0:6 þ 0:4j sinð2ptÞj. We find in (H1 Þ; ½A1  > 0, but ½Ai  < 0ði ¼ 2; 3; 4Þ. Therefore, from Lemma 2.3 system (4.2) has a unique globally asymptotically stable positive periodic solution ðx1 ðtÞ; x2 ðtÞÞ. By numerical simulation, we get 1.43 6 x1 ðtÞ 6 1:56, and 0:9 6 x2 ðtÞ 6 1:26 (see Fig. 4). It is easy to verify that (H2) and (H4) also hold, at the same time in (H3), ½B1  > 0, but ½Bi  < 0 ði ¼ 2; 3; 4Þ. Therefore, from Theorems 3.1 and 3.2, and Corollary 3.1 we obtain that system (4.1) is permanent and have a positively periodic solution. Numerical simulation of these results can be seen in Fig. 5. Contrasting the two situation in Figs. 1, 2 and Figs. 4, 5, we find there is a very interesting phenomenon that is though both the prey and predator species can be permanent in the two situation, the conditions are not identical. In Fig. 1, 2 we fixup the dispersal rate of both prey and predator species, at the same time, the net increasing growth rate of the prey and predator species in both patches are positive on the average value (i.e.,

Fig. 3. Dynamical behavior of system (4.1). Here, we take the initial value ðx0 ; y 0 Þ ¼ ðx10 ; x20 ; y 10 ; y 20 Þ ¼ ð1:0; 1:1; 0:8; 0:9Þ.

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Fig. 4. Dynamical behavior of system (4.2). Here, we take the initial value x0 ¼ ðx10 ; x20 Þ ¼ ð1:42; 0:83Þ.

Fig. 5. Dynamical behavior of system (4.1). Here, we take the initial value ðx0 ; y 0 Þ ¼ ðx10 ; x20 ; y 10 ; y 20 Þ ¼ ð1:05; 0:7; 0:42; 0:18Þ.

½Ai  > 0; ½Bi  > 0; i ¼ 1; 2; 3; 4Þ, however, in Fig. 3, 4, we take a flexible dispersal rate of both prey and predator species and the net increasing growth rate of prey and predator are not positive on the average value in the 2nd-patch (i.e., ½Ai  < 0; ½Bi  < 0ði ¼ 2; 3; 4ÞÞ, but it is positive in the 1st-patch (i.e.,½A1  > 0 and ½B1  > 0Þ. Briefly, the dispersal can influence the persistence of prey and predator species, if the prey and predator species are persistent in one of patch, then due to the effect of dispersal, the prey and predator species will be persistent in other patches.

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