Dispersal permanence of a periodic predator–prey system with Holling type-IV functional response

Dispersal permanence of a periodic predator–prey system with Holling type-IV functional response

Applied Mathematics and Computation 218 (2011) 502–513 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepag...

330KB Sizes 0 Downloads 36 Views

Applied Mathematics and Computation 218 (2011) 502–513

Contents lists available at ScienceDirect

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

Dispersal permanence of a periodic predator–prey system with Holling type-IV functional response Meihua Huang a,⇑, Xuepeng Li b a b

School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, People’s Republic of China School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, People’s Republic of China

a r t i c l e

i n f o

a b s t r a c t In this paper, we study the permanence of a class of periodic predator–prey system with Holling type-IV functional response where the prey disperses in patchy environment with two patches, and provide a sufficient and necessary condition to guarantee permanence of the system. Finally, two examples are presented to illustrate the application of our main results. Ó 2011 Elsevier Inc. All rights reserved.

Keywords: Predator–prey Permanence Holling type-IV Dispersal

1. Introduction Many kinds of predator–prey systems have been studied, especially, permanence of the predator–prey system, for example in [1–6] and the references cited therein. But, it is difficult to get some sufficient and necessary conditions for the permanence of a time dependent predator–prey system. Already, in [6], Cui proposed the following periodic predator–prey system with Beddington–DeAngelis functional response:

 x_ 1 ¼ x1 b1 ðtÞ  a1 ðtÞx1 

 c1 ðtÞy þ DðtÞðx2  x1 Þ; eðtÞ þ bðtÞx1 þ cðtÞy

x_ 2 ¼ x2 ½b2 ðtÞ  a2 ðtÞx2  þ DðtÞðx1  x2 Þ;   c1 ðtÞx1 y_ ¼ y dðtÞ þ  qðtÞy : eðtÞ þ bðtÞx1 þ cðtÞy He obtained a sufficient and necessary condition to guarantee the predator and prey species to be permanent. In this paper, we consider the permanence of the following periodic predator–prey system with Holling type-IV functional response:

 x_ 1 ¼ x1 b1 ðtÞ  a1 ðtÞx1 

 c1 ðtÞy þ DðtÞðx2  x1 Þ; eðtÞ þ bðtÞx1 þ x21

x_ 2 ¼ x2 ½b2 ðtÞ  a2 ðtÞx2  þ DðtÞðx1  x2 Þ;   c2 ðtÞx1  qðtÞy : y_ ¼ y dðtÞ þ 2 eðtÞ þ bðtÞx1 þ x1

ð1:1Þ

where x1 and x2 denote the density of prey species in patch 1 and in patch 2 respectively, and y is the density of predator species that preys on x1  ai(t), bi(t), ci(t)(i = 1, 2), d(t), e(t), b(t), q(t) and D(t) are all positive, x-periodic and continuous for t P 0. ⇑ Corresponding author. E-mail address: [email protected] (M. Huang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.092

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

503

The periodic functions in (1.1) have the following biological meanings. bi(t) is the intrinsic growth rate for prey species in patch i(i = 1, 2); ai(t) represents the self-inhibition coefficient; and D(t) is the diffusion coefficient of prey species from patch i to patch j(i – j, i, j = 1, 2). The death rate of the predator population is proportional to the existing predator population and to its square with coefficients d(t) and q(t), respectively. c1(t) and c2(t) give the coefficients that relate to the conversion rate of the prey biomass (in the patch 1) into predator biomass. c1 ðtÞx1 c1 ðtÞx1 The function eðtÞþbðtÞx 2 represents the functional response of predator to the prey in patch 1. Let uðt; x1 Þ ¼ eðtÞþbðtÞx þx2 , then 1 þx1 1 1 we have @ @x1

uðt; x1 Þ P 0; 0 < x1 ðtÞ 6

@ @x1

uðt; x1 Þ < 0;

pffiffiffiffiffiffiffiffi eðtÞ;

pffiffiffiffiffiffiffiffi x1 ðtÞ > eðtÞ:

ð1:2Þ

The aim of this paper is, by further developing the analysis technique of Cui [6], to obtain a sufficient and necessary condition to guarantee the predator and prey species to be permanent of the system (1.1). The organization of this paper is as follows. In Section 2, we introduce some notations, definitions and lemmas which will be essential to our proofs. In Section 3, we state and prove the main result of this paper. In Section 4, we give two examples which together show the feasibility of our results. We discuss the biological meanings of the main result in Section 5. 2. Notations, definitions and preliminaries In this section, we introduce some definitions and notations and state some results which will be useful in subsequent sections. Let C denote the space of all bounded continuous functions f : R ! R; C 0þ the set of nonnegative f 2 C, and C+ the set of all f 2 C such that f is bounded below by a positive constant. Given f 2 C, we denote

f M ¼ sup f ðtÞ;

f L ¼ inf f ðtÞ tP0

tP0

and define the lower average AL(f) and upper average AM(f) of f by

AL ðf Þ ¼ lim inf ðt  sÞ1

Z

r!1 tsPr

t

f ðsÞds

s

and

AM ðf Þ ¼ lim sup ðt  sÞ1 r!1 tsPr

Z

t

f ðsÞds; s

respectively. If f 2 C is x-periodic, we define the average Ax(f) of f on the time interval [0, x] by

Ax ðf Þ ¼ x1

Z

x

f ðtÞdt:

0

Definition 2.1. The system of differential equations

x_ ¼ Fðt; xÞ;

x 2 Rnþ ;

is said to be permanent if there exists a compact set K in the interior of Rnþ ¼ fðx1 ; x2 ; . . . ; xn Þ 2 Rn : xi P 0; i ¼ 1; 2; . . . ; ng, such that all solutions starting in the interior of Rnþ ultimately enter and remain in K. Lemma 2.1 (see [7]). Let x(t) and y(t) be solution of

x_ ¼ Fðt; xÞ and

y_ ¼ Gðt; yÞ; respectively, where both systems are assumed to have the 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 ½t0 ; t 1   D:

If x(t0) 6 y(t0) then x(t1) 6 y(t1). If F = G and x(t0) < y(t0) then x (t1) < y(t1).To prove the permanence of the species in (1.1), we need the information on the periodic logistic models with and without dispersal. Lemma 2.2 (see [8]). The problem

x_ ¼ x½bðtÞ  aðtÞx;

x 2 Cþ;

has exactly one canonical solution U if a 2 C+, b 2 C and AL(b) > 0.

504

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

For the following dispersal logistic equations:

x_ 1 ¼ x1 ½b1 ðtÞ  a1 ðtÞx1  þ DðtÞðx2  x1 Þ; x_ 2 ¼ x2 ½b2 ðtÞ  a2 ðtÞx2  þ DðtÞðx1  x2 Þ;

ð2:2Þ

Lemma 2.3 (see [9]). Suppose that bi(t), ai(t)(i = 1, 2) and D(t) are positive and x- periodic functions, then (2.2) has a positive and   x-periodic solution x1 ðtÞ; x2 ðtÞ , which is globally asymptotically stable with respect to R2þ n fOg. 3. Main result Theorem 3.1. Suppose that

x1 ðtÞ <

pffiffiffiffiffiffiffiffi eðtÞ;

tP0

ð3:1Þ

holds, then system (1.1) is permanent if and only if

 Ax dðtÞ þ

 c2 ðtÞx1 ðtÞ > 0: eðtÞ þ bðtÞx1 ðtÞ þ x2 1 ðtÞ

ð3:2Þ

  Here x1 ðtÞ; x2 ðtÞ is the globally and asymptotically stable x-periodic solution of (2.2) given by Lemma 2.3. From the proof of Theorem 3.1, we also have the following Corollary 3.1. Corollary 3.1. Assume that (3.1) and

 Ax dðtÞ þ

 c2 ðtÞx1 ðtÞ 60 eðtÞ þ bðtÞx1 ðtÞ þ x2 1 ðtÞ

ð3:3Þ

  hold, where x1 ðtÞ; x2 ðtÞ is the globally and asymptotically stable x-periodic solution of (2.2) given by Lemma 2.3. Then any solution of system (1.1) with positive initial condition satisfies

lim yðtÞ ¼ 0:

t!1

To prove this theorem, we need several propositions. In the rest of this paper we denote (x1(t), x2(t), y(t)) be any solution of (1.1) with positive initial condition. Proposition 3.1. There exist positive constants Mx and My, such that

lim sup xi ðtÞ 6 M x ;

t!1

lim sup yðtÞ 6 M y ;

t!1

i ¼ 1; 2:

ð3:4Þ

Proof. Obviously, R3þ is a positively invariant set of (1.1). Given any positive solution (x1(t), x2(t), y(t)) of (1.1), we have

x_ i 6 xi ½bi ðtÞ  ai ðtÞxi  þ DðtÞðxj  xi Þ;

i; j ¼ 1; 2;

j – i;

on the other hand, the following auxiliary equations:

u_ i ¼ ui ½bi ðtÞ  ai ðtÞui  þ DðtÞðuj  ui Þ;

i; j ¼ 1; 2;

j – i;

has a unique globally asymptotically stable positive x-periodic solution with ui(0) = xi(0), by Lemma 2.1 we have

xi ðtÞ 6 ui ðtÞ;

ð3:5Þ 

x1 ðtÞ; x2 ðtÞ

 . Let (u1(t), u2(t)) be the solution of (3.5)

i ¼ 1; 2 for t P 0:

  Moreover, from the global stability of x1 ðtÞ; x2 ðtÞ , for every given e(0 < e < 1), there exists T0 > 0, such that

ui ðtÞ < xi ðtÞ þ e for t > T 0 ; hence

xi ðtÞ < xi ðtÞ þ e;

i ¼ 1; 2 for t > T 0 :

In addition we have

y_ 6 yðdðtÞ þ

  c2 ðtÞ x1 ðtÞ þ e  qðtÞyÞ; eðtÞ

t P T 0:

By Lemmas 2.1 and 2.2, there exists T1 > T0, such that

yðtÞ < y ðtÞ þ e;

for t > T 1 ;

505

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

where y⁄(t) is the positive and globally asymptotically stable x-periodic solution of the following auxiliary logistic equation:

v_ ¼ v

  c2 ðtÞðx1 ðtÞ þ eÞ  qðtÞv : dðtÞ þ eðtÞ



Denote M x ¼ max06t6x xi ðtÞ þ 1 : i ¼ 1; 2 and My = max06t6x{y⁄(t) + 1}, then (3.4) holds for system (1.1).

h

Proposition 3.2. Suppose (3.2) holds, then there exists a positive constant gx such that

lim sup x1 ðtÞ P gx :

ð3:6Þ

t!1

Proof. Suppose that (3.6) is not true, then there is a sequence fzm g  R3þ , such that

lim sup x1 ðt; zm Þ <

t!1

1 ; m

m ¼ 1; 2; . . . ;

ð3:7Þ

where (x1(t, zm), x2(t, zm), y(t, zm)) is the solution of (1.1) with initial values (x1(0, zm), x2(0, zm), y(0, zm)) = zm. Choose positive e sufficiently small satisfies:

c2 ðtÞe Þ < 0; eðtÞ c1 ðtÞe b1 ðtÞ  > 0; eðtÞ c1 ðtÞe expðaxÞ  a1 ðtÞe; b2 ðtÞ  a2 ðtÞe > 0: /e ðtÞ ¼ min b1 ðtÞ  eðtÞ

Ax ðdðtÞ þ

ð3:8Þ ð3:9Þ ð3:10Þ

n o 2 ðtÞ where a ¼ max06t6x dðtÞ þ ceðtÞ . By (3.7), for the given e > 0, there exists a positive integer N0, such that

lim sup x1 ðt; zm Þ <

t!1

1
ð3:11Þ ðmÞ

for m > N0. In the rest of this proof we always assume that m > N0. Above inequality implies that there exists s1

> 0, such that

x1 ðt; zm Þ < e ðmÞ

for t P s1 , and further

  c ðtÞe _ zm Þ 6 yðt; zm Þ dðtÞ þ 2 yðt;  qðtÞyðt; zm Þ eðtÞ ðmÞ

for t P s1 . By (3.8), any solution v(t) of the following equation:

v_ ¼ v



dðtÞ þ

 c2 ðtÞe  qðtÞv ; eðtÞ

with positive initial condition satisfies

lim v ðtÞ ¼ 0:

t!1

By Lemma 2.1, we have

lim yðt; zm Þ ¼ 0:

t!1

Therefore, there is a

ðmÞ sðmÞ > s1 such that 2

ðmÞ

yðt; zm Þ < e for t P s2 :

ð3:12Þ

This leads to

  c1 ðtÞe x_ 1 ðt; zm Þ P x1 ðt; zm Þ b1 ðtÞ   a1 ðtÞx1 ðt; zm Þ þ DðtÞðx2 ðt; zm Þ  x1 ðt; zm ÞÞx_ 2 ðt; zm Þ eðtÞ ¼ x2 ðt; zm Þ½b2 ðtÞ  a2 ðtÞx2 ðt; zm Þ þ DðtÞðx1 ðt; zm Þ  x2 ðt; zm ÞÞ ðmÞ 2 .

for t P s

Let (u1(t), u2(t)) be any positive solution of the following auxiliary equations:

  c1 ðtÞe u_ 1 ¼ u1 b1 ðtÞ   a1 ðtÞu1 þ DðtÞðu2  u1 Þ; eðtÞ u_ 2 ¼ u2 ½b2 ðtÞ  a2 ðtÞu2  þ DðtÞðu1  u2 Þ:

ð3:13Þ

506

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

  By (3.9) and Lemma 2.3, (3.13) has a unique positive and x-periodic solution u1 ðtÞ; u2 ðtÞ , which is globally asymptotically stable. So we have

xi ðt; zm Þ >

ui ðtÞ ; 2

i ¼ 1; 2

from Lemma 2.1 for sufficiently large t > 0 and m > N0, which is a contradiction with (3.7). This completes the proof.

h

Proposition 3.3. There exists a positive constant cx such that

lim inf qx ðtÞ P cx ;

ð3:14Þ

t!1

where qx(t) = x1(t) + x2(t). Proof. For each positive e in ((3.8)–(3.10)), we choose P = lx(l is a positive integer) sufficiently large such that

M exp

Z 0

P

  c2 ðtÞe dðtÞ þ  qðtÞe dt < e; eðtÞ

ð3:15Þ

where M = max{Mx, My}. Let positive integer m big enough such that

gx m where

< e;

ln m > 2Pf0 ;

ð3:16Þ

c1 ðtÞMy f0 ¼ max b1 ðtÞ þ a1 ðtÞMx þ ; b2 ðtÞ þ a2 ðtÞM x : 0 6 t 6 x eðtÞ

Suppose that (3.14) is not true, then for each m in (3.16) there exists zm 2 R3þ , such that

lim inf qx ðt; zm Þ <

t!1

gx

m2

:

On the other hand, by Proposition 3.2, we have

lim sup qx ðt; zm Þ P lim sup x1 ðt; zm Þ P gx :

t!1

t!1

Hence there are two sequences {sq} and {tq} satisfying the following conditions:

0 < s1 < t 1 < s2 < t2 <    < sq < tq <    ; sq ! 1 as q ! 1 and

qx ðsq ; zm Þ ¼

gx m

;

qx ðtq ; zm Þ ¼

gx m2

;

gx m2

< qx ðt; zm Þ <

gx m

;

t 2 ðsq ; t q Þ:

ð3:17Þ

By Proposition 3.1, for a given integer m > 0, there is a T1 > 0, such that

xi ðt; zm Þ 6 M x ; yðt; zm Þ 6 M y ; fort P T 1

and i ¼ 1; 2:

Because of sq ? 1 as q ? 1, there is a positive integer K, such that sq > T1 as q P K, hence

  c1 ðtÞMy x_ 1 ðt; zm Þ P x1 ðt; zm Þ b1 ðtÞ   a1 ðtÞM x þ DðtÞðx2 ðt; zm Þ  x1 ðt; zm ÞÞ eðtÞ x_ 2 ðt; zm Þ P x2 ðt; zm Þ½b2 ðtÞ  a2 ðtÞM x  þ DðtÞðx1 ðt; zm Þ  x2 ðt; zm ÞÞ

for q P K, so

q_ x ðt; zm Þ P f0 qx ðt; zm Þ for q P K, t 2 [sq, tq]. Integrating it from sq to tq we have

qx ðtq ; zm Þ P qx ðsq ; zm Þ exp

Z

tq

ðf0 Þdt:

sq

By (3.16) and (3.17) we have

tq  sq > 2P where q P K. Note that

xi ðt; zm Þ < e;

i ¼ 1; 2;

t 2 ½sq ; t q 

ð3:18Þ

from (3.16) and (3.17). For the positive e satisfying (3.8, 3.9,3.10) and (3.15), we have the following two circumstances for y(t, zm):

507

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

(i) y(t, zm) P e for all t 2 [sq, sq + P]; (ii) there exists sq1 2 [sq, sq + P], such that y(sq1, zm) < e.If (i) holds, by (3.15) and (3.18) we have

e 6 yðsq þ P; zm Þ 6 yðsq ; zm Þ exp

Z

   Z P c2 ðtÞe c2 ðtÞe dðtÞ þ dðtÞ þ  qðtÞe dt 6 M exp  qðtÞe dt < e; eðtÞ eðtÞ 0

sq þP

sq

which is a contradiction. If (ii) holds,we now claim that

t 2 ðsq1 ; t q :

yðt; zm Þ 6 e expðaxÞ; Otherwise, there exists

ð3:19Þ

sq2 2 ðsq1 ; tq  such that

yðsq2 ; zm Þ > e expðaxÞ:

sq3 2 ðsq1 ; sq2 Þ such that

By the continuity of y(t, zm), there must exist

yðsq3 ; zm Þ ¼ e and

yðt; zm Þ > e for t 2 ðsq3 ; sq2 Þ: Denote P1 the nonnegative integer such that

sq2 2 ðsq3 þ P1 x; sq3 þ ðP1 þ 1Þx, by (3.8) we obtain

 c2 ðtÞe  qðtÞe dt eðtÞ sq3  Z sq3 þP1 x Z sq2 ! c2 ðtÞe ¼ e exp dðtÞ þ þ  qðtÞe dt < e expðaxÞ: eðtÞ sq3 sq3 þP1 x

e expðaxÞ < yðsq2 ; zm Þ < yðsq3 ; zm Þ exp

Z sq2 

dðtÞ þ

This contradiction establishes that (3.19) is true, particularly (3.19) holds for t 2 [sq + P, tq]. By (3.17) and (3.10), we have

gx m2

¼ qx ðt q ; zm Þ P qx ðsq þ P; zm Þ exp

Z

tq

sq þP

/e ðtÞdt >

gx m2

;

which is also a contradiction. This completes the proof. h

Proposition 3.4. There exists positive constants cxi(i = 1, 2) such that

lim inf xi ðtÞ P cxi

t!1

i ¼ 1; 2:

ð3:20Þ

Proof. (3.14) implies that there exists T2 P T1 such that

qx ðtÞ ¼ x1 ðtÞ þ x2 ðtÞ P cx for t P T 2 : Hence,

 x_ 1 ¼ x1 b1 ðtÞ  2DðtÞ  a1 ðtÞx1 

 c1 ðtÞy cM L M 2 1 My þ DðtÞqx ðtÞ P aM Þx1 þ DL cx :¼ F 1 ðx1 Þ 1 x1 þ ðb1  2D  2 eL eðtÞ þ bðtÞx1 þ x1

and

 L M 2 x_ 2 P aM x2 þ DL cx :¼ F 2 ðx2 Þ 2 x2 þ b2  2D for t P T2. The algebraic equation F1(x1) = 0 gives us one positive root L

~x1 ¼

b1  2DM 

cM My 1 eL

þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 cM My L b1  2DM  1eL þ 4DL aM 1 cx 2aM 1

:

Clearly, F1(x1) > 0 for every positive number x1 ð0 < x1 < ~ x1 Þ. Choose cx1 ð0 < cx1 < ~ x1 Þ; x_ 1 jx1 ¼cx1 P F 1 ðcx1 Þ > 0. If x1(T2) P cx1 then it also holds for t P T2; if x(T2) < cx1, then

x_ 1 ðT 2 Þ P inffF 1 ðx1 Þj0 6 x1 < cx1 g > 0; there must exist T3(PT2), such that x1(t) > cx1 for t P T3. Similarly, there exists cx2 > 0 and T4(PT3), such that x2(t) > cx2 for t P T4. This completes the proof. h

508

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

Proposition 3.5. Suppose that (3.2) holds, then there exists a positive constant gy such that

lim sup yðtÞ > gy :

ð3:21Þ

t!1

Proof. By (3.2), we can choose positive constant e0 < e such that

Ax ðwe0 ðtÞÞ > 0;

ð3:22Þ

where

we0 ðtÞ ¼ dðtÞ þ

  c2 ðtÞ x1 ðtÞ  e0      2  qðtÞe0 : eðtÞ þ bðtÞ x1 ðtÞ  e0 þ x1 ðtÞ  e0

 bL eL Consider the following equations with parameter a 0 < a < 2c1 M :

  2ac1 ðtÞ  a1 ðtÞx1 þ DðtÞðx2  x1 Þ; x_ 1 ¼ x1 b1 ðtÞ  eðtÞ x_ 2 ¼ x2 ½b2 ðtÞ  a2 ðtÞx2  þ DðtÞðx1  x2 Þ:

1

ð3:23Þ

By Lemma 2.3, (3.23) has a unique x-periodic solution (x1a(t), x2a(t)), which is globally asymptotically stable. Let   ð~ x1a ðtÞ; ~ x2a ðtÞÞ be the solution of (3.23) with initial condition ~ xia ð0Þ ¼ xi ð0Þ; i ¼ 1; 2, where x1 ðtÞ; x2 ðtÞ be the positive and x-periodic solution of (2.2), then for the given e0, there exists T5 P T4, such that

e0

j~x1a ðtÞ  x1a ðtÞj <

for t P T 5 :

4

  By the continuity of solution to parameter, we have ð~ x1a ðtÞ; ~ x2a ðtÞÞ ! x1 ðtÞ; x2 ðtÞ uniformly in [T5, T5 + x] as a ? 0. Hence for L L

e0 > 0, there exists positive a0 ¼ a0 ðe0 Þ < b2c1 eM such that 1

  ~x1a ðtÞ  x ðtÞ < e0 1 4

for t 2 ½T 5 ; T 5 þ x;

0 < a < a0 :

So we have

  x1a ðtÞ  x ðtÞ 6 j~x1a ðtÞ  x1a ðtÞj þ j~x1a ðtÞ  x ðtÞj < e0 1 1 2 for t 2 [T5, T5 + x]. Since x1a(t) and x1 ðtÞ are all x-periodic, we have

  x1a ðtÞ  x ðtÞ < e0 1 2

for t P 0;

0 < a < a0 :

Choosing constant a1(0 < a1 < a0, 2a1 < e0), then

x1a1 ðtÞ P x1 ðtÞ 

e0 2

;

t P 0:

ð3:24Þ

Suppose that conclusion (3.21) is not true, then there exists Z 2 R3þ , for the positive solution (x1(t), x2(t), y(t)) of (1.1) with initial condition (x1(0), x2(0), y(0)) = Z, we have

lim sup yðtÞ < a1 :

t!1

So there exists T6 P T5 such that

yðtÞ < 2a1 ;

for t P T 6

ð3:25Þ

and hence

  2a1 c1 ðtÞ  a1 ðtÞx1 þ DðtÞðx2  x1 Þ; x_ 1 P x1 b1 ðtÞ  eðtÞ x_ 2 ¼ x2 ½b2 ðtÞ  a2 ðtÞx2  þ DðtÞðx1  x2 Þ: Let (u1(t), u2(t)) be the solution of (3.23) with a = a1 and ui(T6) = xi(T6), i = 1, 2, by Lemma 2.1 we know that

xi ðtÞ P ui ðtÞ;

t P T 6;

i ¼ 1; 2:

By the globally asymptotical stability of ðx1a1 ðtÞ; x2a1 ðtÞÞ, for given e0, there exists T7 P T6 such that

ju1 ðtÞ  x1a1 ðtÞj <

e0 2

for t P T 7 :

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

509

So we have

x1 ðtÞ P u1 ðtÞ > x1a1 ðtÞ 

e0 2

t P T7

;

and hence

x1 ðtÞ > x1 ðtÞ  e0 ;

t P T7:

This implies

_ yðtÞ P we0 ðtÞyðtÞ for t P T 7 : integrating above inequality from T7 to t yields

yðtÞ P yðT 7 Þ exp

Z

t

T7

we0 ðtÞdt:

By (3.22) we know that y(t) ? 1 as t ? 1, which is a contradiction. This completes the proof. h Proposition 3.6. Under assumption (3.2), there exists a positive constant cy such that

lim inf yðtÞ > cy :

ð3:26Þ

t!1

Proof. For each positive e0 in (3.22), we choose P > 0 sufficiently large such that

Z

a

0

we0 ðtÞdt > 0

ð3:27Þ

for a P P. Further, let positive integer m big enough such that

gy mþ1

< a1 ð2a1 < e0 Þ;

lnðm þ 1Þ > 2P max fdðtÞ þ qðtÞM y g:

ð3:28Þ

06t6x

Suppose that (3.26) is not true, then for each m in (3.28) there exists zm 2 R3þ , such that

lim inf yðt; zm Þ <

t!1

gy

:

ðm þ 1Þ2

But

lim sup yðt; zm Þ > gy :

t!1

According to Proposition 3.5. Hence there are two sequences {sq} and {tq} satisfying the following conditions:

0 < s1 < t 1 < s2 < t2 <    < sq < tq <    ; sq ! 1 as q ! 1 and

yðsq ; zm Þ ¼

gx mþ1

;

yðt q ; zm Þ ¼

gx ðm þ 1Þ2

;

gy ðm þ 1Þ2

< yðt; zm Þ <

gy mþ1

;

t 2 ðsq ; t q Þ:

ð3:29Þ

By Proposition 3.1, for a given integer m > 0, there is a T1 > 0, such that

yðt; zm Þ 6 M y ;

for t P T 1 :

Because of sq ? 1 as q ? 1, there is a positive integer K, such that sq > T1 as q P K, hence

_ zm Þ P yðt; zm Þ½dðtÞ  qðtÞMy  yðt; for q P K, t 2 [sq, tq]. Integrating above inequality from sq to tq we get

yðt q ; zm Þ P yðsq ; zm Þ exp

Z

tq



 dðtÞ  qðtÞM y dt:

sq

Combining (3.28) and (3.29) we have

tq  sq > 2P and

tq  sq ! 1; where m ? 1 and q P K.

ð3:30Þ

510

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

Further we have

yðt; zm Þ < 2a1 ;

t 2 ½sq ; tq 

from (3.28). In addition, for t 2 [sq, tq], we have

  2a1 c1 ðtÞ  a1 ðtÞx1 ðt; zm Þ þ DðtÞðx2 ðt; zm Þ  x1 ðt; zm ÞÞ; x_ 1 ðt; zm Þ P x1 ðt; zm Þ b1 ðtÞ  eðtÞ x_ 2 ðt; zm Þ ¼ x2 ðt; zm Þ½b2 ðtÞ  a2 ðtÞx2 ðt; zm Þ þ DðtÞðx1 ðt; zm Þ  x2 ðt; zm ÞÞ:

Let (u1(t), u2(t)) be the solution of (3.23) with a = a1 and ui(sq) = xi(sq, zm), by Lemma 2.1 we have

xi ðt; zm Þ P ui ðtÞ;

t 2 ½sq ; tq :

Further, by Propositions 3.1, 3.4 and sq ? 1 as q ? 1, we can choose K1 > K, such that

cxi 6 xi ðsq ; zm Þ 6 Mx ; i ¼ 1; 2 holds for q P K1. For a = a1, (3.23) has a unique x-periodic solution ðx1a1 ðtÞ; x2a1 ðtÞÞ, which is globally asymptotically stable. Because (3.23) is a periodic system, the periodic solution ðx1a1 ðtÞ; x2a1 ðtÞÞ is uniformly asymptotically stable in R3þ (see [10]). By definitions 11.2 and 11.5 in [10], for the given set X and each e0, there is a corresponding T0 = T0(e0)(>P) such that if (u1(sq), u2(sq)) = (x1(sq, zm), x2(sq, zm)) 2 X for some sq > 0 then

ju1 ðtÞ  x1a1 ðtÞj <

e0 2

for all t P T0 + sq. Hence

u1 ðtÞ P x1a1 ðtÞ 

e0 2

t P T 0 þ sq :

;

Combining (3.24) we have

u1 ðtÞ P x1 ðtÞ  e0 ;

t P T 0 þ sq :

From (3.30), there exists a positive integer N1 P N0, such that tq > sq + 2T0 > sq + 2P for m P N1 and q P K1. So we have

x1 ðt; zm Þ P x1 ðtÞ  e0 ;

t 2 ½sq þ T 0 ; t q 

as m P N1 and q P K1. Hence

_ zm Þ P we0 ðtÞyðt; zm Þ yðt; for t 2 [sq + T0, tq]. Integrating above inequality from sq + T0 to tq yields

yðtq ; zm Þ P yðsq þ T 0 ; zm Þ exp

Z

tq

sq þT 0

we0 ðtÞdt;

that is to say

gy 2

ðm þ 1Þ

P

gy 2

ðm þ 1Þ

exp

Z

tq sq þT 0

we0 ðtÞdt >

gy ðm þ 1Þ2

;

which is a contradiction. This completes the proof. h Combining the Propositions 3.1, 3.2, 3.33.4, 3.5, 3.6, we completes the proof of the sufficiency of Theorem 3.1. To prove the necessity of Theorem 3.1, we will show that

lim yðtÞ ¼ 0

t!1

under the following condition:

Ax ðdðtÞ þ

c2 ðtÞx1 ðtÞ Þ 6 0: eðtÞ þ bðtÞx1 ðtÞ þ x2 1 ðtÞ

In fact, by (3.3) we know that for every given e(0 < e < 1), there exists e1 > 0 and e0 > 0 such that

Ax

!   c2 ðtÞ x1 ðtÞ þ e1 e dðtÞ þ      2  qðtÞe 6  Ax ðqðtÞÞ 6 e0 : 2 eðtÞ þ bðtÞ x1 ðtÞ þ e1 þ x1 ðtÞ þ e1

ð3:31Þ

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

511

Since

x_ 1 6 x1 ½b1 ðtÞ  a1 ðtÞx1  þ DðtÞðx2  x1 Þ; x_ 2 ¼ x2 ½b2 ðtÞ  a2 ðtÞx2  þ DðtÞðx1  x2 Þ: we know that for the given e1 there exists T(1) > 0 such that

for t P T ð1Þ :

x1 ðtÞ 6 x1 ðtÞ þ e1

By (3.31) and (3.1) we have

Ax ðdðtÞ þ

c2 ðtÞx1 ðtÞ  qðtÞeÞ 6 e0 eðtÞ þ bðtÞx1 ðtÞ þ x21 ðtÞ

ð3:32Þ

for t P T(1). Firstly, there must exist T(2) such that y(T(2)) < e. Otherwise, we have

e 6 yðtÞ 6 yðT ð1Þ Þ exp

Z T

t ð1Þ

½dðsÞ þ

c2 ðsÞx1 ðsÞ  qðsÞeds ! 0 as t ! 0: eðsÞ þ bðsÞx1 ðsÞ þ x21 ðsÞ

This implies e 6 0, which is a contradiction. Let

MðeÞ ¼ max

06t6x

dðtÞ þ

c2 ðtÞx1 ðtÞ : e þ qðtÞ eðtÞ þ bðtÞx1 ðtÞ þ x21 ðtÞ

By Proposition 3.1, we know that x1(t) is bounded. So M(e) is also bounded for e 2 [0, 1]. Secondly we have

yðtÞ 6 e expðMðeÞxÞ for t P T ð2Þ : (3)

Otherwise, there exists T

(2)

>T

ð3:33Þ

such that

ð3Þ

yðT Þ > e expðMðeÞxÞ: By the continuity of y(t), there must exist T ð4Þ 2 ðT ð2Þ ; T ð3Þ Þ such that y(T(4)) = e and y(t) > e for t 2 ðT ð4Þ ; T ð3Þ . Let P1 be the nonnegative integer such that T ð3Þ 2 ðT ð4Þ þ P1 x; T ð4Þ þ ðP1 þ 1Þx, by (3.32) we have

 c2 ðtÞx1 ðtÞ dt e  qðtÞ eðtÞ þ bðtÞx1 ðtÞ þ x21 ðtÞ T ð4Þ !   Z T ð4Þ þP1 x Z T ð3Þ c2 ðtÞx1 ðtÞ ¼ e exp dt < e expðMðeÞxÞ: þ e dðtÞ þ  qðtÞ eðtÞ þ bðtÞx1 ðtÞ þ x21 ðtÞ T ð4Þ T ð4Þ þP 1 x

e expðMðeÞxÞ < yðT ð3Þ Þ < yðT ð4Þ Þ exp

Z

T ð3Þ

 dðtÞ þ

which is a contradiction. This implies (3.33) holds. Further by the arbitrariness of e we know that y(t) ? 0 as t ? 1. This completes the proof of Theorem 3.1. 4. Examples Examples 1. Consider the following predator–prey system:

    t 2 þ cos y 10 þ 4ðx2  x1 Þ; x_ 1 ¼ x1 4  2x1  20 þ x1 þ x21 h i x2 x_ 2 ¼ x2 6  þ 4ðx1  x2 Þ; 2     t 2 þ cos x1 1 cos t 10  þ  ð2 þ sin tÞy : y_ ¼ y  10 100 20 þ x1 þ x21

ð4:1Þ

t In this case, corresponding to system (1.1), one has a1 ðtÞ ¼ 2; b1 ðtÞ ¼ 4; a2 ðtÞ ¼ 12 ; b2 ðtÞ ¼ 6; DðtÞ ¼ 4; c1 ðtÞ ¼ c2 ðtÞ ¼ 2 þ cos ; 10 1 t eðtÞ ¼ 20; bðtÞ ¼ 1; dðtÞ ¼ 10 þ cos ; qðtÞ ¼ 2 þ sin t: One could easily see that 100

x_ 1 ¼ 4x2  2x21 ; 1 x_ 2 ¼ 4x1 þ 2x2  x22 2

ð4:2Þ

  has a unique positive periodic solution x1 ðtÞ; x2 ðtÞ ¼ ð4; 8Þ, i.e. the positive periodic solution is the positive equilibrium. By simple computation, one has

 Ax dðtÞ þ

 c2 ðtÞx1 ðtÞ 2 1 ¼  > 0:  2 10 10 eðtÞ þ bðtÞx1 ðtÞ þ x1 ðtÞ

512

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

Using Theorem 3.1, we know that system (4.1) is permanent. Fig. 1 shows the dynamic behavior of the system (4.1) with initial condition (x1(0), x2(0), y(0)) = (15, 10, 5), here t 2 [0, 100]. Examples 2. Consider the following predator–prey system:

    t 2 þ cos y 10 þ 4ðx2  x1 Þ; x_ 1 ¼ x1 4  2x1  20 þ x1 þ x21 h i x2 þ 4ðx1  x2 Þ; x_ 2 ¼ x2 6  2     t 2 þ cos x1 3 cos t 10 y_ ¼ y   ð2 þ sin tÞy :  þ 10 100 20 þ x1 þ x21

Here, we only replace d(t) by

3 10

ð4:3Þ

t þ cos , all the other coefficients of system (4.3) are the same to that of system (4.1). 100

16

x1 x2

14

y

x1(t),x2(t),y(t)

12 10 8 6 4 2

0

20

40

60

80

100

time t Fig. 1. The dynamic behavior of the system (4.1).

5

y

4.5 4 3.5

y(t)

3 2.5 2 1.5 1 0.5 0

0

20

40

60 time t

Fig. 2. The dynamic behavior of the predator.

80

100

M. Huang, X. Li / Applied Mathematics and Computation 218 (2011) 502–513

513

One could easily see that in this case

 Ax dðtÞ þ

 c2 ðtÞx1 ðtÞ 2 3  < 0: ¼ eðtÞ þ bðtÞx1 ðtÞ þ x 21 ðtÞ 10 10

Hence, corresponding to Corollary 3.1, we know that any positive solution of system (4.3) satisfies limt?1y(t) = 0. Fig. 2 shows the dynamic behavior of the predator in system (4.3) with initial condition (x1(0), x2(0), y(0)) = (15, 10, 5), here t 2 [0, 100]. 5. Discussion In this paper, a model which describes a periodic predator–prey system with Holling type-IV functional response is proposed. By Lemma 2.3, system (1.1) without predator y(t) has a unique positive periodic solution which is globally asymptotically stable. Theorem 3.1 says that system (1.1) with y(t) is permanent under (3.2) if the prey dispersal system (2.2) has such a globally asymptotically stable positive x-periodic solution. Otherwise, if (3.2) is not true then the predator goes to extinct by Corollary 3.1. c2 ðtÞx1 ðtÞ In (3.2), the term eðtÞþbðtÞx  ðtÞþx2 ðtÞ describes the growth of the predator by foraging the prey in patch 1, of which quantity is 1  1  specified as x1 ðtÞ. Note that x1 ðtÞ; x2 ðtÞ is a globally asymptotically stable periodic solution in prey dispersal system (2.2) and the predator is confined only in patch 1. Hence condition (3.2) implies that the growth by foraging minus the death for predator is positive on the average. If it is negative, the extinction is inevitable for the predator. Since we have assumed condition (3.1) holds for the functional response of predator to the prey in the patch 1, it is natural to discuss that under the assumption (3.1) not always holds, whether one could obtain the sufficient and necessary condition which ensure the permanence of the system (1.1) or not. The study of this problem is our future work. References [1] J. Cui, L. Chen, Permanent and extinction in logistic and Lotka–Volterra systems with diffusion, J. Math. Anal. Appl. 258 (2001) 512–535. [2] Z. Teng, Uniform persistence of the periodic predator–prey Lotka–Volterra systems, Appl. Anal. 72 (1999) 339–352. [3] R. Xu, L.S. Chen, Persistence and global stability for three-species ratio-dependent predator–prey system with time delays, J. Systems Sci. Math. Sci. 21 (2) (2001) 204–212. [4] W. Wang, Z. Ma, Harmless delays for uniform persistence, J. Math. Anal. Appl. 158 (1991) 256–268. [5] H.I. Freedman, S. Ruan, M. Tang, Uniform persistence near a closed positively invariant set, J. Dyn. Differ. Equa. 6 (1994) 583–600. [6] J. Cui, Dispersal permanence of a periodic predator-prey system with Beddington–DeAngelis functional response, Nonlinear Anal. 64 (2006) 440–456. [7] H.L. Smith, Cooperative systems of differential equation with concave nonlinearities, Nonlinear Anal. 10 (1986) 1037–1052. [8] X.Q. Zhao, The qualitative analysis of N-species Lotka–Volterra periodic competition systems, Math. Comp. Model 15 (1991) 3–8. [9] R. Mahbuba, L.S. Chen, On the nonautonomous Lotka–Volterra competition system with diffusion, Differ. Equa. Dyn. Syst. 2 (1994) 243–253. [10] T. Yoshizawa, Stability by Liapunov’s Second Method, Mathematical Society of Japan, Tokyo, 1966. publication no. 9.