Main Times Series The positive almost periodic solution for Nicholsontype delay systems with linear harvesting terms

The positive almost periodic solution for Nicholson-type delay systemsXingguo Liua, Junxia Mengb,⇑ a College of Business Administration, Hunan University, Changsha, Hunan 410082, PR Chin

The positive almost periodic solution for Nicholson-type delay systems

Xingguo Liua, Junxia Mengb,⇑

a

College of Business Administration, Hunan University, Changsha, Hunan 410082, PR China

b

College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, PR China

a r t i c l e i n f o

Article history:

Received 28 April 2011

Received in revised form 24 September

2011

Accepted 29 September 2011

Available online 18 October 2011

Keywords:

Positive almost periodic solution

Exponential convergence

Nicholson-type delay system

Linear harvesting term

a b s t r a c t

In this paper, we study the existence and exponential convergence of positive almost peri-odic solutions for a class of Nicholson-type delay system with linear harvesting terms Under appropriate conditions, we establish some criteria to ensure that the solutions of this system converge locally exponentially to a positive almost periodic solution Moreover,

we give an example to illustrate our main results

Ó 2011 Elsevier Inc All rights reserved

1 Introduction

In[1], to describe the models of Marine Protected Areas and B-cell Chronic Lymphocytic Leukemia dynamics that belong

to the Nicholson-type delay differential systems, Berezansky et al.[1]considered the dynamics of the following autonomous Nicholson-type delay systems:

x0

1ðtÞ ¼ a1x1ðtÞ þ b1x2ðtÞ þ c1x1ðt sÞex 1 ðtsÞ;

x0

2ðtÞ ¼ a2x2ðtÞ þ b2x1ðtÞ þ c2x2ðt sÞex 2 ðtsÞ;

(

ð1:1Þ with initial conditions:

whereui2 C([s, 0], [0, +1)), ai, bi, ciandsare nonnegative constants, i = 1, 2

Furthermore, Wang et al.[2]showed the existence and exponential convergence of positive almost periodic solutions for the following non-autonomous Nicholson-type delay systems:

0307-904X/$ - see front matter Ó 2011 Elsevier Inc All rights reserved.

q

This work was supported by the Natural Scientific Research Fund of Zhejiang Provincial of PR China (Grant No Y6110436), the Natural Scientific Research Fund of Hunan Provincial of PR China (Grant No 11JJ6006), and the Natural Scientific Research Fund of Hunan Provincial Education Department of

PR China (Grant Nos 11C0916, 11C0915, 11C1186).

⇑Corresponding author Tel./fax: +86 057383643075.

E-mail address: mengjunxia1968@yahoo.com.cn (J Meng).

Contents lists available atSciVerse ScienceDirect

Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a p m

1đtỡ Ử a1đtỡx1đtỡ ợ b1đtỡx2đtỡ ợPm

jỬ1

c1jđtỡx1đt s1jđtỡỡe c 1j đtỡx 1 đts1jđtỡỡ;

x0

2đtỡ Ử a2đtỡx2đtỡ ợ b2đtỡx1đtỡ ợPm

jỬ1

c2jđtỡx2đt s2jđtỡỡe c 2j đtỡx 2 đts2j đtỡỡ;

8

>

<

>

:

đ1:3ỡ

whereai, bi, cij,cij,sij: R1?(0, +1) are almost periodic functions, and i = 1, 2, j = 1, 2, , m

Recently, assuming that a harvesting function is a function of the delayed estimate of the true population, Berezansky

et al.[3]proposed the NicholsonỖs blowflies model with a linear harvesting term:

x0đtỡ Ử dxđtỡ ợ pxđt sỡeaxđtsỡ Hxđt rỡ; d;p;s;a; H; r2 đ0; ợ1ỡ; đ1:4ỡ where Hx(t r) is a linear harvesting term, x(t) is the size of the population at time t, P is the maximum per capita daily egg production,1

ais the size at which the population reproduces at its maximum rate, d is the per capita daily adult death rate, andsis the generation time Moreover, Berezansky et al.[3]pointed out an open problem: How about the dynamic behav-iors of the NicholsonỖs blowflies model with a linear harvesting term

Now, motivated by Berezansky et al.[1], Wang et al.[2], Berezansky et al.[3]a corresponding question arises: How about the existence and convergence of positive almost periodic solutions of Nicholson-type delay differential systems with linear harvesting terms The main purpose of this paper is to give the conditions to ensure the existence and convergence of po-sitive almost periodic solutions of the following non-autonomous Nicholson-type delay systems with linear harvesting terms:

x0

1đtỡ Ử a1đtỡx1đtỡ ợ b1đtỡx2đtỡ ợPm

jỬ1

c1jđtỡx1đt s1jđtỡỡe c 1j đtỡx 1 đts1j đtỡỡ

H1đtỡx1đt r1đtỡỡ;

x0

2đtỡ Ử a2đtỡx2đtỡ ợ b2đtỡx1đtỡ ợPm

jỬ1

c2jđtỡx2đt s2jđtỡỡe c 2j đtỡx 2 đts2j đtỡỡ

H2đtỡx2đt r2đtỡỡ;

8

>

>

>

>

>

>

đ1:5ỡ

whereai, bi, Hi,ri, cij,cij,sij: R1?[0, +1) are almost periodic functions, and i = 1, 2, j = 1, 2, , m

For convenience, we introduce some notations Throughout this paper, given a bounded continuous function g defined on

R1, let g+and gbe defined as

gỬ inf

t2Rgđtỡ; gợỬ sup

t2R

gđtỡ:

It will be assumed that

a

i >0; b

i >0; c

ij >0; riỬ max max

16j6m sợ ij

n o

;rợ i

Denote by RnRnợ

the set of all (nonnegative) real vectors Let

C Ử Cđơr1;0; R1

ỡ  Cđơr2;0; R1

ỡ and CợỬ C ơr1;0; R1

ợ

 C ơr2;0; R1

ợ

:

If xi(t) is defined on [t0 ri,r) with t0,r2 R1and i = 1, 2, then we define xt2 C as xtỬ x1

t;x2 t

where xi

tđhỡ Ử xiđt ợ hỡ for all

h2 [ri, 0] and i = 1, 2 A matrix or vector A P 0 means that all entries of A are greater than or equal to zero A > 0 can be defined similarly For matrices or vectors A and B, A P B (resp A > B) means that A  B P 0 (resp A  B > 0) For vector

X = (x1, x2) 2 R2, we let jXj denote the absolute-value vector given by jXj = (jx1j, jx2j), and define jXk = max16i62jxij

The initial conditions associated with system(1.5)are of the form:

We write xt(t0,u)(x(t; t0,u)) for a solution of the initial value problem(1.5) and (1.7) Also, let [t0,g(u)) be the maximal right-interval of existence of xt(t0,u)

The remaining part of this paper is organized as follows In Section2, we shall give some notations and preliminary results In Section3, we shall derive new sufficient conditions for checking the existence, uniqueness and local exponential convergence of the positive almost periodic solution of(1.5) In Section4, we shall give some example and remark to illustrate our results obtained in the previous sections

2 Preliminary results

In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section3

Definition 2.1 (See[4,5]) Let u(t) : R1?Rnbe continuous in t u(t) is said to be almost periodic on R1, if for anye> 0, the set T(u,e) = {d:ju(t + d)  u(t)j <efor all t 2 R1} is relatively dense, i.e., for anye> 0, it is possible to find a real number l = l(e) > 0, such that for any interval with length l(e), there exists a number d = d(e) in this interval such that ju(t + d)  u(t)j <e, for all

t 2 R1

Definition 2.2 (See[4,5]) Let x 2 Rnand Q(t) be a n  n continuous matrix defined on R1 The linear system

is said to admit an exponential dichotomy on R1if there exist positive constants k,a, projection P and the fundamental solu-tion matrix X(t) of(2.1)satisfying

kXðtÞPX1ðsÞk 6 keaðtsÞ for all t P s;

kXðtÞðI  PÞX1ðsÞk 6 keaðstÞ for all t 6 s:

Set

B ¼ fuju¼ ðu1ðtÞ;u2ðtÞÞ is an almost periodic function on R1g:

For anyu2 B, we define induced module kukB¼ supt2R1kuðtÞk, then B is a Banach space

Lemma 2.1 (See[4,5]) If the linear system(2.1)admits an exponential dichotomy, then almost periodic system

has a unique almost periodic solution x(t), and

xðtÞ ¼

Z t

1

XðtÞPX1ðsÞgðsÞds 

Z þ1 t

Lemma 2.2 (See[4,5]) Let ci(t) be an almost periodic function on R1and

M½ci ¼ lim

T!þ1

1 T

Z tþT t

ciðsÞds > 0; i ¼ 1; 2; ; n:

Then the linear system

x0ðtÞ ¼ diag ðc1ðtÞ; c2ðtÞ; ; cnðtÞÞxðtÞ

admits an exponential dichotomy on R1

Lemma 2.3 Suppose that there exist two positive constants Ei1and Ei2such that

Ei1>Ei2; bþ1E21

a 1

þXm j¼1

cþ 1j

a

1c 1j

1

e<E11;

bþ2E11

a 2

þXm j¼1

cþ 2j

a

2c 2j

1

b1E22

aþ

1

þXm

j¼1

c 1j

aþ 1

E11ecþE11

H

þ

1E11

aþ 1

>E12P 1

min

16j6mc 1j

b2E12

aþ

2

þXm

j¼1

c 2j

aþ 2

E21ecþE21H

þ

2E21

aþ 2

>E22P 1

min

16j6mc 2j

where i = 1, 2 Let

C0

:¼ fuju2 C; Ei2<uiðtÞ < Ei1; for all t 2 ½ri;0; i ¼ 1; 2g:

Moreover, assume that x(t; t0,u) is the solution of(1.5)withu2 C0 Then,

andg(u) = +1

Proof Set x(t) = x(t; t0,u) for all t 2 [t0,g(u)) Let [t0, T) # [t0,g(u)) be an interval such that

we claim that

Assume, by way of contradiction, that(2.9)does not hold Then, one of the following cases must occur

Case i: There exists t12 (t0, T) such that

x1ðt1Þ ¼ E11 and 0 < xiðtÞ < Ei1for all t 2 ½t0 ri;t1Þ; i ¼ 1; 2: ð2:10Þ Case ii: There exists t22 (t0, T) such that

x2ðt2Þ ¼ E21 and 0 < xiðtÞ < Ei1for all t 2 ½t0 ri;t2Þ; i ¼ 1; 2: ð2:11Þ

If Case i holds, calculating the derivative of x1(t), together with(2.4)and the fact that supuP0ueu¼1

e,(1.5) and (2.10)

imply that

0 6 x0

1ðt1Þ ¼ a1ðt1Þx1ðt1Þ þ b1ðt1Þx2ðt1Þ þXm

j¼1

c1jðt1Þx1ðt1s1jðt1ÞÞe c 1j ðt 1 Þx 1 ðt 1 s1j ðt 1 ÞÞ H1ðt1Þx1ðt1r1ðt1ÞÞ

6a

1x1ðt1Þ þ bþ

1E21þXm j¼1

cþ 1j

c 1j

1

e¼a

1 E11þb

þ

1E21

a 1

þXm j¼1

cþ 1j

a

1c 1j

1 e

!

<0;

which is a contradiction and implies that(2.9)holds

If Case ii holds, calculating the derivative of x2(t), together with(2.4)and the fact that supuP0ueu¼1

e,(1.5) and (2.11)

imply that

0 6 x0

2ðt2Þ ¼ a2ðt2Þx2ðt2Þ þ b2ðt2Þx1ðt2Þ þXm

j¼1

c2jðt2Þx2ðt2s2jðt2ÞÞe c 2j ðt 2 Þx 2 ðt 2 s2j ðt 2 ÞÞ H2ðt2Þx2ðt2r2ðt2ÞÞ

6a

2x2ðt2Þ þ bþ

2E11þXm j¼1

cþ 2j

c 2j

1

e¼a

2 E21þb

þ

2E11

a 2

þXm j¼1

cþ 2j

a

2c 2j

1 e

!

<0;

which is a contradiction and implies that(2.9)holds

We next show that

Suppose, for the sake of contradiction, that(2.12)does not hold Then, one of the following cases must occur

Case I: There exists t32 (t0,g(u)) such that

x1ðt3Þ ¼ E12 and xiðtÞ > Ei2for all t 2 ½t0 ri;t3Þ; i ¼ 1; 2: ð2:13Þ Case II: There exists t42 (t0,g(u)) such that

x2ðt4Þ ¼ E22 and xiðtÞ > Ei2for all t 2 ½t0 ri;t4Þ; i ¼ 1; 2: ð2:14Þ

If Case I holds Then, from(2.5), (2.6), (2.9) and (2.13), we get

Ei2<xiðtÞ < Ei1; cþ

ijxiðtÞ Pcþ

ijEi2Pcþ ij

1 min

16j6mc ij

for all t 2 [t0 ri, t3), i = 1, 2, j = 1, 2, , m Calculating the derivative of x1(t), together with (2.5) and the fact that min16u6jueu=jej,(1.5), (2.13) and (2.15)imply that

0 P x0

1ðt3Þ ¼ a1ðt3Þx1ðt3Þ þ b1ðt3Þx2ðt3Þ þXm

j¼1

c1jðt3Þx1ðt3s1jðt3ÞÞe c1jðt 3 Þx 1 ðt 3 s1j ðt 3 ÞÞ H1ðt3Þx1ðt3r1ðt3ÞÞ

¼ a1ðt3Þx1ðt3Þ þ b1ðt3Þx2ðt3Þ þXm c1jðt3Þ

cþ 1j

cþ 1jx1ðt3s1jðt3ÞÞe c 1j ðt 3 Þx 1 ðt 3 s1jðt 3 ÞÞ H1ðt3Þx1ðt3r1ðt3ÞÞ

Pa1ðt3Þx1ðt3Þ þ b1E22þXm

j¼1

c 1j

cþ 1j

cþ 1jx1ðt3s1jðt3ÞÞecþx1 ðt 3 s1j ðt 3 ÞÞ

 Hþ1E11

Paþ

1x1ðt3Þ þ b1E22þXm

j¼1

c 1j

cþ 1j

cþ 1jE11ecþE11

 Hþ1E11¼aþ

1 E12þb



1E22

aþ 1

þXm j¼1

c 1j

aþ 1

E11ecþE11

H

þ

1E11

aþ 1

!

>0;

which is a contradiction and implies that(2.12)holds

If Case II holds, we can show that(2.15)holds for all t 2 [t0 ri, t4),i = 1, 2, j = 1, 2, , m Calculating the derivative of x2(t), together with(2.6)and the fact that min16u6jueu=jej,(1.5), (2.14) and (2.15)imply that

0 P x0

2ðt4Þ ¼ a2ðt4Þx2ðt4Þ þ b2ðt4Þx1ðt4Þ þXm

j¼1

c2jðt4Þx2ðt4s2jðt4ÞÞe c 2j ðt 4 Þx 2 ðt 4 s2jðt 4 ÞÞ H2ðt4Þx2ðt4r2ðt4ÞÞ

¼ a2ðt4Þx2ðt4Þ þ b2ðt4Þx1ðt4Þ þXm

j¼1

c2jðt4Þ

cþ 2j

cþ 2jx2ðt4s2jðt4ÞÞe c2jðt 4 Þx 2 ðt 4 s2j ðt 4 ÞÞ H2ðt4Þx2ðt4r2ðt4ÞÞ

Pa2ðt4Þx2ðt4Þ þ b

2E12þXm j¼1

c 2j

cþ 2j

cþ 2jx2ðt4s2jðt4ÞÞecþx2 ðt 4 s2j ðt 4 ÞÞ

 Hþ

2E21

Paþ

2x2ðt4Þ þ b2E12þXm

j¼1

c 2j

cþ 2j

cþ 2jE21ecþE21 Hþ2E21¼aþ

2 E22þb



2E12

aþ 2

þXm j¼1

c 2j

aþ 2

E21ecþE21H

þ

2E21

aþ 2

!

>0;

which is a contradiction and implies that(2.12)holds

It follows from(2.9) and (2.12)that(2.7)is true From Theorem 2.3.1 in[6], we easily obtaing(u) = +1 This ends the proof ofLemma 2.3 h

3 Main results

Theorem 3.1 Let(2.4)–(2.6)hold Moreover, suppose that

þ

1

a

1

þXm

j¼1

cþ 1j

a

1e2þH

þ 1

a 1

; bþ2

a 2

þXm j¼1

cþ 2j

a

2e2þH

þ 2

a 2

Then, there exists a unique positive almost periodic solution of system(1.5)in the region B⁄= {uju2 B, Ei26ui(t) 6 Ei1, for all

t 2 R1, i = 1, 2}

Proof For any / 2 B, we consider an auxiliary system

x0

1ðtÞ ¼ a1ðtÞx1ðtÞ þ b1ðtÞ/2ðtÞ þPm

j¼1

c1jðtÞ/1ðt s1jðtÞÞe c 1j ðtÞ/ 1 ðts1j ðtÞÞ

H1ðtÞ/1ðt r1ðtÞÞ;

x0

2ðtÞ ¼ a2ðtÞx2ðtÞ þ b2ðtÞ/1ðtÞ þPm

j¼1

c2jðtÞ/2ðt s2jðtÞÞe c 2j ðtÞ/ðts2j ðtÞÞ

H2ðtÞ/2ðt r2ðtÞÞ;

8

>

>

>

>

>

>

>

>

ð3:2Þ

Notice that M[ai] > 0(i = 1, 2), it follows fromLemma 2.2that the linear system

x0

1ðtÞ ¼ a1ðtÞx1ðtÞ;

x0

2ðtÞ ¼ a2ðtÞx2ðtÞ;

(

ð3:3Þ

admits an exponential dichotomy on R Thus, byLemma 2.1, we obtain that the system(3.2)has exactly one almost periodic solution:

x/ðtÞ ¼ x /ðtÞ; x/ðtÞ

¼

Z t

1

e

Rt

s a 1 ðuÞdu

b1ðsÞ/2ðsÞ þXm

j¼1

c1jðsÞ/1ðs s1jðsÞÞe c1jðsÞ/ 1 ðss1j ðsÞÞ H1ðsÞ/1ðs r1ðsÞÞ

! ds;

Z t

1

e

Rt

s a 2 ðuÞdu

b2ðsÞ/1ðsÞ þXm

c2jðsÞ/2ðs s2jðsÞÞe c 2j ðsÞ/ 2 ðss2jðsÞÞ H2ðsÞ/2ðs r2ðsÞÞ

! ds

! : ð3:4Þ

Define a mapping T : B ? B by setting

Tð/ðtÞÞ ¼ x/ðtÞ; 8/2 B:

Since B⁄= {uju2 B, Ei26ui(t) 6 Ei1, for all t 2 R1, i = 1, 2}, it is easy to see that B⁄is a closed subset of B For anyu2 B⁄, from(2.4) and (3.4)and the fact that supuP0ueu¼1

e, we have

x/ðtÞ 6

Z t

1

eRt

s a 1 ðuÞdu

bþ1E21þXm j¼1

1

c1jðsÞec1jðsÞ

! ds;

Zt

1

eRt

s a 2 ðuÞdu

bþ2E11þXm j¼1

1

c2jðsÞec2jðsÞ

! ds

!

6 bþ1E21

a

1

þXm j¼1

cþ 1j

a

1c 1je;

bþ2E11

a 2

þXm j¼1

cþ 2j

a

2c 2je

!

In view of the fact that min16u6jueu=jej, from(2.5), (2.6) and (3.4), we obtain

x/ðtÞ P

Z t

1

eRt

s a 1 ðuÞdu

b1E22þXm j¼1

c1jðsÞ1

cþ 1j

cþ 1j/1ðs s1jðsÞÞecþ/1 ðss1jðsÞÞ

 H1ðsÞ/1ðs r1ðsÞÞ

! ds;

Z t

1

eRt

s a 2 ðuÞdu

b

2E12þXm j¼1

c2jðsÞ1

cþ 2j

cþ 2j/2ðs s2jðsÞÞecþ/2 ðss2jðsÞÞ

 H2ðsÞ/2ðs r2ðsÞÞ

! ds

!



1E22

aþ

1

þXm j¼1

c 1j

aþ 1

E11ecþE11H

þ

1E11

aþ 1

;fracb

2E12aþ

2þXm j¼1

c 2j

aþ 2

E21ecþE21H

þ

2E21

aþ 2

!

>ðE12;E22Þ; for all t 2 R1: ð3:6Þ

This implies that the mapping T is a self-mapping from B⁄to B⁄ Now, we prove that the mapping T is a contraction mapping

on B⁄

In fact, foru,w2 B⁄

, we get sup

t2R jðTðuÞðtÞ  TðwÞðtÞÞ1j; sup

t2R jðTðuÞðtÞ  TðwÞðtÞÞ2j

¼ sup

t2R j

Z t

1

eRt

s a 1 ðuÞdu

ðb1ðsÞðu2ðsÞ  w2ðsÞÞ



þXm j¼1

c1jðsÞðu1ðs s1jðsÞÞe c 1j ðsÞ u 1 ðss1jðsÞÞ

w1ðs s1jðsÞÞe c 1j ðsÞw 1 ðss1j ðsÞÞÞ  H1ðsÞðu1ðs r1ðsÞÞ

w1ðs r1ðsÞÞÞÞdsj; sup

t2R j

Z t

1

eRt

s a 2 ðuÞdu

ðb2ðsÞðu1ðsÞ  w1ðsÞÞ

þXm j¼1

c2jðsÞðu2ðs s2jðsÞÞe c2jðsÞ u2ðss2j ðsÞÞ

w2ðs s2jðsÞÞe c 2j ðsÞw 2 ðss2j ðsÞÞÞ  H2ðsÞðu2ðs r2ðsÞÞ

w2ðs r2ðsÞÞÞÞdsjÞ

¼ sup

t2R

j

Z t

1

e

Rt

s a 1 ðuÞdu

ðb1ðsÞðu2ðsÞ  w2ðsÞÞ



þXm j¼1

c1jðsÞ

c1jðsÞðc1jðsÞu1ðs s1jðsÞÞe c 1j ðsÞ u 1 ðss1j ðsÞÞ

c1jðsÞw1ðs s1jðsÞÞe c1jðsÞw1ðss1jðsÞÞÞ  H1ðsÞðu1ðs r1ðsÞÞ

w1ðs r1ðsÞÞÞÞdsj; supt2Rj

Zt

1

eRt

s a 2 ðuÞdu

ðb2ðsÞðu1ðsÞ

w1ðsÞÞ þXm

j¼1

c2jðsÞ

c2jðsÞðc2jðsÞu2ðs s2jðsÞÞe c 2j ðsÞ u 2 ðss2jðsÞÞ

c2jðsÞw2ðs s2jðsÞÞe c 2j ðsÞw 2 ðss2j ðsÞÞÞ  H2ðsÞðu2ðs r2ðsÞÞ

In view of(1.5), (2.5), (2.6), (3.5), (3.6) and (3.7), from supuP11u

e 2and the inequality jxex yeyj ¼ 1  ðx þ hðy  xÞÞ

exþhðyxÞ 6

1

e2jx  yj where x; y 2 ½1; þ1Þ; 0 < h < 1; ð3:8Þ

we have

t2R

jðTðuÞðtÞ  TðwÞðtÞÞ1j; sup

t2R

jðTðuÞðtÞ  TðwÞðtÞÞ2jÞ

6 bþ1

a

1

ku wkBþ sup

t2R

Z t

1

eRt

s a 1 ðuÞduXm j¼1

cþ 1j

1

e2ju1ðs s1jðsÞÞ  w1ðs s1jðsÞÞjds þH

þ 1

a 1

ku wkB;bþ2

a 2

ku wkB

þ sup

t2R

Z t

1

eRt

s a 2 ðuÞduXm j¼1

cþ 2j

1

e2ju2ðs s2jðsÞÞ  w2ðs s2jðsÞÞjds þH

þ 2

a 2

ku wkB

!

6 bþ1

a

1

þXm j¼1

cþ 1j

a

1e2þH

þ 1

a 1

!

ku wkB; bþ2

a 2

þXm j¼1

cþ 2j

a

2e2þH

þ 2

a 2

!

ku wkB

!

Hence

kTðuÞ  TðwÞkB6max b

þ 1

a 1

þXm j¼1

cþ 1j

a

1e2þH

þ 1

a 1

;bþ 2

a 2

þXm j¼1

cþ 2j

a

2e2þH

þ 2

a 2

ku wkB: Noting that

þ

1

a

1

þXm

j¼1

cþ 1j

a

1e2þH

þ 1

a 1

;bþ2

a 2

þXm j¼1

cþ 2j

a

2e2þH

þ 2

a 2

<1;

it is clear that the mapping T is a contraction on B⁄ Using Theorem 0.3.1 of[7], we obtain that the mapping T possesses a unique fixed pointu⁄

2 B⁄, Tu⁄=u⁄ By(3.2),u⁄satisfies(1.5) Sou⁄is an almost periodic solution of(1.5)in B⁄ The proof

ofTheorem 3.1is now complete h

Theorem 3.2 Let x⁄(t) be the positive almost periodic solution of Eq.(1.5)in the region B⁄ Suppose that(2.4)–(2.6)and(3.1)

hold Then, the solution x(t; t0,u) of(1.5)withu2 C0converges exponentially to x⁄

(t) as t ? +1

Proof Set x(t) = x(t; t0,u) and yiðtÞ ¼ xiðtÞ  x

iðtÞ, where t 2 [t0 ri, +1), i = 1, 2 Then

y0

1ðtÞ ¼ a1ðtÞy1ðtÞ þ b1ðtÞy2ðtÞ þPm

j¼1

c1jðtÞðx1ðt s1jðtÞÞe c 1j ðtÞx1ðts1jðtÞÞ

x

1ðt s1jðtÞÞe c1jðtÞx 

1 ðts1j ðtÞÞÞ  H1ðtÞy1ðt r1ðtÞÞ

y0

2ðtÞ ¼ a2ðtÞy2ðtÞ þ b2ðtÞy1ðtÞ þPm

j¼1

c2jðtÞðx2ðt s2jðtÞÞe c 2j ðtÞx 2 ðts2j ðtÞÞ

x

2ðt s2jðtÞÞe c 2j ðtÞx 

2 ðts2jðtÞÞÞ  H2ðtÞy2ðt r2ðtÞÞ;

8

>

>

>

<

>

>

>

:

ð3:10Þ

Set

CiðuÞ ¼  a

i  u

þ bþ

i þXm j¼1

cþ ij

1

e2euriþ Hþ

Clearly,Ci(u), i = 1, 2, are continuous functions on [0, 1] In view of(3.1), we obtain

Cið0Þ ¼ a

i þ bþ

i þXm j¼1

cþ ij

1

e2þ Hþi <0; i ¼ 1; 2;

we can choose two constantsg> 0 and k 2 (0, 1] such that

CiðkÞ ¼ k a

i

þ bþ

i þXm j¼1

cþ ij

1

We consider the Lyapunov functional

Calculating the upper right derivative of Vi(t)(i = 1, 2) along the solution y(t) of(3.10), we have

DþðV1ðtÞÞ 6 a1ðtÞjy1ðtÞjektþ b1ðtÞjy2ðtÞjektþXm

j¼1

c1jðtÞjx1ðt s1jðtÞÞe c1jðtÞx 1 ðts1j ðtÞÞ

 x

1ðt s1jðtÞÞe c 1j ðtÞx 

1 ðts1j ðtÞÞjektþ H1ðtÞjy1ðt r1ðtÞÞjektþ kjy1ðtÞjekt

¼ ðk a1ðtÞÞjy1ðtÞj þ b1ðtÞjy2ðtÞj þXm

j¼1

c1jðtÞjx1ðt s1jðtÞÞe c1jðtÞx 1 ðts1j ðtÞÞ

"

xðt s ðtÞÞe c 1j ðtÞx 

1 ðts1j ðtÞÞj þ HðtÞjy ðt r ðtÞÞj

DþðV2ðtÞÞ 6 a2ðtÞjy2ðtÞjektþ b2ðtÞjy1ðtÞjektþXm

j¼1

c2jðtÞjx2ðt s2jðtÞÞe c2jðtÞx 2 ðts2j ðtÞÞ

 x

2ðt s2jðtÞÞe c 2j ðtÞx 

2 ðts2j ðtÞÞjektþ H2ðtÞjy2ðt r2ðtÞÞjektþ kjy2ðtÞjekt

¼ ðk a2ðtÞÞjy2ðtÞj þ b2ðtÞjy1ðtÞj þXm

j¼1

c2jðtÞjx2ðt s2jðtÞÞe c2jðtÞx 2 ðts2j ðtÞÞ

"

x

2ðt s2jðtÞÞe c 2j ðtÞx 

2 ðts2jðtÞÞj þ H2ðtÞjy1ðt r2ðtÞÞj

Let maxi¼1;2fekt 0ðmaxt2½t 0 r i ;t 0 juiðtÞ  x

iðtÞj þ 1Þg :¼ M We claim that

Otherwise, one of the following cases must occur

Case 1: There exists T1> t0such that

Case 2: There exists T2> t0such that

If Case 1 holds, together with(2.7), (3.8)and(3.14), (3.17)implies that

0 6 Dþ

ðV1ðT1Þ  MÞ ¼ DþðV1ðT1ÞÞ 6 ðk ½ a1ðT1ÞÞjy1ðT1Þj þ b1ðT1Þjy2ðT1Þj

þXm

j¼1

c1jðT1Þjx1ðT1s1jðT1ÞÞe c1jðT 1 Þx 1 ðT 1 s1jðT 1 ÞÞ x

1ðT1s1jðT1ÞÞe c1jðT 1 Þx 

1 ðT 1 s1jðT 1 ÞÞj þ H1ðT1Þjy1ðT1r1ðT1ÞÞj

#

ek 1

¼ ðk a1ðT1ÞÞjy1ðT1Þj þ b1ðT1Þjy2ðT1Þj þXm

j¼1

c1jðT1Þ

c1jðT1Þjc1jðT1Þx1ðT1s1jðT1ÞÞe c 1j ðT 1 ÞxðT 1 s1jðT 1 ÞÞ

"

c1jðT1Þx

1ðT1s1jðT1ÞÞe c 1j ðT 1 Þx  ðT 1 s1j ðT 1 ÞÞj þ H1ðT1Þjy1ðT1r1ðT1ÞÞji

ek 1

6ðk a1ðT1ÞÞjy1ðT1Þjek 1þ b1ðT1Þjy2ðT1Þjek 1þXm

j¼1

c1jðT1Þ1

e2jy1ðT1s1jðT1ÞÞjekðT 1 s1j ðT 1 ÞÞeks1j ðT 1 Þ

þH1ðT1Þjy1ðT1r1ðT1ÞÞjekðT 1 r1 ðT 1 ÞÞekr1 ðT 1 Þ6

ka 1

þ bþ

1þXm j¼1

cþ 1j

1

e2ekr 1þ Hþ

1ekr 1

Thus,

0 6 k a

1

þ bþ

1þXm j¼1

cþ 1j

1

e2ekr 1þ Hþ1ekr 1; which contradicts with(3.12) Hence,(3.16)holds

If Case 2 holds, together with(2.7), (3.8)and(3.15), (3.18)implies that

0 6 Dþ

ðV2ðT2Þ  MÞ ¼ DþðV2ðT2ÞÞ 6 ðk ½ a2ðT2ÞÞjy2ðT2Þj þ b2ðT2Þjy1ðT2Þj

þXm

j¼1

c2jðT2Þjx2ðT2s2jðT2ÞÞe c2jðT 2 Þx 2 ðT 2 s2j ðT 2 ÞÞ x

2ðT2s2jðT2ÞÞe c2jðT 2 Þx  ðT 2 s2j ðT 2 ÞÞj þ H2ðT2Þjy2ðT2r2ðT2ÞÞj

#

ek 2

¼ ðk a2ðT2ÞÞjy2ðT2Þj þ b2ðT2Þjy1ðT2Þj þXm

j¼1

c2jðT2Þ

c2jðT2Þjc2jðT2Þx2ðT2s2jðT2ÞÞe c2jðT 2 Þx 2 ðT 2 s2j ðT 2 ÞÞ

"

c2jðT2Þx

2ðT2s2jðT2ÞÞe c 2j ðT 2 Þx 

2 ðT 2 s2j ðT 2 ÞÞj þ H2ðT2Þjy2ðT2r2ðT2ÞÞji

ek 2

6ðk a2ðT2ÞÞjy2ðT2Þjek 2þ b2ðT2Þjy1ðT2Þjek 2þXm

j¼1

c2jðT2Þ1

e2jy2ðT2s2jðT2ÞÞjekðT 2 s2j ðT 2 ÞÞeks2j ðT 2 Þ

þH2ðT2Þjy2ðT2r2ðT2ÞÞjekðT 2 r2 ðT 2 ÞÞekr2 ðT 2 Þ6

ka 2

þ bþ

2þXm j¼1

cþ 2j

1

e2ekr 2þ Hþ2ekr 2

Thus,

0 6 k a

2

þ bþ2þXm j¼1

cþ 2j

1

e2ekr2

þ Hþ2ekr2; which contradicts with(3.12) Hence,(3.16)holds It follows that

This completes the proof h

4 Example and remark

In this section, we give an example to demonstrate the results obtained in previous sections

Example 4.1 Consider the following Nicholson-type delay system with linear harvesting terms:

x0

1ðtÞ ¼  18 þ cos 2t

x1ðtÞ þ 0:00001 þ 0:000005 sin 2t

ee3x2ðtÞ

þ ee1 9:5 þ 0:005j sin ffiffiffi

2 p tj

x1t  e2j sin tj

ex 1ðte 2j sin tjÞ;

þ ee1 9:5 þ 0:005j sin ffiffiffi

5 p tj

x1 t  e2j cos pffiffi3

tj

ex 1ðte 2j cospffiffi3 tjÞ

 ð0:000001 cos2tÞee3x1 t  e2j cos pffiffi3

tj

x0

2ðtÞ ¼  18 þ sin 2t

x2ðtÞ þ 0:00001 þ 0:000005 cos 2t

ee3x1ðtÞ

þ ee1 9:5 þ 0:005j cos ffiffiffi

2

p tj

x2t  e2j cos tj

ex 2ðte 2j cos tjÞ;

þ ee1 9:5 þ 0:005j sin ffiffiffi

6

p tj

x2 t  e2j cos pffiffi7

tj

ex 2ðte 2j cospffiffi7 tjÞ

 0:000001 cos 4t

ee3x2 t  e2j cos pffiffi3

tj

;

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

ð4:1Þ

i ¼ 18; aþ

i ¼ 19; cþ

ij¼c

ij¼ 1; b

i ¼ 0:00001ee3; bþ

i ¼ 0:000015ee3; c

ij¼ 9:5ee1; cþ

ij ¼ 9:505ee1;Hþ

0:000001ee3; ri¼ max max16j6m sþ

ij

n o

; rþ i

¼ e2;

bi

aþ

i

þX2

j¼1

c

ije1cþij e

aþ i

H

þ

ie

aþ i

¼19 þ 0:00001e

e3 0:000001ee2

bþie

a

i

þX2

j¼1

cþ ij

a

ic ij

1

e¼

bþie

a i

þX2 j¼1

cþ ij

a i

1

e¼

0:000015ee2þ 19:01ee2

and

þ

1

a

1

þXm

j¼1

cþ 1j

a

1e2þH

þ 1

a 1

;bþ2

a 2

þXm j¼1

cþ 2j

a

2e2þH

þ 2

a 2

¼0:000016e

e2þ 19:01ee2

where i, j = 1, 2 Let Ei1= e and Ei2= 1 for i = 1, 2 Then,(4.2)–(4.4)imply that the Nicholson-type delay differential system

(4.1)satisfies(2.4)–(2.6) and (3.1) Hence, fromTheorems 3.1 and 3.2, system(4.1)has a positive almost periodic solution

xðtÞ 2 B¼ fuju2 B; 1 6uiðtÞ 6 e; for all t 2 R; i ¼ 1; 2g:

Moreover, ifu2 C0= {uju2 C, 1 <ui(t) < e, for all t 2 [e2, 0], i = 1, 2}, then x(t; t0,u) converges exponentially to x⁄(t) as

t ? +1

Remark 4.1 To the best of our knowledge, few authors have considered the problems of positive almost periodic solution of Nicholson-type delay system with linear harvesting terms Therefore, all the results in[1–3,8]and the references therein cannot be applicable to prove that all the solutions of(4.1)with initial valueu2 C0converge exponentially to the positive almost periodic solution Moreover, if H1(t)  H2(t)  0, we can find that the main results of[2]are special ones ofTheorems 3.2with Ei1= e and Ei2= 1 for i = 1, 2 This implies that the results of this paper are new and they complement previously known results

Acknowledgements

The authors thank the referees very much for the helpful comments and suggestions

[1] L Berezansky, L Idels, L Troib, Global dynamics of Nicholson-type delay systems with applications, Nonlinear Anal Real World Appl 12 (1) (2011) 436– 445.

[2] W Wang, L Wang, W Chen, Existence and exponential stability of positive almost periodic solution for Nicholson-type delay systems, Nonlinear Anal Real World Appl 12 (4) (2011) 1938–1949.

[3] L Berezansky, E Braverman, L Idels, Nicholson’s blowflies differential equations revisited: main results and open problems, Appl Math Model 34 (2010) 1405–1417.

[4] A.M Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol 377, Springer, Berlin, 1974.

[5] C.Y He, Almost Periodic Differential Equation, Higher Education Publishing House, Beijing, 1992 (in Chinese).

[6] J.K Hale, S.M Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

[7] J.K Hale, Ordinary Differential Equations, Krieger, Malabar, Florida, 1980.

[8] F Long, Positive Almost Periodic Solution for a Class of Nicholson’s Blowflies Model with a Linear Harvesting Term, Nonlinear Anal Real World Appl 13 (2012) 686–693.