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

Applied Mathematical Modelling 36 (2012) 3289–3298

Contents lists available at SciVerse ScienceDirect

Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm

The positive almost periodic solution for Nicholson-type delay systems
with linear harvesting terms q
Xingguo Liu a, Junxia Meng b,⇑
a
b

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

a b s t r a c t
In this paper, we study the existence and exponential convergence of positive almost periodic 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.

Keywords:
Positive almost periodic solution
Exponential convergence
Nicholson-type delay system
Linear harvesting term

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:

(

x01 ðtÞ ¼ Àa1 x1 ðtÞ þ b1 x2 ðtÞ þ c1 x1 ðt À sÞeÀx1 ðtÀsÞ ;
x02 ðtÞ ¼ Àa2 x2 ðtÞ þ b2 x1 ðtÞ þ c2 x2 ðt À sÞeÀx2 ðtÀsÞ ;

ð1:1Þ

with initial conditions:

xi ðsÞ ¼ ui ðsÞ;

s 2 ½Às; 0Š;

ui ð0Þ > 0;

ð1:2Þ

where ui 2 C([Às, 0], [0, +1)), ai, bi, ci and s are 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:

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: (J. Meng).

0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2011.09.087


3290

X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

8
m
P
>
Àc1j ðtÞx1 ðtÀs1j ðtÞÞ
0
>
;
>
< x1 ðtÞ ¼ Àa1 ðtÞx1 ðtÞ þ b1 ðtÞx2 ðtÞ þ c1j ðtÞx1 ðt À s1j ðtÞÞe

j¼1

m
P
>
>
Àc ðtÞx ðtÀs ðtÞÞ
0
>
: x2 ðtÞ ¼ Àa2 ðtÞx2 ðtÞ þ b2 ðtÞx1 ðtÞ þ c2j ðtÞx2 ðt À s2j ðtÞÞe 2j 2 2j ;

ð1:3Þ

j¼1

1

where ai, bi, cij, cij, sij : R ? (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 Nicholson’s blowflies model with a linear harvesting term:

x0 ðtÞ ¼ ÀdxðtÞ þ pxðt À sÞeÀaxðtÀsÞ À Hxðt À rÞ;

d; p; s; a; H;

r 2 ð0; þ1Þ;

ð1:4Þ

where Hx(t À 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, 1a is the size at which the population reproduces at its maximum rate, d is the per capita daily adult death rate,
and s is the generation time. Moreover, Berezansky et al. [3] pointed out an open problem: How about the dynamic behaviors of the Nicholson’s 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 positive almost periodic solutions of the following non-autonomous Nicholson-type delay systems with linear harvesting
terms:

8
m
P
>
Àc1j ðtÞx1 ðtÀs1j ðtÞÞ
0
>
>
> x1 ðtÞ ¼ Àa1 ðtÞx1 ðtÞ þ b1 ðtÞx2 ðtÞ þ c1j ðtÞx1 ðt À s1j ðtÞÞe
>
j¼1
>
>
>
<
ÀH1 ðtÞx1 ðt À r1 ðtÞÞ;
m
P
> 0
>
>
x2 ðtÞ ¼ Àa2 ðtÞx2 ðtÞ þ b2 ðtÞx1 ðtÞ þ c2j ðtÞx2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ
>

>
>
j¼1
>
>
:
ÀH2 ðtÞx2 ðt À r2 ðtÞÞ;

ð1:5Þ

where ai, 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 gÀ be defined as

g À ¼ inf gðtÞ;
t2R

g þ ¼ sup gðtÞ:
t2R

It will be assumed that

&

n

o

'

aÀi > 0; bÀi > 0; cÀij > 0; ri ¼ max max sþij ; rþi > 0; i ¼ 1; 2:
16j6m

ð1:6Þ

À Á
Denote by Rn Rnþ the set of all (nonnegative) real vectors. Let





C ¼ Cð½Àr 1 ; 0Š; R1 Þ Â Cð½Àr 2 ; 0Š; R1 Þ and C þ ¼ C ½Àr1 ; 0Š; R1þ  C ½Àr 2 ; 0Š; R1þ :
À
Á
If xi(t) is defined on [t0 À ri, r) with t0, r 2 R1 and i = 1, 2, then we define xt 2 C as xt ¼ x1t ; x2t where xit ðhÞ ¼ xi ðt þ hÞ for all
h 2 [Àri, 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 À B P 0 (resp. A À 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:

xt0 ¼ u; u ¼ ðu1 ; u2 Þ 2 C þ

and ui ð0Þ > 0;

i ¼ 1; 2:

ð1:7Þ

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 rightinterval of existence of xt(t0, u).

The remaining part of this paper is organized as follows. In Section 2, we shall give some notations and preliminary
results. In Section 3, 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 Section 4, 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
Section 3.


X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

3291

Definition 2.1 (See [4,5]). Let u(t) : R1 ? Rn be continuous in t. u(t) is said to be almost periodic on R1, if for any e > 0, the set
T(u,e) = {d:ju(t + d) À u(t)j < e for all t 2 R1} is relatively dense, i.e., for any e > 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) À u(t)j < e, for all
t 2 R1.

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

x0 ðtÞ ¼ Q ðtÞxðtÞ

ð2:1Þ
1

is said to admit an exponential dichotomy on R if there exist positive constants k, a, projection P and the fundamental solution matrix X(t) of (2.1) satisfying
ÀaðtÀsÞ

kXðtÞPX À1 ðsÞk 6 ke

for all t P s;

ÀaðsÀtÞ

kXðtÞðI À PÞX À1 ðsÞk 6 ke

for all t 6 s:

Set

B ¼ fuju ¼ ðu1 ðtÞ; u2 ðtÞÞ is an almost periodic function on R1 g:
For any u 2 B, we define induced module kukB ¼ supt2R1 kuðtÞk, 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

x0 ðtÞ ¼ Q ðtÞx þ gðtÞ

ð2:2Þ

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

xðtÞ ¼

Z

t

XðtÞPX À1 ðsÞgðsÞds À

Z

þ1

XðtÞðI À PÞX À1 ðsÞgðsÞ ds:

ð2:3Þ

t

À1

Lemma 2.2 (See [4,5]). Let ci(t) be an almost periodic function on R1 and

M½ci Š ¼ lim

T!þ1

Z

1
T

tþT

ci ðsÞds > 0;

i ¼ 1; 2; . . . ; n:

t

Then the linear system

x0 ðtÞ ¼ diag ðÀc1 ðtÞ; Àc2 ðtÞ; . . . ; Àcn ðtÞÞxðtÞ
admits an exponential dichotomy on R1.
Lemma 2.3. Suppose that there exist two positive constants Ei1 and Ei2 such that

Ei1 > Ei2 ;
bÀ1 E22
þ
1

a

bÀ2 E12
þ
2

a

þ

a

À
1

m
X
cÀ1j
j¼1

þ

bþ1 E21

þ
1

a

m cÀ
X
2j
j¼1

þ
2

a

m
X
cþ1j 1
< E11 ;
À À
a1 c1j e
j¼1

þ

Àcþ
E
1j 11

E11 e

Àcþ
E
2j 21

E21 e

À

À

Hþ1 E11

aþ1
Hþ2 E21

aþ2

bþ2 E11

a

À
2

> E12 P

þ

m
X
cþ2j 1
< E21 ;
À À
a2 c2j e
j¼1

ð2:4Þ

1
;
min cÀ1j

ð2:5Þ

1
;
min cÀ2j

ð2:6Þ

16j6m

> E22 P

16j6m

where i = 1, 2. Let

C 0 :¼ fuju 2 C; Ei2 < ui ðtÞ < Ei1 ; for all t 2 ½Àr i ; 0Š; i ¼ 1; 2g:
Moreover, assume that x(t; t0, u) is the solution of (1.5) with u 2 C0. Then,

Ei2 < xi ðt; t 0 ; uÞ < Ei1 ; for all t 2 ½t 0 ; gðuÞÞ;
and g(u) = +1.

i ¼ 1; 2

ð2:7Þ


3292

X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

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

0 < xi ðtÞ for all t 2 ½t0 ; TÞ;

i ¼ 1; 2;

ð2:8Þ

we claim that

0 < xi ðtÞ < Ei1 for all t 2 ½t0 ; TÞ;

i ¼ 1; 2:

ð2:9Þ

Assume, by way of contradiction, that (2.9) does not hold. Then, one of the following cases must occur.
Case i: There exists t1 2 (t0, T) such that

x1 ðt1 Þ ¼ E11 and 0 < xi ðtÞ < Ei1 for all t 2 ½t 0 À r i ; t1 Þ;

i ¼ 1; 2:

ð2:10Þ

i ¼ 1; 2:

ð2:11Þ

Case ii: There exists t2 2 (t0, T) such that

x2 ðt2 Þ ¼ E21 and 0 < xi ðtÞ < Ei1 for all t 2 ½t 0 À r i ; t2 Þ;

If Case i holds, calculating the derivative of x1(t), together with (2.4) and the fact that supuP0 ueÀu ¼ 1e , (1.5) and (2.10)
imply that

0 6 x01 ðt 1 Þ ¼ Àa1 ðt 1 Þx1 ðt1 Þ þ b1 ðt1 Þx2 ðt 1 Þ þ

m
X

c1j ðt 1 Þx1 ðt 1 À s1j ðt 1 ÞÞeÀc1j ðt1 Þx1 ðt1 Às1j ðt1 ÞÞ À H1 ðt 1 Þx1 ðt 1 À r1 ðt1 ÞÞ

j¼1
À
1 x1 ðt 1 Þ

6 Àa

þ

bþ1 E21

m cþ
m
X
cþ1j 1
bþ E21 X
1j 1
¼ aÀ1 ÀE11 þ 1 À þ
þ
À
À À
c1j e
a1
a1 c1j e
j¼1
j¼1

!
< 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 supuP0 ueÀu ¼ 1e , (1.5) and (2.11)
imply that

0 6 x02 ðt 2 Þ ¼ Àa2 ðt 2 Þx2 ðt2 Þ þ b2 ðt2 Þx1 ðt 2 Þ þ

m
X

c2j ðt 2 Þx2 ðt 2 À s2j ðt 2 ÞÞeÀc2j ðt2 Þx2 ðt2 Às2j ðt2 ÞÞ À H2 ðt 2 Þx2 ðt 2 À r2 ðt2 ÞÞ

j¼1

6 ÀaÀ2 x2 ðt 2 Þ þ bþ2 E11 þ

m cþ
m
X
cþ2j 1
bþ E11 X
2j 1
¼ aÀ2 ÀE21 þ 2 À þ
À
À À
c2j e
a2
a2 c2j e
j¼1
j¼1

!
< 0;

which is a contradiction and implies that (2.9) holds.
We next show that

xi ðtÞ > Ei2 ; for all t 2 ðt0 ; gðuÞÞ;

i ¼ 1; 2:

ð2:12Þ

Suppose, for the sake of contradiction, that (2.12) does not hold. Then, one of the following cases must occur.
Case I: There exists t3 2 (t0, g(u)) such that

x1 ðt3 Þ ¼ E12

and xi ðtÞ > Ei2 for all t 2 ½t0 À r i ; t 3 Þ;

i ¼ 1; 2:

ð2:13Þ

Case II: There exists t4 2 (t0, g(u)) such that

x2 ðt4 Þ ¼ E22 and xi ðtÞ > Ei2 for all t 2 ½t 0 À r i ; t4 Þ;

i ¼ 1; 2:

ð2:14Þ

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

Ei2 < xi ðtÞ < Ei1 ;

cþij xi ðtÞ P cþij Ei2 P cþij

1
P 1;
min cÀij

ð2:15Þ

16j6m

for all t 2 [t0 À ri, t3), i = 1, 2, j = 1, 2, . . . , m. Calculating the derivative of x1(t), together with (2.5) and the fact that
min16u6jueÀu = jeÀj, (1.5), (2.13) and (2.15) imply that

0 P x01 ðt3 Þ ¼ Àa1 ðt 3 Þx1 ðt 3 Þ þ b1 ðt3 Þx2 ðt3 Þ þ

m
X

c1j ðt3 Þx1 ðt 3 À s1j ðt 3 ÞÞeÀc1j ðt3 Þx1 ðt3 Às1j ðt3 ÞÞ À H1 ðt 3 Þx1 ðt 3 À r1 ðt3 ÞÞ

j¼1

¼ Àa1 ðt 3 Þx1 ðt 3 Þ þ b1 ðt 3 Þx2 ðt 3 Þ þ

m
X
c1j ðt3 Þ
j¼1

cþ1j

cþ1j x1 ðt3 À s1j ðt3 ÞÞeÀc1j ðt3 Þx1 ðt3 Às1j ðt3 ÞÞ À H1 ðt3 Þx1 ðt3 À r1 ðt3 ÞÞ


3293

X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

P Àa1 ðt 3 Þx1 ðt 3 Þ þ bÀ1 E22 þ

m cÀ
X
1j
j¼1

þ
1 x1 ðt 3 Þ

P Àa

þ

bÀ1 E22

þ

m
X

cÀ1j
þ
1j
j¼1

c

þ

þ
1j

c

cþ1j x1 ðt3 À s1j ðt3 ÞÞeÀc1j x1 ðt3 Às1j ðt3 ÞÞ À Hþ1 E11

Àcþ
E
þ
1j 11
1j E11 e

c

À

Hþ1 E11

þ
1

¼a

ÀE12 þ

bÀ1 E22

aþ1

þ

m
X
cÀ1j
j¼1

aþ1

E11 e

Àcþ
E
1j 11

À

Hþ1 E11

!

aþ1

> 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 À ri, t4),i = 1, 2, j = 1, 2, . . . , m. Calculating the derivative of x2(t),
together with (2.6) and the fact that min16u6jueÀu = jeÀj, (1.5), (2.14) and (2.15) imply that

0 P x02 ðt4 Þ ¼ Àa2 ðt 4 Þx2 ðt 4 Þ þ b2 ðt4 Þx1 ðt4 Þ þ

m
X

c2j ðt4 Þx2 ðt4 À s2j ðt 4 ÞÞeÀc2j ðt4 Þx2 ðt4 Às2j ðt4 ÞÞ À H2 ðt 4 Þx2 ðt 4 À r2 ðt4 ÞÞ

j¼1

¼ Àa2 ðt4 Þx2 ðt4 Þ þ b2 ðt 4 Þx1 ðt 4 Þ þ

m
X
c2j ðt4 Þ

cþ2j

j¼1

P Àa2 ðt 4 Þx2 ðt 4 Þ þ bÀ2 E12 þ

m cÀ
X
2j

cþ2j

j¼1
þ
2 x2 ðt 4 Þ

P Àa

þ

bÀ2 E12

m cÀ
X
2j

þ

j¼1

cþ2j

cþ2j x2 ðt4 À s2j ðt4 ÞÞeÀc2j ðt4 Þx2 ðt4 Às2j ðt4 ÞÞ À H2 ðt4 Þx2 ðt4 À r2 ðt4 ÞÞ
þ

cþ2j x2 ðt4 À s2j ðt4 ÞÞeÀc2j x2 ðt4 Às2j ðt4 ÞÞ À Hþ2 E21

Àcþ
E
þ
2j 21
2j E21 e

c

À

Hþ2 E21

þ
2

¼a

ÀE22 þ

bÀ2 E12

aþ2

þ

m cÀ
X
2j
j¼1

aþ2

E21 e

Àcþ
E
2j 21

À

Hþ2 E21

aþ2

!
> 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 obtain g(u) = +1. This ends the
proof of Lemma 2.3. h

3. Main results

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

(
max

bþ1

aÀ1

þ

m
X
cþ1j
j¼1

aÀ1 e2

Hþ1

þ

;

bþ2

aÀ1 aÀ2

þ

m
X

cþ2j
j¼1

aÀ2 e2

þ

Hþ2

)

aÀ2

< 1:

ð3:1Þ

Then, there exists a unique positive almost periodic solution of system (1.5) in the region B⁄ = {uju 2 B, Ei2 6 ui(t) 6 Ei1, for all
t 2 R1, i = 1, 2}.
Proof. For any / 2 B, we consider an auxiliary system

8
m
P
>
>
x01 ðtÞ ¼ Àa1 ðtÞx1 ðtÞ þ b1 ðtÞ/2 ðtÞ þ c1j ðtÞ/1 ðt À s1j ðtÞÞeÀc1j ðtÞ/1 ðtÀs1j ðtÞÞ
>
>
>

j¼1
>
>
>
>
>
<
ÀH1 ðtÞ/1 ðt À r1 ðtÞÞ;

ð3:2Þ

m
>
P
>
>
>
x02 ðtÞ ¼ Àa2 ðtÞx2 ðtÞ þ b2 ðtÞ/1 ðtÞ þ c2j ðtÞ/2 ðt À s2j ðtÞÞeÀc2j ðtÞ/ðtÀs2j ðtÞÞ
>
>
>
j¼1
>
>
>
:
ÀH2 ðtÞ/2 ðt À r2 ðtÞÞ;

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

(

x01 ðtÞ ¼ Àa1 ðtÞx1 ðtÞ;

ð3:3Þ

x02 ðtÞ ¼ Àa2 ðtÞx2 ðtÞ;

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

x ðtÞ ¼

À

x/1 ðtÞ; x/2 ðtÞ

Á

Z

¼

t

À

e

Rt
s

a1 ðuÞdu

b1 ðsÞ/2 ðsÞ þ

m
X

À1

Z

t

À1

e

À

j¼1

Rt
s

a2 ðuÞdu

b2 ðsÞ/1 ðsÞ þ

m
X
j¼1

!
Àc1j ðsÞ/1 ðsÀs1j ðsÞÞ

c1j ðsÞ/1 ðs À s1j ðsÞÞe

À H1 ðsÞ/1 ðs À r1 ðsÞÞ ds;
! !

c2j ðsÞ/2 ðs À s2j ðsÞÞeÀc2j ðsÞ/2 ðsÀs2j ðsÞÞ À H2 ðsÞ/2 ðs À r2 ðsÞÞ ds : ð3:4Þ


3294

X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

Define a mapping T : B ? B by setting

Tð/ðtÞÞ ¼ x/ ðtÞ;

8/ 2 B:

Since B = {uju 2 B, Ei2 6 ui(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 any u 2 B⁄,
from (2.4) and (3.4) and the fact that supuP0 ueÀu ¼ 1e, we have
⁄

/

x ðtÞ 6

6

! Z
! !
Rt
m
t
X
1
1
À
a2 ðuÞdu
þ
e s
þ
e s
b2 E11 þ
c ðsÞ ds;
c ðsÞ ds
c1j ðsÞe 1j
c2j ðsÞe 2j
À1
À1
j¼1
j¼1
!

m
m
cþ1j bþ2 E11 X
cþ2j
bþ1 E21 X
< ðE11 ; E21 Þ; for all t 2 R1 :
þ
þ
;
aÀ1
aÀ1 cÀ1j e aÀ2
aÀ2 cÀ2j e
j¼1
j¼1
Z

t

À

Rt

a1 ðuÞdu

m
X

bþ1 E21

In view of the fact that min16u6jueÀu = jeÀj, from (2.5), (2.6) and (3.4), we obtain

/

x ðtÞ P

Z

t

e

À

Rt
s

a1 ðuÞdu

À1

Z

t

e

À

Rt
s

þ

m
X

c1j ðsÞ

j¼1

a2 ðuÞdu

À1

P

bÀ1 E22

bÀ2 E12

m
X

þ

c2j ðsÞ

j¼1

bÀ1 E22

aþ1

þ

m cÀ
X
1j

Àcþ
E
1j 11

þ E11 e

j¼1

a1

À

1

cþ2j

cþ1j

þ
1j /1 ðs À

c

þ
2j /2 ðs À

c

Hþ1 E11

aþ1

1

s1j ðsÞÞe

s2j ðsÞÞe

Àcþ
/ ðsÀs1j ðsÞÞ
1j 1

Àcþ
/ ðsÀs2j ðsÞÞ
2j 2

; fracbÀ2 E12 aþ2 þ

m cÀ
X
2j
j¼1

!

À H1 ðsÞ/1 ðs À r1 ðsÞÞ ds;
! !

À H2 ðsÞ/2 ðs À r2 ðsÞÞ ds
Àcþ
E
2j 21

þ E21 e

a2

ð3:5Þ

À

Hþ2 E21

!
> ðE12 ; E22 Þ; for all t 2 R1 :

aþ2

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, for u, w 2 B⁄, we get


 
Z
sup jðTðuÞðtÞ À TðwÞðtÞÞ1 j; sup jðTðuÞðtÞ À TðwÞðtÞÞ2 j ¼ sup j
t2R

t2R

t2R

þ

m
X

t

À

e

Rt
s

a1 ðuÞdu

À1

ðb1 ðsÞðu2 ðsÞ À w2 ðsÞÞ

c1j ðsÞðu1 ðs À s1j ðsÞÞeÀc1j ðsÞu1 ðsÀs1j ðsÞÞ

j¼1

Àw1 ðs À s1j ðsÞÞeÀc1j ðsÞw1 ðsÀs1j ðsÞÞ Þ À H1 ðsÞðu1 ðs À r1 ðsÞÞ
Z t
Rt
À
a ðuÞdu
e s 2
ðb2 ðsÞðu1 ðsÞ À w1 ðsÞÞ
Àw1 ðs À r1 ðsÞÞÞÞdsj; sup j
t2R

þ

m
X

À1

c2j ðsÞðu2 ðs À s2j ðsÞÞeÀc2j ðsÞu2 ðsÀs2j ðsÞÞ

j¼1

Àw2 ðs À s2j ðsÞÞeÀc2j ðsÞw2 ðsÀs2j ðsÞÞ Þ À H2 ðsÞðu2 ðs À r2 ðsÞÞ
Àw2 ðs À r2 ðsÞÞÞÞdsjÞ

Z t

Rt
À
a ðuÞdu
e s 1
ðb1 ðsÞðu2 ðsÞ À w2 ðsÞÞ
¼ sup j
t2R

À1

m
X
c1j ðsÞ
þ
ðc1j ðsÞu1 ðs À s1j ðsÞÞeÀc1j ðsÞu1 ðsÀs1j ðsÞÞ
c
1j ðsÞ
j¼1

Àc1j ðsÞw1 ðs À s1j ðsÞÞeÀc1j ðsÞw1 ðsÀs1j ðsÞÞ Þ À H1 ðsÞðu1 ðs À r1 ðsÞÞ
Z t
Rt
À
a ðuÞdu
Àw1 ðs À r1 ðsÞÞÞÞdsj; supt2R j
e s 2
ðb2 ðsÞðu1 ðsÞ
À1
m
X

c2j ðsÞ
ðc ðsÞu2 ðs À s2j ðsÞÞeÀc2j ðsÞu2 ðsÀs2j ðsÞÞ
Àw1 ðsÞÞ þ
c2j ðsÞ 2j
j¼1

Àc2j ðsÞw2 ðs À s2j ðsÞÞeÀc2j ðsÞw2 ðsÀs2j ðsÞÞ Þ À H2 ðsÞðu2 ðs À r2 ðsÞÞ
Àw2 ðs À r2 ðsÞÞÞÞdsjÞ:
In view of (1.5), (2.5), (2.6), (3.5), (3.6) and (3.7), from supuP1

1Àu
 u¼
e

1
e2

and the inequality



1 À ðx þ hðy À xÞÞ
jx À yj 6 1 jx À yj where x; y 2 ½1; þ1Þ;
jxeÀx À yeÀy j ¼ 

exþhðyÀxÞ
e2

we have

ð3:7Þ

0 < h < 1;

ð3:8Þ


X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

ðsup jðTðuÞðtÞ À TðwÞðtÞÞ1 j; sup jðTðuÞðtÞ À TðwÞðtÞÞ2 jÞ
t2R
t2R
Z t
Rt
m
X
bþ1
1
Hþ
bþ
À
a ðuÞdu
6
ku À wkB þ sup
e s 1
cþ1j 2 ju1 ðs À s1j ðsÞÞ À w1 ðs À s1j ðsÞÞjds þ À1 ku À wkB ; 2À ku À wkB
À
e
a1
a1

a2
t2R
À1
j¼1
!
Z t
Rt
þ
m
X
1
H
À
a ðuÞdu
e s 2
cþ2j 2 ju2 ðs À s2j ðsÞÞ À w2 ðs À s2j ðsÞÞjds þ À2 ku À wkB
þ sup
e
a2
t2R
À1
j¼1 !
!
!
þ
þ
þ
þ
m
m

þ
þ
b1 X c1j
H1
b2 X c2j
H2
ku À wkB ;
ku À wkB :
þ
þ
þ
þ
6
aÀ1 j¼1 aÀ1 e2 aÀ1
aÀ2 j¼1 aÀ2 e2 aÀ2
Hence

(
kTðuÞ À TðwÞkB 6 max

bþ1

aÀ1

max

bþ1

aÀ1

þ

)
m
cþ2j
Hþ1 bþ2 X
Hþ2
ku À wkB :
þ
;
þ
þ
aÀ1 e2 aÀ1 aÀ2 j¼1 aÀ2 e2 aÀ2

þ

j¼1

m
X
cþ1j
j¼1

ð3:9Þ

m
X
cþ1j

Noting that

(

3295

m
X
cþ2j
Hþ bþ
Hþ
þ À1 ; 2À þ
þ À2
À 2
À 2
a1 e a1 a2
a2 e a2
j¼1

)
< 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 point u⁄ 2 B⁄, Tu⁄ = u⁄. By (3.2), u⁄ satisfies (1.5). So u⁄ is an almost periodic solution of (1.5) in B⁄. The proof
of Theorem 3.1 is 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) with u 2 C0 converges exponentially to x⁄(t) as t ? +1.
Proof. Set x(t) = x(t; t0, u) and yi ðtÞ ¼ xi ðtÞ À xÃi ðtÞ, where t 2 [t0 À ri, +1), i = 1, 2. Then

8
m

P
>
>
>
y01 ðtÞ ¼ Àa1 ðtÞy1 ðtÞ þ b1 ðtÞy2 ðtÞ þ c1j ðtÞðx1 ðt À s1j ðtÞÞeÀc1j ðtÞx1 ðtÀs1j ðtÞÞ
>
>
>
j¼1
>
>
>
Ã
<
ÀxÃ1 ðt À s1j ðtÞÞeÀc1j ðtÞx1 ðtÀs1j ðtÞÞ Þ À H1 ðtÞy1 ðt À r1 ðtÞÞ
m
P
> 0
>
>
y2 ðtÞ ¼ Àa2 ðtÞy2 ðtÞ þ b2 ðtÞy1 ðtÞ þ c2j ðtÞðx2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ
>
>
>
j¼1
>
>
>
Ã
:

ÀxÃ2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ Þ À H2 ðtÞy2 ðt À r2 ðtÞÞ;

ð3:10Þ

Set

À

Á

Ci ðuÞ ¼ À aÀi À u þ bþi þ

m
X

cþij

j¼1

1 uri
e þ Hþi euri ;
e2

u 2 ½0; 1Š;

i ¼ 1; 2:

ð3:11Þ

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

Ci ð0Þ ¼ ÀaÀi þ bþi þ

m
X
j¼1

cþij

1
þ Hþi < 0;
e2

i ¼ 1; 2;

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

À

Á

Ci ðkÞ ¼ k À aÀi þ bþi þ

m
X
j¼1

cþij

1 kri

e þ Hþi ekri < Àg < 0;
e2

i ¼ 1; 2:

ð3:12Þ

We consider the Lyapunov functional

V 1 ðtÞ ¼ jy1 ðtÞjekt ;

V 2 ðtÞ ¼ jy2 ðtÞjekt :

ð3:13Þ

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

Dþ ðV 1 ðtÞÞ 6 Àa1 ðtÞjy1 ðtÞjekt þ b1 ðtÞjy2 ðtÞjekt þ

m
X

c1j ðtÞjx1 ðt À s1j ðtÞÞeÀc1j ðtÞx1 ðtÀs1j ðtÞÞ

j¼1
Ã

À xÃ1 ðt À s1j ðtÞÞeÀc1j ðtÞx1 ðtÀs1j ðtÞÞ jekt þ H1 ðtÞjy1 ðt À r1 ðtÞÞjekt þ kjy1 ðtÞjekt
"
m

X
¼ ðk À a1 ðtÞÞjy1 ðtÞj þ b1 ðtÞjy2 ðtÞj þ
c1j ðtÞjx1 ðt À s1j ðtÞÞeÀc1j ðtÞx1 ðtÀs1j ðtÞÞ
j¼1
Ã
Ã
ÀxÃ1 ðt À s1j ðtÞÞeÀc1j ðtÞx1 ðtÀs1j ðtÞÞ j þ H1 ðtÞjy1 ðt À r1 ðtÞÞj ekt ;

for all t > t0 ;

ð3:14Þ


3296

X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

and

Dþ ðV 2 ðtÞÞ 6 Àa2 ðtÞjy2 ðtÞjekt þ b2 ðtÞjy1 ðtÞjekt þ

m
X

c2j ðtÞjx2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ

j¼1
Ã

À xÃ2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ jekt þ H2 ðtÞjy2 ðt À r2 ðtÞÞjekt þ kjy2 ðtÞjekt

"
m
X
¼ ðk À a2 ðtÞÞjy2 ðtÞj þ b2 ðtÞjy1 ðtÞj þ
c2j ðtÞjx2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ
j¼1

Ã
Ã
ÀxÃ2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ j þ H2 ðtÞjy1 ðt À r2 ðtÞÞj ekt ;

for all t > t 0 :

ð3:15Þ

Let maxi¼1;2 fekt0 ðmaxt2½t0 Àri ;t0 Š jui ðtÞ À xÃi ðtÞj þ 1Þg :¼ M. We claim that

V i ðtÞ ¼ jyi ðtÞjekt < M

for all t > t 0 ;

i ¼ 1; 2:

ð3:16Þ

Otherwise, one of the following cases must occur.
Case 1: There exists T1 > t0 such that

V 1 ðT 1 Þ ¼ M

and V i ðtÞ < M

for all t 2 ½t 0 À r i ; T 1 Þ;

i ¼ 1; 2:

ð3:17Þ

i ¼ 1; 2:

ð3:18Þ

Case 2: There exists T2 > t0 such that

V 2 ðT 2 Þ ¼ M

and V i ðtÞ < M

for all t 2 ½t 0 À r i ; T 2 Þ;

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

0 6 Dþ ðV 1 ðT 1 Þ À MÞ ¼ Dþ ðV 1 ðT 1 ÞÞ 6 ½ðk À a1 ðT 1 ÞÞjy1 ðT 1 Þj þ b1 ðT 1 Þjy2 ðT 1 Þj
þ
"

m
X

Àc1j ðT 1 Þx1 ðT 1 Às1j ðT 1 ÞÞ

c1j ðT 1 Þjx1 ðT 1 À s1j ðT 1 ÞÞe

À

xÃ1 ðT 1

Àc1j ðT 1 ÞxÃ1 ðT 1 Às1j ðT 1 ÞÞ

À s1j ðT 1 ÞÞe

#
j þ H1 ðT 1 Þjy1 ðT 1 À r1 ðT 1 ÞÞj ekT 1

j¼1

m
X
c1j ðT 1 Þ
jc1j ðT 1 Þx1 ðT 1 À s1j ðT 1 ÞÞeÀc1j ðT 1 ÞxðT 1 Às1j ðT 1 ÞÞ
c
ðT
Þ
1
1j
j¼1
i
Ã
Àc1j ðT 1 ÞxÃ1 ðT 1 À s1j ðT 1 ÞÞeÀc1j ðT 1 Þx ðT 1 Às1j ðT 1 ÞÞ j þ H1 ðT 1 Þjy1 ðT 1 À r1 ðT 1 ÞÞj ekT 1

¼ ðk À a1 ðT 1 ÞÞjy1 ðT 1 Þj þ b1 ðT 1 Þjy2 ðT 1 Þj þ

m
X

1
jy ðT 1 À s1j ðT 1 ÞÞjekðT 1 Às1j ðT 1 ÞÞ eks1j ðT 1 Þ
e2 1
"
#
m
X
À
Á
1
6 k À aÀ1 þ bþ1 þ
cþ1j 2 ekr1 þ Hþ1 ekr1 M:
e
j¼1

6 ðk À a1 ðT 1 ÞÞjy1 ðT 1 ÞjekT 1 þ b1 ðT 1 Þjy2 ðT 1 ÞjekT 1 þ

c1j ðT 1 Þ

j¼1

þH1 ðT 1 Þjy1 ðT 1 À r1 ðT 1 ÞÞjekðT 1 Àr1 ðT 1 ÞÞ ekr1 ðT 1 Þ

ð3:19Þ

Thus,
m
X
À
Á
1
0 6 k À aÀ1 þ bþ1 þ
cþ1j 2 ekr1 þ Hþ1 ekr1 ;
e
j¼1

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 ðV 2 ðT 2 Þ À MÞ ¼ Dþ ðV 2 ðT 2 ÞÞ 6 ½ðk À a2 ðT 2 ÞÞjy2 ðT 2 Þj þ b2 ðT 2 Þjy1 ðT 2 Þj
þ
"

m
X

Àc2j ðT 2 Þx2 ðT 2 Às2j ðT 2 ÞÞ

c2j ðT 2 Þjx2 ðT 2 À s2j ðT 2 ÞÞe

À

xÃ2 ðT 2

Àc2j ðT 2 Þxà ðT 2 Às2j ðT 2 ÞÞ

À s2j ðT 2 ÞÞe

#
j þ H2 ðT 2 Þjy2 ðT 2 À r2 ðT 2 ÞÞj ekT 2

j¼1

m
X
c2j ðT 2 Þ
jc2j ðT 2 Þx2 ðT 2 À s2j ðT 2 ÞÞeÀc2j ðT 2 Þx2 ðT 2 Às2j ðT 2 ÞÞ
c
ðT
Þ
2
2j
j¼1
i
Ã
Àc2j ðT 2 ÞxÃ2 ðT 2 À s2j ðT 2 ÞÞeÀc2j ðT 2 Þx2 ðT 2 Às2j ðT 2 ÞÞ j þ H2 ðT 2 Þjy2 ðT 2 À r2 ðT 2 ÞÞj ekT 2

¼ ðk À a2 ðT 2 ÞÞjy2 ðT 2 Þj þ b2 ðT 2 Þjy1 ðT 2 Þj þ

þH2 ðT 2 Þjy2 ðT 2 À r2 ðT 2 ÞÞjekðT 2 Àr2 ðT 2 ÞÞ ekr2 ðT 2 Þ
Thus,

m
X

1
jy2 ðT 2 À s2j ðT 2 ÞÞjekðT 2 Às2j ðT 2 ÞÞ eks2j ðT 2 Þ
2
e
j¼1
"
#
m
X
À
Á
1
6 k À aÀ2 þ bþ2 þ
cþ2j 2 ekr2 þ Hþ2 ekr2 M:
e
j¼1

6 ðk À a2 ðT 2 ÞÞjy2 ðT 2 ÞjekT 2 þ b2 ðT 2 Þjy1 ðT 2 ÞjekT 2 þ

c2j ðT 2 Þ

ð3:20Þ


X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

3297

m
X

À
Á
1
0 6 k À aÀ2 þ bþ2 þ
cþ2j 2 ekr2 þ Hþ2 ekr2 ;
e
j¼1

which contradicts with (3.12). Hence, (3.16) holds. It follows that

jyi ðtÞj < MeÀkt

for all t > t 0 ; i ¼ 1; 2:

ð3:21Þ

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:



À
Á
2
À 18 þ cos2 t x1 ðtÞ þ 0:00001 þ 0:000005 sin t eeÀ3 x2 ðtÞ

pffiffiffi  À

Á
2j sin tj
Þ;
þ eeÀ1 9:5 þ 0:005j sin 2tj x1 t À e2j sin tj eÀx1 ðtÀe
pffi




pffiffi
pffiffiffi
2j cos 3tj
Þ
þ eeÀ1 9:5 þ 0:005j sin 5tj x1 t À e2j cos 3tj eÀx1 ðtÀe

pffiffi 
À ð0:000001 cos2 tÞeeÀ3 x1 t À e2j cos 3tj


À
Á
2
>
>
x02 ðtÞ ¼ À 18 þ sin t x2 ðtÞ þ 0:00001 þ 0:000005 cos2 t eeÀ3 x1 ðtÞ
>
>
>
>


>
pffiffiffi  À
Á
2j cos tj
>
Þ;
>
þ eeÀ1 9:5 þ 0:005j cos 2tj x2 t À e2j cos tj eÀx2 ðtÀe
>
>
>
>
pffi

 
>
pffiffi 
p
ffiffiffi
>
2j cos 7tj
>
Þ
>
þ eeÀ1 9:5 þ 0:005j sin 6tj x2 t À e2j cos 7tj eÀx2 ðtÀe
>
>
>
>



pffiffi
>
À
Á
>
:
À 0:000001 cos4 t eeÀ3 x2 t À e2j cos 3tj ;
8
>
x01 ðtÞ ¼
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>

>
>
>
<

ð4:1Þ

À
þ
À
eÀ3
eÀ3
eÀ1
eÀ1
Obviously,
aÀi ¼ n18; aþi ¼ 19;
; bþ
; cÀ
; cþ
; Hþ
n coij ¼ co
i ¼
i ¼ 0:000015e
ij ¼ 1; bi ¼ 0:00001e
ij ¼ 9:5e
ij ¼ 9:505e
þ
2
;
¼

e
0:000001eeÀ3 ; r i ¼ max max16j6m sþ
r
;
ij
i
þ

bÀi

þ þ

ai

þ
i

a

j¼1

bþi e

a

2 c À e1Àcij e
X
ij

À

i

þ

À

Hþi e
þ
i

a

¼

19 þ 0:00001eeÀ3 À 0:000001eeÀ2
> 1;
19

2
2 cþ
X
cþij 1 bþi e X
0:000015eeÀ2 þ 19:01eeÀ2
ij 1
¼ À þ
¼
À À
À
e

e
a
c
a
a
18
i ij
i
i
j¼1
j¼1

ð4:2Þ

ð4:3Þ

and

(
max

bþ1

aÀ1

þ

m
X
cþ1j

j¼1

m
cþ2j
Hþ1 bþ2 X
Hþ2
þ
;
þ
þ
aÀ1 e2 aÀ1 aÀ2 j¼1 aÀ2 e2 aÀ2

)
¼

0:000016eeÀ2 þ 19:01eeÀ2
< 1:
18e

ð4:4Þ

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, from Theorems 3.1 and 3.2, system (4.1) has a positive almost periodic solution

xà ðtÞ 2 Bà ¼ fuju 2 B; 1 6 ui ðtÞ 6 e;

for all t 2 R; i ¼ 1; 2g:

0

Moreover, if u 2 C = {uju 2 C, 1 < ui(t) < e, for all t 2 [Àe2, 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 value u 2 C0 converge 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 of Theorems
3.2 with 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.


3298

X. Liu, J. Meng / Applied Mathematical Modelling 36 (2012) 3289–3298

References
[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.