Permanence and periodic solutions for a class of delay Nicholson’s blowflies models

Applied Mathematical Modelling 37 (2013) 1537–1544

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

Permanence and periodic solutions for a class of delay Nicholson’s blowflies models Xinhua Hou a, Lian Duan b,⇑, Zuda Huang c a

Hunan Industry Polytechnic, Changsha, Hunan 410208, PR China College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, PR China c College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, Hunan 415000, PR China b

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 26 October 2011 Received in revised form 6 March 2012 Accepted 10 April 2012 Available online 24 April 2012 Keywords: Nicholson’s blowflies model Permanence Positive periodic solution Coincidence degree Nonlinear density-dependent mortality term

This paper covers the dynamic behaviors for a class of Nicholson’s blowflies models with a nonlinear density-dependent mortality term. By using inequality analyze technique and coincidence degree theory, some sufficient conditions are determined that guarantee the permanence of the model and the existence of positive periodic solutions. Moreover, we give an example to illustrate our main results. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction The following nonlinear delay differential equation

N0 ðtÞ ¼ dðtÞNðtÞ þ

m X cj ðtÞNðt  sj ðtÞÞecj ðtÞNðtsj ðtÞÞ

ð1:1Þ

j¼1

has been introduced by Gurney et al. [1] to describe the population of the Australian sheep-blowfly and to agree with the experimental data obtained in [2]. Its dynamics was later studied in [3–8] and the references cited therein. In particular, Chen [7] and Li and Du [8] established some sufficient conditions on the existence of positive periodic solutions for the model (1.1). Recently, as pointed out in Berezansky et al. [9], a new study indicates that a linear model of density-dependent mortality will be most accurate for populations at low densities, and marine ecologists are currently in the process of constructing new fishery models with nonlinear density-dependent mortality rates. Therefore, Berezansky et al. [9] proposed an open problem: reveal the dynamic behaviors of the Nicholson’s blowflies model with a nonlinear density-dependent mortality term as follows:

N0 ðtÞ ¼ DðNÞ þ PNðt  sÞeNðtsÞ ;

⇑ Corresponding author. Tel.:/fax: +86 057383643075. E-mail addresses: [email protected] (X. Hou), [email protected] (L. Duan), [email protected] (Z. Huang). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.04.018

ð1:2Þ

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X. Hou et al. / Applied Mathematical Modelling 37 (2013) 1537–1544 N

where P is a positive constant and function D might have one of the following forms: DðNÞ ¼ aN=ðN þ bÞ or DðNÞ ¼ a  be

.

N

Furthermore, Wang [10] studied the existence of positive periodic solutions for the model (1.2) with DðNÞ ¼ a  be . However, to the best of our knowledge, few authors have considered the dynamics of Nicholson’s blowflies model (1.2) with DðNÞ ¼ aN=ðN þ bÞ. Thus, it is worthwhile to continue to investigate the dynamics of (1.2) in this case. Since the coefficients and delays in differential equations of population and ecology problems are usually time-varying in the real world, the model (1.2) can be naturally extended to the following nonlinear delay differential equation:

N0 ðtÞ ¼ 

m X aðtÞNðtÞ þ cj ðtÞNðt  sj ðtÞÞecj ðtÞNðtsj ðtÞÞ ; bðtÞ þ NðtÞ j¼1

ð1:3Þ

where aðtÞ; bðtÞ; cj ðtÞ and cj ðtÞ are all continuous functions bounded above and below by positive constants, and sj ðtÞ P 0 are bounded continuous functions, j ¼ 1; 2; . . . ; m. It is obvious that when DðNÞ ¼ aN=ðN þ bÞ, (1.2) is a special case of (1.3). The main purpose of this paper is to give the conditions ensuring the permanence and existence of positive periodic solutions for Nicholson’s blowflies model (1.3). For convenience, we introduce some notations. In the following part of this paper, given a bounded continuous function g defined on R, let g þ and g  be defined as

g þ ¼ sup gðtÞ;

g  ¼ inf gðtÞ: t2R

t2R

It will be assumed that

r ¼ max sþj ; 16j6m

cs ¼ max cþj ; ci ¼ min cj : 16j6m

ð1:4Þ

16j6m

Throughout this paper, let Rþ denote nonnegative real number space, C ¼ Cð½r; 0; RÞ be the continuous functions space equipped with the usual supremun norm jj  jj, and let C þ ¼ Cð½r; 0; Rþ Þ. If xðtÞ is continuous and defined on ½r þ t 0 ; rÞ with t0 ; r 2 R, then we define xt 2 C where xt ðhÞ ¼ xðt þ hÞ for all h 2 ½r; 0. It is biologically reasonable to assume that only positive solutions of model (1.3) are meaningful and therefore admissible. Much can be learned by considering admissible initial conditions

xt 0 ¼ u ;

u 2 C þ and uð0Þ > 0:

ð1:5Þ

We write xt ðt0 ; uÞðxðt; t0 ; uÞÞ for an admissible solution of the admissible initial value problem (1.3) and (1.5). Also, let ½t0 ; gðuÞÞ be the maximal right-interval of existence of xt ðt 0 ; uÞ. The remaining part of this paper is organized as follows. In Section 2, we shall derive the sufficient conditions for checking the permanence of the model (1.3). In Section 3, we shall present some sufficient conditions for ensuring the existence of the positive periodic solutions of the model (1.3). In Section 4, we shall give an example and a remark to illustrate our results obtained in the previous sections.

2. The permanence Definition 2.1. The model (1.3) with initial conditions (1.5) is said to be permanent, if there are two positive constants k and K such that

k 6 lim inf xðt; t0 ; uÞ 6 lim sup xðt; t0 ; uÞ 6 K: t!þ1

t!þ1

Lemma 2.1. Let sup

Pm

t2R

cj ðtÞ j¼1 aðtÞcj ðtÞe

< 1. Then, the solution xt ðt 0 ; uÞ 2 C þ for all t 2 ½t0 ; gðuÞÞ, the set of fxt ðt 0 ; uÞ : t 2 ½t0 ; gðuÞÞg is

bounded, and gðuÞ ¼ þ1. Moreover, xðt; t 0 ; uÞ > 0 for all t P t0 . Proof. Since u 2 C þ , using Theorem 5.2.1 in [11, p. 81], we have xt ðt0 ; uÞ 2 C þ for all t 2 ½t 0 ; gðuÞÞ. Let xðtÞ ¼ xðt; t0 ; uÞ. From (1.3) and the fact that

x0 ðtÞ ¼ 

aðtÞx bðtÞþx

6 aðtÞx for all t 2 R; x P 0, we get bðtÞ

m m X X aðtÞxðtÞ aðtÞ cj ðtÞxðt  sj ðtÞÞecj ðtÞxðtsj ðtÞÞ P  cj ðtÞxðt  sj ðtÞÞecj ðtÞxðtsj ðtÞÞ : þ xðtÞ þ bðtÞ þ xðtÞ j¼1 bðtÞ j¼1

ð2:1Þ

In view of xðt 0 Þ ¼ uð0Þ > 0, integrating (2.1) from t 0 to t, we have

xðtÞ P e



Rt

aðuÞ du t0 bðuÞ



xðt 0 Þ þ e

Rt

aðuÞ du t0 bðuÞ

Z

t

t0

for all t 2 ½t 0 ; gðuÞÞ.

R s aðv Þ X m dv e t0 bðv Þ cj ðsÞxðs  sj ðsÞÞecj ðsÞxðssj ðsÞÞ ds > 0 j¼1

ð2:2Þ

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X. Hou et al. / Applied Mathematical Modelling 37 (2013) 1537–1544

For each t 2 ½t 0  r; gðuÞÞ, we define

  MðtÞ ¼ max n : n 6 t; xðnÞ ¼ max xðsÞ : t 0 r6s6t

We now show that xðtÞ is bounded on ½t 0 ; gðuÞÞ. In the contrary case, observe that MðtÞ ! gðuÞ as t ! gðuÞ, we have

lim xðMðtÞÞ ¼ þ1:

ð2:3Þ

t!gðuÞ

But xðMðtÞÞ ¼ max xðsÞ, and so x0 ðMðtÞÞ P 0 for all MðtÞ > t 0 . Thus, t 0 r6s6t

0

0 6 x ðMðtÞÞ ¼ 

m X aðMðtÞÞxðMðtÞÞ cj ðMðtÞÞxðMðtÞ  sj ðMðtÞÞÞecj ðMðtÞÞxðMðtÞsj ðMðtÞÞÞ þ bðMðtÞÞ þ xðMðtÞÞ j¼1

and consequently, m X aðMðtÞÞxðMðtÞÞ cj ðMðtÞÞxðMðtÞ  sj ðMðtÞÞÞecj ðMðtÞÞxðMðtÞsj ðMðtÞÞÞ ; 6 bðMðtÞÞ þ xðMðtÞÞ j¼1

ð2:4Þ

where MðtÞ > t 0 . Let fT n gþ1 n¼1 be a sequence such that

lim T n ¼ gðuÞ;

n!þ1

lim xðMðT n ÞÞ ¼ þ1:

ð2:5Þ

n!þ1

1 In view of (2.4) and the fact that sup uecj ðtÞu ¼ c ðtÞe , we get j

uP0

m m X X aðMðT n ÞÞxðMðT n ÞÞ 1 cj ðMðT n ÞÞxðMðT n Þ  sj ðMðT n ÞÞÞecj ðMðT n ÞÞxðMðT n Þsj ðMðT n ÞÞÞ 6 cj ðMðT n ÞÞ 6 bðMðT n ÞÞ þ xðMðT n ÞÞ j¼1 c ðMðT n ÞÞe j j¼1

and m m X X xðMðT n ÞÞ cj ðMðT n ÞÞ cj ðtÞ 6 6 sup : bðMðT n ÞÞ þ xðMðT n ÞÞ j¼1 aðMðT n ÞÞcj ðMðT n ÞÞe t2R j¼1 aðtÞcj ðtÞe

ð2:6Þ

Letting n ! þ1, (2.5) and (2.6) imply that

1 6 sup

m X

t2R j¼1

cj ðtÞ ; aðtÞcj ðtÞe

which contradicts the fact that sup t2R

Pm

cj ðtÞ j¼1 aðtÞcj ðtÞe

< 1. This implies that xðtÞ is bounded on ½t0 ; gðuÞÞ. From Theorem 2.3.1 in

[12], we easily obtain gðuÞ ¼ þ1. This ends the proof of Lemma 2.1. h Theorem 2.1. Let

sup

m X

t2R j¼1

m X cj ðtÞ cj ðtÞbðtÞ > 1: < 1 and inf t2R aðtÞcj ðtÞe aðtÞ j¼1

ð2:7Þ

Then, the model (1.3) with initial conditions (1.5) is permanent. Proof. Let xðtÞ ¼ xðt; t 0 ; uÞ. From Lemma 2.1, the set of fxt ðt0 ; uÞ : t 2 ½t 0 ; þ1Þg is bounded, and there exists a positive constant K such that

0 < xðtÞ 6 K

for all t > t0 :

ð2:8Þ

It follows that

lim sup xðtÞ 6 K:

ð2:9Þ

t!þ1

We next prove that there exists a positive constant k such that

lim inf xðtÞ P k:

ð2:10Þ

t!þ1

Suppose, for the sake of contradiction, lim inf xðtÞ ¼ 0. For each t P t 0 , we define t!þ1

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X. Hou et al. / Applied Mathematical Modelling 37 (2013) 1537–1544

  mðtÞ ¼ max n : n 6 t; xðnÞ ¼ min xðsÞ : t 0 6s6t

Observe that mðtÞ ! þ1 as t ! þ1 and that

lim xðmðtÞÞ ¼ 0:

ð2:11Þ

t!þ1

However, xðmðtÞÞ ¼ min xðsÞ, and so x0 ðmðtÞÞ 6 0 for all mðtÞ > t0 . According to (1.3), we have t 0 6s6t

0 P x0 ðmðtÞÞ ¼  P

m X aðmðtÞÞxðmðtÞÞ þ cj ðmðtÞÞxðmðtÞ  sj ðmðtÞÞÞecj ðmðtÞÞxðmðtÞsj ðmðtÞÞÞ bðmðtÞÞ þ xðmðtÞÞ j¼1

m aðmðtÞÞxðmðtÞÞ X cj ðmðtÞÞxðmðtÞ  sj ðmðtÞÞÞecj ðmðtÞÞxðmðtÞsj ðmðtÞÞÞ þ bðmðtÞÞÞ j¼1

and consequently, m X aðmðtÞÞxðmðtÞÞ cj ðmðtÞÞxðmðtÞ  sj ðmðtÞÞÞecj ðmðtÞÞxðmðtÞsj ðmðtÞÞÞ P bðmðtÞÞÞ j¼1

P cj ðmðtÞÞxðmðtÞ  sj ðmðtÞÞÞecj ðmðtÞÞxðmðtÞsj ðmðtÞÞÞ ;

ð2:12Þ

where mðtÞ > t0 ; j ¼ 1; 2; . . . ; m. This, together with (2.11) implies that

lim xðmðtÞ  sj ðmðtÞÞÞ ¼ 0;

j ¼ 1; 2; . . . ; m:

t!þ1

ð2:13Þ

Noting that the continuities and boundedness of the functions aðtÞ; bðtÞ and cj ðtÞ, we can select a sequence ftn gþ1 n¼1 such that

lim t n ¼ þ1;

n!þ1

lim xðmðtn ÞÞ ¼ 0;

n!þ1

lim

n!þ1

cj ðmðtn ÞÞbðmðt n ÞÞ ¼ aj ; aðmðtn ÞÞ

j ¼ 1; 2; . . . ; m:

ð2:14Þ

In view of (2.12), we get m m X X aðmðtn ÞÞ xðmðt n Þ  sj ðmðt n ÞÞÞecj ðmðtn ÞÞxðmðtn Þsj ðmðtn ÞÞÞ xðmðt n Þ  sj ðmðtn ÞÞÞec xðmðtn Þsj ðmðtn ÞÞÞ cj ðmðtn ÞÞ cj ðmðt n ÞÞ P P bðmðtn ÞÞ xðmðtn ÞÞ xðmðt n Þ  sj ðmðt n ÞÞÞ j¼1 j¼1 s

¼

m X s cj ðmðt n ÞÞec xðmðtn Þsj ðmðtn ÞÞÞ j¼1

and

1P

m X cj ðmðtn ÞÞbðmðt n ÞÞ

aðmðtn ÞÞ

j¼1

s

ec xðmðtn Þsj ðmðtn ÞÞÞ :

ð2:15Þ

Letting n ! þ1, (2.13)–(2.15) imply that

1P

m X

m m X X cj ðmðt n ÞÞbðmðt n ÞÞ cj ðmðt n ÞÞbðmðt n ÞÞ cj ðtÞbðtÞ s ; lim ec xðmðtn Þsj ðmðtn ÞÞÞ ¼ lim P inf n!þ1 n!þ1 n!þ1 t2R aðmðt aðmðt aðtÞ ÞÞ ÞÞ n n j¼1 j¼1 j¼1

lim

which contradicts to (2.7). Hence, (2.9) holds. This completes the proof of Theorem 2.1. h

3. Existence of positive periodic solutions In this section, we shall study the existence of positive periodic solution of system (1.3). The method to be used in this paper involves the applications of the continuation theorem of the coincidence degree. We introduce some concepts and results concerning the coincidence degree as follows. Let X and Z be real Banach spaces, L : DomL  X ! Z be a linear mapping, and N : X ! Z be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if dimKerL ¼ CodimImL < þ1 and ImL is closed in Z. If L is a Fredholm mapping of index zero, there exist continuous projectors P : X ! X, and Q : Z ! Z such that ImP ¼ KerL; KerQ ¼ ImL ¼ ImðI  Q Þ. Lemma 3.1 (Continuation Theorem [13] ). Let X and Z be two Banach spaces. Suppose that L : DðLÞ  X ! Z is a Fredholm operator with index zero and N : X ! Z is L-compact on X, where X is an open subset of X. Moreover, assume that all the following conditions are satisfied: (1) Lx – kNx, for all x 2 @ X \ DðLÞ; k 2 ð0; 1Þ; (2) Nx R ImL, for all x 2 @ X \ KerL;

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X. Hou et al. / Applied Mathematical Modelling 37 (2013) 1537–1544

(3) The Brouwer degree

degfQN; X \ KerL; 0g – 0: Then equation Lx ¼ Nx has at least one solution in domL \ X. Theorem 3.1. Suppose that (2.7) holds, and a; b; cj ; cj ; sj are T-periodic functions for j ¼ 1; 2; . . . ; m. Set

A¼2

Z

T

0

aðtÞ dt; bðtÞ

B¼

m Z X

T

cj ðtÞdt;

ln

0

j¼1

2B > A: A

ð3:1Þ

Then (1.3) has a positive T-periodic solution. Proof. Set NðtÞ ¼ exðtÞ , then (1.3) can be rewritten as

x0 ðtÞ ¼ 

m X xðtsj ðtÞÞ aðtÞ þ cj ðtÞexðtsj ðtÞÞxðtÞcj ðtÞe :¼ Dðx; tÞ: xðtÞ bðtÞ þ e j¼1

ð3:2Þ

Then, to prove Theorem 3.1, it suffices to show that Eq. (3.1) has at least one T-periodic solution. Let X ¼ Z ¼ fx 2 CðR; RÞ : xðt þ TÞ ¼ xðtÞ for all t 2 Rg be Banach spaces equipped with the norm jj  jj, where jjxjj ¼ max jxðtÞj. t2½0;T

For any x 2 X, because of periodicity, it is easy to see that Dðx; Þ 2 CðR; RÞ is T-periodic. Let

L : DðLÞ ¼ fx 2 X : x 2 C 1 ðR; RÞg 3 x # x0 2 Z; 1 T

P:X 3x#

1 T

Q :Z3z#

Z

T

xðsÞds 2 X; 0

Z

T

zðsÞds 2 Z;

0

N : X 3 x # Dðx; Þ 2 Z: RT It is easy to see that ImL ¼ fxjx 2 Z; 0 xðsÞds ¼ 0g; KerL ¼ R; ImP ¼ KerL and KerQ ¼ ImL. Thus, the operator L is a Fredholm operator with index zero. Furthermore, denoting by L1 P : ImL ! DðLÞ \ KerP the inverse of LjDðLÞ\KerP , we have

L1 P yðtÞ ¼ 

1 T

Z

T

Z

0

t

yðsÞds dt þ

Z

0

t

yðsÞds:

ð3:3Þ

0

To apply Lemma 3.1, we first claim that N is L-compact on X, where X is a bounded open subset of X. From (3.3), it follows that

QNx ¼

1 T

Z

T

NxðtÞdt ¼

0

L1 P ðI  Q ÞNx ¼

Z

1 T

Z

T

" 

0

t

NxðsÞds 

0

t T

# m X xðtsj ðtÞÞ aðtÞ xðtsj ðtÞÞxðtÞcj ðtÞe dt; þ c ðtÞe j bðtÞ þ exðtÞ j¼1

Z 0

T

NxðsÞds 

1 T

Z 0

T

Z

t

NxðsÞds dt þ

0

1 T

Z 0

T

Z

ð3:4Þ

t

QNxðsÞds dt:

ð3:5Þ

0

1 Obviously, QN and L1 P ðI  Q ÞN are continuous. It is not difficult to show that LP ðI  Q ÞNðXÞ is compact for any open bounded set X  X by using the Arzela-Ascoli theorem. Moreover, QNðXÞ is clearly bounded. Thus, N is L-compact on X with any open bounded set X  X. Considering the operator equation Lx ¼ kNx; k 2 ð0; 1Þ, we have

x0 ðtÞ ¼ kDðx; tÞ:

ð3:6Þ

Assume that x 2 X is a solution of (3.6) for some k 2 ð0; 1Þ. Then

Z 0

T

  Z TX Z T  X m m xðtsj ðtÞÞ  xðtsj ðtÞÞ aðtÞ  cj ðtÞexðtsj ðtÞÞxðtÞcj ðtÞe dt ¼ dt dt ¼  cj ðtÞexðtsj ðtÞÞxðtÞcj ðtÞe xðtÞ   j¼1 0 j¼1 0 bðtÞ þ e  Z T Z T  aðtÞ  aðtÞ   ¼ dt: bðtÞ þ exðtÞ dt < 0 0 bðtÞ

It follows from (3.6) and (3.7) that

ð3:7Þ

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X. Hou et al. / Applied Mathematical Modelling 37 (2013) 1537–1544

Z

T

jx0 ðtÞjdt 6 k

Z

0

   Z T Z T  X m  aðtÞ  xðtsj ðtÞÞ  aðtÞ  xðtsj ðtÞÞxðtÞcj ðtÞe  dt < 2 dt þ k dt ¼ A: c ðtÞe   j   xðtÞ   j¼1 bðtÞ þ e 0 0 bðtÞ

T

0

ð3:8Þ

Since x 2 X, there exist n; g 2 ½0; T such that

xðgÞ ¼ max xðtÞ;

xðnÞ ¼ min xðtÞ; t2½0;T

t2½0;T

and x0 ðnÞ ¼ x0 ðgÞ ¼ 0:

ð3:9Þ

It follows from (3.7) and (3.8) that

A ¼ 2

Z

T

Z

aðtÞ dt > bðtÞ

0

T

0

s xðgÞ

¼ BexðnÞxðgÞc e

aðtÞ dt ¼ bðtÞ þ exðtÞ

Z 0

T

m m Z X X xðtsj ðtÞÞ s xðgÞ cj ðtÞexðtsj ðtÞÞxðtÞcj ðtÞe dt P exðnÞxðgÞc e j¼1

j¼1

T

cj ðtÞdt

0

;

which implies that

xðnÞ < ln

A þ xðgÞ þ cs exðgÞ : 2B

Using (3.8) yields

xðtÞ 6 xðnÞ þ

Z

T

jx0 ðtÞjdt < ln 0

A þ xðgÞ þ cs exðgÞ þ A: 2B

In particular,

xðgÞ < xðnÞ þ

Z

T

jx0 ðtÞjdt < ln

0

A þ xðgÞ þ cs exðgÞ þ A: 2B

It follows that

xðgÞ > ln



  2B ln  A : A cs 1

Again from (3.8), we have

xðtÞ P xðgÞ 

Z

T

jx0 ðtÞjdt > ln



0

  2B A  A :¼ H1 : ln A c 1

ð3:10Þ

s

Since x0 ðnÞ ¼ 0, from (3.6), we obtain m X xðnsj ðnÞÞ aðnÞ ¼ cj ðnÞexðnsj ðnÞÞxðnÞcj ðnÞe : xðnÞ bðnÞ þ e j¼1

ð3:11Þ

Hence, from (3.11) and the fact that sup ueu ¼ 1e, we have uP0

m m X X xðns ðnÞÞ exðnÞ exðnÞ cj ðnÞ cj ðtÞ 6 ¼ cj ðnÞexðnsj ðnÞÞ ecj ðnÞe j 6 sup : þ xðnÞ xðnÞ aðnÞ aðtÞ bðnÞ þ e c ðnÞ cj ðtÞe b þe t2R j j¼1 j¼1

Noting that

u bþ þu

ð3:12Þ

is strictly monotone increasing on ½0; þ1Þ and

m X u cj ðtÞ ; ¼ 1 > sup t2R j¼1 aðtÞcj ðtÞe uP0 b þ u

sup

þ



it is clear that there exists a constant k > 0 such that m X u cj ðtÞ > sup b þu t2R j¼1 aðtÞcj ðtÞe



for all u 2 ½k ; þ1Þ:

þ

ð3:13Þ

In view of (3.12) and (3.13), we get 

exðnÞ 6 k



and xðnÞ 6 ln k :

ð3:14Þ 

Then, we can choose a sufficiently large positive constant H2 > ln k such that

xðtÞ < H2

and

þ

ln b < H2 :

Let H > maxfjH1 j; H2 g be a fix constant such that

ð3:15Þ

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X. Hou et al. / Applied Mathematical Modelling 37 (2013) 1537–1544

eH >

  Z T 2B with H þ ln K ¼ aðtÞdt K ci 0 1

and define X ¼ fx 2 X : jjxjj < Hg. Then (3.9) and (3.14) imply that there is no k 2 ð0; 1Þ and x 2 @ X such that Lx ¼ kNx. When x 2 @ X \ KerL ¼ @ X \ R; x ¼ H. Then

QNðHÞ > 0 and QNðHÞ < 0:

ð3:16Þ

Otherwise, if QNðHÞ 6 0, it follows from (3.4) that

A ¼ 2

Z

T

aðtÞ dt > bðtÞ

0

Z

T 0

aðtÞ dt P bðtÞ þ eH

Z

T

m m Z X X H s H cj ðtÞecj ðtÞe dt P ec e

0

j¼1

j¼1

T

s H

cj ðtÞdt ¼ Bec e ;

0

which implies

 H P ln

1

cs

ln

    2B 1 2B > ln s ln A  A ¼ H1 : A A c

This is a contradiction and implies that QNðHÞ > 0. If QNðHÞ P 0, it follows from (3.4) that

K H e ¼ 2

Z 0

T

aðtÞ dt < 2eH

Z 0

T

aðtÞ dt 6 bðtÞ þ eH

Z 0

T

m X

H

i H

cj ðtÞecj ðtÞe dt 6 ec e

j¼1

m Z X j¼1

T

i H

cj ðtÞdt ¼ Bec e :

0

Consequently,

eH <

  2B ; H þ ln ci K 1

a contradiction to the choice of H. Thus, QNðHÞ < 0. Furthermore, define continuous function Hðx; lÞ by setting

1 Hðx; lÞ ¼ ð1  lÞx þ l T

Z 0

T

"

# m X aðtÞ cj ðtÞx dt:  þ cj ðtÞe bðtÞ þ ex j¼1

It follows from (3.16) that xHðx; lÞ – 0 for all x 2 @ X \ kerL. Hence, using the homotopy invariance theorem, we obtain

# ( Z " ) m X 1 T aðtÞ cj ðtÞx dt; X \ kerL; 0 ¼ degfx; X \ kerL; 0g – 0: degfQN; X \ kerL; 0g ¼ deg  þ cj ðtÞe T 0 bðtÞ þ ex j¼1 In view of all the discussions above, we conclude from Lemma 3.1 that Theorem 3.1 is proved. h

4. An example In this section we present an example to illustrate our results. Example 4.1. Consider the delayed periodic Nicholson’s blowflies models with a nonlinear density-dependent mortality term:

    4pþj sin tj Nðte4pþsin t Þ ð2 þ sin tÞNðtÞ 1 e4p þ þ 1 ð4 þ cos tÞN t  e4pþsin t ee 2 þ sin t þ NðtÞ 2 4     4pþj cos tj Nðte4pþcos t Þ 1 e4p þ : þ 1 ð4 þ cos tÞN t  e4pþcos t ee 2 4

N0 ðtÞ ¼ 

ð4:1Þ

Then, Eq. (4.1) with initial conditions (1.5) is permanent, and there exists at least one positive 2p-periodic solution of (4.1). Proof. By (4.1), we have

aðtÞ ¼ bðtÞ ¼ 2 þ sin t;

c1 ðtÞ ¼ c2 ðtÞ ¼

 4p  e þ 1 ð4 þ cos tÞ; 4

c1 ðtÞ ¼ e4pþj sin tj ; c2 ðtÞ ¼ e4pþj cos tj ; s1 ðtÞ ¼ e4pþsin t ; s1 ðtÞ ¼ e4pþcos t ; then

1544

X. Hou et al. / Applied Mathematical Modelling 37 (2013) 1537–1544

A¼2

Z 2p aðtÞ dt ¼ 4p; bðtÞ 0

a1 ¼ a2 ¼ 1;

cþ1 ¼ cþ2 ¼

B¼

Z 2p

c1 ðtÞdt þ

0

Z 2p

c2 ðtÞdt ¼ 8p þ 2pe4p ;

0

  5 e4p þ1 ; 2 4

c1 ¼ c2 ¼ e4p ; r ¼ e4pþ1 :

Clearly,

ln

2B ¼ lnðe4p þ 4Þ > 4p ¼ A; A

inf t2R

2 X cj ðtÞbðtÞ j¼1

aðtÞ

 4p  e þ 1 ð4 þ cos tÞ > 1; t2R 4

¼ inf

and 2 X sup t2R j¼1

þ

2 X cj cj ðtÞ 5 5 þ 4pþ1 < 1; 6 ¼   aðtÞcj ðtÞe 4e e a c e j¼1 j

it means that conditions in Theorems 2.1 and 3.1 hold. Hence, Eq. (4.1) with initial conditions (1.5) is permanent. Moreover, there exists at least one positive 2p-periodic solution of (4.1). h Remark 4.1. Eq. (4.1) is a kind of delayed periodic Nicholson’s blowflies models with two time-varying delays and a nonlinear density-dependent mortality term, but as far as we know there are not results can be applicable to (4.1) to obtain the existence of 2p-periodic solutions. This implies that the results of this paper are essentially new.

Acknowledgments The authors would like to express their sincere appreciation to the anonymous referee for the valuable comments which have led to an improvement in the presentation of the paper. This work was supported by the Construct Program of the Key Discipline in Hunan Province (Mechanical Design and Theory), the Scientific Research Fund of Hunan Provincial Natural Science Foundation of PR China (Grant No. 11JJ6006), the Natural Scientific Research Fund of Hunan Provincial Education Department of PR China (Grants Nos. 11C0916, 11C0915, 11C1186), the Natural Scientific Research Fund of Zhejiang Provincial of PR China (Grants Nos. Y6110436, Y12A010059), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of PR China (Grant No. Z201122436). References [1] W.S.C. Gurney, S.P. Blythe, R.M. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980) 17–21. [2] A.J. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool. 2 (1954) 9–65. [3] K. Cook, P. van den Driessche, X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999) 332–352. [4] M.R.S. Kulenovic´, G. Ladas, Y. Sficas, Global attractivity in Nicholson’s blowflies, Appl. Anal. 43 (1992) 109–124. [5] T.S. Yi, X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case, J. Differ. Equat. 245 (11) (2008) 3376–3388. [6] X. Hou, L. Duan, New results on periodic solutions of delayed Nicholson’s blowflies models, Electron. J. Qual. Theory Differ. Equat. 24 (2012) 1–11. [7] Y. Chen, Periodic solutions of delayed periodic Nicholson’s blowflies models, Can. Appl. Math. Q. 11 (2003) 23–28. [8] J. Li, C. Du, Existence of positive periodic solutions for a generalized Nicholson’s blowflies model, J. Comput. Appl. Math. 221 (2008) 226–233. [9] L. Berezansky, E. Braverman, L. Idels, Nicholson’s blowflies differential equations revisited: main results and open problems, Appl. Math. Model. 34 (2010) 1405–1417. [10] W. Wang, Positive periodic solutions of delayed nicholsons blowflies models with a nonlinear density-dependent mortality term, Appl. Math. Model. (2011), http://dx.doi.org/10.1016/j.apm.2011.12.001. [11] H.L. Smith, Monotone Dynamical Systems, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 1995. [12] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [13] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin-Heidelberg-New York, 1977.