Applied Mathematics and Computation 232 (2014) 1262–1268

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Nonlinear analysis of a delayed stage-structured predator–preymodel with impulsive effect and environment pollution q

http://dx.doi.org/10.1016/j.amc.2014.01.0030096-3003/� 2014 Elsevier Inc. All rights reserved.

q This work is supported by the National Natural Science Foundation of China (No. 11371164), NSFC-Talent Training Fund of Henan (U13041Henan Science and Technology Department (122300410398 and 132300410084).⇑ Corresponding author at: Junior College, Zhejiang Wanli University, Ningbo 315100, Zhejiang, PR China.

E-mail addresses: zhaozhong8899@163.com (Z. Zhao), wxb2012@163.com, wuxb2012@163.com (X. Wu).

Zhong Zhao a, Xianbin Wu a,b,⇑a Department of Mathematics, Huanghuai University, Zhumadian 463000, Henan, PR Chinab Junior College, Zhejiang Wanli University, Ningbo 315100, Zhejiang, PR China

a r t i c l e i n f o a b s t r a c t

Keywords:Stage structureGlobal attractivityPermanence

In this paper, we investigate a stage-structured predator–prey model with impulsive poi-soning and environment pollution. Sufficient conditions which guaranteed the global attr-activity of predator-extinction periodic solution and permanence of the system areobtained. We believe that the results will provide reliable tactic basis for the practical pestmanagement.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Pests outbreak often cause serious ecological and economic problems. People adopt all kinds of measures to control thepests outbreak. One important method for pest control is chemical pesticides that kill the pest directly, usually by exposing itto lethal substances or unsuitable environmental conditions. It is a convenient and effective method of pest control.However, pesticide pollution is also recognized as a major health hazard to human beings and to natural enemies. Pesticideappears in environment first, then it is absorbed by organism, which leads to a large impact on the people health. Hence, inthis paper, we consider the above effects and introduce the pollution model to model the process of pest control problemsand study its dynamics, which is different from the previous pest control model which neglected the effect of the sprayingpesticides on the natural enemy [1,2].

On the other hand, in 1973 Ayala et al. [3] conducted experiments on fruit fly dynamics to test the validity of ten modelsof competitions. One of the models accounting best for the experimental results is given by

_x1 ¼ r1x1 1� x1K1

� �h1� a12

x2K2

� �;

_x2 ¼ r2x2 1� x2K2

� �h2� a21

x1K1

� �:

According to the above biological background, we consider the following model with age structure for predator:

04) and

Z. Zhao, X. Wu / Applied Mathematics and Computation 232 (2014) 1262–1268 1263

_x1ðtÞ ¼ x1ðtÞða� bxa1ðtÞÞ �

a1x1ðtÞx3ðtÞ1þAx1ðtÞ

;

_x2ðtÞ ¼ b1x1ðtÞx3ðtÞ1þAx1ðtÞ

� b2x2ðtÞ � e�b2s1 b1x1ðt�s1Þx3ðt�s1Þ1þAx1ðt�s1Þ

� r2cðtÞx2ðtÞ;

_x3ðtÞ ¼ e�b2s1 b1x1ðt�s1Þx3ðt�s1Þ1þAx1ðt�s1Þ

� dx3ðtÞ � r3cðtÞx3ðtÞ;

_cðtÞ ¼ �Qc;

9>>>>>>>=>>>>>>>;

t – ns;

x1ðnsþÞ ¼ ð1� hÞx1ðnsÞ;

x2ðnsþÞ ¼ x2ðnsÞ;

x3ðnsþÞ ¼ x3ðnsÞ;

Dc ¼ l;

9>>>>>=>>>>>;

t ¼ ns; n 2 N ¼ 1;2; . . . ;

ð/1ðnÞ;/2ðnÞ;/3ðnÞ;/4ðnÞÞ 2 Cð½�s1; 0�;R3þÞ; /ið0Þ > 0; i ¼ 1;2;3;

8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

ð1Þ

where x1ðtÞ denotes the density of prey, x2ðtÞ; x3ðtÞ represent the immature and mature predator densities respectively. cðtÞ isthe concentration of the toxicant in the environment at time t. l is the input concentration of the toxicant.s1; a; b;a; a1; b1; b2;A, and d are positive constants. s1 represents a constant time to maturity. a is the intrinsic growth rateof prey, b is the coefficient of intra-specific competition. a provides a nonlinear measures of intra-specific interference, b1

a1

is the rate of conversion of nutrients into the reproduction rate of the mature predator, b2 is the death rate of immature pred-ator and d is the death rate of mature predator. hð0 < h < 1Þ represents partial impulsive harvest to preys by catching orpesticides.

Noting that the variables x2ðtÞ do not appear in the first and third equations of system (1), hence we only need to considerthe subsystem of (1) as follows:

_x1ðtÞ ¼ x1ðtÞða� bxa1ðtÞÞ �

a1x1ðtÞx3ðtÞ1þAx1ðtÞ

;

_x3ðtÞ ¼ e�b2s1 b1x1ðt�s1Þx3ðt�s1Þ1þAx1ðt�s1Þ

� dx3ðtÞ � r3cðtÞx3ðtÞ;

_cðtÞ ¼ �Qc;

9>>>=>>>;

t – ns;

x1ðnsþÞ ¼ ð1� hÞx1ðnsÞ;

x3ðnsþÞ ¼ x3ðnsÞ;

Dc ¼ l;

9>>=>>; t ¼ ns;

n 2 N ¼ 1;2; . . . ;

ð/1ðnÞ;/3ðnÞ;/4ðnÞÞ 2 Cð½�s1;0�;R3þÞ; /ið0Þ > 0; i ¼ 1;3;4;

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

ð2Þ

the initial conditions for (1) are

ð/1ðnÞ;/3ðnÞ;/4ðnÞÞ 2 Cð½�s1;0�;R2þÞ;/ið0Þ > 0; i ¼ 1;3;4: ð3Þ

2. Some important lemmas

Before we have the main results, we need to give some lemmas which will be used for our proof.

Lemma 2.1. Let ð/1ðtÞ;/2ðtÞ;/3ðtÞ;/4ðtÞÞ > 0 for �s1 < t < 0. Then any solution of system (1) is strictly positive.The proof is similar to [4], hence we omit it.

Lemma 2.2. There exists a constant M > 0 such that x1ðtÞ 6 M; x2ðtÞ 6 M; x3ðtÞ 6 M for each solution ðx1ðtÞ; x2ðtÞ; x3ðtÞ; cðtÞÞ of(1) with all t large enough.

The proof is similar to [5], hence we omit it.

Lemma 2.3 [6]. Consider the following equation:

_xðtÞ ¼ a1xðt �xÞ � a2xðtÞ; ð4Þ

where a1; a2;x > 0 for �x 6 t 6 0. We have

(i) If a1 < a2, then limt!1xðtÞ ¼ 0,(ii) If a1 > a2, then limt!1xðtÞ ¼ 1.

1264 Z. Zhao, X. Wu / Applied Mathematics and Computation 232 (2014) 1262–1268

Finally, we give basic properties about the following subsystem of (2).

Lemma 2.4. Consider the following impulsive system:

_x1ðtÞ ¼ x1ðtÞða� bxa1ðtÞÞ; t – ns;

x1ðnsþÞ ¼ ð1� hÞx1ðnsÞ; t ¼ ns;

(ð5Þ

where a > 0; b > 0;a > 0;0 < h < 1. Then there exists a unique positive periodic solution x�1ðtÞ, which is globally asymptoticallystable.x�1ðtÞ ¼ b

aþ x�1�a � b

a

� �e�aaðt�nsÞ� �1

a, where x�1 ¼bð1�hÞ�að1�e�aasÞað1�ð1�hÞ�aÞe�aas

h i�1a.

Proof. From (5), we have

dðx�a1 eaatÞdt

¼ baeaat : ð6Þ

Integrating (6) on ðns; ðnþ 1Þs�, we have

x�a1 ðtÞ ¼ x�a

1 ðnsþÞe�aaðt�nsÞ þ bað1� e�aaðt�nsÞÞ;

from the second equation of (5), we obtain

x1 ðnþ 1Þsþð Þ ¼ ð1� hÞ½x�a1 nsþð Þe�aas� þ b

a1� e�aasð Þ

��1a

: ð7Þ � 1

Obviously, difference system (7) has an equilibrium x�1 ¼bð1�hÞ�a 1�e�aasð Þa 1�ð1�hÞ�að Þe�aas

�a

, which implies that system (5) has unique as�periodic solution

x�1ðtÞ ¼baþ x�1

�a � ba

� �e�aaðt�nsÞ

��1a

; ns < t 6 ðnþ 1Þs:

In the following, we will prove that x�1ðtÞ is globally asymptotically stable. By Lemma 2.3, we find any solution of system(3) is ultimately upper bounded, so we need only to prove that

limt!1jx1ðtÞ � x�1ðtÞj ¼ 0:

Since jx�a1 ðtÞ � x�1

�aðtÞj ¼ jx�a1 ð0Þ � x�1

�að0Þje�aat6 jx�a

1 ð0Þ � x�1�að0Þje�aaðn�1Þs, then limt!1e�aaðn�1Þs ¼ 0, thus limt!1jx�a

1 ðtÞ�x�1�aðtÞj ¼ limt!1j

x�1 tð Þa�xa1 tð Þ

x�1 tð Þaxa1 tð Þ j ¼ 0, and limt!1jx�1 tð Þa � xa

1 tð Þj ¼ 0, which implies limt!1jx�1 tð Þ � x1 tð Þj ¼ 0. h

Lemma 2.5 [7]. Consider the following impulsive system:

_cðtÞ ¼ �QcðtÞ; t – ns;Dc ¼ l; t ¼ ns:

�ð8Þ

System (8) has a positive periodic solution c�ðtÞ and for every solution cðtÞ of (8), jcðtÞ � c�ðtÞj ! 0 as t !1, wherec�ðtÞ ¼ le�Qðt�nsÞ

1�e�Qs .

By Lemmas 2.4 and 2.5, we can obtain system (2) has a predator-extinction periodic solution ðx�1ðtÞ;0; c�ðtÞÞ fort 2 ðns; ðnþ 1Þs�;n 2 N.

3. Global attractivity

Theorem 3.1. Let ðx1ðtÞ; x2ðtÞ; x3ðtÞ; cðtÞÞ be any solution of system (1). If b1e�b2s1 dð1þAdÞ

eQs�1dðeQs�1Þþr3l

< 1 holds, then the predator-

extinction periodic solution ðx�1ðtÞ;0;0; c�ðtÞÞ of system (1) is globally attractive, where d ¼ að1�ð1�hÞ�ae�aasÞbð1�e�aasÞ

h i1a.

Proof. It is clear that the global attraction of predator-extinction solution ðx�1ðtÞ;0;0; c�ðtÞÞ of system (1) is equivalent to theglobal attraction of predator-extinction solution ðx�1ðtÞ;0; c�ðtÞÞ of system (2). So we only devote to system (2).

Let ðx1ðtÞ; x3ðtÞ; cðtÞÞ be any solution of system (2) with initial condition (3). Since

b1e�b2s1 dð1þ AdÞ

eQs � 1dðeQs � 1Þ þ r3l

< 1;

we can choose e and e0 sufficiently small such that

Z. Zhao, X. Wu / Applied Mathematics and Computation 232 (2014) 1262–1268 1265

b1e�b2s1 ðdþ eÞð1þ Aðdþ eÞÞ < dþ r3l

eQs � 1� r3e0: ð9Þ

Note that dx1ðtÞdt 6 x1ðtÞða� bxa

1ðtÞÞ, x1ðtþÞ ¼ ð1� hÞx1ðtÞ for ns < t 6 ðnþ 1Þs, then we consider the following impulsive dif-ferential inequalities:

dx1ðtÞdt 6 x1ðtÞða� bxa

1ðtÞÞ; t – ns;x1ðtþÞ ¼ ð1� hÞx1ðtÞ; t ¼ ns;

(ð10Þ

by using the comparison theorem, we have

limt!1

sup x1ðtÞ 6að1� ð1� hÞ�ae�aasÞ

bð1� e�aasÞ

�1a

:

Hence, there exists a positive integer n1 and an arbitrarily small positive constant e > 0 such that for all t P n1s,

x1ðtÞ 6að1� ð1� hÞ�ae�aasÞ

bð1� e�aasÞ

�1a

þ e ¼: g: ð11Þ

From 11) and the second equation of (2), we get that, for t > n1sþ s1.

_x3ðtÞ 6 e�b2s1b1gx3ðt � s1Þ

1þ Ag� dx3ðtÞ � r3cðtÞx3ðtÞ:

According to Lemma 2.5, there exist t > n01sþ s1(n01 2 N) and e0 > 0 such that

leQs � 1

� e0 < c�ðtÞ � e0 6 cðtÞ: ð12Þ

Consider the following comparison equation

_zðtÞ ¼ e�b2s1b1gzðt � s1Þ

1þ Ag� dzðtÞ � r3

leQs � 1

� e0� �

zðtÞ: ð13Þ

According to Lemma 2.3 and (9), we obtain that

limt!1

zðtÞ ¼ 0:

Since x3ðtÞ ¼ zðsÞ ¼ /3ðsÞ > 0 for all s 2 ½�s1; 0Þ, by the comparison theorem in differential equation and the positivity ofsolution, we have that x3ðtÞ ! 0 as t !1.

Without loss of generality, we may assume that 0 < x3ðtÞ < e1 for all t P 0, by the first equation of (2), we have

x1ðtÞða� bxa1ðtÞÞ � a1x1ðtÞe1 6 _x1ðtÞ 6 x1ðtÞða� bxa

1ðtÞÞ: ð14Þ

Then we have z1ðtÞ 6 x1ðtÞ 6 z2ðtÞ and z1ðtÞ ! z�1ðtÞ; z2ðtÞ ! z�2ðtÞ, as t !1, while z�1ðtÞ and z�2ðtÞ are the periodic solutionsof

dz1ðtÞdt ¼ z1ðtÞða� bza

1ðtÞÞ � a1z1ðtÞe1; t – ns;z1ðtþÞ ¼ ð1� hÞz1ðtÞ; t ¼ nsz1ð0þÞ ¼ x1ð0þÞ

8><>: ð15Þ

and

dz2ðtÞdt ¼ z2ðtÞða� bza

2ðtÞÞ; t – ns;z2ðtþÞ ¼ ð1� hÞz2ðtÞ; t ¼ ns;z2ð0þÞ ¼ x1ð0þÞ:

8><>: ð16Þ

For ns 6 t 6 ðnþ 1Þs,

z�1ðtÞ ¼b

a� a1e1þ z�1

�a � ba� a1e1

� �e�aða�a1e1Þðt�nsÞ

��1a

;

where z�1 ¼bð1�hÞ�að1�eaða�a1e1 ÞsÞ

ða�a1e1Þð1�ð1�hÞ�aÞe�aða�a1e1Þs

h i�1a.

Therefore, for any e2 > 0, there exists an integer k4, n > k4 such that

z�1ðtÞ � e2 < x1ðtÞ < x�1ðtÞ þ e2:

Let e1 ! 0, so we have

x�1ðtÞ � e2 < x1ðtÞ < x�1ðtÞ þ e2

1266 Z. Zhao, X. Wu / Applied Mathematics and Computation 232 (2014) 1262–1268

for t large enough, which implies x1ðtÞ ! x�1ðtÞ as t !1. According to Lemma 2.5, we have cðtÞ ! c�ðtÞ as t !1. This com-pletes the proof. h

4. Permanence

Theorem 4.1. Suppose b1e�b2s1 x�1ð1þAx�1Þ

1�e�Qs

dð1�e�QsÞþr3l> 1, then there is a positive constant q such that each positive solution

ðx1ðtÞ; x3ðtÞ; cðtÞÞ of (2) satisfies x3ðtÞP q for t large enough, where x�1 ¼bð1�hÞ�að1�e�aasÞað1�ð1�hÞ�ae�aasÞ

h i�1a.

Proof. Suppose that ðx1ðtÞ; x3ðtÞ; cðtÞÞ is any positive solution of system (2) with initial conditions (3). The second equation ofsystem (2) may be rewritten as follows:

_x3ðtÞ ¼ b1e�b2s1x1ðtÞ

1þ Ax1ðtÞ� d� r3cðtÞ

�x3ðtÞ � b1e�b2s1

ddt

Z t

t�s1

x1ðhÞx3ðhÞ1þ Ax1ðhÞ

dh: ð17Þ

Define

VðtÞ ¼ x3ðtÞ þ b1e�b2s1

Z t

t�s1

x1ðhÞx3ðhÞ1þ Ax1ðhÞ

dh: ð18Þ

Calculating the derivative of VðtÞ along the solution of (2), it follows from (17) that

dVdt¼ ½b1e�b2s1

x1ðtÞ1þ Ax1ðtÞ

� d� r3cðtÞ�x3ðtÞ: ð19Þ

Since b1e�b2s1 x�1dð1þAx�1Þ

> 1 holds, then we choose two positive constants m�3; e3 small enough such that

b1e�b2s1rdð1þ ArÞ > 1; ð20Þ � 1

where r ¼ bð1�hÞ�að1�e�aða�a1m�

3Þs1 Þ

ða�a1m�3Þð1�ð1�hÞ�ae�aða�a1m�

3Þs1 Þ

�a

� e3, 0 < m�3 <b1d

e�b2s1 x�11þAx�1

� db1

� �.

For any positive constant t0, we claim the inequality x3ðtÞ 6 m�3 cannot hold for all t P t0. Otherwise, there is a positiveconstant t0 such that x3ðtÞ < m�3 for all t P t0. From the first and third equation of system (2), we have

dx1ðtÞdt P x1ðtÞða� bxa

1ðtÞ � a1m�3Þ; t – ns;x1ðtþÞ ¼ ð1� hÞx1ðtÞ; t ¼ ns:

(ð21Þ

Then we have z�3ðtÞ 6 x1ðtÞ, where z�3ðtÞ is a positive periodic solution of

dz3ðtÞdt ¼ z3ðtÞða� bza

3ðtÞ � a1m�3Þ; t – ns;z3ðtþÞ ¼ ð1� hÞz3ðtÞ; t ¼ ns:

(ð22Þ

From (22), we have

z�3ðtÞ ¼b

a� a1m�3þ z�3

�a � ba� a1m�3

� �e�aða�a1m�3Þðt�nsÞ

��1a

; ns < t 6 ðnþ 1Þs; ð23Þ

� 1

where z�3 ¼bð1�hÞ�að1�e

�aða�a1m�3ÞsÞ

ða�a1m�3Þð1�ð1�hÞ�ae�aða�a1m�

3ÞsÞ

�a

. By using comparison theorem of impulsive differential equation, there exists

T1ð> T1 þ s1Þ such that the following inequality holds for t P T 0,

x1ðtÞP z�3ðtÞ � e3: ð24Þ

Thus

x1ðtÞP z�3 � e3 ¼: r ð25Þ

for all t P T 0.From (19) and (25), we have

dVðtÞdt

Pb1e�b2s1r

1þ Ar� d� r3cðtÞ

�x3ðtÞ ð26Þ

for all t P T 0.Again according to Lemma 2.5, for any e03 > 0 small enough, there exists a positive constant T 00 such that

Z. Zhao, X. Wu / Applied Mathematics and Computation 232 (2014) 1262–1268 1267

cðtÞ 6 l1� e�Qs þ e03 ¼: M1; t > T 00: ð27Þ

Let t1 ¼maxfT 0; T 00g, hence, we have

dVðtÞdt

P ðdþ r3M1Þb1e�b2s1r

1þ Ar1

dþ r3M1� 1

�x3ðtÞ; t > t1: ð28Þ

Set

xl3 ¼ min

t2½t1 ;t1þs1 �x3ðtÞ;

we will show that x3ðtÞP xl3 for all t > t1. Suppose the contrary, there is a T0 > 0 such that x3ðtÞP xl

3 fort1 6 t 6 t1 þ T0 þ s1; x3ðt1 þ T0 þ s1Þ ¼ xl

3 and _x3ðt1 þ T0 þ s1Þ 6 0. However, the second equation of system (2) and (28)imply that

_x3ðt1 þ T0 þxÞP b1e�b2s1x1ðt1 þ T0Þx3ðt1 þ T0Þ

1þ Ax1ðt1 þ T0Þ� dx3ðt1 þ T0 þ s1Þ � r3M1x3ðt1 þ T0 þ s1Þ

Pb1e�b2s1r

1þ Ar� d� r3M1

� �xl

3 > 0; ð29Þ

which is a contradiction to _x3ðt1 þ T0 þ s1Þ 6 0. Thus x3ðtÞP xl3 for all t > t1. As a consequence (29) leads to

dVðtÞdt P b1e�b2s1 r

1þAr � d� r3M1

� �xl

3, for all t > t1, which implies t !1;VðtÞ ! 1 as t !1. This contradicts

VðtÞ 6 M 1þ b1e�b2s1 M2s11þAM

� �. Hence, for any t0 > 0, it is impossible that x3ðtÞ < m�3 for all t > t0.

Following, we are left to consider two cases:

(i) x3ðtÞP m�3 for all large t.(ii) x3ðtÞ oscillates about m�3 for all t large enough.

Define q ¼minfm�32 ; q1g and q1 ¼ m�3e�ðdþr3MÞs1 . We hope to show that x3ðtÞP q for all large t. The conclusion is evident in

(i). For (ii), let t� > 0 and n > 0 satisfy x3ðt�Þ ¼ x3ðt� þ nÞ ¼ m�3 and x3ðtÞ < m�3 for all t� < t < t� þ n, where t� is sufficientlylarge such that x1ðtÞP r for t� < t < t� þ n.

Since x3ðtÞ is uniformly continuous and bounded and x3ðtÞ is not effected by impulses. Hence, there is a T (0 < t < s1 and T

is dependent of the choice of t�) such that x3ðtÞ >m�32 for t� < t < t� þ T. If n < T , there is nothing to prove. Let us consider the

case T < n < s1. Since _x3ðtÞ > �dx3ðtÞ � r3Mx3ðtÞ and x3ðt�Þ ¼ m�3, it is clear that x3ðtÞP q1 for t 2 ½t�; t� þ s1�. Then,proceeding exactly as the proof for the above claim. We see that x3ðtÞP q1 for t 2 ½t� þ s1; t� þ n�. Because the kind of intervalt 2 ½t�; t� þ n� is chosen in an arbitrary way (we only need t� to be large). We concluded x3ðtÞP q for all large t in the secondcase. In view of our above discussion, the choice of q is independent of the positive solution, and we proved that any positivesolution of (2) satisfies x3ðtÞP q for all sufficiently large t. This completes the proof of the theorem. h

Theorem 4.2. Supposeb1e�b2s1 x�1ð1þAx�1Þ

1�e�Qs

dð1�e�QsÞþr3l> 1, then system (1) is permanent.

Proof. Let ðx1ðtÞ; x2ðtÞ; x3ðtÞ; cðtÞÞ be any solution of system (1). From the first equation of system (2) and Theorem 4.1, wehave

dx1ðtÞdt

P x1ðtÞða� a1M � bx1 tð ÞaÞ: ð30Þ

By the same argument as those in the proof of Theorem 4.1, we have that

limt!1

x1ðtÞP p; ð31Þ

where p ¼ bð1�hÞ�að1�e�aða�a1MÞsÞða�a1MÞð1�ð1�hÞ�ae�aða�a1MÞsÞ

h i�1a � e3. In view of Theorem 4.1, the second equation of system (1) becomes

dx2ðtÞdt

Pb1pq

1þ Apð1� e�b2s1 Þ � b2x2ðtÞ � r2Mx2ðtÞ; ð32Þ

it is easy to obtain

lim x2ðtÞP q;

where q ¼ b1pqð1þApÞðb2þr2MÞ ð1� e�b2s1 Þ. From (12), for t enough large, we have cðtÞP c�ðtÞ � e0 > l

eQs�1� e0 ¼: m > 0

By Theorem 4.1 and the above discussion, system (1) is permanent. The proof of Theorem 4.2 is complete. h

1268 Z. Zhao, X. Wu / Applied Mathematics and Computation 232 (2014) 1262–1268

5. Discussion

In this paper, we introduce a time delay and pulse into the prey-predator model with the impulsive pollution, and the-oretically analyze the influence of impulsive catching for the prey and impulsive pollution for the predator on predator-

extinction and permanence of populations. By Theorem 3.1, we know that b1e�b2s1 dð1þAdÞ

eQs�1dðeQs�1Þþr3l

< 1, the predator-eradication

periodic solution is globally attractive. The predator population becomes extinction and the prey population fluctuates peri-odically. In practice, from the principle of ecosystem balance and in order to save resources, we need to only to control thepest population under the economic threshold level and not to eradicate natural enemy totally, and hope pest populationand natural enemy population can coexist when the pest do not bring about immense economic losses. As a result, we inves-

tigate the permanence of system (2) in Section 4, we know that b1e�b2s1 x�1ð1þAx�1Þ

1�e�Qs

dð1�e�QsÞþr3l> 1, the system (2) is permanent.

References

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