A Pest Management SI Model With Periodic Biological and Chemical Control Concern

Applied Mathematics and Computation 183 (2006) 1018–1026

www.elsevier.com/locate/amc

A pest management SI model with periodic biologicaland chemical control concern q

Jianjun Jiao a,b,*, Lansun Chen a

a Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of Chinab School of Mathematics and Statistics, Guizhou College of Finance and Economics, Guiyang 550004, People’s Republic of China

Abstract

In this work, we consider an SI model for pest management, with concerns about impulsive releases of infective pestsand pesticides sprays. We prove that all solutions of

0096-3

doi:10

q SupEcono

* CoRepub

E-m

S0ðtÞ ¼ rSðtÞ 1� SðtÞþhIðtÞK

� �� bSðtÞI2ðtÞ; t 6¼ ns;

I 0ðtÞ ¼ bSðtÞI2ðtÞ � wIðtÞ; t 6¼ ns;

DSðtÞ ¼ �l1SðtÞ; t ¼ ns;

DIðtÞ ¼ �l2IðtÞ þ l; t ¼ ns; n ¼ 1; 2; . . . ;

8>>>><>>>>: ðIÞ

are uniformly ultimately bounded and there exists globally asymptotic stability periodic solution of pest-extinction when

ln 11�l1

> rs� rlhð1�expð�wsÞÞKwð1�ð1�l2Þ expð�wsÞÞ �

bl2ð1�expð�2wsÞÞ2wð1�ð1�l2Þ expð�wsÞÞ2 is satisfied, and the condition for permanence of system (I) is also ob-

tained. It is concluded that the approach of combining impulsive infective releasing with impulsive pesticide spraying pro-

vides reliable tactic basis for practical pest management.� 2006 Elsevier Inc. All rights reserved.

Keywords: Impulsive; Infective; Chemical control; Pest- extinction

1. Introduction

According to reports of the Food and Agriculture Organization, the warfare between man and pests hassustained for thousands of years. With the development of society and progress of science and technology,human have adopted some advanced and modern weapons, for instance chemical pesticides, biological pesti-cides, remote sensing and measure, computers, atomic energy etc., where some brilliant achievements have

003/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

.1016/j.amc.2006.06.070

ported by National Natural Science Foundation of China (10171117) and the Doctoral Fund of Guizhou College of Finance andmic (in China).rresponding author. Address: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’slic of China.ail addresses: [email protected] (J. Jiao), [email protected] (L. Chen).

mailto:[email protected]

mailto:[email protected]

J. Jiao, L. Chen / Applied Mathematics and Computation 183 (2006) 1018–1026 1019

been obtained. However, the warfare is not over, and will continue. A great deal of and a large variety of pes-ticides were used to control pests. In all, pesticides are useful because they can quickly kill a significant portionof a pest population and sometimes provide the only feasible method for preventing economic loss. However,pesticide pollution is also recognized as a major health hazard to human beings and beneficial insects. Length-ening the period of pesticides spraying may reduce the cost of pest management. In this paper, a goodapproach is given which combines pesticides efficacy tests with biological research.

The use of bacteria, fungi and viruses is potentially one of the most important approaches in pest control.For example, Bacillus thuringiensis, which is available in commercial preparations, is used in the control of alarge number of pests [1–3,8,9]. An advantage of using insect pathogens is that they are safe to man and areusually safe to beneficial insects.

There is a vast amount of literature on the applications of microbial disease to suppress pests [3,4,7,9,10],and many good articles [15–25] devote to disease transmission, but there are a few papers on a mathematicalmodel of the dynamical of microbial disease in pest control [5,6,11,31]. In this paper, we will introduce addi-tional infective pests, which are obtained in the laboratory, into a natural SI system with spaying pesticides forpest control. We shall examine the strategy of combining periodic releasing of infective pests with periodicspraying of pesticides in a more flexible manner.

2. Model formulation

The basic SI model is

S0ðtÞ ¼ �bSðtÞIðtÞ;

I 0ðtÞ ¼ bSðtÞIðtÞ � wIðtÞ;

(ð2:1Þ

where b > 0 is called the transmission coefficient, w > 0 is called the death coefficient of I(t), S(t) denotes thenumber of susceptible insects and I(t) denotes the number of infective insects. In our models we assume that allnewborns are susceptible, and the basic model considered in [12] follows a model of the epidemic under a con-trol variable:

S0ðtÞ ¼ �bSðtÞIðtÞ;

I 0ðtÞ ¼ bSðtÞIðtÞ � wIðtÞ þ uðtÞ;

(ð2:2Þ

where u(t), which is a control variable, denotes the rate of pests infected in the laboratory, and there are someother conditions for the above system, but we consider that the variable u(t) is difficult in practice. And thesusceptible pests S(t) will not go to extinction with regard to human beings and some mass residing animals,Anderson et al. pointed out that standard incidence is more suitable than bilinear incidence [26–28]. Levinet al. have adopted a incidence form like bSqIp or bSqIp

N which depends on different infective diseases and envi-ronments [29,30]. So we develop (2.2) by introducing a constant periodic releasing of the infective pests andspaying pesticides at a fixed moment, that is, we consider the following impulsive differential equation with afixed moment:

S0ðtÞ ¼ rSðtÞ 1� SðtÞþhIðtÞK

� �� bSðtÞI2ðtÞ; t 6¼ ns;

I 0ðtÞ ¼ bSðtÞI2ðtÞ � wIðtÞ; t 6¼ ns;

DSðtÞ ¼ �l1SðtÞ; t ¼ ns;

DIðtÞ ¼ �l2IðtÞ þ l; t ¼ ns; n ¼ 1; 2; . . . ;

8>>>>>><>>>>>>:ð2:3Þ

where r > 0 is the intrinsic growth rate of pests, K > 0 is the pests capacity of environment, DI(t) = I(t+) � I(t),0 < h < 1, l P 0 is the released amount of infective pests at t = ns, n 2 Z+, 1 > l1 P 0, 1 > l2 P 0 respectively,which represents the portion of susceptible and infective pests due to spraying pesticides at t = ns, n 2 Z+ andZ+ = {1,2, . . .}, s is the period of the impulsive effect, that is, we can use a combination of biological andchemical tactics to eradicate pests or keep the pest population below the damage level.

1020 J. Jiao, L. Chen / Applied Mathematics and Computation 183 (2006) 1018–1026

3. Qualitative analysis for model (2.3)

The solution of (2.3), denote by x(t) = (S(t), I(t))T, is a piecewise continuous function x : Rþ ! R2þ, x(t) is

continuous on (ns, (n + 1)s], n 2 Z+ and xðnsþÞ ¼ limt!nsþxðtÞ exists. Obviously the global existence anduniqueness of solutions of (2.3) is guaranteed by the smoothness properties of f, which denote the mappingdefined by the right-side of system (2.3) (see [13]). Before we state the main results, we need to give some lem-mas which will be used in the next section. Since (S 0(t) = 0 whenever S(t) = 0, I 0(t) = 0 whenever I(t) = 0,t 5 ns, and I(ns+) = I(ns) + l, l P 0. So we have

Lemma 1. Suppose x(t) is a solution of (2.3) with x(0+) P 0, then x(t) P 0 for t P 0, and further x(t) > 0, t P 0

for x(0+) > 0.

Lemma 2 (see [13, p. 23, Lemma 2.2]). Let the function m 2 PC 0[R+,R] satisfies the inequalities

m0ðtÞ 6 pðtÞmðtÞ þ qðtÞ; t P t0; t 6¼ tk; k ¼ 1; 2; . . . ;

mðtþk Þ 6 dkmðtkÞÞ þ bk; t ¼ tk;

�ð3:1Þ

where p,q 2 PC[R+,R] and dk P 0, bk are constants, then

mðtÞ 6 mðt0ÞY

t0<tk<t

dk exp

Z t

t0

pðsÞds� �

þX

t0<tk<t

Ytk<tj<t

dj exp

Z t

t0

pðsÞds� � !

bk

þZ t

t0

Ys<tk<t

dk exp

Z t

spðrÞdr

� �qðsÞds; t P t0:

Now, we show that all solutions of (2.3) are uniformly ultimately bounded.

Lemma 3. There exists a constant M > 0 such that S(t) 6M, I(t) 6M for each solution (S(t), I(t)) of (2.3) with

all t large enough.

Proof. Define V(t) = S(t) + I(t), then t 5 ns. We have

DþV ðtÞ þ wV ðtÞ ¼ ðr þ wÞSðtÞ � rS2ðtÞK� rhSðtÞIðtÞ

K6 ðr þ wÞSðtÞ � rS2ðtÞ

K6 M0;

where M0 ¼ KðrþwÞ24r , when t = ns, V(ns+) = (1 � l1)S(ns) + (1 � l2)I(ns) + l 6 V(ns) + l. By Lemma 2, for

t 2 (ns, (n + 1)s], we have

V ðtÞ 6 V ð0Þ expð�wtÞ þZ t

0

M0 expð�wðt � sÞÞdsþX

0<ns<t

l expð�wðt � nsÞÞ

¼ V ð0Þ expð�wtÞ þM0

wð1� expð�wtÞÞ þ l

expð�wðt � sÞÞ � expð�wðt � ðnþ 1ÞsÞÞ1� expðwsÞ

< V ð0Þ expð�wtÞ þM0

wð1� expð�wtÞÞ þ l expð�wðt � sÞÞ

1� expðwsÞ þ l expðwsÞexpðwsÞ � 1

! M0

wþ l expðwsÞ

expðwsÞ � 1as t!1:

So V(t) is uniformly ultimately bounded. Hence, by the definition of V(t), there exists a constant M > 0 suchthat S(t) 6M, I(t) 6M for t large enough. The proof is complete. h

If S(t) = 0, we have the following subsystem of (2.3):

I 0ðtÞ ¼ �wIðtÞ; t 6¼ ns;

IðnsþÞ ¼ ð1� l2ÞIðnsÞ þ l; t ¼ ns; n ¼ 1; 2; . . .

�ð3:2Þ


Obviously gIðtÞ ¼ l expð�wðt�nsÞÞ1�ð1�l2Þ expð�wsÞ ; t 2 ðns; ðnþ 1Þs�; n 2 Zþ; gIð0þÞ ¼ l

1�ð1�l2Þ expð�wsÞ is a positive periodic solu-

tion of (3.2). Since the solution of (3.2) is IðtÞ ¼ ð1� l2Þ gIð0þÞ � l1�ð1�l2Þ expð�wtÞ

� �expð�wtÞ þ gIðtÞ;

t 2 ðns; ðnþ 1Þs�; n 2 Zþ, so we derive

Lemma 4. System (3.2) has a positive periodic solution gIðtÞ and for every solution I(t) of (3.2) we have IðtÞ ! gIðtÞas t!1.

From the above discussion we know that (2.3) has a pest-extinction periodic solution

ð0; gIðtÞÞ ¼ 0;l expð�wðt � nsÞÞ

1� ð1� l2Þ expð�wsÞ

� �ð3:3Þ

for ns < t 6 (n + 1)s, and gIðnsþÞ ¼ gIð0þÞ ¼ l1�ð1�l2Þ expð�wsÞ ; n 2 Zþ.

Theorem 1. Let (S(t), I(t)) be any solution of (2.3), if

ln1

1� l1

> rs� rlhð1� expð�wsÞÞKwð1� ð1� l2Þ expð�wsÞÞ �

bl2ð1� expð�2wsÞÞ2wð1� ð1� l2Þ expð�wsÞÞ2

ð3:4Þ

holds, then pest-extinction solution ð0; gIðtÞÞ of (2.3) is globally asymptotically stable.

Proof. First, we prove the local stability. Defining s(t) = S(t), iðtÞ ¼ IðtÞ � gIðtÞ, we have the following linearlysimilar system for (2.3) which concerns one periodic solution ð0; gIðtÞÞ:

dsðtÞdt

diðtÞdt

!¼

r � rhKgIðtÞ � b gIðtÞ2 0

�b gIðtÞ2 �w

!sðtÞiðtÞ

� �:

It is easy to obtain the fundamental solution matrix

UðtÞ ¼ expR t

0ðr � rh

KgIðsÞ � b gIðtÞ2Þds

� �0

� expð�wtÞ

!:

There is no need to calculate the exact form of * as it is not required in the analysis that follows. The linear-ization of the third and fourth equations of (2.3) is

sðnsþÞiðnsþÞ

� �¼

1� l1 0

0 1� l2

� �sðnsÞiðnsÞ

� �:

The stability of the periodic solution ð0; gIðtÞÞ is determined by the eigenvalues of

M ¼1� l1 0

0 1� l2

� �/ðsÞ;

where

k1 ¼ ð1� l2Þ expð�wsÞ < 1; k2 ¼ ð1� l1Þ exp

Z s

0

r � rhKgIðsÞ � b gIðtÞ2� �

ds� �

according to the Floquet theory [14], if jk2j < 1, i.e.

ln1

1� l1

> rs� rlhð1� expð�wsÞÞKwð1� ð1� l2Þ expð�wsÞÞ �


holds, then ð0; gIðtÞÞ is locally stable.In the following, we will prove the global attraction. Choose a e > 0 such that

q ¼ ð1� l1Þ exp

Z s

0

r � rhKðgIðtÞ � eÞ � bðgIðtÞ � eÞ2

� �dt

� �< 1:


From the second equation of (2.3) we notice that dIðtÞdt P �wIðtÞ, so we consider the following impulsive dif-

ferential equation:

dyðtÞdt ¼ �wyðtÞ; t 6¼ ns;

yðtþÞ ¼ ð1� l2ÞyðtÞ þ l; t ¼ ns;

yð0þÞ ¼ Ið0þÞ:

8><>: ð3:5Þ

From Lemma 2 and comparison theorem of impulsive equation (see Theorem 3.1.1 in [13]), we haveI(t) P y(t) and yðtÞ ! gIðtÞ as t!1. Then

IðtÞP yðtÞP gIðtÞ � e ð3:6Þ

for all t large enough. For convenience we may assume (3.6) holds for all t P 0, From (2.3) and (3.6) we get

dSðtÞdt 6 rSðtÞ 1� rh

K ðgIðtÞ � eÞ � bðgIðtÞ � eÞ2� �

; t 6¼ ns;

SðnsþÞ ¼ ð1� l1ÞSðnsÞ; t ¼ ns; n ¼ 1; 2; . . .

(ð3:7Þ

So Sððnþ 1ÞsÞ 6 SðnsþÞð1� l2Þ expR ðnþ1Þs

ns r � rhK ðIðsÞ � eÞ � bðgIðtÞ � eÞ2

� �ds

� �. Hence S(ns) 6 S(0+)qn and

S(ns)! 0 as n!1, therefore S(t)! 0 as t!1.

Next we prove that IðtÞ ! gIðtÞ as t!1, For 0 < e 6 w, there must exist a t0 > 0 such that 0 < S(t) < e forall t P t0. Without loss of generality, we may assume that 0 < S(t) < e for all t P 0, then for system (2.3) wehave

�wIðtÞ 6 dIðtÞdt6 ð�wþ eÞIðtÞ ð3:8Þ

then we have z1(t) 6 I(t) 6 z2(t) and z1ðtÞ ! gIðtÞ; z2ðtÞ ! gIðtÞ as t!1. While z1(t) and z2(t) are the solutionsof

dz1ðtÞdt ¼ �wz1ðtÞ; t 6¼ ns;

z1ðtþÞ ¼ ð1� l2Þz1ðtÞ þ l; t ¼ ns;

z1ð0þÞ ¼ Ið0þÞ

8><>: ð3:9Þ

and

dz2ðtÞdt ¼ ð�wþ eÞz2ðtÞ; t 6¼ ns;

z2ðtþÞ ¼ ð1� l2Þz2ðtÞ þ l; t ¼ ns;

z2ð0þÞ ¼ Ið0þÞ;

8><>: ð3:10Þ

respectively, gz2ðtÞ ¼ l expðð�wþeÞðt�nsÞÞ1�ð1�l2Þ expðð�wþeÞsÞ for ns < t 6 (n + 1)s. Therefore, for any e1 > 0, there exists a t1, t > t1

such that
gIðtÞ � e1 < IðtÞ < gz2ðtÞ þ e1:
Let e! 0, so we have

gIðtÞ � e1 < IðtÞ < gIðtÞ þ e1

for t large enough, which implies IðtÞ ! gIðtÞ as t!1. This completes the proof. h

The next part is the investigation of the permanence of the system (2.3). Before starting our theorem, wegive the following definition.

Definition 3.1. System (2.3) is said to be permanent if there are constants m, M > 0 (independent of initialvalue) and a finite time T0 such that for all solutions (S(t), I(t)) with all initial values S(0+) > 0, I(0+) > 0,m 6 S(t) 6M, m 6 I(t) 6M holds for all t P T0. Here T0 may depend on the initial values (S(0+), (I(0+)).


Theorem 2. Let (S(t), I(t)) be any solution of (2.3). If

ln1

1� l1

< rs� rlhð1� expð�wsÞÞKwð1� ð1� l2Þ expð�wsÞÞ �


ð3:11Þ

holds, then the system (2.3) is permanent.

Proof. Suppose (S(t), I(t)) is a solution of (2.3) with S(0) > 0, I(0) > 0. By Lemma 3, we have proved thereexists a constant M > 0 such that S(t) 6M, I(t) 6M for t large enough. We may assume S(t) 6M,I(t) 6M and M >

ffiffirb

qfor t P 0. From (3.6) we know IðtÞ > gIðtÞ � e2 for all t large enough and e2 > 0, so

IðtÞP l expð�wsÞ1�expð�wsÞ � e2 ¼ m2 for t large enough. Thus, we only need to find m1 > 0 such that S(t) P m1 for t large

enough, we will do it in the following two steps.(1) By condition (3.11), we can select m3 > 0, e1 > 0 small enough such that m3 <

wb ; d ¼

bm3M < w; r ¼ rs� rK m3s� rh

K þ be21

� s� l

w�drhK þ 2�1

� 1�eð�wþdÞs

1�ð1�l2Þeð�wþdÞs þ bl2ð1�e2ð�wþdÞsÞ2ð�wþdÞð1�ð1�l2Þeð�wþdÞsÞ > 0, we will

prove S(t) < m3 cannot hold for t P 0. Otherwise,

dIðtÞdt 6 ð�wþ dÞIðtÞ; t 6¼ ns;

IðtþÞ ¼ ð1� l2ÞIðtÞ þ l; t ¼ ns:

(ð3:12Þ

By Lemmas 3 and 4, we have I(t) 6 z(t) and zðtÞ ! zðtÞ; t!1, where z(t) is the solution of

dzðtÞdt ¼ ð�wþ dÞzðtÞ; t 6¼ ns;

zðtþÞ ¼ zðtÞ þ l; t ¼ ns;

zð0þÞ ¼ Ið0þÞ

8>><>>: ð3:13Þ

and gz2ðtÞ ¼ l expðð�wþdÞðt�nsÞÞ1�ð1�l2Þ expðð�wþdÞsÞ, t 2 (ns, (n + 1)s]. Therefore, there exists a T1 > 0 such that

IðtÞ 6 zðtÞ 6 zðtÞ þ e1

and

dSðtÞdt P SðtÞ r � r

K m3 � rhK ðzðtÞ þ e1Þ � bðzðtÞ þ e1Þ2

� �; t 6¼ ns;

SðtþÞ ¼ ð1� l1ÞSðtÞ; t ¼ ns; n ¼ 1; 2; . . .

(ð3:14Þ

for t P T1. Let N1 2 N and N1s > T1. Integrating (3.14) on (ns, (n + 1)s), n P N1, we have

Sððnþ 1ÞsÞP ð1� l1ÞSðnsþÞ exp

Z ðnþ1Þs

nsr � r

Km3 �

rhKðzðtÞ þ e1Þ � bðzðtÞ þ e1Þ2

� �dt

� �¼ ð1� l1ÞSðnsÞer

then S((N1 + k)s) P (1 � l1)kS(N1s+)ekr!1, as k!1, which is a contradiction to the boundedness of

S(t). Hence there exists a t1 > 0 such that S(t) P m3.(2) If S(t) P m3 for t P t1, then our aim is obtained. Hence, we need only to consider those solutions which

leave region R ¼ fðSðtÞ; IðtÞÞ 2 R2þ : SðtÞ < m3g and reenter it again. Let t� ¼ inf tPt1

fSðtÞ < m3g, then there aretwo possible cases for t*. h

Case 1. t* = ns, n1 2 Z+, then S(t) P m3 for t 2 [t1, t*) and S(t*) = m3. And (1 � l1)m3 6 S(t+) 6(1 � l1)S(t) 6 m3. Select n2,n3 2 N, such that

n2s > T 2 ¼ln e1

Mþl

� ��wþ d

;

ð1� l1Þn2 en3ren2r1s > ð1� l1Þ

n2 en3reðn2þ1Þr1s > 1;


where r1 ¼ r � rm3

K � hK r � rh

K M � bM2 < 0. Let T = n1s + n2s. We claim that there must be a t2 2 [t*, t* + T]such that S(t2) > m3. Otherwise, consider (3.13) with z(t*+) = I(t*+). We have

zðtÞ ¼ zðt�þÞ � l1� ð1� l2Þeð�wþdÞs

� �eð�wþdÞðt�t�Þ þ zðtÞ;

t 2 (ns, (n + 1)s], n1 6 n 6 n1 + n2 + n3, then

jzðtÞ � zðtÞj < ðlþMÞeð�wþdÞðt�ðn1þ1ÞsÞ < e1

and IðtÞ 6 zðtÞ 6 zðtÞ þ e1, t* + n2s 6 t 6 t* + T, which implies (3.14) holds for t* + n2s 6 t 6 t* + T. As instep1, we have

Sðt� þ T ÞP ðSðt� þ n2ÞsÞen3r:

The first equation of (2.3) gives

dSðtÞdt P SðtÞ r � r

K m3 � rhK M þ bM2

� ¼ r1SðtÞ; t 6¼ ns;

SðtþÞ ¼ ð1� l1ÞSðtÞ; t ¼ ns:

(ð3:15Þ

Integrating (3.15) on [t*, t* + n2s], we have

Sðt� þ n2sÞP m3ð1� l1Þn2 er1n2s:

Thus we have

Sðt� þ T ÞP m3ð1� l1Þn2 er1n2sen3r > m3

which is a contradiction. Let �t ¼ inf tPt�fSðtÞP m3g, thus Sð�tÞP m3 for t 2 ½t�;�t�. We have SðtÞPð1� l1Þ

n2þn3 Sðt�Þer1ðt�t�Þ P ð1� l1Þn2þn3 m3er1ðn2þn3Þ ¼ m1 for t P �t. So we have S(t) P m1. The same arguments

can be continued since Sð�tÞP m3. Hence S(t) P m1 for all t P t1.

Case 2. t 5 ns, n 2 Z+, then S(t) P m3 for t P t01, and S(t*) = m3. Let t� ¼ inf tPt01fSðtÞ < m3g, then S(t) P m3

for t 2 ½t01; t�Þ and S(t*) = m3. Suppose t� 2 ½n01s; ðn01 þ 1ÞsÞ; n01 2 N . We claim that there must be at02 2 ½ðn01 þ 1Þs; ðn01 þ 1Þsþ T �. Consider (3.13) with zððn01 þ 1ÞsþÞ ¼ Iððn01 þ 1ÞsþÞ. We have

zðtÞ ¼ zððn1 þ 1ÞsþÞ � l1� ð1� l2Þeð�wþdÞs

� �eð�wþdÞðt�ðn0

1þ1ÞsÞ þ zðtÞ;

t 2 (ns, (n + 1)s], n01 þ 1 6 n 6 n01 þ 1þ n2 þ n3, then

jzðtÞ � zðtÞj < ðlþMÞeð�wþdÞðt�ðn01þ1ÞsÞ < e1

and IðtÞ 6 zðtÞ 6 zðtÞ þ e1, ðn01 þ 1þ n2Þs 6 t 6 ðn01 þ 1þ T Þs, which implies (3.14) holds for ðn01þ1þ n2Þs 6 t 6 ðn01 þ 1Þsþ T . As in step1, we have

Sððn01 þ 1þ n2 þ n3ÞsÞP ð1� l1Þn3 Sððn01 þ 1þ n2ÞsÞen3r:

The first equation of (2.3) gives

dSðtÞdt

P SðtÞ r � rK

m3 �rhKþ b

� �M

� �¼ r1SðtÞ:

Integrating it on ½t�; ðn01 þ 1þ n2Þs�, we have

Sððn01 þ 1þ n2ÞsÞP ð1� l1Þn2þ1m3er1ðn2þ1Þs:

Then

Sððn01 þ 1þ n2 þ n3ÞsÞP m3er1ðn2þ1Þsen3r > m3

which is a contradiction. Let �t0 ¼ inf tPt�fSðtÞP m3g, thus Sð�t0ÞP m3 for t 2 ½t�;�t�. We have SðtÞPSðt�Þer1ðt�t�Þ P m3er1ð1þn2þn3Þ ¼ m1 for t P �t0. The same arguments can be continued since Sð�t0ÞP m3. HenceS(t) P m1 for all t P t1, The proof is complete.


Remark 3.1. Let f ðsÞ ¼ rs� rlhð1�expð�wsÞÞKwð1�ð1�l2Þ expð�wsÞÞ �

bl2ð1�expð�2wsÞÞ2wð1�ð1�l2Þ expð�wsÞÞ2 � ln 1

1�l1, we easily know f ð0Þ ¼

� ln 11�l1

< 0; f ðsÞ ! 1 as s!1, and

f 00ðsÞ ¼ rll2hw2ð1� ð1� l2Þe�wsÞð1� ð1� l2Þe�wsÞ3

þ 2e�2w þ e�3wf3ð1� l2Þ � 2ð1� l2Þ2ews þ ð1� l2Þe2wg

ð1� ð1� l2Þe�wsÞ4bwl2 > 0:

So f(s) = 0 has a unique positive root, denoted by smax. From Theorems 1 and 2 we know smax is a threshold.If s < smax, then the pest-extinction solution ð0; gIðtÞÞ of system (2.3) is globally asymptotically stable. Ifs > smax, then the system (2.3) is permanent.

Remark 3.2. If l = 0, i.e. there are no periodic releasing infective pests. Thus, we can easily obtain thats1 ¼ ln 1

1�l1is the threshold. Obviously, smax > s1, i.e. impulsively releasing infective pests may lengthen the

period of spraying pesticides and therefore reduce the cost of pests control.

4. Discussion

In this paper, according to the fact, an SI model with impulsive transmitting infective pests and sprayingpesticides at fixed moment is proposed and investigated, we analyzed the pest-extinction periodic solutionof such a system which is global asymptotic stability, and obtained the condition for the permanence of system(2.3). There are some interesting problems: how to optimize the amount of spraying pesticides? What is thedetermining time of impulsive releasing infective pests depending on the state of insect pest? How this impul-sive releasing infective pest affect the system? In the real world, for the insect pest is with seasonal damages,should impulsive releasing infective be considered on finite interval? We will continue to study these problemsin the future.

References

[1] L.A. Falcon, Use of Bacteria for Microbial Control of Insects, Academic Press, New York, NY, 1971.[2] H.D. Burges, N.W. Hussey, Microbial Control of Insects and Mites, Academic Press, New York, NY, 1971, pp. 67–95.[3] L.A. Falcon, Problems associated with the use of arthropod viruses in pest control, Ann. Rev. Entomol. 21 (1976) 305–324.[4] F. Fenner, F.N. Ratcliffe, Myxomatosis, Cambridge University Press, 1965, pp. 379.[5] R.M. Anderson, R.M. May, Regulation and stability of host-parasity population interactions, I. Regulatory processes, J. Anim. Ecol.

47 (1978) 219–247.[6] B.S. Goh, The potential utility of control theory to pest management, Proc. Ecol. Soc. 6 (1971) 84–89.[7] P.E. Davis, K. Myers, J.B. Hoy, Biological control among vertebrates, in: C.B. Huffaker, P.S. Messenger (Eds.), Theory and Practice

of Biological Control, Plenum Press, New York, NY, 1976, pp. 501–519.[8] N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Griffin, London, 1975, pp. 413.[9] H.D. Burges, N.W. Hussey, Microbial Control of Insects and Mites, Academic Press, New York, NY, 1971, pp. 861.

[10] Y. Tanada, Epizootiology of insect diseases, in: P. Debach (Ed.), Biological Control of Insect Pests and Weeds, Chapman and Hall,London, 1964, pp. 548–578.

[11] N. Gilbert, A.P. Gutierrez, B.D. Frazer, R.E. Jones, Ecological Relationships, W.H. Freeman and Co., San Francisco, CA, 1976.[12] Bean Sangoh, Management and Analysis of Biological Populations, Elsevier Scientific Press Company, 1980.[13] V. Lakshmikantham, D.D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.[14] D. Bainov, P. Simeonov, Impulsive differential equations: periodic solutions and applications, Pitman Monographs Surveys Pure

Appl. Math. 66 (1993).[15] H.J. Barclay, Models for pest control using predator release, habitat management and pesticide release in combination, J. Appl. Ecol.

19 (1982) 337–348.[16] J.C. Paneyya, A mathematical model of periodically pulse chemotherapy: tumor recurrence and metastasis in a competition

environment, Bull. Math. Biol. 58 (1996) 425–447.[17] A. d’Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Biol. 179 (2002) 57–72.[18] J.C. Van Lanteren, Integrated pest management in protected crops, in: D. Dent (Ed.), Integrated Pest Management, Chapman and

Hall, London, 1995.[19] M.G. Roberts, R.R. Kao, The dynamics of an infectious disease in a population with birth pulse, Math. Biol. 149 (2002) 23–36.[20] Y.N. Xiao, L.S. Chen, A ratio-dependent predator–prey model with disease in the prey, Appl. Math. Comput. 131 (2002) 397–414.[21] Y.N. Xiao, L.S. Chen, An SIS epidemic model with stage structure and a delay, Acta Math. Appl. English Ser. 16 (2002) 607–618.[22] Y.N. Xiao, L.S. Chen, F.V.D. Bosh, Dynamical behavior for stage-structured SIR infectious disease model, Nonlinear Anal.: RWA 3

(2002) 175–190.


[23] Y.N. Xiao, L.S. Chen, On an SIS epidemic model with stage-structure, J. Syst. Sci. Complexity 16 (2003) 275–288.[24] Z.H. Lu, S.J. Gang, L.S. Chen, Analysis of an SI epidemic with nonlinear transmission and stage structure, Acta Math. Sci. 4 (2003)

440–446.[25] H. Hethcote, The mathematics of infectious disease, SIAM Rev. 42 (2002) 599–653.[26] R. Anderson, R. May, Population Biological of Infectious Diseases, Springer, Berlin, Heidelberg, New York, 1982.[27] R. Anderson, R. May, Infectious Diseases of Human: Dynamics and Control, Oxford University Press, 1991.[28] M.C.M. De Jong, O. Diekmann, J.A.P. Heesterbeek, How dose transmission depend on population size? in: D. Mollison (Ed.),

Human Infectious Diseases, Epidemic Models, Cambridge University Press, Cambridge, UK, 1995, pp. 84–94.[29] Wei-min, S.A. Levin, Y. Lwasa, Influence of nonlinear incidence rates upon the behavior of SIRS Epidemiological models, J. Math.

Biol. 25 (1987) 359–380.[30] Wei-min, H.W. Hethcote, S.A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 23

(1986) 187–240.[31] K. Wickwire, Mathematical models for the control of pests and infectious diseases: a survey, Theor. Popul. Biol. 8 (1977) 182–238.

Documents

A Pest Management SI Model With Periodic Biological and Chemical Control Concern