the dynamics of plant disease models with continuous and...

Journal of Theoretical Biology 266 (2010) 29–40

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Journal of Theoretical Biology

0022-51

doi:10.1

$This

200904

and N

2007CB� Corr

E-m

Lizhenq

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

The dynamics of plant disease models with continuous andimpulsive cultural control strategies$

Xinzhu Meng a,b, Zhenqing Li a,�

a State Key Laboratory of Vegetation and Environmental Change, Institute of Botany, Chinese Academy of Sciences, Beijing 100093, PR Chinab College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266510, PR China

a r t i c l e i n f o

Article history:

Received 27 October 2009

Received in revised form

23 January 2010

Accepted 27 May 2010Available online 9 June 2010

Keywords:

Plant disease models

Impulsive cultural control

Permanence

Periodic solution

Bifurcation

93/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jtbi.2010.05.033

project was supported by China Postdocto

60570), the National Natural Science Foundat

ational Basic Research Program of Ch

108904-4, 2006CB403207-4).

esponding author.

ail addresses: [email protected] (X. M

[email protected], [email protected] (Z. Li).

a b s t r a c t

Plant disease mathematical models including continuous cultural control strategy and impulsive

cultural control strategy are proposed and investigated. This novel theoretical framework could result

in an objective criterion on how to control plant disease transmission by replanting of healthy plants

and removal of infected plants. Firstly, continuous replanting of healthy plants and removing of infected

plants is taken. The existence and stability of disease-free equilibrium and positive equilibrium are

studied and continuous cultural control strategy is given. Secondly, plant disease model with impulsive

replanting of healthy plants and removing of infected plants is also considered. Using Floquet’s theorem

and small amplitude perturbation, the sufficient conditions under which the infected plant free periodic

solution is locally stable are obtained. Moreover, permanence of the system is investigated. Under

certain parameter spaces, it is shown that a nontrivial periodic solution emerges via a supercritical

bifurcation. Finally, our findings are confirmed by means of numerical simulations. The modeling

methods and analytical analysis presented can serve as an integrating measure to identify and design

appropriate plant disease control strategies.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Plant viruses or pathogens are an important constraint to cropproduction worldwide, and cause serious losses in yield andquality of cultivated plants. Therefore, plant disease are widelyregarded. The 10th International Plant Virus EpidemiologySymposium on the theme ‘Controlling Epidemics of Emergingand Established Plant Virus Diseases—the Way Forward’ was heldin India. In this symposium, many novel and interesting view-points and ways on plant virus epidemiology were offered andelaborated. Several plant diseases caused by plant viruses incassava (Manihot esculenta) and sweet potato (Ipomoea batatas)are among the main constraints to sustainable production of thesevegetatively propagated staple food crops in lesser-developedcountries (Thresh and Cooter, 2005; Dubern, 1994; Gibson et al.,2004). Epidemics of many polycyclic plant diseases caused byascomycete fungal pathogens are initiated by wind-borne ascos-pores transported into the crop in autumn (Inman et al., 1999)

ll rights reserved.

ral Science Foundation (no.

ion of China (no. 30870397)

ina (973 Program) (nos.

eng),

and subsequently develop further through cycles of splashdispersed conidia (Madden et al., 1996; Evenhuis et al., 1997).Sharka is one of the most severe diseases damaging stone-fruittrees including apricot, cherry, peach and plum. The causativeagent is plum pox potyvirus (PPV) that is naturally transmitted byaphids. No effective measures have been found to restrict PPVoutbreaks and it has spread across European borders to theMiddle East, North Africa, and North and South America. It seemsthat the best measure is to remove the diseased stone-fruit trees.Therefore, farmers have been evolving practices for combating thevarious plagues suffered by crops and trees, and growing under-standing of the interactions between pathogen and host hasenabled us to develop a wide array of measures for the control ofspecific plant diseases.

Such experiences have led to the development of integratedmanagement concepts for virus diseases that combine available hostresistance, cultural, chemical and biological control measures.Examples of how epidemiological information can be used todevelop effective integrated disease management (IDM) strategiesfor diverse situations have been described (Jones, 2001, 2004). IDMinvolves the selection and application of a wide range of controlstrategies that minimize losses and maximize returns. A culturalcontrol strategy including replanting, and/or removing diseasedplants is a widely accepted treatment for plant epidemics whichinvolves periodic inspections followed by removal of the detectedinfected plants (Fishman and Marcus, 1984; Jeger et al., 2004).

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X. Meng, Z. Li / Journal of Theoretical Biology 266 (2010) 29–4030

Mathematical models of plant–virus (or pathogens) diseaseepidemics were developed to provide a detailed exposition onhow to describe, analyze, and predict epidemics of plant diseasefor the ultimate purposes of developing and testing controlstrategies and tactics for crop protection (Grilli and Holt, 2000;Chan and Jeger, 1994; Holt and Chancellor, 1997). A simple modelfor plant disease with a continuous cultural control strategy, suchas replanting and removing, is as follows:

dSðtÞ

dt¼ r�bSðtÞIðtÞ

1þaSðtÞ�mSðtÞþoIðtÞ,

dIðtÞ

dt¼bSðtÞIðtÞ

1þaSðtÞ�ðdþrþoÞIðtÞ,

8>>><>>>: ð1:1Þ

where S(t), I(t) denote the number of susceptible and infectedplants, respectively. b is the transmission rate, a denotespotentially density dependent, m either denotes harvest time orthe end of reproductive lifetime of plants, r represents the totalrate at which the susceptible plants enter the system, r is theremoval rate for the infected plants, o is the recovery rate of thecured diseased plants, and the infected plants suffer an extradisease-related death with constant rate d. The detailed biologicalbackground and model development can be found in Chan andJeger (1994). Further, a model for the temporal spread of anepidemic in a closed plant population with periodic removals ofinfected plants has been considered by Fishman et al. (1983) withan application to the spread of citrus tristeza virus disease. Theirmodel helped in evaluating policies of controlling the disease andcould be also modified to simulate other plant epidemics withperiodic treatments. Fishman et al. (1983) proposed two types ofmodel with Logistic growth and periodic removing of infectedtrees, aimed at eradicating them. Therefore, one of the mainpurposes of the models described in this paper is to extend thedeterministic model (1.1), constructed for a close plant populationwith periodic replanting and removing controlling strategy, andanalyze its dynamical behavior by using the theories of impulsivedifferential equations. The results imply that the infected plantscan be completely eradicated if the period of control strategyapplication is sufficiently large, i.e. the infected plant free periodicsolution is stable. However, a good plant disease control programshould reduce the infected plants to levels acceptable to thepublic. That is, we do not need to eradicate the infected plants butto remove the infected plants under an Economic Threshold (ET).

Systems with sudden perturbations lead to impulsive differ-ential equations, which have been studied intensively andsystematically in Bainov and Simeonov (1993), and Lakshmikan-tham et al. (1989). The authors in Liu and Chen (2003), Wang et al.(2008), Jiao et al. (2009), Xiang et al. (2009), Li et al. (2009), Jiaoand Chen (2008), Meng et al. (2009), and Liu et al. (2007)introduced some impulsive differential equations in populationdynamics and exhibited complex behavior of impulsive equations.We extend model (1.1) by implementing periodic replanting ofhealthy plants or removing infected plants at a critical time. Thatis, the continuous replanting of healthy plants and removal ofinfected plants in model (1.1) are replaced by discrete andperiodic pulse cultural control strategies, which is more realistic.We also need to consider the following plant disease model withpulse replanting and removing strategies at fixed moments.

dSðtÞ

dt¼�

bSðtÞIðtÞ

1þaSðtÞ�mSðtÞþoIðtÞ,

dIðtÞ

dt¼bSðtÞIðtÞ

1þaSðtÞ�ðdþoÞIðtÞ,

tant,

Sðtþ Þ ¼ SðtÞþr,

Iðtþ Þ ¼ ð1�rÞIðtÞ,t¼ nt,

8>>>>>>>>><>>>>>>>>>:ð1:2Þ

where t is a fixed positive constant and denotes the period of theimpulsive effect, n¼ f1,2, . . .g and 0rro1.

In the following we will consider system (1.1) with continuousreplanting and removing and system (1.2) with impulsivereplanting and removing, respectively. The aim of this work isto study the dynamical behaviors of the plant disease models withcontinuous and pulsed input for the susceptible plants andremoval for the infected plants, and investigate how they affectsthe dynamical behaviors of unforced continuous system.

The organization of this paper is as follows. In Section 2, weinvestigate dynamical behaviors of system (1.1) with continuousreplanting and removing and give plant disease continuouscontrol strategies. In this section, stability of two steady statesof system (1.1) is investigated. In Section 3, we investigatedynamical behaviors of system (1.2) with impulsive replantingand removing and give plant disease impulsive control strategies.In this section, stability of disease-free periodic solution andpermanence of system (1.2) are investigated, respectively. InSection 4, we investigate existence of positive periodic solutionand bifurcation of system (1.2). Finally, we give numericalanalysis and biological conclusions to show our main results.

2. Plant disease continuous control for system (1.1)

In this section, we consider system (1.1) with continuousreplanting and removing and without impulsive effect. Obviously,we have SðtÞZ0, xðtÞZ0 for Sð0ÞZ0, xð0ÞZ0,tZ0. Define N(S, I)¼S(t) + I(t),

dNðtÞ

dt¼ r�mSðtÞ�ðrþdÞIðtÞ

rr�g1ðSðtÞþ IðtÞÞ,

NðS,IÞrNð0Þe�g1tþrg1

ð1�e�g1tÞ-rg1

for t-1,

where g1 ¼minfm,ðrþdÞg. Hence, system (1.1) is uniformlybounded.

An equilibrium point of system (1.1) satisfies the system

r�bSðtÞIðtÞ

1þaSðtÞ�mSðtÞþoIðtÞ ¼ 0,

bSðtÞIðtÞ

1þaSðtÞ�ðdþrþoÞIðtÞ ¼ 0:

8>>><>>>: ð2:1Þ

It can be seen that system (2.1) has a disease-free equilibrium ofthe form E0 ¼ ðr=m,0Þ. We start by analyzing the behavior of thesystem (1.1) near E0. The characteristic equation of the lineariza-tion of (1.1) E0 is

det

�m�l o� brmþar

0br

mþar�ðdþrþoÞ�l

0BBB@1CCCA¼ 0:

It is easy to see that l1 ¼�mo0, and l2 ¼ br=ðmþarÞ�ðdþrþoÞo0 is equivalent to

brmþaro ðdþrþoÞ: ð2:2Þ

So the boundary steady state E0 is locally stable if (2.2) holds.Define the threshold value

R0 ¼br

ðmþarÞðdþrþoÞ:

The above results show that the disease-free equilibrium E0 islocally asymptotically stable if R0o1, and E0 is unstable if R041.If R0o1, then system (1.1) has a unique steady state E0. It followsfrom the uniform boundedness and Fig. 1 that the disease-free

(1.1)

0

0.05

0.1

0.15

0.2

0.25

0.3

I

t

(1.2)

1

2

3

4

5

S

0t

(1.3)

0.857

0.8575

0.858

0.8585

0.859

I

1.998S

1.9985 1.999 1.9995 2

10 20 30 40

0 10 20 30 40

Fig. 1. Illustration of basic behavior of solutions of the model (1.1), where

parameters are fixed as follows r¼ 1, b¼ 1:2, a¼ 1, m¼ 0:2, o¼ 0:1, d¼0.3, the

initial value is (S0, I0)¼(0.5, 0.3). (a) Time series of I(t) with r¼0.8; (b) time series

of S(t) with r¼0.8; (c) phase portrait of S(t) and I(t) with r¼0.4. Suppose the unit of

the plants density S and I is thousand and the unit of time is day.

X. Meng, Z. Li / Journal of Theoretical Biology 266 (2010) 29–40 31

equilibrium E0 is globally asymptotically stable provided R0o1.Conversely, if R041, system (1.1) has a unique positiveequilibrium E*(S*, I*), where

S� ¼dþrþo

b�aðdþrþoÞand I� ¼

brdþr

1�1

R0

� �:

The Jacobian matrix at E*(S*, I*) is

�m� bI�

ð1þaS�Þ2o� bS�

1þaS�

bI�

ð1þaS�Þ2bS�

1þaS��ðdþrþoÞ

0BBB@1CCCA:

After a few computations, we know that the real parts ofeigenvalues are negative if and only if R041. Hence, as long asR041 holds, the positive equilibrium of system (1.1) is locallyasymptotically stable.

Further, let P(S,I), Q(S,I) denote the right side of (1.1),respectively, then we have

PðS,IÞ ¼ r� bSðtÞIðtÞ

1þaSðtÞ�mSðtÞþoIðtÞ, QðS,IÞ ¼

bSðtÞIðtÞ

1þaSðtÞ�ðdþrþoÞIðtÞ:

Set Dulac function

BðS,IÞ ¼ I�1:

Then we have

@BP

@Sþ@BQ

@I¼�

bð1þaSÞ2

�mI�1o0:

Hence, there is not limit-cycle in R2+, so, the positive equilibrium

E*(S*, I*) is globally asymptotically stable (see Fig. 1c).

Theorem 2.0. (i) The system (1.1) has a unique equilibrium E0

which is globally asymptotically stable if R0o1, (ii) If R041, system

(1.1) has a unique positive equilibrium E*(S*, I*) which is globally

asymptotically stable, and E0 is unstable.

We have developed and extended the epidemic model withcontinuous cultural control to include the impulsive culturalcontrol strategy such as replanting and/or removing diseasedplants. Therefore, we give our main results as follows.

3. Plant disease impulsive control for system (1.2)

By the basic theories of impulsive differential equations(Bainov and Simeonov, 1993; Lakshmikantham et al., 1989), thesolution of system (3.1) is unique and piecewise continuous inðnt,ðnþ1Þt�,nAN for any initial values in R+

2 .

3.1. Disease-free periodic solution

Lemma 3.0. There exists a constant L40 such that SðtÞ,IðtÞrL for

each solution of (1.2) with t large enough.

Proof. Let V(t)¼S(t)+ I(t). Since 0oro1,Iðntþ Þo IðntÞ, thusVðntþ ÞrVðntÞþr. For V(t), the following inequalities hold fortZ0,

_V ðtÞr�g2ðSðtÞþ IðtÞÞ ¼�g2VðtÞ, tant,

Vðntþ ÞrVðntÞþr, t¼ nt,

where g2 ¼minfm,dg. Hence, we have

VðtÞrVð0Þe�g2tþr

1�e�g2t-

r1�e�g2t

for t-1:

So V(t) is uniformly ultimately bounded. Hence, by the definitionof V(t), for any e40 there exists a constant L¼ r=ð1�e�g2tÞþesuch that SðtÞ,IðtÞrL for each solution of (1.2) with t large enough.The proof is completed. &

In the following we shall prove the disease-free periodicsolution is stable if it exists. For this purpose, we give firstly

(2.1)

0

0.05

0.1

0.15

0.2

0.25

0.3

I

0t

(2.2)

5.2

5.4

5.6

5.8

6

S

90t

(2.3)

0

0.002

0.004

0.006

0.008

I

5.2S

5.4 5.6 5.8 6

100 110 120 130 140 150 160

10 20 30 40


parameters are fixed as follows r¼ 1, b¼ 1:2, a¼ 1, m¼ 0:2, o¼ 0:1, d¼0.3, r¼0.8,

t¼ 0:5 the initial value is (S0, I0)¼(0.5, 0.3). (a) time series of I(t); (b) time series of

S(t); (c) phase portrait of S(t) and I(t). Suppose the unit of the plants density S and I

is thousand and the unit of time is day.


some basic properties of the following subsystem:

dSðtÞ

dt¼�mSðtÞ, tant,

Sðtþ Þ ¼ SðtÞþr, t¼ nt:

8<: ð3:1Þ

We can find a unique positive periodic solution gSðtÞ ¼rexpð�mðt�ntÞÞ=ð1�expð�mtÞÞ,tAðnt,ðnþ1Þt�. Similar to Liu andChen (2003), it can be shown that gSðtÞ is globally asymptoticallystable by using stroboscopic map. As a consequence, system (3.1)always has a disease-free periodic solution ðgSðtÞ,0Þ. The localstability of the periodic solution may be determined byconsidering the behavior of small amplitude perturbations ofthe solution.

The Jacobi matrix A(t) at ðgSðtÞ,0Þ and matrix B are as follows:

AðtÞ ¼

�m o� bgSðtÞ1þagSðtÞ

0bgSðtÞ

1þagSðtÞ�d�o

2666664

3777775, B¼1 0

0 1�r

� �:

We can calculate the monodromy matrix MðtÞ

MðtÞ ¼ BeR t

0AðtÞ dt

¼

e�m �

0 ð1�rÞexpR t

0

bgSðtÞ1þagSðtÞdt�ðdþoÞt

!264375,

whereR t

0 bgSðtÞ=ð1þagSðtÞÞdt¼ b=amlnð1�e�mtþarÞ=ð1�e�mtþ

are�mtÞ, and there is no need to calculate the exact form of *since it is not required for the following analyses. The Floquetmultipliers are l1 ¼ expð�mtÞ and

l2 ¼ ð1�rÞexp

Z t

0

bgSðtÞ1þagSðtÞ �d�o !

dt

!,

respectively. Obviously, l1o1, so the stability of ðgSðtÞ,0Þ isdecided by whether

l2 ¼ ð1�rÞexp

Z t

0


dt

!o1,

that is, when

bam½ðdþoÞt�lnð1�rÞ�

ln1�e�mtþar

1�e�mtþare�mto1,

the disease-free solution is locally stable.Define the threshold value

R1 ¼b

am½ðdþoÞt�lnð1�rÞ�ln

1�e�mtþar1�e�mtþare�mt

:

Theorem 3.0. If R1o1, then the disease-free periodic solution

ðgSðtÞ,0Þ of system (1.2) is locally stable.

We can show the disease-free periodic solution ðgSðtÞ,0Þ ofsystem (1.2) is globally asymptotically stable by numericalsimulations (see Fig. 2), but it is very difficult to provetheoretically the global asymptotical stability now. Hence, wewill do it in the future.

3.2. Impulsive control strategy with Economic Threshold

The result of Theorem 3.0 implies that the infected plants canbe completely eradicated if the impulsive period of controlstrategy application is sufficiently small or the removal rate forthe infected plants is sufficiently large, i.e. the infected plant freeperiodic solution is stable. However, a good plant disease controlprogram should reduce the infected plants to levels acceptable tothe public. That is, we do not need to eradicate the infected plants

but to remove the infected plants under an Economic Threshold(ET). Therefore, we consider following how to remove the infectedplants under an Economic Threshold (ET). In practice, on the onehand it is usually required in agriculture and forestry that we


control the pest density to be lower than the economic injurylevel in a finite given time before harvest. On the other hand thecontrol is considered in an infinite time.

We first consider the ET strategy for a finite time interval now.Using Lemma 3.0, it follows from the second and fourth

equations of (1.2) that

dI

dtr

bL

1þaL�ðdþoÞ

� �IðtÞ, tant,

Iðtþ Þ ¼ ð1�rÞIðtÞ, t¼ nt:

8><>: ð3:2Þ

Consider the following impulsive differential equation

du

dt¼

bL

1þaL�ðdþoÞ

� �uðtÞ, tant,

uðtþ Þ ¼ ð1�rÞuðtÞ, t¼ nt,

uð0Þ ¼ Ið0Þ40:

8>>><>>>: ð3:3Þ

Suppose that m times of impulsive removing the infected plantsare taken (m is a positive integer) in a given time T0, andimpulsive period t satisfies 0oto2t � � �omtrT0, then themodel (3.3) becomes

du

dt¼ suðtÞ, tant,

uðtþ Þ ¼ ð1�rÞuðtÞ, t¼ nt,

uð0Þ ¼ h, uðT0ÞrET,

8>>><>>>: ð3:4Þ

where s¼ bL=ð1þaLÞ�ðdþoÞ40, 0rtrT0, h and ET are positiveconstants. Without loss of generality, by real-life biologicalmeanings we assume that u0ðT0Þ4ET when r¼0.

We can calculate the solution of the first two equations ofsystem (3.4)

uðtÞ ¼ uð0Þðð1�rÞestÞnesðt�ntÞ, tAðnt,ðnþ1Þt�:

Then, we have

uðT0Þ ¼ uð0Þðð1�rÞestÞnesðT0�ntÞ, T0A ðnt,ðnþ1Þt�: ð3:5Þ

In order to remove the infected plants under an EconomicThreshold (ET), suppose ð1�rÞesto1. We consider two cases,T0=t¼ nþ1 and noT0=tonþ1.

Case (I): Suppose T0=t¼ nþ1.From (3.5), we get that

uðT0Þ ¼ uð0Þðð1�rÞestÞnest: ð3:6Þ

From (3.6) and the third equation of system (3.4), we have

uðT0Þ ¼ hðð1�rÞestÞnestrET ,

and then

hðð1�rÞestÞT0=t�1est

ETr1 or nZn� ¼ 1þ

lnðETe�sth�1Þ

lnðð1�rÞestÞ:

Case (II): Suppose noT0=tonþ1.Similarly, we can obtain

hðð1�rÞestÞ½T0=t�esðT0�½T0=t�tÞ

ETr1, or nZn� ¼

lnðETe�sth�1Þ

lnðð1�rÞestÞ

here ½T0=t� ¼ n denotes integer partition of T0=t.

Theorem 3.1. Suppose ð1�rÞesto1, where s¼ bL=ð1þaLÞ�

ðdþoÞ40.

(i)
When T0=t¼ nþ1, if
R2 ¼hðð1�rÞestÞT0=t�1est

ETr1, or nZn� ¼ 1þ

lnðETe�sth�1Þ

lnðð1�rÞestÞ,

then (3.4) has a solution which satisfies the boundary value

problem.

(ii)
When noT0=tonþ1, if
R2 ¼hðð1�rÞestÞ½T0=t�esðT0�½T0=t�tÞ

ETr1, or nZn� ¼

lnðETe�sth�1Þ

lnðð1�rÞestÞ,

then (3.4) has a solution which satisfies the boundary value

problem.

Using the comparison theorem of impulsive differentialequations, we can obtain that if uðT0ÞrET, then IðT0ÞrET . Thisimplies we can control the infected plants under an EconomicThreshold (ET) at t¼T0. Therefore, we give an important result asfollows.

Theorem 3.2. Suppose ð1�rÞesto1, where s¼ bL=ð1þaLÞ�

ðdþoÞ40.

(i)
When T0=t¼ nþ1, if
R2 ¼Ið0Þðð1�rÞestÞT0=t�1est

ETr1, or nZn� ¼ 1þ

lnðETe�sth�1Þ

lnðð1�rÞestÞ,

then the infected plants are controlled under an Economic

Threshold (ET) at t¼T0.
(ii) When noT0=tonþ1, if
R2 ¼Ið0Þðð1�rÞestÞ½T0=t�esðT0�½T0=t�tÞ

ETr1, or nZn� ¼

lnðETe�sth�1Þ

lnðð1�rÞestÞ,

then the infected plants are controlled under an Economic

Threshold (ET) at t¼T0.

We next consider the ET strategy in the whole time interval.By (3.5), if ð1�rÞesto1, here s¼ bL=ð1þaLÞ�ðdþoÞ40, then

limn-1uðT0Þ ¼ 0, i.e. limt-1uðtÞ ¼ 0. By the comparison theoremof impulsive differential equations, we can obtain thatlimt-1IðtÞ ¼ 0oET.

Using Lemma 3.0, it follows from the second and fourthequations of (1.2) that

dI

dtrbL

a �ðdþoÞIðtÞ, tant,

Iðtþ Þ ¼ ð1�rÞIðtÞ, t¼ nt:

8<:By using impulse differential inequalities, we have

IðtÞr Iðn1tþ ÞY

n1tonto t

ð1�rÞexp

Z t

n1t�ðdþoÞds

� �

þ

Z t

n1t

Ysonto t

ð1�rÞexp

Z t

s�ðdþoÞdy

� �bL

a ds

¼ Iðn1tþ Þð1�rÞ½t=t�e�ðdþoÞðt�n1tÞ þ1

dþobL

a e�ðdþoÞt

�ð1�rÞ½t=t��n1þ1

ðeðdþoÞt�1Þen1ðdþoÞt�ð1�rÞðeðdþoÞt�1ÞeðdþoÞt½t=t�

1�r�eðdþoÞt

"

þeðdþoÞt�eðdþoÞt½t=t�i

r Iðn1tÞen1tðdþoÞe�ðdþoÞtþbL

aðdþoÞ 1�reðdþoÞtð½t=t�þ1�t=tÞ

eðdþoÞt�1þr

� �r Iðn1tÞen1tðdþoÞe�ðdþoÞtþ

bL

aðdþoÞ 1�r

eðdþoÞt�1þr

� �,

which implies

limsupt-1

IðtÞrbL

aðdþoÞ 1�r

eðdþoÞt�1þr

� �:

If bL=aðdþoÞ½1�r=ðeðdþoÞt�1þrÞ�rET , i.e. bL=aðdþoÞET½1�r=

ðeðdþoÞt�1þrÞ�r1, then we can control the infected plants underan Economic Threshold (ET) in an infinite time interval.


Theorem 3.3. Assume ð1�rÞesto1 or

R3 ¼bL

aðdþoÞET1�

r

eðdþoÞt�1þr

� �r1,

then the infected plants are controlled under an Economic Threshold

(ET) in an infinite time interval.

According to the above analysis for Theorems 3.2 and 3.3, it isclearly to see that the conditions of Theorem 3.2 can deduce theresults of Theorem 3.3, but the conditions of Theorem 3.3 cannotdeduce the results of Theorem 3.2.

3.3. Permanence

Theorem 3.4. If R141, then system (1.2) is permanent.

Proof. Suppose X(t)¼(S(t),I(t)) any solution of system (1.2)with Xð0Þ40. From Lemma 3.0, we have SðtÞrL,IðtÞrL for all t

large enough, hence, _SðtÞZ�ðbLþmÞSðtÞ for tant andSðntþ Þ ¼ SðntÞþr for t¼ nt. Therefore, there exists a constantm1 ¼ re�ðbLþmÞt=ð1�e�ðbLþmÞtÞ�e40 for e40 small enough suchthat SðtÞZm1 with t large enough. Thus we only need to findm240 such that IðtÞZm2 for t large enough.

From R141, we can choose e140 and m�240 to be small

enough such that Z¼ ð1�rÞexpðR ðnþ1Þt

nt ½bðezðtÞ�e1Þ=1þaðezðtÞ�e1Þ�

ðdþoÞ�dtÞ41, where ezðtÞ ¼ rexpð�ðb m�2þmÞðt�ntÞÞ=1�exp

ð�ðbm�2þmÞtÞ,tAðnt,ðnþ1Þt�. We claim that IðtÞom�2 cannot hold

for all tZ0. Otherwise,

dSðtÞ

dtZ�ðbm�2þmÞSðtÞ, tant,

Sðntþ Þ ¼ SðntÞþr, t¼ nt:

8<:So we have SðtÞZzðtÞ and zðtÞ-ezðtÞ,t-1, where z(t) is the

solution of

dz

dt¼�ðbm�2þmÞzðtÞ, tant,

zðtþ Þ ¼ zðtÞþr, t¼ nt,

Sð0þ ÞZzð0þ Þ40

8>>><>>>: ð3:7Þ

and ezðtÞ ¼ rexpð�ðbm�2þmÞðt�ntÞÞ=ð1�expð�ðbm�2þmÞtÞÞ, tAðnt,

ðnþ1Þt�.Therefore, there exists a t140 such that

SðtÞZzðtÞ4ezðtÞ�e1,

and

dI

dtZ

bðezðtÞ�e1Þ

1þaðezðtÞ�e1Þ�ðdþoÞ

� �IðtÞ, tant,

Iðtþ Þ ¼ ð1�rÞIðtÞ, t¼ nt

8><>: ð3:8Þ

for tZt1. Let NAZþ and NtZt1. Integrating (3.8) on

ðnt,ðnþ1Þt�, nZN, we have

Iððnþ1ÞtÞZð1�rÞIðntÞexp

Z ðnþ1Þt

nt

bðezðtÞ�e1Þ

1þaðezðtÞ�e1Þ�ðdþoÞ

� �dt

!¼ IðntÞZ:

Then IððNþnÞtÞZ IðNtÞZn-1 as n-1, which is a contradiction

to the boundedness of I(t). Hence there exists a t140 such that

Iðt1ÞZm�2. If Iðt1ÞZm�2 for all tZt1, and let m2¼m2* then our aim

is obtained. Otherwise, let t� ¼ inf t4 t1fIðtÞom�2g, there are two

possible cases for t*.

Case (I): t� ¼ n1t,n1AZþ . Then IðtÞZm�2 for tA ½t1,t�� and

Iðt�þ Þom�2. By the definition of m1, we can have bm1�d�oo0

and choose n2,n3AZþ such that

n2t4�lnðe1=ð1þez0Þ

m�2bþm, ð3:9Þ

ð1�rÞn2 expbm1

1þam1�ðdþoÞ

� �n2t

� �Zn3

4 ð1�rÞn2 expbm1

1þam1�ðdþoÞ

� �ðn2þ1Þt

� �Zn3

41: ð3:10Þ

Let tu ¼ n2tþn3t, there exists a t2Aðt�,t�þt� such that Iðt2Þ4m�2.

Otherwise, consider (3.7) with zðt�þ Þ ¼ zðt�Þþr, for tAðnt,ðnþ1Þt�and n1rnrn1þn2þn3, we have

zðtÞ ¼ zðt�þ Þexpf�ðm�2bþmÞðt�t�Þg,

¼ ezðtÞþðzðt�þ Þ�ez0Þexpf�ðm�2bþmÞðt�t�Þg:

Then we obtain jzðtÞ�ezðtÞjo ð1þez0Þexpf�ðm�2bþmÞðt�t�Þgoe1.

Hence, for t�þn2trtrt�þtu, we have ezðtÞ�e1ozðtÞrSðtÞ. From

(3.8) and for t�þn2trtrt�þtu, as in above step 1, we have

Iðt�þtuÞZ Iðt�þn2tÞZn3 . From system (1.2), we get

dI

dtZ

bm1

1þam1�ðdþoÞ

� �I, tant, nAN,

Iðtþ Þ ¼ ð1�rÞIðtÞ, t¼ nt, nAN:

8><>: ð3:11Þ

Integrating (3.11) on ½t�,t�þn2t�, we have Iðt�þn2tÞZð1�rÞn2 m�2expf½bm1=ð1þam1Þ�ðdþoÞ�n2tg. Hence Iðt�þtuÞZð1�rÞn2 m�2expf½bm1=ð1þam1Þ�ðdþoÞ�n2tgZn3 4m�2, which is a

contradiction.

Let et ¼ inf t4 t� fIðtÞ4m�2g, then for tAðt�,etÞ,IðtÞrm�2 and

IðetÞ ¼m�2, since I(t) is left continuous and Iðtþ Þ ¼ ð1�rÞIðtÞr IðtÞ

when t¼ nt. For tAðt�,etÞ, suppose tAðt�þðk�1Þt,t�þkt�,kAZþand krn2þn3, from (3.11) we have

IðtÞZ Iðt�þ Þð1�rÞk�1exp ðk�1Þbm1

1þam1�ðdþoÞ

� �t

� ��exp

bm1

1þam1�ðdþoÞ

� �ðt�ðt�þðk�1ÞtÞÞ

� �Z ð1�rÞkm�2exp k

bm1

1þam1�ðdþoÞ

� �t

� �Z ð1�rÞn2þn3 m�2exp ðn2þn3Þ

bm1

1þam1�ðdþoÞ

� �t

� �:

Let mu

2 ¼ ð1�rÞn2þn3 m�2expfðn2þn3Þ½bm1=ð1þam1Þ�ðdþoÞ�tg,hence, we have IðtÞZmu

2 for tA ðt�,etÞ. For t4et , the same

arguments can be continued since IðetÞZm�2.

Case (II): t�ant,nAZþ . Then IðtÞZm�2 for tA ½t1,t�� and

I(t*)¼m2*, suppose t�A ðn4t,ðn4þ1ÞtÞ,n4AZþ . There are two

possible cases for tAðt�,ðn4þ1ÞtÞ.Case (IIa): IðtÞrm�2 for all tAðt�,ðn4þ1ÞtÞ. We claim that

there must be a tu2A ½ðn4þ1Þt,ðn4þ1Þtþtu� such that Iðtu2Þ4m�2.

Otherwise consider (3.2) with zððn4þ1Þtþ ÞrSððn4þ1Þtþ Þ,we get

zðtÞ ¼ zððn4þ1Þtþ Þexpf�ðm�2bþmÞðt�ðn4þ1Þtg,

¼ ezðtÞþðzððn4þ1Þtþ Þ�ez0Þexpf�ðm�2bþmÞðt�ðn4þ1ÞtÞg,

for tAðnt,ðnþ1Þt� and n4þ1rnrn4þ1þn2þn3. A similar argu-

ment as in step case (I) we get

Iððn4þ1þn2þn3ÞtÞZ Iððn4þ1þn2ÞtÞZn3 : ð3:12Þ

(3.1)

0.05

0.1

0.15

0.2

0.25

0.3I

5t

(3.2)

0.1

0.15

0.2

0.25

0.3

I

0 5t

1 2 3 4

0 1 2 3 4

Fig. 3. Illustration of basic behavior of solutions of (3.4) for the boundary value

problem, where parameters are fixed as follows r¼ 1, b¼ 1:2, a¼ 1, m¼ 0:2,

o¼ 0:1, d¼0.3, ET¼0.1 the initial value is (S0, I0)¼(0.5, 0.3). (a) Time series of I(t)

with t¼ 1, r¼0.5; (b) time series of I(t) with t¼ 0:7145, r¼0.4. Suppose the unit of

the plants density S and I is thousand and the unit of time is day.


Since IðtÞrm�2 for tAðt�,ðn4þ1ÞtÞ, (3.11) holds on ½t�,ðn4þ1þ

n2þn3Þt�, so we have

Iððn4þ1þn2ÞtÞZð1�rÞn2 m�2exp ðn2þ1Þbm1

1þam1�ðdþoÞ

� �t

� �:

Thus, Iððn4þ1þn2þn3ÞtÞZ ð1�rÞn2 m�2expfðn2þ1Þ½bm1=ð1þam1Þ�

ðdþoÞ�tgZn3 4m�2, which is a contradiction. Let t ¼ inf t4 t�

fIðtÞ4m�2g, then IðtÞrm�2 for tAðt�,tÞ and IðtÞ ¼m�2. For tAðt�,tÞ,

suppose tAðn4tþðku�1Þt,n4tþkut�, kuAN,kur1þn2þn3, we have

IðtÞZ ð1�rÞku�1m�2exp ku

bm1

1þam1�ðdþoÞ

� �t

� �Z ð1�rÞn2þn3 m�2exp ð1þn2þn3Þ

bm1

1þam1�ðdþoÞ

� �t

� �:

Let

m2 ¼ ð1�rÞn2þn3 m�2expfð1þn2þn3Þ½bm1=ð1þam1Þ�ðdþoÞ�tgomu

2, hence, we have IðtÞZm2 for tAðt�,etÞ. For t4et , the same

arguments can be continued since IðetÞZm2.

Case (IIb): There exists a tA ðt�,ðn4þ1ÞtÞ such that IðtÞ4m�2. Let

t�� ¼ inf t4 t� fIðtÞ4m�2g, then IðtÞrm�2 for tAðt�,t��Þ and Iðt��Þ ¼m�2.

For tAðt�,t��Þ, (3.11) holds true, integrating (3.11) on (t*,t**),

we have

IðtÞZ Iðt�Þexpbm1

1þam1�ðdþoÞ

� �ðt�t�Þ

� �Zm�2exp

bm1

1þam1�ðdþoÞ

� �t

� �4m�24m2:

Since Iðt��ÞZm�2 for t4t��, the same arguments can be continued.

Hence IðtÞZm2 for all tZt1. The proof is completed. &

4. Positive periodic solution and bifurcation

By Theorems 3.0 and 3.1, we know that if R1o1, then thedisease-free periodic solution ðgSðtÞ,0Þ of system (1.2) is stable, andif R141 then the disease-free periodic solution ðgSðtÞ,0Þ of system(1.2) is not stable and the system is permanent. In the following,we shall study the loss of stability phenomenon and prove that itis due to the onset of nontrivial periodic solutions obtained via asupercritical bifurcation in the limiting case, that is, R1¼1. Hence,we shall employ a fixed point argument. We denote by Fðt,U0Þ thesolution of the (unperturbed) system consisting of the first twoequations of (1.2) for the initial state U0¼(u0

1, u02), and

F¼ ðF1,F2Þ. We define the mapping Y1,Y2 : R2-R2 by

Y1ðx1,x2Þ ¼ x1þr, Y2ðx1,x2Þ ¼ ð1�rÞx2

and the mapping F1,F2 : R2-R2 by

F1ðx1,x2Þ ¼�bx1x2

1þax1�mx1þox2, F2ðx1,x2Þ ¼

bx1x2

1þax1�ðdþoÞx2:

Furthermore, let us define C : ½0,1Þ � R2-R2 by

Cðt,U0Þ ¼YðFðt,U0ÞÞ, Cðt,U0Þ ¼ ðC1ðt,U0Þ,C2ðt,U0ÞÞ,

where Y¼ ðY1,Y2Þ. It is easy to see that C is actually thestroboscopic mapping associated to the system (1.2), which putsin correspondence the initial data at 0+ with the subsequent stateof the system Cðtþ ,U0Þ at tþ , where t is the stroboscopic timesnapshot. We reduce the problem of finding a periodic solution of(1.2) to a fixed problem. Here, U is a periodic solution of period tfor (1.2) if and only if its initial value U(0)¼U0 is a fixed point forCðt,�Þ. In order to establish the existence of nontrivial periodicsolutions of (1.2), we need to prove the existence of the nontrivialfixed points of C.

Next, we deal with problem of the bifurcation of nontrivialperiodic solution of system (1.2), near ðgSðtÞ,0Þ. Assume thatX0¼(x0,0) is a starting point for the trivial periodic solutionðgSðtÞ,0Þ, where x0 ¼

eSð0þ Þ. To find a nontrivial periodic solution ofperiod T with initial value X, we need to solve the fixed pointproblem X ¼CðT ,XÞ, or denoting T ¼ tþeT ,X ¼ X0þ

eX , X0þeX ¼

CðtþeT ,X0þeX Þ.

Define

NðeT ,eX Þ ¼ X0þeX�CðtþeT ,X0þ

eX Þ ¼ ðN1ðeT ,eX Þ,N2ð

eT ,eX ÞÞ: ð4:1Þ

At the fixed point NðeT ,eX Þ ¼ 0, we denote

DXNð0,ð0,0ÞÞ ¼au

0 bu

0

cu0 du

0

" #:

By formally deriving the equation

d

dtFðt,X0Þ ¼ FðFðt,X0ÞÞ:

This characterized the dynamics of the unperturbed flowassociated to the first two equations in (1.2), one obtains that

d

dt½DXFðt,X0Þ� ¼DXFðFðt,X0ÞÞDXFðt,X0Þ: ð4:2Þ


This relation will be integrated in what follows in order tocompute the components of DXFðt,X0Þ explicitly. Firstly, it is clearthat Fðt,X0Þ ¼ ðF1ðt,X0Þ,0Þ. Then we deduce that (4.2) takes theparticular form

d

dt

@F1

@x1

@F1

@x2

@F2

@x1

@F2

@x2

2666437775ðt,X0Þ ¼

�m o� bgSðtÞ1þagSðtÞ

0bgSðtÞ

1þagSðtÞ�d�o

2666664

3777775@F1

@x1

@F1

@x2

@F2

@x1

@F2

@x2

2666437775ðt,X0Þ:

ð4:3Þ

Then the initial condition for (4.3) at t¼0 is

DXFð0,X0Þ ¼ E2, ð4:4Þ

where E2 is the identity matrix in M2(R). This implies the initialcondition @F2ð0,X0Þ=@x1 ¼ 0. It follows that

@F2ðt,X0Þ

@x1¼ exp

Z t

0

bgSðyÞ1þagSðyÞ �d�o !

dy

!@F2ð0,X0Þ

@x1¼ 0, for tZ0:

ð4:5Þ

From (4.3) one obtains that

d

dt

@F1ðt,X0Þ

@x1

� �¼�m @F1ðt,X0Þ

@x1

� �,

d

dt

@F1ðt,X0Þ

@x2

� �¼�m @F1ðt,X0Þ

@x2þ o� bgSðtÞ

1þagSðtÞ !

@F2ðt,X0Þ

@x2,

R1-tau

0.6

0.7

0.8

0.9

1

1.1

0.5tau

(4.1)

1 1.5 2 2.5 3

(4.3)

Fig. 4. The relationship between the parameters and R1. (a) R1�t; (b) R1�r; (c) R1�t and

of time is day.

d

dt

@F2ðt,X0Þ

@x2

� �¼


@F2ðt,X0Þ

@x2:

According to the initial condition, we have

@F1ðt,X0Þ

@x1¼ expð�mtÞ,

@F1ðt,X0Þ

@x2¼ expð�mtÞ

Z t

0o� bgSðsÞ

1þagSðsÞ !

@F2

@x2ðs,X0ÞexpðmsÞds,

@F2ðt,X0Þ

@x2¼ exp

Z t

0

bgSðsÞ1þagSðsÞ �d�o !

ds

!:

From (4.1), one obtains that

DXNð0,ð0,0ÞÞ ¼ E2�BDXFðt,X0Þ:

This implies

DXNð0,ð0,0ÞÞ ¼au

0 bu

0

0 du

0

" #,

where

au

0 ¼ 1�expð�mtÞ40,

bu

0 ¼�expð�mtÞ

Z t

0o� bgSðsÞ

1þagSðsÞ !

@F2

@x2ðs,X0ÞexpðmsÞds,

R1-r

0.5

1

1.5

2

2.5

0r

(4.2)

(4.4)

0.2 0.4 0.6 0.8

R1-omega

0.6

0.7

0.8

0.9

1

1.1

1.2

omega0 0.2 0.4 0.6 0.8 1

r; (d) R1�o. Suppose the unit of the plants density S and I is thousand and the unit

(5.1)

0.22

0.24

0.26

I


du

0 ¼ 1�ð1�rÞexp

Z t

0

bgSðsÞ1þagSðsÞ �d�o !

ds

!:

The necessary condition for the bifurcation of nontrivial periodicsolutions near ðgSðtÞ,0Þ is then det[DXN(0,(0,0))]¼0. SinceDXN(0,(0,0)) is an upper triangular matrix and 1�expð�mtÞ40,it consequently follows that du

0 ¼ 0 i.e. R1¼1 is necessary for thebifurcation. It now remains to show that this necessary conditionis also sufficient. This assertion represents the statement of thefollowing theorem, which is our main result.

Theorem 4.1. A supercritical bifurcation occurs at R1¼1, in the

sense that there is e40 such that for all 0oeoe there is a stable

positive nontrivial periodic solution of (1.2) with period tþe.

Theorem 4.1 is further proved in Appendix A.

0.14

0.16

0.18

0.2

t(5.2)

2.2

2.4

2.6

2.8

3

S

150t

(5.3)

0.14

0.16

0.18

0.2

0.22

0.24

0.26

I

2.2S

2.4 2.6 2.8 3

152 154 156 158 160 162 164

150 152 154 156 158 160 162 164


parameters are fixed as follows r¼ 1, b¼ 1:2, a¼ 1, m¼ 0:2, o¼ 0:1, d¼0.3, r¼0.5,

t¼ 1:5 the initial value is (S0, I0)¼(0.5, 0.3). (a) time series of I(t); (b) time series of

S(t); (c) phase portrait of S(t) and I(t). Suppose the unit of the plants density S and I

is thousand and the unit of time is day.

5. Numerical analysis and biological conclusions

Cultural control is one of the methods of IDM which is aimed ateither curing or reducing the epidemic plant potential to reducethe damage or encouraging the healthy growth of the host plant.The goal of plant disease management is to reduce the economicdamage caused by plant diseases. Plant disease managementprocedures are frequently determined by disease forecasting ordisease modeling. It is very important to establish economicthresholds for the plant disease management scheme and tomodel approaches for predicting epidemics of plant disease (Jegeret al., 2004; Holt and Chancellor, 1997; Zhang and Holt, 2004).Therefore, in the present paper we formulate two different plantdisease models to estimate and analyze the effects of a culturalcontrol strategy on disease control. We give the threshold R0 of theinfected plants-eradication for system (1.1) with continuouscultural control and the threshold R1 of the infected plants-eradication for system (1.2) with impulsive cultural control.Moreover, we give also Economic Threshold R2 which implies thatwe do not need to eradicate the infected plants but to remove theinfected plants under an Economic Threshold (ET). The resultsobtained here imply that we can control the period of controlstrategy application ðR1o1Þ such that the infected plant freeperiodic solution is logically stable. This means the infected plantscould be completely eradicated if we implement the culturalcontrol strategies relatively sparsely. The modeling methods andresults obtained here partially extended and generalized Fish-man’s works (Fishman and Marcus, 1984) on plant disease control,and all such models can be used to develop and investigate morerealistic plant disease models. Furthermore, we obtain theEconomic Thresholds R2 and R3 which mean not to eradicate theinfected plants but to remove the infected plants under anEconomic Threshold (ET) in a finite time and an infinite time.

At first, we formulate and analyze the plant disease modelincluding continuous cultural control such as curing, replantingand/or removing diseased plants. The global asymptotical stabilityof disease-free equilibrium and positive equilibrium is investi-gated. The threshold ðR0o1Þ of the infected plants-eradication forsystem (1.1) with continuous cultural control is obtained, which isalso demonstrated in Figs. 1a and b, here R0 ¼ 0:8333o1. Thismeans the infected plants could be completely eradicated if weimplement the cultural control strategies relatively sparsely.Fig. 1c shows that when R0 ¼ 1:25041, positive equilibriumE*(S*,I*) is globally asymptotically stable. Secondly, we havedeveloped and extended the epidemic model with continuouscultural control to include the impulsive cultural control strategysuch as replanting and/or removing diseased plants. We investi-gate the stability of the infected plant free periodic solution andthe permanence of system (1.2). From Theorems 3.0 and 3.3, we

obtain that the disease-free periodic solution ðgSðtÞ,0Þ is locallyasymptotically stable (in Fig. 2) if R1o1, and that the system (1.2)is permanent if R141. Therefore, R1¼1 is a critical value. Usingthe bifurcation theorem, we show that once a threshold conditionis reached, a stable nontrivial periodic solution emerges via a


supercritical bifurcation, which is confirmed in Fig. 5. If keepingmodel (1.1)’s notations, r should be substituted by rt in model(1.2). Similarly, r should be substituted by 1�e�rt. This would thenallow one to compare the two models. From Figs. 1 and 2,althought¼ 0:5 in Fig. 2 under the other same parameters, we can computethe R0 ¼ 0:83339o1 and R1 ¼ 0:83338o1, which theoreticallyshows impulsive removing diseased plants is more efficient andmore economical than continuous removing. Theorem 3.2 meansthat if R2r1 or pulse times nZn�, then the infected plants arecontrolled under an Economic Threshold (ET) at fixed time t¼T0,which is also demonstrated in Fig. 3. In Fig. 1b, whenIðT0ÞrET ¼ 0:1, if r¼0.5, then n*¼5 and if r¼0.4, then n*¼7.This implies that when the infected plants are controlled under anEconomic Threshold (ET) at fixed time t¼T0, the amount of onetime impulsive removing diseased plants is larger, the impulsiveremoving times are fewer. In Fig. 4, we can easily see that R1 isdirectly proportional to t value and inversely proportional to the r

and o values, which implies to reduce the impulsive periodof removing diseased plants or increase the amount of impulsiveremoving diseased plants or increase the amount of curedplants in order to control the infected plants under an EconomicThreshold. Theorems 3.3 and 4.1 show the system (1.2) is per-manent and has positive periodic solution, here R1 ¼ 1:07041.This is confirmed in Fig. 5.

In conclusion, we give a mathematical conclusion as follows:

(I)
Impulsive removing diseased plants is more efficient andmore economical than continuous removing.
(II)
One can appropriately control the amount of impulsiveremoving diseased plants and the impulsive period ofremoving to control infected plants under an EconomicThreshold in a finite time and an infinite time.
Acknowledgements

We would like to thank the referees and the editor for theircareful reading of the original manuscript and many valuablecomments and suggestions that greatly improved the presenta-tion of this paper.

Appendix A

The proof of Theorem 4.1. According to the above notations, weobtain that

dimðKer½DXNð0,ð0,0ÞÞ�Þ ¼ 1

and a basis ð�bu

0=au

0,1Þ in Ker[DXN(0,(0,0))]. Then the equationNðeT ,eX Þ ¼ 0 is equivalent to N1ð

eT ,gY0þwE0Þ ¼ 0,N2ðeT ,gY0þwE0Þ ¼ 0,

here E0 ¼ ð1,0Þ,Y0 ¼ ð�bu

0=au

0,1Þ,eX ¼ gY0þwE0 represents the directsum decomposition of eX using the projections ontoKer[DXN(0,(0,0))] (the central manifold) and Im[DXN(0,(0,0))] (thestable manifold).

Define

f1ðeT ,g,wÞ ¼N1ð

eT ,gY0þwE0Þ,f2ðeT ,g,wÞ ¼N2ð

eT ,gY0þwE0Þ:

Then

@f1

@wð0,0,0Þ ¼

@N1

@x1ð0,ð0,0ÞÞ ¼ au

0a0:

Hence, by the implicit function theorem, one may solve the

equation f1ðeT ,g,wÞ ¼ 0 near (0,0,0) with respect to w as a function

of eT and g, and find w¼wðeT ,gÞ such that w(0,0)¼0 and

f1ðeT ,g,wðeT ,gÞÞ ¼N1ð

eT ,gY0þwðeT ,gÞE0Þ ¼ 0:

Hence, we have

@w

@g ð0,0Þ ¼@N1ð0,0Þ

@x1

� ��1 @N1ð0,0Þ

@x2þ

bu

0

au

0

¼ 0:

Then NðeT ,eX Þ ¼ 0 if and only if

f2ðeT ,gÞ ¼N2

eT , �bu

0

au

0

gþwðeT ,gÞ,g ! !

¼ 0: ðA:1Þ

Eq. (A.1) is called the determining equation and the number of its

solutions are equal to the number of periodic solutions of (1.2).

Denote

gðeT ,gÞ ¼ f2ðeT ,g,wÞ: ðA:2Þ

It is clear to see that g(0, 0)¼N2(0,(0,0))¼0. We determine the

Taylor expansion of g around (0,0). For this, we compute the first

order partial derivatives @g=@eT ð0,0Þ and @g=@gð0,0Þ.

The first order partial derivative of g with respect to g is

@g

@gðeT ,gÞ ¼ @

@gðg�C2ðtþeT ,X0þgY0þwðeT ,gÞE0ÞÞ

¼ 1�@F2

@x1ðtþeT ,X0þgY0þwðeT ,gÞE0Þ �

bu

0

au

0

þ@wðeT ,gÞ@g

!

þ@F2

@x2ðtþeT ,X0þgY0þwðeT ,gÞE0Þ

�:

Note that

@F2

@x1ðtþeT ,X0þgY0þwðeT ,gÞE0Þj

ðeT ,gÞ ¼ ð0,0Þ¼ 0

and

du

0 ¼ 1�@F2

@x2ðt,X0Þj

ðeT ,gÞ ¼ ð0,0Þ¼ 0:

Then we have @g=@gð0,0Þ ¼ 0. Similarly, the first order partial

derivative of g with respect to eT is

@g

@eT ðeT ,gÞ ¼ @

@eT ðg�C2ðtþeT ,X0þgY0þwðeT ,gÞE0ÞÞ

¼�@F2

@eT ðtþeT ,X0þgY0þwðeT ,gÞE0Þ

�@F2

@x1ðtþeT ,X0þgY0þwðeT ,gÞE0Þ:

Note that

@F2

@eT ðtþeT ,X0þgY0þwðeT ,gÞE0ÞjðeT ,gÞ ¼ ð0,0Þ

¼ 0

and

@F2

@x1ðtþeT ,X0þgY0þwðeT ,gÞE0Þj

ðeT ,gÞ ¼ ð0,0Þ¼ 0:

Hence, one obtains @g=@eT ð0,0Þ ¼ 0.

Second partial derivatives of g

Denote

A¼@2g

@eT 2ð0,0Þ, B¼

@2g

@eT@g ð0,0Þ, C ¼@2g

@g2ð0,0Þ:

Take

x1ðeT Þ ¼ tþeT , x2ð

eT ,gÞ ¼ x0�bu

0

au

0

þwðeT ,gÞ, x3ðgÞ ¼ g:

Now we calculate these quantities in terms of the parameters of

the equation.


Firstly, we have

@2g

@eT 2ðeT ,gÞ ¼ @2

@eT 2ðx3�Y23Fðx1,x2,x3ÞÞð

eT ,gÞ

¼�@2Y2

@x21

@F1ðx1,x2,x3Þ

@eT þ@F1ðx1,x2,x3Þ

@x1

@w

@eT� �2

�@2Y2

@x1@x2

@F2ðx1,x2,x3Þ

@eT @F1ðx1,x2,x3Þ


@x1

@w

@eT� �

�@2Y2

@x1@x2

@F2

@x1

@w

@eT @F1ðx1,x2,x3Þ


@x1

@w

@eT� �

�@Y2

@x1

@2F1ðx1,x2,x3Þ

@eT 2þ2

@2F1ðx1,x2,x3Þ

@x1@eT @w

@eT !

�@Y2

@x1

@2F1ðx1,x2,x3Þ

@x21

@w

@eT� �2

þ@F1ðx1,x2,x3Þ

@x1

@w

@eT !

�@2Y2

@x1@x2

@F1ðx1,x2,x3Þ

@eT @F2ðx1,x2,x3Þ


@x1

@w

@eT� �

�@2Y2

@x1@x2

@F1ðx1,x2,x3Þ

@eT @w

@eT @F2ðx1,x2,x3Þ


@x1

@w

@eT� �

�@2Y2

@x22

@F2ðx1,x2,x3Þ


@x1

@w

@eT� �2

�@Y2

@x2

@2F2ðx1,x2,x3Þ

@eT 2þ2

@2F2ðx1,x2,x3Þ

@x1@eT @w

@eT !

�@Y2

@x2

@2F2ðx1,x2,x3Þ

@x21

@w

@eT� �2

þ@F2ðx1,x2,x3Þ

@x1

@w

@eT !

:

Note that @Y2=@x1 � 0 for any ðeT ,gÞ and @2Y2=@x22 ¼ @F2=@x2 ¼

@F2=@eT ¼ @2F2=@x1@eT ¼ @F2=@x1 ¼ @2F2=@x2

1 ¼ 0 for ðeT ,gÞ ¼ ð0,0Þ,

then

A¼�ð1�rÞ@2F2ðt,x0Þ

@eT 2:

Moreover, @2F2ðt,X0Þ=@eT 2¼ 0, then we have A¼0.

Secondly, we have

@2g

@g2ðeT ,gÞ ¼� @

2Y2

@x21

@F1ðx1,x2,x3Þ

@x1�

bu

0

au

0

þ@wðeT ,gÞ@g

!þ@F1ðx1,x2,x3Þ

@x2

!2

�@2Y2

@x1@x2

@F1ðx1,x2,x3Þ

@x1�

bu

0

au

0

þ@wðeT ,gÞ@g

!�@F1ðx1,x2,x3Þ

@x2

!

�@F2ðx1,x2,x3Þ

@x2�

bu

0

au

0

þ@wðeT ,gÞ@g

!

�@2Y2

@x1@x2

@F1ðx1,x2,x3Þ

@x1�

bu

0

au

0

þ@wðeT ,gÞ@g

! !@F2ðx1,x2,x3Þ

@x2

�@2Y2

@x1@x2

@F1ðx1,x2,x3Þ

@x2

@F2ðx1,x2,x3Þ

@x2

�@Y2

@x1

@2F1ðx1,x2,x3Þ

@x21

�bu

0

au

0

þ@wðeT ,gÞ@g

!2

�2@Y2

@x1

@2F1ðx1,x2,x3Þ

@x1@x2�

bu

0

au

0

þ@wðeT ,gÞ@g

!

�@Y2

@x1

@F1ðx1,x2,x3Þ

@x1

@2wðeT ,gÞ@g2

þ@2F1ðx1,x2,x3Þ

@x22

!

�@2Y2

@x1@x2

@F2ðx1,x2,x3Þ

@x1�

bu

0

au

0

þ@wðeT ,gÞ@g

!�@F2ðx1,x2,x3Þ

@x2

!

�@F1ðx1,x2,x3Þ

@x2�

bu

0

au

0

þ@wðeT ,gÞ@g

!

�@2Y2

@x1@x2

@F2ðx1,x2,x3Þ

@x1�

bu

0

au

0

þ@wðeT ,gÞ@g

! !@F1ðx1,x2,x3Þ

@x2

�@2Y2

@x1@x2

@F2ðx1,x2,x3Þ

@x2

@F1ðx1,x2,x3Þ

@x2

�@2Y2

@x21

@F2ðx1,x2,x3Þ

@x1�

bu

0

au

0

þ@wðeT ,gÞ@g

!þ@F2ðx1,x2,x3Þ

@x2

!2

�@Y2

@x2

@2F2ðx1,x2,x3Þ

@x21

�bu

0

au

0

þ@wðeT ,gÞ@g

!2

�2@Y2

@x2

@2F2ðx1,x2,x3Þ

@x1@x2�

bu

0

au

0

þ@wðeT ,gÞ@g

!

�@Y2

@x2

@F2ðx1,x2,x3Þ

@x1

@2wðeT ,gÞ@g2

þ@2F2ðx1,x2,x3Þ

@x22

!:

Since @wð0,0Þ=@g¼ 0, then C ¼ ð1�rÞ½2ðbu

0=au

0Þð@2F2ðt,X0Þ=@x1@x2Þ�

@2F2ðt,X0Þ=@x22�. Therefore, for determining C, one needs to

calculate the terms @2F2ðt,X0Þ=@x1@x2 and @2F2ðt,X0Þ=@x22.

We calculate:

d

dt

@2F2ðt,X0Þ

@x1@x2

� �¼

bgSðtÞ1þagSðtÞ �ðdþoÞ !

@2F2ðt,X0Þ

@x1@x2

þb

ð1þagSðtÞÞ2 @F1ðt,X0Þ

@x1

@F2ðt,X0Þ

@x2:

Note that

@2F2ð0,X0Þ

@x1@x2¼ 0:

Hence we have

@2F2ðt,X0Þ

@x1@x2¼ exp

Z t

0

bgSðsÞ1þagSðsÞ �ðdþoÞ !

ds

!

�

Z t

0

b

ð1þagSðsÞÞ2 @F1ðs,X0Þ

@x1ds: ðA:3Þ

Similarly, we have

d

dt

@2F2ðt,X0Þ

@x22

!¼

bgSðtÞ1þagSðtÞ �ðdþoÞ !

@2F2ðt,X0Þ

@x22

þb

ð1þagSðtÞÞ2 @F1ðt,X0Þ

@x2

@F2ðt,X0Þ

@x2:

And note that

@2F2ð0,X0Þ

@x22

¼ 0:

Hence we have

@2F2ðt,X0Þ

@x22

¼ exp

Z t

0


ds

!

�

Z t

0

b


@x2ds: ðA:4Þ

Substituting (A.3) and (A.4) into C yields

C ¼ ð1�rÞ 2bu

0

au

0

@2F2ðt,X0Þ

@x1@x2�@2F2ðt,X0Þ

@x22

" #

¼ ð1�rÞ 2bu

0

au

0

exp

Z t

0

bgSðsÞ1þagSðsÞ �ðdþoÞ

!ds

!"

�

Z t

0

b


@x1ds

�exp

Z t

0

bgSðsÞ1þagSðsÞ �ðdþoÞ

!ds

!�

Z t

0

b


@x2ds

#:


Similarly, we also have

B¼�ð1�rÞ@2F2ðt,X0Þ

@x1@x2

1

au

0

@F1ðt,X0Þ

@eT þ@2F2ðt,X0Þ

@eT@x2

" #

¼�ð1�rÞ exp

Z t

0


ds

!"

�

Z t

0

b


@x1dsgSðtÞau

0

þbgSðtÞ

1þagSðtÞ �ðdþoÞ !

� exp

Z t

0


ds

!#

¼ ð1�rÞ �exp

Z t

0


ds

!"

�

Z t

0

b


@x1dsgSðtÞau

0

�bgSðtÞ

1þagSðtÞ �ðdþoÞ !

� exp

Z t

0


ds

!#:

Since R1¼1, i.e.Z t

0

bgSðsÞ1þagSðsÞ ds¼ ðdþoÞt:

And since

@F1

@x1o0,

bgSðtÞ1þagSðtÞ o 1

t

Z t

0

bgSðsÞ1þagSðsÞ ds,

then one obtains B40.

Then we have

A¼@2g

@eT 2ð0,0Þ ¼ 0, B¼

@2g

@eT@g ð0,0Þ40, C ¼@2g

@g2ð0,0Þ,

and then

gðeT ,gÞ ¼ BgeTþCg2

2þOðeT ,gÞðeT 2

þg2Þ:

By denoting eT ¼ hg (where h¼ hðgÞ), we obtain that (A.1) is

equivalent to

BhþCh2

2þOðhg,gÞð1þh2Þ ¼ 0: ðA:5Þ

In the following, we discuss the solutions of Eq. (A.5) with

respect to h to consider two cases:

Case I: If C ¼ @2g=@g2ð0,0Þ40, by denoting eT ¼ hg (where

h¼ hðgÞ), we obtain that (A.1) is equivalent to

Ch2

2þBhþOðhg,gÞð1þh2Þ ¼ 0: ðA:6Þ

Since B40, Eq. (A.6) is solvable with respect to h as a function of

g. Moreover, here h��2B=Co0.

Case II: If C ¼ @2g=@g2ð0,0Þo0, by denoting eT ¼ hg (where

h¼ hðgÞ). Similarly we have h��2B=C40.

This means that there is a supercritical bifurcation to a

nontrivial periodic solution near a period t which satisfies the

sufficient condition of the bifurcation for R1¼1.

It is noteworthy that since this periodic solution appears via a

supercritical bifurcation, the nontrivial periodic solution is stable.

Therefore, there exists e40 such that for all 0ogoe there is a

stable positive nontrivial periodic solution of (1.2) with period

tþeT ðgÞ which starts in X0gY0þwðeT ðgÞ,gÞE0. Here, X0, Y0, E0, w andeT are as defined above. The proof is completed. &

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