# Bifurcation Control of a Delayed Fractional Mosaic Disease Model forJatropha curcas with Farming Awareness

Authors: Shouzong Liu; Mingzhan Huang; Juan Wang
Journal: Complexity
 (2020)

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## Abstract

In this paper, the bifurcation control of a fractional-order mosaic virus infection model forJatropha curcas with farming awareness and an execution delay is investigated. By analyzing the associated characteristic equation, Hopf bifurcation induced by the execution delay is studied for the uncontrolled system. Then, a time-delayed controller is introduced to control the occurrence of Hopf bifurcation. Our study implies that bifurcation dynamics is significantly affected by the change of the fractional order, the feedback gain and the extended feedback delay provided that the other parameters are fixed. A series of numerical simulations is performed, which not only verifies our theoretical results but also reveals some specific features. Numerically, we find that the Hopf bifurcation gradually occurs in advance with the increase of the fractional order, and there exist extreme points for the feedback gain and the extended feedback delay which can minimize the bifurcation value.


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## Body

## 1. Introduction

Jatropha curcas is a plant that is commonly seen in tropical and subtropical areas. It can grow well in marginal and poor soil, grows fast, and lives relatively long. Meanwhile, it produces seeds with an oil content of about 27% to 40%. The oil burns with clear smoke-free flame and can be used to produce biodiesel fuel [1]. Therefore, as one of the most suitable alternative renewable energy resources, it is widely planted in many countries [2, 3]. However, many studies showed that Jatropha plants are hosts of mosaic virus which is a main cause of the occurrence of viruses in Jatropha curcas [4, 5]. Early reports about virus infections of Jatropha plants indicated that the occurrence of Cassava mosaic virus in India causes a high disease incidence from 25 to 47%. This constitutes a major obstacle to the large scale planting of Jatropha [6]. When the Jatropha plant is infected with mosaic virus, its fruit will be attacked and the yield and the quality of oil may be severely affected.Awareness of the disease is really necessary for farmers who plantJatropha curcas because they can take action to prevent or mitigate the problem in time. Effective knowledge can provide proper control measures when the virus infection outbreaks. It is advisable to educate the Jatropha curcas growers with agricultural information about agronomic practices and plant protection measures [7]. Common platforms for mass media such as newspapers, magazines, radio, television, and the Internet are main ways to propagandize correct and relevant information about this crop and its disease. There are also new technologies adopted in agricultural awareness programs [8, 9]. These are very important for the grower who is engaged in the crop production. For example, Yang et al. exploited the levels of knowledge and awareness of the side effects of insecticides to the environment and human health in regions with different farming modes and found that farmers can improve their agronomic practice to reduce environmental hazard and protect human health after properly raising awareness [10]. Le Bellec et al. studied the collaboration mechanism of growers, researchers, and other factors which can alleviate the problem we face in crop management [11]. Basir et al. proposed a mathematical model to investigate the impact of awareness programs on the protection of the Jatropha curcas plant against possible epidemic spread of the mosaic disease [3].Generally speaking, whenJatropha curcas growers realize the harm of virus infection, they need to take some time before they can take appropriate measures to prevent or reduce the occurrence of this disease. So, it is more plausible to consider an execution time delay when building such kind of virus infection model. Basir et al. analyzed how execution time delay in accepting the awareness campaign can affect the ultimate dynamics of the mosaic disease [2], while the authors in [3] investigated the Hopf bifurcation phenomenon of an epidemic model with awareness programs due to the execution time delay. Basir et al. [12] proposed a mathematical model and analyzed the effect of awareness programs on the control of pest in agricultural practice. However, these studies are all aimed at integer-order models. In this work, we further analyze the stability and Hopf bifurcation of a fractional-order virus infection model with execution delay of awareness programs.In recent decades, fractional calculus theory has been widely applied in biology, optical and thermal system, materials science, electromagnetic field theory, mechanical mechanics, and so on [13, 14]. It has been found that fractional calculus can accurately describe the rules and development process of some phenomena in natural science. Furthermore, it is found that fractional-order differential system has the advantages of simple modeling, clear parameter meaning, and accurate description for some materials and processes with memory and genetic characteristics [15–17]. Hence, fractional calculus has become a new mathematical tool favored by researchers, and more and more study of practical problems introduces the theory of fractional calculus and remarkable achievements have been made [18–30].For a given nonlinear system, a controller aiming at modifying the bifurcation behavior is usually designed, so as to achieve some desirable dynamical behaviors, which is often called bifurcation control [31, 32]. According to the different objectives of bifurcation control, different control strategies are formulated, such as PD control, time-delayed feedback control, and hybrid control. For example, Xiao et al. [22] adopted a PD control method to control Hopf bifurcations in delayed fractional-order small-world networks, while Lu et al. [32] analysed the stability and bifurcation of a fractional-order single-gene regulatory model under a PD control law. Xu et al. [28] applied two time-delayed feedback controllers to regulate a fractional-order chaotic Genesio-Tesi model, and Huang et al. [33] designed a hybrid controller for the first time to control the Hopf bifurcation of a network model.As an effective control scheme, time-delayed feedback control was proposed by constructing a control force using the difference between the current state and its delay value, i.e.,xt−xt−δ, and once the system reaches a stable state, the control force vanishes [34]. The main advantage of this scheme is that it is noninvasive and it does not require a reference system because the control is generated from the information of the system itself. Besides, this scheme can be easily implemented in the actual system. Hence, in recent years, the bifurcation controls of many dynamic systems in many fields are studied by introducing time-delayed feedback controllers.In this paper, delayed fractional mosaic disease models forJatropha curcas with farming awareness are proposed. We will study the stability and bifurcation of the system and discuss the bifurcation control by introducing a time-delayed feedback controller.This paper is organized as follows. In Section2, some basic materials regarding fractional calculus are presented. The delayed fractional mosaic disease models with and without control are proposed in Section 3. In Section 4, detailed analysis of bifurcation phenomena for the two systems is carried out. Series of numerical simulations are carried out in Section 5, which not only confirm the theoretical results we have obtained but also are complementary to those results with specific features. The paper ends with a brief conclusion.


## 2. Preliminaries

In this section, we will briefly introduce some notations and definitions about fractional calculus theory, which will be useful in the following discussion. There are several kinds of definitions of fractional derivatives proposed in previous research, such as Riemann–Liouville fractional derivative, Grünwald–Letnikov fractional derivative, Caputo fractional derivative, Weyl fractional derivative, and Marchaud fractional derivative [35]. It is important to note that Caputo fractional derivative has the advantage of requiring the initial conditions to be easily derived from the controlled system, which make it easier to apply to practical problems [32]. Therefore, in this paper, we only study Caputo fractional derivative.Definition 1.
(see [36]). The Caputo fractional-order derivative with fractional order α for a continuous function pt:R+⟶Rn is defined by(1)DCt0,tαpt=1Γm−α∫t0tt−τm−α−1pmτdτ,where 0≤m−1≤α<m,m∈Z+ and Γ· is the Gamma function. The constant α is the value of the fractional order.
Especially, when0<α≤1, DCt0,tαpt=1/Γm−α∫t0tt−ταp′τdτ.Remark 1.
Our work is based on the Caputo derivative, and for convenience, in this work, we denote the Caputo fractional-order derivative operatorDCt0,tαpt by Dαpt and suppose that α∈0,1.Definition 2.
(see [37]). Consider the following n-dimension fractional-order system:(2)Dαxt=fxt,where xt=x1t,x2t,…,xnt and ft=f1xt,f2xt,…,fnxt. The equilibrium point x∗=x1∗,x2∗,…,xn∗ is defined by the algebraic equation fix1,x2,…,xn=0,i=1,2,…,n.The stability of the solution of the aboven-dimension system is defined in a lot of literature studies. For more details, readers can refer to [38].


## 3. Model Formulation

The authors in [2] studied an integral order model of virus infection for Jatropha curcas with farming awareness. An execution delay is incorporated, which measures from the moment that the relevant information is available to the moment that farmers take action with these knowledge. They mainly explored the impact of the time delay on the dynamical behaviors.Motivated by the work in [2], in this paper, we consider the virus infection model with fractional-order and the execution delay; then, we propose the following delayed fractional-order virus infectious model:(3)DαPSt=rPSt1−PSt+PItK−aPStVt,DαPIt=aPStVt−mPIt−nMt−τPIt,DαVt=bPIt−cVt−dMt−τVt,DαMt=γ+βPIt−ηMt,with PS0≥0, PI0≥0, V0≥0, M0≥0, and 0<α≤1.The interpretations of the variables and parameters are listed in Table1.Table 1
Descriptions of parameters in (3).
ParameterDescriptionPStThe healthy plant biomass at timetPItThe infected plant biomass at timetVtThe number of infected vectors at timetMtThe density of aware people at timetrThe intrinsic growth rate of healthy plantsKThe carrying capacity of healthy plantsaThe contact rate between healthy plants and infected vectorsmThe disease-induced death rate of infected plantsnThe harvesting rate of infected plants by aware peoplebThe appearance rate of infected vectorcThe death rate of infected vectorsdThe removal rate of vectors by awareγThe rate of global awarenessβThe recruitment rate of aware peopleηThe fading rate of awarenessτThe execution delayTo control the bifurcation caused by the execution delay, we introduce a time-delayed feedback controller as follows:(4)kPSt−PSt−δ.Then, we gain the following controlled system:(5)DαPSt=rPSt1−PSt+PItK−aPStVt+kPSt−PSt−δ,DαPIt=aPStVt−mPIt−nMt−τPIt,DαVt=bPIt−cVt−dMt−τVt,DαMt=γ+βPIt−ηMt.Remark 2.
k<0 is the negative feedback gain and δ>0 stands for the delay of feedback control. In the field of ecological control, with the aim of enhancing the stability performance, the farmers may harvest or transplant some plants on the basis of past data PSt−δ.We will study the existence of Hopf bifurcation and explore the impact of time delay, the fractional order, the feedback gain, and the feedback delay on the occurrence of the bifurcation.


## 4. Main Results

In this section, we firstly study the stability of the coexistence equilibrium and the Hopf bifurcation caused by the time delay for the uncontrolled system (3). Then, the impact of the fractional order and the feedback control on the occurrence of the bifurcation is investigated for the controlled system (5).The system (3) has three equilibria: the plant-vector-free equilibrium E1=0,0,0,γ/η, the disease-free equilibrium E2=K,0,0,γ/η, and the coexistence equilibrium E∗=PS∗,PI∗,V∗,M∗. By direct calculation, we can easily obtain(6)PS∗=m+nM∗c+dM∗ab,PI∗=ηM∗−γβ,V∗=bPI∗c+dM∗,and M∗ is the positive root of the following equation:(7)fx=a1x3+a2x2+a3x+a4=0,where(8)a1=βrnd2>0,a2=rdadη+2cβn+βdm>0,a3=βr2cdm+c2n−kabd+abrcη−dγ+Kaη,a4rβcmc−Kab−abγrc+Ka.Obviously, ifa4<0, the above cubic polynomial equation has a unique positive root which is denoted by x=M∗. Then, we give the following result.Proposition 1.
Assume that(9)H1:a4<0andηM∗>γ.
Then, system (3) has a unique coexistence equilibrium E∗=PS∗,PI∗,V∗,M∗.For convenience, let(10)m1=m+nM∗,m2=c+dM∗,m3=m1+m2,m4=m+nM∗+aV∗,m5=m+nM∗+rKPS∗,m6=m1m2,and these denotations will be used in the following discussion.


### 4.1. Bifurcation Analysis of the Uncontrolled System (3)

In this section, by choosing the execution delayτ as a bifurcation parameter, we analyse the existence of Hopf bifurcation and the critical value of the time delay at which a Hopf bifurcation occurs is obtained.By linearizing (3) at E∗, we get the following Jacobian matrix:(11)JE∗=−rKPS∗−rKPS∗−aPS∗0aV∗−m−nM∗aPS∗−nPI∗e−sτ0b−c−dM∗−dV∗e−sτ0β0−η.Thus, we have the following characteristic equation:(12)sα+rKPS∗rKPS∗aPS∗0−aV∗sα+m+nM∗−aPS∗nPI∗e−sτ0−bsα+c+dM∗dV∗e−sτ0−β0sα+η=0.Through simplification, we can obtain(13)s4α+A1s3α+A2s2α+A3sα+A4+A5s2α+A6sα+A7e−sτ=0,where(14)A1=η+m2+m5,A2=m6+rKPS∗m2+m4−abPS∗+ηm2+m5,A3=m2m4rKPS∗−abPS∗rKPS∗−aV∗+ηm2m5+m4rKPS∗−abPS∗,A4=ηm2m4rKPS∗−abPS∗rKPS∗−aV∗,A5=βnPI∗,A6=βnPI∗m2+rKPS∗+adPS∗V∗,A7=βPS∗m2nrKPI∗+adV∗rKPS∗−aV∗.Suppose thatτ=0, then equation (13) becomes(15)s4α+A1s3α+A2+A5s2α+A3+A6sα+A4+A7=0.Assume that(16)H2:A4+A7>0,A1A2+A5−A3+A6>0,A1A2+A5−A3+A6A3+A6−A12A4+A7>0.Then, by the Routh–Hurwitz criterion, we have the following result.Proposition 2.
Forτ=0, if assumptions H1 and H2 are satisfied, then the roots of (15) are real and negative. Therefore, the coexistence equilibrium E∗ is locally asymptotically stable.To study the phenomenon of Hopf bifurcation, we assume thats=ωiω>0 is a root of (13), then, we can have(17)ω4αcos2απ+i2sinαπ+A1ω3αcos3απ2+isin3απ2+A2ω2αcosαπ+isinαπ+A3ωαcosαπ2+isinαπ2+A4+A5ω2αcosαπ+isinαπ+A6ωαcosαπ2+isinαπ2+A7cosωτ−isinωτ=0.Separating the real and imaginary parts, we obtain(18)C1cosωτ+C2sinωτ=−C3,C2cosωτ−C1sinωτ=−C4,where(19)C1=A5ω2αcosαπ+A6ωαcosαπ2+A7,C2=A5ω2αsinαπ+A6ωαsinαπ2,C3=ω4αcos2απ+A1ω3αcos3απ2+A2ω2αcosαπ+A3ωαcosαπ2+A4,C4=ω4αsin2απ+A1ω3αsin3απ2+A2ω2αsinαπ+A3ωαsinαπ2.It follows from (18) that(20)sinωτ=C1C4−C2C3C12+C22,cosωτ=−C1C3+C2C4C12+C22.Sincesin2ωτ+cos2ωτ=1, we can obtain(21)C32+C42=C12+C22.By simple calculation, we deduce(22)λ8+2A1cosαπ2λ7+A12+2A2cosαπλ6+2A3cos3απ2+2A1A2cosαπ2λ5+A22+2A1A3cosαπ+2A4cos2απ−A52λ4+2A1A3−A5A6cosαπ2+2A1A4cos3απ2λ3+A32+2A2A4−A5A7cosαπ−A62λ2+2A3A4−A6A7cosαπ2λ+A42−A72=0,where λ=ωα.Assume that(23)H3:A42−A72<0.Then, (18) has at least one positive real root ω0. Denote(24)τi=1ω0arccosC1C3+C2C4C12+C22+2iπ,i=0,1,2,….Define(25)τ0=minτi,i=0,1,2,….To derive conditions for the existence of Hopf bifurcation, we make the following hypothesis:(26)H4:Φ1Ψ1+Φ2Ψ2Ψ12+Ψ22≠0,where Φ1,Φ2,Ψ1,andΨ2 are defined by equation (30).Lemma 1.
Letsτ=φτ+iωτ be a root of the characteristic equation (13) near τ=τi meeting φτi=0,ωτi=ω0; then, the transversality condition(27)Redsdττ=τ0,ω=ω0≠0,holds.Proof.
Differentiating both sides of (13) with respect to τ, we obtain(28)dsdτ=ΦsΨs,where Φs=A5s2α+A6sα+A7se−sτ and(29)Ψs=4αs4α−1+3αA1s3α−1+2αA2s2α−1+αA3sα−1+2αA5s2α−1+αA6sα−1−A5s2α+A6sα+A7τe−sτ.
Let(30)Φω0iτ=τ0=Φ1+iΦ2,Ψω0iτ=τ0=Ψ1+iΨ2,where(31)Φ1=n1sinω0τ0−n2cosω0τ0,Φ2=n1cosω0τ0+n2sinω0τ0,Ψ1=n3+n5cosω0τ0+n6sinω0τ0,Ψ2=n4−n5sinω0τ0+n6cosω0τ0,and ni,i=1,2,…,6 are defined by Appendix A.
By straightforward computation, it can be derived from (28) that(32)Redsdττ=τ0,ω=ω0=Φ1Ψ1+Φ2Ψ2Ψ12+Ψ22≠0.
The proof is completed.Theorem 1.
Suppose assumptionsH1−H4 hold; then,(1)
The coexistence equilibriumE∗ of system (3) is locally asymptotically stable for τ∈0,τ0(2)
System (3) undergoes Hopf bifurcation at E∗ when τ=τ0


### 4.2. Bifurcation Control of System (5)

In this section, we exploit the bifurcation control problem of system (5) through the time-delayed feedback controller.Analogously, by linearizing (5) at E∗, we get the Jacobian matrix:(33)JE∗=−rKPS∗+k1−e−sδ−rKPS∗−aPS∗0aV∗−m−nM∗aPS∗−nPI∗e−sτ0b−c−dM∗−dV∗e−sτ0β0−η,and the corresponding characteristic equation:(34)s4α+B1s3α+B2s2α+B3sα+B4+B5s2α+B6sα+B7e−sτ=0,where(35)B1=A1−k1−e−sδ,B2=A2−m3+ηk1−e−sδ,B3=A3−m6+m3η−abPS∗k1−e−sδ,B4=A4−ηm6−abPS∗k1−e−sδ,B5=A5,B6=A6−βnPI∗k1−e−sδ,B7=A7−m2nPI∗+adPS∗V∗βk1−e−sδ.Assume thats=ωiω>0 is a root of (11); then, we obtain(36)ω4αcos2απ+isin2απ+A1−k1−cosδω+isinδωω3αcos3απ2+isin3απ2+A2−m3+ηk1−cosδω−im3+ηksinδωω2αcosαπ+isinαπ+A3−m6+m3η−abPS∗k1−cosδω−im6+m3η−abPS∗ksinδωωα×cosαπ2+isinαπ2+A4−ηm6−abPS∗k1−cosδω−isinδω+A5ω2αcosαπ+isinαπ+A6−βnPI∗k1−cosδω−iβnPI∗ksinδω×ωαcosαπ2+isinαπ2+A7−m2nPI∗+adPS∗V∗βk1−cosδω−im2nPI∗+adPS∗V∗βksinδωcosωτ−isinωτ=0.Separating the real and imaginary parts, we obtain(37)D1cosωτ+D2sinωτ=−D3,D2cosωτ−D1sinωτ=−D4,where(38)D1=φ1cosαπ+φ2cosαπ2+φ3sinαπ2+φ4,D2=φ1sinαπ+φ2sinαπ2−φ3cosαπ2−φ5,D3=ω4αcos2απ+φ6cos3απ2+φ7sin3απ2+φ8cosαπ+φ9sinαπ+φ10cosαπ2+φ11sinαπ2+φ12,D4=ω4αsin2απ+φ6sin3απ2−φ7cos3απ2+φ8sinαπ−φ9cosαπ+φ10sinαπ2−φ11cosαπ2−φ13,and φi,i=1,2,…,13 are defined by Appendix B.It follows from (40) that(39)sinωτ=D1D4−D2D3D12+D22,cosωτ=−D1D3+D2D4D12+D22.Similarly, we can obtain(40)D32+D42=D12+D22.Assume that equation (40) has at least one positive real root ω1; then, it follows from the second equation of (39) that(41)τδi=1ω1arccosD1D3+D2D4D12+D22+2iπ,i=0,1,2,….Define(42)τδi=minτδi,i=0,1,2,….To derive conditions for the existence of Hopf bifurcation, we make the following hypothesis:(43)H4:Φ1Ψ1+Φ2Ψ2Ψ12+Ψ22≠0,whereΦ˜1,Φ˜2,Ψ˜1,andΨ˜2 are defined by equation (47).Lemma 2.
Letsτ=φτ+iωτ be a root of the characteristic equation (34) near τ=τδi meeting φτδi=0,ωτδi=ω1; then, the transversality condition(44)Redsdττ=τδ0,ω=ω1≠0,holds.Proof.
Differentiating both sides of (39) with respect to τ, we obtain(45)dsdτ=Φ˜sΨ˜s,where Φ˜s=B5s2α+B6sα+B7se−sτ and(46)Ψ˜s=4αs4α−1+3αB1s3α−1+2αB2s2α−1+αB3sα−1+2αB5s2α−1+αB6sα−1−B5s2α+B6sα+B7τe−sτ−kδe−sδs3α+m3+ηs2α+m6+m3η−abPS∗sα+ηm6−abPS∗+βnPI∗sα+m2nPI∗+adPS∗V∗βe−sτ.
Let(47)Φ˜ω1iτ=τδ0=Φ˜1+iΦ˜2,Ψω1iτ=τδ0=Ψ˜1+iΨ˜2,where Φ˜1andΦ˜2 are the real and imaginary parts of Φ˜ω1i, respectively, while Ψ˜1andΨ˜2 are the real and imaginary parts of Ψ˜ω1i, respectively.
By straightforward computation, it can be derived from (45) that(48)Redsdττ=τδ0,ω=ω1=Φ˜1Ψ˜1+Φ˜2Ψ˜2Ψ˜12+Ψ˜22≠0.
The proof is completed.Theorem 2.
Suppose assumptionsH1,H2,andH5 hold; then,(1)
The coexistence equilibriumE∗ of system (5) is locally asymptotically stable for τ∈0,τδ0(2)
System (5) undergoes Hopf bifurcation at E∗ when τ=τδ0


## 4.1. Bifurcation Analysis of the Uncontrolled System (3)

In this section, by choosing the execution delayτ as a bifurcation parameter, we analyse the existence of Hopf bifurcation and the critical value of the time delay at which a Hopf bifurcation occurs is obtained.By linearizing (3) at E∗, we get the following Jacobian matrix:(11)JE∗=−rKPS∗−rKPS∗−aPS∗0aV∗−m−nM∗aPS∗−nPI∗e−sτ0b−c−dM∗−dV∗e−sτ0β0−η.Thus, we have the following characteristic equation:(12)sα+rKPS∗rKPS∗aPS∗0−aV∗sα+m+nM∗−aPS∗nPI∗e−sτ0−bsα+c+dM∗dV∗e−sτ0−β0sα+η=0.Through simplification, we can obtain(13)s4α+A1s3α+A2s2α+A3sα+A4+A5s2α+A6sα+A7e−sτ=0,where(14)A1=η+m2+m5,A2=m6+rKPS∗m2+m4−abPS∗+ηm2+m5,A3=m2m4rKPS∗−abPS∗rKPS∗−aV∗+ηm2m5+m4rKPS∗−abPS∗,A4=ηm2m4rKPS∗−abPS∗rKPS∗−aV∗,A5=βnPI∗,A6=βnPI∗m2+rKPS∗+adPS∗V∗,A7=βPS∗m2nrKPI∗+adV∗rKPS∗−aV∗.Suppose thatτ=0, then equation (13) becomes(15)s4α+A1s3α+A2+A5s2α+A3+A6sα+A4+A7=0.Assume that(16)H2:A4+A7>0,A1A2+A5−A3+A6>0,A1A2+A5−A3+A6A3+A6−A12A4+A7>0.Then, by the Routh–Hurwitz criterion, we have the following result.Proposition 2.
Forτ=0, if assumptions H1 and H2 are satisfied, then the roots of (15) are real and negative. Therefore, the coexistence equilibrium E∗ is locally asymptotically stable.To study the phenomenon of Hopf bifurcation, we assume thats=ωiω>0 is a root of (13), then, we can have(17)ω4αcos2απ+i2sinαπ+A1ω3αcos3απ2+isin3απ2+A2ω2αcosαπ+isinαπ+A3ωαcosαπ2+isinαπ2+A4+A5ω2αcosαπ+isinαπ+A6ωαcosαπ2+isinαπ2+A7cosωτ−isinωτ=0.Separating the real and imaginary parts, we obtain(18)C1cosωτ+C2sinωτ=−C3,C2cosωτ−C1sinωτ=−C4,where(19)C1=A5ω2αcosαπ+A6ωαcosαπ2+A7,C2=A5ω2αsinαπ+A6ωαsinαπ2,C3=ω4αcos2απ+A1ω3αcos3απ2+A2ω2αcosαπ+A3ωαcosαπ2+A4,C4=ω4αsin2απ+A1ω3αsin3απ2+A2ω2αsinαπ+A3ωαsinαπ2.It follows from (18) that(20)sinωτ=C1C4−C2C3C12+C22,cosωτ=−C1C3+C2C4C12+C22.Sincesin2ωτ+cos2ωτ=1, we can obtain(21)C32+C42=C12+C22.By simple calculation, we deduce(22)λ8+2A1cosαπ2λ7+A12+2A2cosαπλ6+2A3cos3απ2+2A1A2cosαπ2λ5+A22+2A1A3cosαπ+2A4cos2απ−A52λ4+2A1A3−A5A6cosαπ2+2A1A4cos3απ2λ3+A32+2A2A4−A5A7cosαπ−A62λ2+2A3A4−A6A7cosαπ2λ+A42−A72=0,where λ=ωα.Assume that(23)H3:A42−A72<0.Then, (18) has at least one positive real root ω0. Denote(24)τi=1ω0arccosC1C3+C2C4C12+C22+2iπ,i=0,1,2,….Define(25)τ0=minτi,i=0,1,2,….To derive conditions for the existence of Hopf bifurcation, we make the following hypothesis:(26)H4:Φ1Ψ1+Φ2Ψ2Ψ12+Ψ22≠0,where Φ1,Φ2,Ψ1,andΨ2 are defined by equation (30).Lemma 1.
Letsτ=φτ+iωτ be a root of the characteristic equation (13) near τ=τi meeting φτi=0,ωτi=ω0; then, the transversality condition(27)Redsdττ=τ0,ω=ω0≠0,holds.Proof.
Differentiating both sides of (13) with respect to τ, we obtain(28)dsdτ=ΦsΨs,where Φs=A5s2α+A6sα+A7se−sτ and(29)Ψs=4αs4α−1+3αA1s3α−1+2αA2s2α−1+αA3sα−1+2αA5s2α−1+αA6sα−1−A5s2α+A6sα+A7τe−sτ.
Let(30)Φω0iτ=τ0=Φ1+iΦ2,Ψω0iτ=τ0=Ψ1+iΨ2,where(31)Φ1=n1sinω0τ0−n2cosω0τ0,Φ2=n1cosω0τ0+n2sinω0τ0,Ψ1=n3+n5cosω0τ0+n6sinω0τ0,Ψ2=n4−n5sinω0τ0+n6cosω0τ0,and ni,i=1,2,…,6 are defined by Appendix A.
By straightforward computation, it can be derived from (28) that(32)Redsdττ=τ0,ω=ω0=Φ1Ψ1+Φ2Ψ2Ψ12+Ψ22≠0.
The proof is completed.Theorem 1.
Suppose assumptionsH1−H4 hold; then,(1)
The coexistence equilibriumE∗ of system (3) is locally asymptotically stable for τ∈0,τ0(2)
System (3) undergoes Hopf bifurcation at E∗ when τ=τ0


## 4.2. Bifurcation Control of System (5)

In this section, we exploit the bifurcation control problem of system (5) through the time-delayed feedback controller.Analogously, by linearizing (5) at E∗, we get the Jacobian matrix:(33)JE∗=−rKPS∗+k1−e−sδ−rKPS∗−aPS∗0aV∗−m−nM∗aPS∗−nPI∗e−sτ0b−c−dM∗−dV∗e−sτ0β0−η,and the corresponding characteristic equation:(34)s4α+B1s3α+B2s2α+B3sα+B4+B5s2α+B6sα+B7e−sτ=0,where(35)B1=A1−k1−e−sδ,B2=A2−m3+ηk1−e−sδ,B3=A3−m6+m3η−abPS∗k1−e−sδ,B4=A4−ηm6−abPS∗k1−e−sδ,B5=A5,B6=A6−βnPI∗k1−e−sδ,B7=A7−m2nPI∗+adPS∗V∗βk1−e−sδ.Assume thats=ωiω>0 is a root of (11); then, we obtain(36)ω4αcos2απ+isin2απ+A1−k1−cosδω+isinδωω3αcos3απ2+isin3απ2+A2−m3+ηk1−cosδω−im3+ηksinδωω2αcosαπ+isinαπ+A3−m6+m3η−abPS∗k1−cosδω−im6+m3η−abPS∗ksinδωωα×cosαπ2+isinαπ2+A4−ηm6−abPS∗k1−cosδω−isinδω+A5ω2αcosαπ+isinαπ+A6−βnPI∗k1−cosδω−iβnPI∗ksinδω×ωαcosαπ2+isinαπ2+A7−m2nPI∗+adPS∗V∗βk1−cosδω−im2nPI∗+adPS∗V∗βksinδωcosωτ−isinωτ=0.Separating the real and imaginary parts, we obtain(37)D1cosωτ+D2sinωτ=−D3,D2cosωτ−D1sinωτ=−D4,where(38)D1=φ1cosαπ+φ2cosαπ2+φ3sinαπ2+φ4,D2=φ1sinαπ+φ2sinαπ2−φ3cosαπ2−φ5,D3=ω4αcos2απ+φ6cos3απ2+φ7sin3απ2+φ8cosαπ+φ9sinαπ+φ10cosαπ2+φ11sinαπ2+φ12,D4=ω4αsin2απ+φ6sin3απ2−φ7cos3απ2+φ8sinαπ−φ9cosαπ+φ10sinαπ2−φ11cosαπ2−φ13,and φi,i=1,2,…,13 are defined by Appendix B.It follows from (40) that(39)sinωτ=D1D4−D2D3D12+D22,cosωτ=−D1D3+D2D4D12+D22.Similarly, we can obtain(40)D32+D42=D12+D22.Assume that equation (40) has at least one positive real root ω1; then, it follows from the second equation of (39) that(41)τδi=1ω1arccosD1D3+D2D4D12+D22+2iπ,i=0,1,2,….Define(42)τδi=minτδi,i=0,1,2,….To derive conditions for the existence of Hopf bifurcation, we make the following hypothesis:(43)H4:Φ1Ψ1+Φ2Ψ2Ψ12+Ψ22≠0,whereΦ˜1,Φ˜2,Ψ˜1,andΨ˜2 are defined by equation (47).Lemma 2.
Letsτ=φτ+iωτ be a root of the characteristic equation (34) near τ=τδi meeting φτδi=0,ωτδi=ω1; then, the transversality condition(44)Redsdττ=τδ0,ω=ω1≠0,holds.Proof.
Differentiating both sides of (39) with respect to τ, we obtain(45)dsdτ=Φ˜sΨ˜s,where Φ˜s=B5s2α+B6sα+B7se−sτ and(46)Ψ˜s=4αs4α−1+3αB1s3α−1+2αB2s2α−1+αB3sα−1+2αB5s2α−1+αB6sα−1−B5s2α+B6sα+B7τe−sτ−kδe−sδs3α+m3+ηs2α+m6+m3η−abPS∗sα+ηm6−abPS∗+βnPI∗sα+m2nPI∗+adPS∗V∗βe−sτ.
Let(47)Φ˜ω1iτ=τδ0=Φ˜1+iΦ˜2,Ψω1iτ=τδ0=Ψ˜1+iΨ˜2,where Φ˜1andΦ˜2 are the real and imaginary parts of Φ˜ω1i, respectively, while Ψ˜1andΨ˜2 are the real and imaginary parts of Ψ˜ω1i, respectively.
By straightforward computation, it can be derived from (45) that(48)Redsdττ=τδ0,ω=ω1=Φ˜1Ψ˜1+Φ˜2Ψ˜2Ψ˜12+Ψ˜22≠0.
The proof is completed.Theorem 2.
Suppose assumptionsH1,H2,andH5 hold; then,(1)
The coexistence equilibriumE∗ of system (5) is locally asymptotically stable for τ∈0,τδ0(2)
System (5) undergoes Hopf bifurcation at E∗ when τ=τδ0


## 5. Numerical Simulation

In this section, two numerical examples are provided to verify our theoretical results of systems (3) and (5).


### 5.1. Example 1

In the following, we will investigate the stability of the coexistence equilibriumE∗ and the existence of Hopf bifurcation for system (3). Model parameters are chosen from [3, 39–41] (refer to Table 2).Table 2
Model parameter values from [3, 39–41].
ParameterValueUnitr0.05day−1a0.005kgvector−1day−1n0.005day−1c0.12day−1γ0.03day−1η0.015day−1K50kg·plant−1M0.03day−1b0.8day−1d0.005day−1β0.05day−1By simple computation, we get the unique coexistence equilibrium:(49)E∗=39.3954,2.9012,13.0133,11.6706.Selecting the fractional order asα=0.95, we easily obtain that ω0=0.0215 and the critical value τ0=52.5989. According to Theorem 1, the coexistence equilibrium E∗ is asymptotically stable when τ=45<τ0, which is shown in Figure 1. Figure 2 shows that the coexistence equilibrium E∗ is unstable when τ=53>τ0 and a Hopf bifurcation occurs.Figure 1
The time series of system (3) when α=0.95andτ=45<τ0.
(a)(b)(c)(d)Figure 2
The time series of system (3) when α=0.95andτ=53>τ0.
(a)(b)(c)(d)In addition, we explore the impact of the fractional orderα on the bifurcation value τ0 for the uncontrolled system (3). From Figure 3, we can see that the bifurcation value τ0 is very sensitive to the change of the fractional order α. With the increase of the order, the bifurcation value decreases rapidly, which implies that the stability region of system (3) becomes smaller. In particular, compared to the corresponding integral order, the fractional system has a larger stability region.Figure 3
The impact of the fractional orderα on the bifurcation value τ0 for the uncontrolled system (3).Besides, by selectingτ=15, we investigate the convergence rate of the system. From Figure 4, we find that, with the increase of the order α, the convergence rate of system (3) becomes slower.Figure 4
The time series of system (3) when α = 0.98, α = 0.92, α = 0.86, and τ=15.
(a)(b)(c)(d)


### 5.2. Example 2

Now, we study the bifurcation control problem of uncontrolled system (3) by introducing the time-delayed feedback controller. Model parameters of controlled system (5) are chosen the same as in Table 2, while the feedback gain and the feedback delay are selected as k=−2 and δ=4.8, respectively. When the order is chosen as α=0.95, we can obtain that ω1=0.0213 and the corresponding critical value τδ0=32.8007. According to Theorem 2, the coexistence equilibrium E∗ is asymptotically stable when τ=28<τδ0 (see Figure 5), while it is unstable when τ=33>τδ0 and a Hopf bifurcation occurs (see Figure 6).Figure 5
The time series of system (5) when α = 0.95 and τ=28<τδ0.
(a)(b)(c)(d)Figure 6
The time series of system (5) when α = 0.95 and τ=33>τδ0.
(a)(b)(c)(d)We also explore the impact of the fractional orderα on the bifurcation value τδ0 for the controlled system (5). From Figure 7, we can see that the bifurcation value decreases rapidly with the increase of the order, which means that the stability region of system (5) also becomes smaller. Similarly, with the rise of the order α, the convergence rate of system (5) slows down (see Figure 8).Figure 7
The impact of the fractional orderα on the bifurcation value τδ0 for the controlled system (5).Figure 8
The time series of system (5) when α = 0.98, α = 0.92, α = 0.86, k = −2, δ=4.8, and τ=15.
(a)(b)(c)(d)Furthermore, we investigate the impact of the feedback gaink and the feedback delay δ on the bifurcation value τδ0. By fixing α=0.95,δ=4.8, and τ=30, we find that there is a minimum point around k=−2 (see Figure 9). Specifically, with the increase of the feedback gain k, the stability region of system (5) first slowly reduces to a minimum, then gradually increases to a high level. Three representative values k=−0.5, k=−2, and k=−6 are chosen in Figure 10, and we can see that system (5) displays the slowest convergence rate when k=−2 (which is the nearest to the minimum point).Figure 9
The impact of the feedback gaink on the bifurcation value τδ0 for the controlled system (5).Figure 10
The time series of system (5) when α = 0.95, k = −0.5, k = −2, k = −6, δ=4.8, and τ=30.
(a)(b)(c)(d)In addition, by fixingα=0.95,k=−2, and τ=30, we can see from Figure 11 that there is also a minimum point around δ=3.5. Specifically, with the increase of the feedback delay δ, the stability region of system (5) first quickly reduces to a minimum and then slowly increases. Three representative values δ=0.5, δ=4, and δ=10 are chosen, and we can see that system (5) converges the slowest when δ=4 which is the nearest to the minimum point (see Figure 12).Figure 11
The impact of the extended feedback delayδ on the bifurcation value τδ0 for the controlled system (5).Figure 12
The time series of system (5) when α = 0.98, k=−2, δ=0.5, δ=4, δ=10, and τ=30.
(a)(b)(c)(d)


## 5.1. Example 1

In the following, we will investigate the stability of the coexistence equilibriumE∗ and the existence of Hopf bifurcation for system (3). Model parameters are chosen from [3, 39–41] (refer to Table 2).Table 2
Model parameter values from [3, 39–41].
ParameterValueUnitr0.05day−1a0.005kgvector−1day−1n0.005day−1c0.12day−1γ0.03day−1η0.015day−1K50kg·plant−1M0.03day−1b0.8day−1d0.005day−1β0.05day−1By simple computation, we get the unique coexistence equilibrium:(49)E∗=39.3954,2.9012,13.0133,11.6706.Selecting the fractional order asα=0.95, we easily obtain that ω0=0.0215 and the critical value τ0=52.5989. According to Theorem 1, the coexistence equilibrium E∗ is asymptotically stable when τ=45<τ0, which is shown in Figure 1. Figure 2 shows that the coexistence equilibrium E∗ is unstable when τ=53>τ0 and a Hopf bifurcation occurs.Figure 1
The time series of system (3) when α=0.95andτ=45<τ0.
(a)(b)(c)(d)Figure 2
The time series of system (3) when α=0.95andτ=53>τ0.
(a)(b)(c)(d)In addition, we explore the impact of the fractional orderα on the bifurcation value τ0 for the uncontrolled system (3). From Figure 3, we can see that the bifurcation value τ0 is very sensitive to the change of the fractional order α. With the increase of the order, the bifurcation value decreases rapidly, which implies that the stability region of system (3) becomes smaller. In particular, compared to the corresponding integral order, the fractional system has a larger stability region.Figure 3
The impact of the fractional orderα on the bifurcation value τ0 for the uncontrolled system (3).Besides, by selectingτ=15, we investigate the convergence rate of the system. From Figure 4, we find that, with the increase of the order α, the convergence rate of system (3) becomes slower.Figure 4
The time series of system (3) when α = 0.98, α = 0.92, α = 0.86, and τ=15.
(a)(b)(c)(d)


## 5.2. Example 2

Now, we study the bifurcation control problem of uncontrolled system (3) by introducing the time-delayed feedback controller. Model parameters of controlled system (5) are chosen the same as in Table 2, while the feedback gain and the feedback delay are selected as k=−2 and δ=4.8, respectively. When the order is chosen as α=0.95, we can obtain that ω1=0.0213 and the corresponding critical value τδ0=32.8007. According to Theorem 2, the coexistence equilibrium E∗ is asymptotically stable when τ=28<τδ0 (see Figure 5), while it is unstable when τ=33>τδ0 and a Hopf bifurcation occurs (see Figure 6).Figure 5
The time series of system (5) when α = 0.95 and τ=28<τδ0.
(a)(b)(c)(d)Figure 6
The time series of system (5) when α = 0.95 and τ=33>τδ0.
(a)(b)(c)(d)We also explore the impact of the fractional orderα on the bifurcation value τδ0 for the controlled system (5). From Figure 7, we can see that the bifurcation value decreases rapidly with the increase of the order, which means that the stability region of system (5) also becomes smaller. Similarly, with the rise of the order α, the convergence rate of system (5) slows down (see Figure 8).Figure 7
The impact of the fractional orderα on the bifurcation value τδ0 for the controlled system (5).Figure 8
The time series of system (5) when α = 0.98, α = 0.92, α = 0.86, k = −2, δ=4.8, and τ=15.
(a)(b)(c)(d)Furthermore, we investigate the impact of the feedback gaink and the feedback delay δ on the bifurcation value τδ0. By fixing α=0.95,δ=4.8, and τ=30, we find that there is a minimum point around k=−2 (see Figure 9). Specifically, with the increase of the feedback gain k, the stability region of system (5) first slowly reduces to a minimum, then gradually increases to a high level. Three representative values k=−0.5, k=−2, and k=−6 are chosen in Figure 10, and we can see that system (5) displays the slowest convergence rate when k=−2 (which is the nearest to the minimum point).Figure 9
The impact of the feedback gaink on the bifurcation value τδ0 for the controlled system (5).Figure 10
The time series of system (5) when α = 0.95, k = −0.5, k = −2, k = −6, δ=4.8, and τ=30.
(a)(b)(c)(d)In addition, by fixingα=0.95,k=−2, and τ=30, we can see from Figure 11 that there is also a minimum point around δ=3.5. Specifically, with the increase of the feedback delay δ, the stability region of system (5) first quickly reduces to a minimum and then slowly increases. Three representative values δ=0.5, δ=4, and δ=10 are chosen, and we can see that system (5) converges the slowest when δ=4 which is the nearest to the minimum point (see Figure 12).Figure 11
The impact of the extended feedback delayδ on the bifurcation value τδ0 for the controlled system (5).Figure 12
The time series of system (5) when α = 0.98, k=−2, δ=0.5, δ=4, δ=10, and τ=30.
(a)(b)(c)(d)


## 6. Conclusion

In this paper, a fractional-order mosaic virus infection model forJatropha curcas with farming awareness and an execution delay was studied based on the model in [2]. Compared with the studies, we not only generalized the integral order system to a fractional-order system which may describe the mechanism of the disease transmission more accurately but also investigated the Hopf bifurcation control through a time-delayed feedback controller. We mainly studied the existence of Hopf bifurcation and explored the impact of time delay, the fractional order, the feedback gain, and the feedback delay on the occurrence of bifurcation phenomenon.By choosing the execution delay as a bifurcation parameter, Hopf bifurcation of the uncontrolled system was firstly studied. The stability of the coexistence equilibrium and the bifurcation criteria were obtained. Then, by introducing a time-delayed feedback controller, the bifurcation control problem of the uncontrolled system was investigated in detail. A series of numerical simulations were performed, which not only verified our theoretical results but also revealed some specific features. Numerically, we exploited the impact of the time delay, the fractional order, the feedback gain, and the feedback delay on the occurrence of Hopf bifurcation. We found that, for both uncontrolled and controlled systems, when the execution delay is small enough (less than the critical values), the systems are stable, while they lose their stability and Hopf bifurcations occur when the execution delay exceeds the critical values. Besides, our study showed that the bifurcation values are very sensitive to the change of the fractional order. With the increase of the order, the bifurcation values of the two systems decrease rapidly, which implies that the stability region of systems becomes smaller. Furthermore, we investigated the impact of the feedback gain and the extended feedback delay on the bifurcation value for the controlled system, and we found the existence of the extreme points for the feedback gain and the extended feedback delay which can minimize the bifurcation value.According to our study, the execution time delay can lead to the fluctuation of the population quantity ofJatropha plants, which means that the population quantity is at some unreasonable level. By introducing a time-delayed feedback controller, we can stabilize the population by harvesting or transplanting new plants according to the current and past population level (at time t and t−δ). If the population quantity of the healthy plant at t−δ is more than the current level, we harvest some plants. Otherwise, we transplant some new plants. Our numerical simulations show that there exist extreme points for the feedback gain k and the feedback delay δ which can both minimize the bifurcation value, so we can select values of the parameters k and δ which are far away from those extreme points, then the population size tends to be stabilized. Specifically, we prefer to choose the feedback gain k which is greater than the minimum point and choose the feedback delay δ which is less than the minimum point because these values are more conducive to a stable population level.For the control of system (3), we can also add a controller to the other equations or add more than one controller at a time, and comparisons between different controllers would be very meaningful. Besides, introducing artificial impulsive control behaviors into system (3) is also interesting. We will continue these studies in our future work.



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