# Sliding Dynamics of a Filippov Forest-Pest Model with Threshold Policy Control

Authors: Lirong Liu; Changcheng Xiang; Guangyao Tang; Yuan Fu
Journal: Complexity
 (2019)

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## Abstract

A novel Filippov forest-pest system with threshold policy control (TPC) is established while an economic threshold (ET) is used to guide switching. The aim of our work is to address how to reasonably and successfully control pests by means of sliding dynamics for the Filippov system. On the basis of the above considerations, conditions for the existence and stability of equilibria of subsystems are addressed, and the sliding segments and several types of equilibria of the proposed system are defined. These equilibria include the regular/virtual equilibrium, pseudoequilibrium, boundary equilibrium, and tangent point. Further, not only are the relations between nullclines and equilibria of the Filippov system discussed, but the relations between pseudoequilibrium, nullclines, and the sliding segment are discussed. More importantly, four cases of sliding bifurcations of the Filippov system with respect to different types of equilibria of subsystems are investigated, and the corresponding biological implications concerning integrated pest management (IPM) are analyzed. Our results show that the points of intersection between nullclines are equilibria of the system, and the two endpoints of the sliding segment are on the nullclines. It is also verified that the pseudoequilibrium is the point of intersection of the sliding segment and nullclines of the Filippov system, and the pseudoequilibrium exists on the sliding segment. More interestingly, sliding dynamics analysis reveals that the Filippov system has sliding limit cycles, a bistable state and a stable refuge equilibrium point, and the optimal time and strategy for controlling pests are provided.


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## Body

## 1. Introduction

Because insect infestations increase forest mortality, which in turn affects the carbon cycle and causes air pollution [1], many scholars have paid much attention to the effects of disturbances on forests [2–7]. Disturbances are classified as natural and anthropogenic disturbances. Natural disturbances include wildfires, insect infestations, floods, droughts, and bad weather while anthropogenic disturbances include deforestation, the logging of timber, and the spraying of pesticides [4]. Hence, it is vital that the effects of disturbances on the forest are solved. To do this, the age structures of trees in forests where pests exist have been investigated, and strategies for forest management and fire protection have been developed by considering the size of fires, fire control methods, and the behaviors of planted trees [6–10]. However, few researchers have addressed the insect infestations from a biological point of view.The dynamics of numerous real-world systems can be modeled using discontinuous differential equations [11–15]. As an example, Filippov pest control models have been proposed to investigate sliding bifurcations [16]. Yang and Liao studied the Filippov Hindmarsh–Rose neuron model for several sliding bifurcation phenomena, and the threshold policy control (TPC) was considered [17]. Wang et al. proposed a Filippov epidemic model in discussing the effects of factors on controlling epidemic diseases [18]. Moreover, research on the related Filippov system has focused on bifurcation analysis [16,19–22]. Tan et al. discussed the sliding bifurcation analysis for a Filippov predator-prey system [23]. Qu and Li investigated the sliding phenomenon in Filippov dynamical systems based on Chua’s circuit [24]. Therefore, combining integrated pest management (IPM), a Filippov system with TPC (i.e., a discontinuous piecewise smooth system [25]) is established to reasonably and successfully control pests. It is a challenging but important work to determine the functional response of pests [26–28].IPM is a well-known strategy of controlling pests effectively [29–35], which includes biological strategies (i.e., releasing natural enemies), cultural strategies (i.e., acquiring or capturing pests artificially), and chemical strategies (i.e., spraying a certain dosage of pesticides) that are used for pest control to avoid exceeding an economic threshold (ET) [36]. In other words, the three types of strategies aim to reduce the number of pests below the economic injury level (EIL), i.e., the environment is not destroyed [37]. TPC, a control strategy, is implemented once the number of pests reaches and exceeds the ET [38–42].Recently, Xiao and Bosch have built a model with respect to the effects of pest on biologically based technologies for controlling pest (BBTs), which revealed that biological control has two effects including stabilization and destabilization [43]. Stern et al. have verified that integrated control was the most effective where chemical treatment was carried out for successful pest control and pest eradication was not necessary to be considered, and an ET was applied to determine the use of pesticides [36]. Liang and Tang have analyzed the key factors (i.e., optimum timing, the dosage of pesticide, and ET) which affect pest management by employing different impulsive differential equations [44]. A pest-natural enemy system has been proposed by Liang et al., and pesticide resistance was taken into account, which aimed to determine the number of releasing natural enemies to ensure pest eradication [45]. The present paper focuses on the control of pests (beetles), and the sliding dynamics of a Filippov system based on the tree-beetle model is discussed.First, a novel Filippov forest-pest model is proposed by combining IPM and TPC. Then, conditions for the existence and stability of equilibria of subsystems are addressed, which is helpful to study the dynamics of the proposed system. Further, the sliding segments and several types of equilibria of the system are defined, and these equilibria include the regular/virtual equilibrium, pseudoequilibrium, boundary equilibrium, and tangent point. The relations between nullclines and equilibria of the Filippov system are discussed, which reveal that the points of intersection between nullclines are these equilibrium points of the system. Moreover, the relations between pseudoequilibrium, nullclines and the sliding segment are also discussed. The results verify that pseudoequilibrium is the point of intersection of the sliding segment and nullclines of the Filippov system, and the pseudoequilibrium exists on the sliding segment.The sliding bifurcations of the Filippov system are addressed for different values ofET. Results show that the value of ET affects the control range and also the time that we decide to take action to prevent the density of the pest population from reaching the value of ET. Therefore, it is important to choose the optimal strategy when the pest density reaches the ET. More importantly, the Filippov system has sliding limit cycles, a bistable state, and a stable refuge equilibrium point (i.e., when the pest population with relatively small intrinsic growth rate, it will be stable at the small density).The remainder of the paper is organized as follows. In Section2, the model description and preliminaries of the Filippov system are introduced. In Section 3, equilibria and sliding dynamics including the conditions for the existence and stability of equilibria of subsystems are addressed. Moreover, the sliding regions and equilibria of the Filippov system are discussed. In Section 4, the sliding bifurcations of the Filippov system with respect to different types of equilibria of subsystems are presented, and the corresponding biological implications related to pest control are analyzed. Conclusions drawn from the results of our work are presented in Section 5.


## 2. Model Description and Preliminaries

### 2.1. Model Description

To explore the effects of insect infestations on the forest, the classical tree-beetle model was considered [4]:(1)V˙t=rvVt1−Vtkv−mB,B˙t=rbBt1−Btkb−αB2t1+βB2t,where the number of susceptible trees is denoted by Vt, Bt represents the population density of mountain pine beetles per tree, rv and rb are the intrinsic growth rates of Vt and Bt, respectively, and their carrying capacity is denoted by kv and kb, respectively, α is a positive constant that depends upon trees, and β is also a positive constant that determines the scale at which beetle density generates saturation.Note that the beetles can be treated as herbivore type insects, which consume trees [46–48]. The density growth of trees is subject to a logistic self-interaction in the absence of beetles where the forest does not have any defense against insect predation. In particular, a functional response (i.e., the Holling type-III functional response) is assumed [26]. The functional response of the predator is an important part of the prey-predator relationship, which is the impact of the predator (pest) per unit time on the change in prey (forest) density. Types of functional response include Holling types I-III [26, 49], the ratio-dependence type [27, 50], and the Hassell-Varley type [28].When the beetles are introduced in the first equation of model (1), a linear function is assumed. It means that the number of trees decreases because the beetles feed on trees. So a novel tree-beetle model is obtained as(2)V˙t=rvVt1−Vtkv−δBtVt,B˙t=rbBt1−Btkb+ηδBtVt−αB2t1+βB2t,where δ represents the rate that every beetle eats tree; η is the conversion coefficient of the beetles; ηδBtVt denotes the number of the beetles increases after they eat trees; and Vt and Bt can be treated as the forest (tree) and pest (beetle). Thus, model (2) is called the forest-pest model. To a certain extent, they can also be seen as prey and predator.Furthermore, we letq1 represent the proportion of pests (prey) that is captured, transferred, or killed by using the cultural and chemical strategies. Therefore, the control model for Bt>ET is written as(3)V˙t=rvVt1−Vtkv−δBtVt,B˙t=rbBt1−Btkb+ηδBtVt−αB2t1+βB2t−q1Bt,where the ET is a switching threshold, i.e., system (2) is satisfied if Bt<ET while system (3) is satisfied if Bt>ET.In the next subsection, some preliminaries of the Filippov system are provided to understand this paper easily, which prepares for the later sections.


### 2.2. Preliminaries

On the basis of IPM strategies and TPC, models (2) and (3) are combined and rewritten as [11, 51](4)V˙t=rvVt1−Vtkv−δBtVt,B˙t=rbBt1−Btkb+ηδBtVt−αB2t1+βB2t−εq1Bt,with(5)ε=0,Bt<ET,1,Bt>ET.Models (4) and (5) are related to TPC. More details on the Filippov system have been given in the literature [52, 53].LetHZ=Bt−ET with column vector Z=V,BT and(6)FS1Z=rvV1−Vkv−δBV,rbB1−Bkb+ηδBV−αB21+βB2T,FS2Z=rvV1−Vkv−δBV,rbB1−Bkb+ηδBV−αB21+βB2−q1BT.Then, system (6) is rewritten as a Filippov system [52, 53]:(7)Z˙t=FS1Z,Z∈S1,FS2Z,Z∈S2.Moreover, the discontinuity boundary set is defined; that is,Σ=Z∈R+2∣HZ=0, which divides R+2 into two regions, i.e.,(8)S1=Z∈R+2∣HZ<0,S2=Z∈R+2∣HZ>0.In this paper, Filippov system (7) in region S1 or S2 is, respectively, referred to as subsystem S1 or subsystem S2.Let(9)σZ=HZZ,FS1ZHZZ,FS2Z,where HZ is a nonvanishing gradient of the smooth scale function on Σ and ⋅ denotes the standard scalar product. Then, sliding regions are defined as(10)ΣS=Z∈Σ∣σZ≤0.Meanwhile, the following regions onΣ are distinguished:(i)
Escaping region: ifHZZ,FS1Z<0 and HZZ,FS2Z>0(ii)
Sliding region: ifHZZ,FS1Z>0 and HZZ,FS2Z<0(iii)
Sewing region: ifHZZ,FS1ZHZZ,FS2Z>0The essential definitions for different types of equilibria of Filippov system (7) are taken from the literature [54, 55].Definition 1.
Z
∗ is called a regular equilibrium of Filippov system (7) if FS1Z∗=0 and HZ∗<0, or FS2Z∗=0 and HZ∗>0. Z∗ is called a virtual equilibrium of Filippov system (7) if FS1Z∗=0 and HZ∗>0, or FS2Z∗=0 and HZ∗<0.Definition 2.
Z
∗ is called a pseudoequilibrium if it is an equilibrium on the sliding segment of Filippov system (7), i.e., 1−λZFS1Z∗+λZFS2Z∗=0, HZ∗=0, and 0<λZ<1, where(11)λZ=HZZ,FS1ZHZZ,FS1Z−FS2Z.Definition 3.
Z
∗ is called a boundary equilibrium of Filippov system (7) if FS1Z∗=0 with HZ∗=0 or FS2Z∗=0 with HZ∗=0.Definition 4.
Z
∗ is called a tangent point of Filippov system (7) if Z∗∈ΣS and HZZ∗,FS1Z∗=0 or HZZ∗,FS2Z∗=0.


## 2.1. Model Description

To explore the effects of insect infestations on the forest, the classical tree-beetle model was considered [4]:(1)V˙t=rvVt1−Vtkv−mB,B˙t=rbBt1−Btkb−αB2t1+βB2t,where the number of susceptible trees is denoted by Vt, Bt represents the population density of mountain pine beetles per tree, rv and rb are the intrinsic growth rates of Vt and Bt, respectively, and their carrying capacity is denoted by kv and kb, respectively, α is a positive constant that depends upon trees, and β is also a positive constant that determines the scale at which beetle density generates saturation.Note that the beetles can be treated as herbivore type insects, which consume trees [46–48]. The density growth of trees is subject to a logistic self-interaction in the absence of beetles where the forest does not have any defense against insect predation. In particular, a functional response (i.e., the Holling type-III functional response) is assumed [26]. The functional response of the predator is an important part of the prey-predator relationship, which is the impact of the predator (pest) per unit time on the change in prey (forest) density. Types of functional response include Holling types I-III [26, 49], the ratio-dependence type [27, 50], and the Hassell-Varley type [28].When the beetles are introduced in the first equation of model (1), a linear function is assumed. It means that the number of trees decreases because the beetles feed on trees. So a novel tree-beetle model is obtained as(2)V˙t=rvVt1−Vtkv−δBtVt,B˙t=rbBt1−Btkb+ηδBtVt−αB2t1+βB2t,where δ represents the rate that every beetle eats tree; η is the conversion coefficient of the beetles; ηδBtVt denotes the number of the beetles increases after they eat trees; and Vt and Bt can be treated as the forest (tree) and pest (beetle). Thus, model (2) is called the forest-pest model. To a certain extent, they can also be seen as prey and predator.Furthermore, we letq1 represent the proportion of pests (prey) that is captured, transferred, or killed by using the cultural and chemical strategies. Therefore, the control model for Bt>ET is written as(3)V˙t=rvVt1−Vtkv−δBtVt,B˙t=rbBt1−Btkb+ηδBtVt−αB2t1+βB2t−q1Bt,where the ET is a switching threshold, i.e., system (2) is satisfied if Bt<ET while system (3) is satisfied if Bt>ET.In the next subsection, some preliminaries of the Filippov system are provided to understand this paper easily, which prepares for the later sections.


## 2.2. Preliminaries

On the basis of IPM strategies and TPC, models (2) and (3) are combined and rewritten as [11, 51](4)V˙t=rvVt1−Vtkv−δBtVt,B˙t=rbBt1−Btkb+ηδBtVt−αB2t1+βB2t−εq1Bt,with(5)ε=0,Bt<ET,1,Bt>ET.Models (4) and (5) are related to TPC. More details on the Filippov system have been given in the literature [52, 53].LetHZ=Bt−ET with column vector Z=V,BT and(6)FS1Z=rvV1−Vkv−δBV,rbB1−Bkb+ηδBV−αB21+βB2T,FS2Z=rvV1−Vkv−δBV,rbB1−Bkb+ηδBV−αB21+βB2−q1BT.Then, system (6) is rewritten as a Filippov system [52, 53]:(7)Z˙t=FS1Z,Z∈S1,FS2Z,Z∈S2.Moreover, the discontinuity boundary set is defined; that is,Σ=Z∈R+2∣HZ=0, which divides R+2 into two regions, i.e.,(8)S1=Z∈R+2∣HZ<0,S2=Z∈R+2∣HZ>0.In this paper, Filippov system (7) in region S1 or S2 is, respectively, referred to as subsystem S1 or subsystem S2.Let(9)σZ=HZZ,FS1ZHZZ,FS2Z,where HZ is a nonvanishing gradient of the smooth scale function on Σ and ⋅ denotes the standard scalar product. Then, sliding regions are defined as(10)ΣS=Z∈Σ∣σZ≤0.Meanwhile, the following regions onΣ are distinguished:(i)
Escaping region: ifHZZ,FS1Z<0 and HZZ,FS2Z>0(ii)
Sliding region: ifHZZ,FS1Z>0 and HZZ,FS2Z<0(iii)
Sewing region: ifHZZ,FS1ZHZZ,FS2Z>0The essential definitions for different types of equilibria of Filippov system (7) are taken from the literature [54, 55].Definition 1.
Z
∗ is called a regular equilibrium of Filippov system (7) if FS1Z∗=0 and HZ∗<0, or FS2Z∗=0 and HZ∗>0. Z∗ is called a virtual equilibrium of Filippov system (7) if FS1Z∗=0 and HZ∗>0, or FS2Z∗=0 and HZ∗<0.Definition 2.
Z
∗ is called a pseudoequilibrium if it is an equilibrium on the sliding segment of Filippov system (7), i.e., 1−λZFS1Z∗+λZFS2Z∗=0, HZ∗=0, and 0<λZ<1, where(11)λZ=HZZ,FS1ZHZZ,FS1Z−FS2Z.Definition 3.
Z
∗ is called a boundary equilibrium of Filippov system (7) if FS1Z∗=0 with HZ∗=0 or FS2Z∗=0 with HZ∗=0.Definition 4.
Z
∗ is called a tangent point of Filippov system (7) if Z∗∈ΣS and HZZ∗,FS1Z∗=0 or HZZ∗,FS2Z∗=0.


## 3. Equilibria and Sliding Dynamics

This section discusses conditions for the existence and stability of equilibria of subsystemsS1 and S2. The sliding segment/region and equilibria of Filippov system (7) are addressed, which prepares for Section 4.


### 3.1. Existence of Equilibria for SubsystemsS1 and S2

To discuss the types of equilibria and sliding bifurcations of Filippov system (7), the existence of equilibria for subsystems S1 and S2 needs to be solved. For subsystem S1, let V˙t=0 and B˙t=0. Then, the equilibrium E1∗(V1∗, B1∗) satisfies(12)B3t+a1B2t+b1Bt+c1=0,where a1=−kbrbrv−ηδkbkvrv/rbrv+δ2ηkbkv, b1=rbrv+δ2ηkbkv+αkbrv/rbrvβ+δ2ηkbkvβ, and c1=−kbrbrv−ηδkbkvrv/rbrvβ+δ2ηkbkvβ. Let Q1=a12−3b1/9, R1=−2a13−9a1b1+27c1/54, M1=R12−Q13, and θ1=arccosR1/Q13/2.(i)
IfM1<0, then (12) has three fixed solutions given by(13)B1i=−2Q1cosθ1+2kπ3−a13,k=−1,0,1;i=1,2,3,V1i=kv−δB1ikvrv.(ii)
IfM1>0, then (12) has a unique equilibrium point given by(14)B1=−R1+M11/3+−R1−M11/3−a13,V1=kv−δB1kvrv.(iii)
IfM1=0, then (12) has two equilibrium points. We do not discuss this case in the present paper.In particular, the equilibriumE1∗ must be positive, i.e., V1i>0,B1i>0 i=1,2,3. Analogously, the existence conditions of equilibria for subsystem S2 can be obtained.


### 3.2. Stability of Equilibria for SubsystemsS1 and S2

In general, the Jacobian matrix is employed to explore the stability of equilibriumE1∗V1∗,B1∗ for subsystem S1, i.e.,(15)J1∗=rv−2rvV1∗kv−δB1∗−δV1∗ηδB1∗rb−2rbB1∗kb+ηδV1∗−2αB1∗1+βB1∗22.Then, the characteristic equation isλ2+p1λ+q1=0, where(16)p1=−rv+2rvV1∗kv+δB1∗−rb+2rbB1∗kb−ηδV1∗+2αB1∗1+βB1∗22,(17)q1=rv−2rvV1∗kv−δB1∗rb−2rbB1∗kb+ηδV1∗−2αB1∗1+βB1∗22+ηδ2B1∗V1∗.If inequalitiesp1>0 and q1>0 are satisfied, then E1∗ is asymptotically stable. Similarly, the conditions for the stability of equilibria of subsystem S2 are addressed.


### 3.3. Sliding Segment and Region

From Definition 2, the equation is obtained:(18)λZ=rb1−Bt/kb+ηδVt−αBt/1+βB2tq1.The sliding regions are determined by solving0≤λZ≤1 with respect to Vt. Thus, the algebraic equations need to be considered, i.e.,(19)rb1−ETkb+ηδVt−αET1+βET2=0,rb1−ETkb+ηδVt−αET1+βET2−q1=0.Then, solving equation (19) with respect to Vt yields two real roots:(20)V1=αET/1+βET2−rb1−ET/kbηδ,V2=q1+αET/1+βET2−rb1−ET/kbηδ.Because of the relation betweenV1 and V2 (V1<V2), the sliding segment of Filippov system (7) can be defined as(21)ΣS=V,B∣V1≤V≤V2,B=ET.


### 3.4. Equilibria of the Filippov System

Nullclines of both subsystemsS1 and S2 are related to the existence of equilibria, which is conducive to sliding bifurcation analysis and to estimating the existence of equilibria of Filippov system (7). For the two subsystems, nullclines are determined by using equations V˙t=0 and B˙t=0 as(22)fS1=fS2=kvrvrv−δBt,gS1=1ηδ−rb1−Btkb+αBt1+βB2t,gS2=1ηδq1−rb1−Btkb+αBt1+βB2t.Different types of equilibria of Filippov system (7) were defined in Section 2. Several types of equilibria for Filippov system (7) are defined, which include the regular equilibrium, virtual equilibrium, pseudoequilibrium, boundary equilibrium, and a special point called the tangent point. These equilibria are denoted by ER, EV, EP, EB, and ET. For Filippov system (7), more detailed definitions of the equilibria mentioned are in the following section.


#### 3.4.1. Regular/Virtual Equilibrium

According to Definition 1,Zji = (Vji, Bji) (j=1,2;i=1,2,3) is a regular equilibrium of Filippov system (7) if FSjZji=0, for subsystem S1 (i.e., j=1), B1i<ET, or for subsystem S2 (i.e., j=2), B2i>ET. These equilibria are denoted by E1iR and E2iR, respectively. Zji = (Vji, Bji) (j=1,2;i=1,2,3) is a virtual equilibrium of Filippov system (7) if FSjZji=0, for subsystem S1 (i.e., j=1), B1i>ET, or for subsystem S2 (i.e., j=2), B2i<ET. These equilibria are denoted by E1iV and E2iV, respectively.


#### 3.4.2. Pseudoequilibrium

According to Definition 2, by employing the Utkin equivalent control method [51] and letting HZ=B˙t=0, equations with respect to ε are obtained:(23)ε=rb1−ET/kb+ηδV−αET/1+βET2q1.Thus, the equationV˙t=rvVt1−Vt/kv−δBtVt determines the dynamics on ΣS. If pseudoequilibrium EPVP,ET satisfies V˙t=0 (i.e., VP=kvrv−δBt/rv and Bt=ET), then EPkvrv−δET/rv,ET∈ΣS. Therefore, VP must satisfy the inequality V1≤VP≤V2, i.e., if the inequalities(24)kbkvδ2ηETkbkvδη+rbkb−ET−αETkb/1+βET2<rv<kbkvδ2ηETkbkvδη−q1+rbkb−ET−αETkb/1+βET2,hold, then Filippov system (7) has a pseudoequilibrium EP.


#### 3.4.3. Boundary Equilibrium

According to Definition 3, the boundary equilibrium of Filippov system (7) satisfies(25)rvVt1−Vtkv−δBtVt=0,rb1−Btkb+ηδVt−αBt1+βB2t−q1=0,Bt=ET,with ε=0 or 1. If ε=0, the boundary equilibrium(26)E1B=αET/1+βET2−rb1−ET/kbηδ,ET,is obtained. If ε=1, we have the boundary equilibrium(27)E2B=q1+αET/1+βET2−rb1−ET/kbηδ,ET.


#### 3.4.4. Tangent Point

According to Definition 4, the tangent pointETVT,ET on ΣS satisfies(28)rb1−Btkb+ηδVt−αBt1+βB2t−q1=0,Bt=ET.Solving equations (28) with respect to Vt and Bt yields(29)E1T=αET/1+βET2−rb1−ET/kbηδ,ET,E2T=q1+αET/1+βET2−rb1−ET/kbηδ,ET.


## 3.1. Existence of Equilibria for SubsystemsS1 and S2

To discuss the types of equilibria and sliding bifurcations of Filippov system (7), the existence of equilibria for subsystems S1 and S2 needs to be solved. For subsystem S1, let V˙t=0 and B˙t=0. Then, the equilibrium E1∗(V1∗, B1∗) satisfies(12)B3t+a1B2t+b1Bt+c1=0,where a1=−kbrbrv−ηδkbkvrv/rbrv+δ2ηkbkv, b1=rbrv+δ2ηkbkv+αkbrv/rbrvβ+δ2ηkbkvβ, and c1=−kbrbrv−ηδkbkvrv/rbrvβ+δ2ηkbkvβ. Let Q1=a12−3b1/9, R1=−2a13−9a1b1+27c1/54, M1=R12−Q13, and θ1=arccosR1/Q13/2.(i)
IfM1<0, then (12) has three fixed solutions given by(13)B1i=−2Q1cosθ1+2kπ3−a13,k=−1,0,1;i=1,2,3,V1i=kv−δB1ikvrv.(ii)
IfM1>0, then (12) has a unique equilibrium point given by(14)B1=−R1+M11/3+−R1−M11/3−a13,V1=kv−δB1kvrv.(iii)
IfM1=0, then (12) has two equilibrium points. We do not discuss this case in the present paper.In particular, the equilibriumE1∗ must be positive, i.e., V1i>0,B1i>0 i=1,2,3. Analogously, the existence conditions of equilibria for subsystem S2 can be obtained.


## 3.2. Stability of Equilibria for SubsystemsS1 and S2

In general, the Jacobian matrix is employed to explore the stability of equilibriumE1∗V1∗,B1∗ for subsystem S1, i.e.,(15)J1∗=rv−2rvV1∗kv−δB1∗−δV1∗ηδB1∗rb−2rbB1∗kb+ηδV1∗−2αB1∗1+βB1∗22.Then, the characteristic equation isλ2+p1λ+q1=0, where(16)p1=−rv+2rvV1∗kv+δB1∗−rb+2rbB1∗kb−ηδV1∗+2αB1∗1+βB1∗22,(17)q1=rv−2rvV1∗kv−δB1∗rb−2rbB1∗kb+ηδV1∗−2αB1∗1+βB1∗22+ηδ2B1∗V1∗.If inequalitiesp1>0 and q1>0 are satisfied, then E1∗ is asymptotically stable. Similarly, the conditions for the stability of equilibria of subsystem S2 are addressed.


## 3.3. Sliding Segment and Region

From Definition 2, the equation is obtained:(18)λZ=rb1−Bt/kb+ηδVt−αBt/1+βB2tq1.The sliding regions are determined by solving0≤λZ≤1 with respect to Vt. Thus, the algebraic equations need to be considered, i.e.,(19)rb1−ETkb+ηδVt−αET1+βET2=0,rb1−ETkb+ηδVt−αET1+βET2−q1=0.Then, solving equation (19) with respect to Vt yields two real roots:(20)V1=αET/1+βET2−rb1−ET/kbηδ,V2=q1+αET/1+βET2−rb1−ET/kbηδ.Because of the relation betweenV1 and V2 (V1<V2), the sliding segment of Filippov system (7) can be defined as(21)ΣS=V,B∣V1≤V≤V2,B=ET.


## 3.4. Equilibria of the Filippov System

Nullclines of both subsystemsS1 and S2 are related to the existence of equilibria, which is conducive to sliding bifurcation analysis and to estimating the existence of equilibria of Filippov system (7). For the two subsystems, nullclines are determined by using equations V˙t=0 and B˙t=0 as(22)fS1=fS2=kvrvrv−δBt,gS1=1ηδ−rb1−Btkb+αBt1+βB2t,gS2=1ηδq1−rb1−Btkb+αBt1+βB2t.Different types of equilibria of Filippov system (7) were defined in Section 2. Several types of equilibria for Filippov system (7) are defined, which include the regular equilibrium, virtual equilibrium, pseudoequilibrium, boundary equilibrium, and a special point called the tangent point. These equilibria are denoted by ER, EV, EP, EB, and ET. For Filippov system (7), more detailed definitions of the equilibria mentioned are in the following section.


### 3.4.1. Regular/Virtual Equilibrium

According to Definition 1,Zji = (Vji, Bji) (j=1,2;i=1,2,3) is a regular equilibrium of Filippov system (7) if FSjZji=0, for subsystem S1 (i.e., j=1), B1i<ET, or for subsystem S2 (i.e., j=2), B2i>ET. These equilibria are denoted by E1iR and E2iR, respectively. Zji = (Vji, Bji) (j=1,2;i=1,2,3) is a virtual equilibrium of Filippov system (7) if FSjZji=0, for subsystem S1 (i.e., j=1), B1i>ET, or for subsystem S2 (i.e., j=2), B2i<ET. These equilibria are denoted by E1iV and E2iV, respectively.


### 3.4.2. Pseudoequilibrium

According to Definition 2, by employing the Utkin equivalent control method [51] and letting HZ=B˙t=0, equations with respect to ε are obtained:(23)ε=rb1−ET/kb+ηδV−αET/1+βET2q1.Thus, the equationV˙t=rvVt1−Vt/kv−δBtVt determines the dynamics on ΣS. If pseudoequilibrium EPVP,ET satisfies V˙t=0 (i.e., VP=kvrv−δBt/rv and Bt=ET), then EPkvrv−δET/rv,ET∈ΣS. Therefore, VP must satisfy the inequality V1≤VP≤V2, i.e., if the inequalities(24)kbkvδ2ηETkbkvδη+rbkb−ET−αETkb/1+βET2<rv<kbkvδ2ηETkbkvδη−q1+rbkb−ET−αETkb/1+βET2,hold, then Filippov system (7) has a pseudoequilibrium EP.


### 3.4.3. Boundary Equilibrium

According to Definition 3, the boundary equilibrium of Filippov system (7) satisfies(25)rvVt1−Vtkv−δBtVt=0,rb1−Btkb+ηδVt−αBt1+βB2t−q1=0,Bt=ET,with ε=0 or 1. If ε=0, the boundary equilibrium(26)E1B=αET/1+βET2−rb1−ET/kbηδ,ET,is obtained. If ε=1, we have the boundary equilibrium(27)E2B=q1+αET/1+βET2−rb1−ET/kbηδ,ET.


### 3.4.4. Tangent Point

According to Definition 4, the tangent pointETVT,ET on ΣS satisfies(28)rb1−Btkb+ηδVt−αBt1+βB2t−q1=0,Bt=ET.Solving equations (28) with respect to Vt and Bt yields(29)E1T=αET/1+βET2−rb1−ET/kbηδ,ET,E2T=q1+αET/1+βET2−rb1−ET/kbηδ,ET.


## 3.4.1. Regular/Virtual Equilibrium

According to Definition 1,Zji = (Vji, Bji) (j=1,2;i=1,2,3) is a regular equilibrium of Filippov system (7) if FSjZji=0, for subsystem S1 (i.e., j=1), B1i<ET, or for subsystem S2 (i.e., j=2), B2i>ET. These equilibria are denoted by E1iR and E2iR, respectively. Zji = (Vji, Bji) (j=1,2;i=1,2,3) is a virtual equilibrium of Filippov system (7) if FSjZji=0, for subsystem S1 (i.e., j=1), B1i>ET, or for subsystem S2 (i.e., j=2), B2i<ET. These equilibria are denoted by E1iV and E2iV, respectively.


## 3.4.2. Pseudoequilibrium

According to Definition 2, by employing the Utkin equivalent control method [51] and letting HZ=B˙t=0, equations with respect to ε are obtained:(23)ε=rb1−ET/kb+ηδV−αET/1+βET2q1.Thus, the equationV˙t=rvVt1−Vt/kv−δBtVt determines the dynamics on ΣS. If pseudoequilibrium EPVP,ET satisfies V˙t=0 (i.e., VP=kvrv−δBt/rv and Bt=ET), then EPkvrv−δET/rv,ET∈ΣS. Therefore, VP must satisfy the inequality V1≤VP≤V2, i.e., if the inequalities(24)kbkvδ2ηETkbkvδη+rbkb−ET−αETkb/1+βET2<rv<kbkvδ2ηETkbkvδη−q1+rbkb−ET−αETkb/1+βET2,hold, then Filippov system (7) has a pseudoequilibrium EP.


## 3.4.3. Boundary Equilibrium

According to Definition 3, the boundary equilibrium of Filippov system (7) satisfies(25)rvVt1−Vtkv−δBtVt=0,rb1−Btkb+ηδVt−αBt1+βB2t−q1=0,Bt=ET,with ε=0 or 1. If ε=0, the boundary equilibrium(26)E1B=αET/1+βET2−rb1−ET/kbηδ,ET,is obtained. If ε=1, we have the boundary equilibrium(27)E2B=q1+αET/1+βET2−rb1−ET/kbηδ,ET.


## 3.4.4. Tangent Point

According to Definition 4, the tangent pointETVT,ET on ΣS satisfies(28)rb1−Btkb+ηδVt−αBt1+βB2t−q1=0,Bt=ET.Solving equations (28) with respect to Vt and Bt yields(29)E1T=αET/1+βET2−rb1−ET/kbηδ,ET,E2T=q1+αET/1+βET2−rb1−ET/kbηδ,ET.


## 4. Numerical Simulation

In this section, four cases of sliding bifurcations for Filippov system (7) with different equilibria of subsystems S1 and S2 are investigated, which are summarized in Table 1.Table 1
The summary of numerical simulation.
Case
Subsystem
Description of the equilibria
A
S
1
An unstable equilibrium point, stable limit cycles
An unstable equilibrium point, stable limit cycles
&S2
A bistable state
Only one stable refuge equilibrium point
B
S
1
Only one stable equilibrium point
Only one stable equilibrium point
&S2
A bistable state
A stable refuge equilibrium point
C
S
1
Three equilibrium points (two stable equilibrium points)
&S2
A bistable state
A stable refuge equilibrium point
D
S
1 & S2
Three equilibrium points (only a stable refuge equilibrium point)


### 4.1. Case A

SubsystemS1 has an unstable equilibrium point, and there exist stable limit cycles. There are three cases of subsystem S2:(i)
SubsystemS2 has only an unstable equilibrium point and stable limit cycles(ii)
SubsystemS2 is a bistable state(iii)
SubsystemS2 has only one stable equilibrium pointFigure1(a) shows that there is only one unstable equilibrium E11 and stable limit cycles in subsystem S1. Figure 1(b) shows that subsystem S2 is similar to subsystem S1 in that there is only one unstable equilibrium E21 and stable limit cycles. Figure 1(c) shows that there are three equilibria (i.e., E21, E22 and E23) in subsystem S2. E21 and E23 are stable, i.e., subsystem S2 is a bistable state. Figure 1(d) shows that there is only one stable equilibrium E23, a stable refuge equilibrium point, in subsystem S2.Figure 1
Existence of equilibria for Case A of Filippov system (7). Parameters are rb=1, kb=20, η=0.06, δ=0.5, α=6, β=1, rv=3, and kv=100. (a) q1=0; (b) q1=0.15; (c) q1=0.35; (d) q1=0.5.
(a)
(b)
(c)
(d)The sliding mode phenomena in the three subcases of Case A are discussed as follows.Case A.1: subsystemS2 has only one unstable equilibrium point and stable limit cycles.Figure2(a) shows the sliding bifurcation of the Filippov system when the value of ET varies in the range [3.8, 5.8]. V1 and V2 are two endpoints of the sliding segment. Moreover, sliding limit cycles appear. Figures 2(b)–2(d) present the three cases of Figure 2(a). In Figure 2(b), if B11>B21>ET (ET=3.9), Filippov system (7) has limit cycles and stabilizes to the larger limit cycle and virtual equilibria E11V of subsystem S1. In Figure 2(c), if B11>ET>B21 (ET=4.8), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment with EP. In Figure 2(d), if ET>B11>B21 (ET=5.9), there are limit cycles. Filippov system (7) stabilizes to the larger limit cycle and virtual equilibria E21V of subsystem S2.The results show that the Filippov system has limit cycles regardless of the initial values (Vt and Bt) and stabilizes to the larger limit cycle. In other words, when q1=0.2, the pest population density stabilizes in a relatively small range, i.e., no pest outbreaks will occur. Other interesting problems can be considered, for example, how to decide the time and choose strategies for controlling pests, how to determine the equilibria of the Filippov system, and how to find regions of pseudoequilibrium when there is sliding bifurcation. To solve these problems, sliding bifurcation diagrams with nullclines of the Filippov system for different situations are investigated.Case A.2: subsystemS2 is a bistable state.In Figure3, V1V2 is the sliding segment. In this section, for the Filippov system, red, black and magenta dashed lines, respectively, indicate nullclines fS1/fS2, gS1, and gS2. The points of intersection between the nullclines are equilibria of the Filippov system. In Figure 3(a), when the ET is set as different values in the range [2, 9], the Filippov system shows sliding bifurcation and limit cycles. The difference compared with Figure 2(a) is that there is only one larger limit cycle. Subsystem S1 has the equilibrium E11 while subsystem S2 has the three equilibria E21, E22, and E23. In particular, E21 and E23 are stable. Therefore, subsystem S2 is a bistable state. Figures 3(b)–3(d) present the three cases of Figure 3(a), i.e., if B11>B21>ET>B22>B23 (ET=3), regardless of the initial values, the Filippov system is stable at E21. E21 is regular and denoted by E21R (see Figure 3(b)). If B11>ET>B21>B22>B23 (ET=4.7), the Filippov system has a pseudoequilibrium EP but no sliding limit cycle, and it stabilizes on the V1V2 with EP (see Figure 3(c)). If ET>B11>B21>B22>B23 (ET=6.45), the Filippov system has only one sliding limit cycle and stabilizes to this limit cycle (see Figure 3(d)). Therefore, when q1=0.45, the pest population density will be stable in a larger range than in the case of q1=0.2 as shown in Figure 1.We conclude that the equilibria (E11, E21, E22, and E23) are the points of intersection of nullclines (i.e., fS1/fS2, gS1, and gS2) for the Filippov system. The endpoints (V1 and V2) of the sliding segment are located on two nullclines (i.e., gS1 and gS2). The pseudoequilibrium is the point of intersection of the sliding segment and nullclines of the Filippov system. In addition, the pseudoequilibrium exists on the sliding segment (E11E21). These findings are also confirmed in the following discussions.Case A.3: subsystemS2 has only one stable refuge equilibrium point.Figure4(a) presents the sliding bifurcation of the Filippov system for the different values of ET in the range [0.5, 7.8]. Subsystem S2 has only one stable equilibrium E21. Figures 4(b)–4(d) present the three cases of Figure 4(a). If B11>ET>B21 (ET=1.54), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP. There exists EP on the E11E21 (see Figure 4(b)). If B11>B21>ET (ET=0.5), the ET is sufficiently small for the Filippov system to be stable at a refuge equilibrium point (see Figure 4(c)). If ET>B11>B21 (ET=6.76), regardless of the initial values, there is a sliding limit cycle and the Filippov system stabilizes to the limit cycle as shown in Figure 4(d). Hence, when q1=0.6, the pest population density will be stable in a smaller range than in the case of q1=0.45 as shown in Figure 3.Figure 2
Sliding bifurcation for Case A.1 of Filippov system (7). Subsystem S2 has only one unstable equilibrium point, and stable limit cycles exist. q1=0.2, the other parameters are the same as those in Figure 1. (a) ET∈3.8,5.8; (b) ET=3.9; (c) ET=4.8; (d) ET=5.9.
(a)
(b)
(c)
(d)Figure 3
Sliding bifurcation for Case A.2 of Filippov system (7). Subsystem S2 is a bistable state. q1 = 0.45, the other parameters are the same as those in Figure 1. (a) ET∈2,9; (b) ET=3; (c) ET=4.7; (d) ET=6.45.
(a)
(b)
(c)
(d)Figure 4
Sliding bifurcation for Case A.3 of Filippov system (7). Subsystem S2 has only one stable refuge equilibrium point. q1 = 0.6, the other parameters are the same as those in Figure 1. (a) ET∈0.5,7.8; (b) ET=1.54; (c) ET=0.5; (d) ET=6.76.
(a)
(b)
(c)
(d)Therefore, whenq1 = 0.45, the range of pests that can be controlled is largest. At the same time, the strategy is the most suitable for controlling pests.


### 4.2. Case B

SubsystemS1 has a stable equilibrium point while there are three cases of subsystem S2.(i)
SubsystemS2 has only one stable equilibrium point(ii)
SubsystemS2 has two stable equilibrium points (i.e., Subsystem S2 is a bistable state)(iii)
SubsystemS2 has only one stable refuge equilibrium pointIn Figure5(a), only one stable equilibrium E11 emerges in subsystem S1. In Figure 5(b), subsystem S2 has only one stable equilibrium E21. In Figure 5(c), subsystem S2 has two stable equilibria E21 and E23, which is a bistable state. In Figure 5(d), subsystem S2 has a stable equilibrium E23 that becomes a stable refuge equilibrium point with relatively small number of pests.Figure 5
Existence of equilibria for Case B of Filippov system (7). Parameters are fixed as rb=0.5, kb=10, η=0.4, δ=0.15, α=5.7, β=1, rv=2, and kv=45. (a) q1=0; (b) q1=0.05; (c) q1=0.5; (d) q1=0.9.
(a)
(b)
(c)
(d)The sliding mode phenomena in the three subcases of Case B are discussed as follows:Case B.1: subsystemS2 has a stable equilibrium point.Figure6(a) shows that there is sliding bifurcation of Filippov system (7) when the ET is set as the different values in [2, 9.07]. There is only one stable equilibrium E11 of subsystem S1 and a stable equilibrium E21 of subsystem S2. The three cases of Figure 6(a) are presented in Figures 6(b)–6(d). If B11>B21>ET (ET=4), the Filippov system stabilizes at the equilibrium E21, which is regular and denoted by E21R (see Figure 6(b)). If ET>B11>B21 (ET=8), the Filippov system stabilizes at E11. E11 is virtual and defined by E11R (see Figure 6(c)). If B11>ET>B21 (ET=5), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP (see Figure 6(d)). In particular, there exists a pseudoequilibrium on the E11E21. While there is no limit cycle in this case, there are no pest outbreaks and the system stabilizes within a relatively small range.Case B.2: subsystemS2 is a bistable state.Figure7(a) shows the sliding bifurcation of the Filippov system for the value of ET varying in the range [4.2, 11.8]. Subsystem S1 has only one stable equilibrium E11, subsystem S2 is a bistable state where E21 and E23 are stable. Figures 7(b)–7(d) presents the three cases of Figure 7(a). In Figure 7(b), if B11>B21>ET>B22>B23 (ET=5.29), the Filippov system always stabilizes at E21, which is real and denoted by E21R. In Figure 7(c), if B11>ET>B21>B22>B23 (ET=8.54), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP. There is a pseudoequilibrium on the E11E21. In Figure 7(d), if ET>B11>B21>B22>B23 (ET=11.8), Filippov system (7) is stable at E11 with different initial values. E11 is regular and denoted by E11R. In this case, when q1=0.5, the pest population density will stabilize within a relatively large range.Case B.3: subsystemS2 has only one stable refuge equilibrium point.Figure8(a) presents the sliding mode phenomena for the different values of ET (ET∈0.1,12). Subsystem S1 has only one stable equilibrium E11 while subsystem S2 has only one stable equilibrium E21, which is a stable refuge equilibrium point. Figures 8(b)–8(d) present the three cases of Figure 8(a). In Figure 8(b), if B11>B21>ET (ET=0.1), no matter how large the initial values are, the Filippov system stabilizes at E21 (i.e., a stable refuge equilibrium point). In Figure 8(c), if B11>ET>B21 (ET=6.9), the Filippov system has a pseudoequilibrium EP and stabilizes on the V1V2 (i.e., the sliding segment) with EP. Moreover, EP emerges on the E11E21. In Figure 8(d), if ET=B11>B21 (ET=10.6), the equilibrium E11 becomes a stable refuge equilibrium point. The Filippov system stabilizes on the V1V2 with E11. E11 is virtual and denoted by E11V. When q1=0.9, the pest population density stabilizes within a smaller range than in the case of q1=0.5 as shown in Figure 7. This means that the proper strategy can better control pests.Figure 6
Sliding bifurcation for Case B.1 of Filippov system (7). Both subsystem S1 and subsystem S2 have one stable equilibrium point. q1=0.6, the other parameters are rb=0.6, kb=10, η=0.06, δ=0.5, α=3, β=1, rv=3, and kv=150. (a) ET∈2,9.07; (b) ET=4; (c) ET=8; (d) ET=5.
(a)
(b)
(c)
(d)Figure 7
Sliding bifurcation for Case B.2 of Filippov system (7). Subsystem S1 has one stable equilibrium point and subsystem S2 is a bistable state. q1=0.5 and the other parameters are fixed as rb=0.5, kb=10, η=0.4, δ=0.15, α=5.7, β=1, rv=2, and kv=45. (a) ET∈4.2,11.8; (b) ET=5.29; (c) ET=8.54; (d) ET=11.8.
(a)
(b)
(c)
(d)Figure 8
Sliding bifurcation for Case B.3 of Filippov system (7). Subsystem S1 has a stable equilibrium point. Subsystem S2 has only one stable refuge equilibrium point. q1=0.9, the other parameters are the same as those in Figure 7. (a) ET∈0.1,12; (b) ET=0.1; (c) ET=6.9; (d) ET=10.6.
(a)
(b)
(c)
(d)


### 4.3. Case C

SubsystemS1 has three equilibrium points, two of which are stable. There are two subcases for subsystem S2.(i)
SubsystemS2 is a bistable state(ii)
SubsystemS2 has a stable refuge equilibrium pointThe sliding mode phenomena in the two subcases of CaseC are discussed as follows:Case C.1: SubsystemS2 is bistable.Figure9(a) presents the sliding bifurcation of the Filippov system for the different values of ET in the range [6.4, 13.6]. Both subsystem S1 and subsystem S2 have three equilibrium points, namely E11, E12, E13, E21, E22 and E23. E11, E13, E21 and E23 are stable. Subsystems S1 and S2 are bistable states. The three cases of Figure 9(a) are presented in Figures 9(b)–9(d). If ET>B11>B21>B22>B12>B13>B23 (ET=15), the Filippov system stabilizes at E11, which is called a regular equilibrium of subsystem S1 and denoted by E11R (see Figure 9(b)). If B11>ET>B21>B22>B12>B13>B23 (ET=10.5), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP (see Figure 9(c)). If B11>B21>ET>B22>B13>B23 (ET=6.4), the Filippov system stabilizes at E21, which is called a regular equilibrium of subsystem S2 and denoted by E21R (see Figure 9(d)). Therefore, in this case, if the density of the pest population is high, the pest population can be well controlled; if the initial values is relatively small, the Filippov system stabilize at a refuge equilibrium point.Case C.2: SubsystemS2 has only one stable refuge equilibrium point.Figure10(a) shows the sliding bifurcation of the Filippov system under the different values of ET in the range [5.3, 13.8]. Subsystem S1 has three equilibrium points, namely E11, E12 and E13. Subsystem S2 has an equilibrium E21. There is a pseudoequilibrium on the E11E12. Figure 10(b) shows that a pseudoequilibrium EP emerges if B11>ET>B12>B13>B21 (ET=6.5). When the value of B is large, the Filippov system is stable on the V1V2 with EP. When the value of B is small, the Filippov system stabilizes at a refuge equilibrium point. Therefore, when q1=0.6, the pest population density can be controlled in a larger range than in the case of q1=0.4 as shown in Figure 9.Figure 9
Sliding bifurcation for Case C.1 of Filippov system (7). Both subsystem S1 and subsystem S2 are bistable states. q1=0.4, parameters are rb=0.8, kb=15, η=0.4, δ=0.1, α=6, β=1, rv=1.5, and kv=40. (a) ET∈6.4,11.6; (b) ET=15; (c) ET=10.5; (d) ET=6.4.
(a)
(b)
(c)
(d)Figure 10
Sliding bifurcation for Case C.2 of Filippov system (7). Subsystem S2 has only one refuge equilibrium point. q1=0.6, the other parameters are the same as those in Figure 9. (a) ET∈5.3,13.8; (b) ET=6.5.
(a)
(b)


### 4.4. Case D

SubsystemS1 and subsystem S2 have three equilibrium points. Only one is a stable refuge equilibrium point.In Figure11(a), subsystem S1 has the three equilibria E11, E12, and E13. E11 and E12 are unstable and colored black. Only the red E13 is stable. In Figure 11(b), subsystem S1 has the three equilibria E21, E22 and E23. The black E21 and E22 are unstable, and only the smallest red dot E23 is stable.Figure 11
Existence of equilibria for Case D of Filippov system (7). Parameters are fixed as rb=0.1, kb=10, η=0.15, σ=0.2, α=6, β=1, rv=2, and kv=100. (a) q1=0; (b) q1=0.4.
(a)
(b)The sliding mode phenomena of Case D are discussed as follows.Figure12(a) shows the sliding bifurcation of the Filippov system for the ET in the range [3.6, 16.4]. Both subsystem S1 and subsystem S2 have three equilibrium points. Only one is a stable refuge equilibrium point. Figures 12(b)–12(d) present the three cases of Figure 12(a). If B11>ET>B21>B22>B12>B13>B23 (ET=9.1), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP (see Figure 12(b)). If ET>B11>B21>B22>B12>B13>B23 (ET=12.7), Filippov system (7) has a stable sliding limit cycle and stabilizes on it (see Figure 12(c)). Figure 12(d) has no limit cycle in contrast with Figure 12(c). Therefore, regardless of their initial values, the Filippov system stabilizes at a stable refuge equilibrium point as shown in Figures 12(b)–12(d). In this case, when q1=0.4, the pest population remains stable within a relatively large range.Figure 12
Sliding bifurcation for Case D of Filippov system (7). q1=0.4, the other parameters are the same as those in Figure 11. (a) ET∈3.6,16.4; (b) ET=5.7; (c) ET=9.1; (d) ET=12.7.
(a)
(b)
(c)
(d)Overall, in Cases A–D, the value ofq1 is large and the number of beetles is controlled within a certain range that is better. Taking spraying pesticides as an example, from the perspective of economic costs, there is no desire to use more pesticides. In addition, from a biomathematical point of view, the spraying of pesticides (i.e., a chemical measure) is performed frequently to kill beetles, which creates pollution. Therefore, the results show that it is crucial to choose a optimal strategy (or the dosage of pesticide) if the pest density reaches the ET.


## 4.1. Case A

SubsystemS1 has an unstable equilibrium point, and there exist stable limit cycles. There are three cases of subsystem S2:(i)
SubsystemS2 has only an unstable equilibrium point and stable limit cycles(ii)
SubsystemS2 is a bistable state(iii)
SubsystemS2 has only one stable equilibrium pointFigure1(a) shows that there is only one unstable equilibrium E11 and stable limit cycles in subsystem S1. Figure 1(b) shows that subsystem S2 is similar to subsystem S1 in that there is only one unstable equilibrium E21 and stable limit cycles. Figure 1(c) shows that there are three equilibria (i.e., E21, E22 and E23) in subsystem S2. E21 and E23 are stable, i.e., subsystem S2 is a bistable state. Figure 1(d) shows that there is only one stable equilibrium E23, a stable refuge equilibrium point, in subsystem S2.Figure 1
Existence of equilibria for Case A of Filippov system (7). Parameters are rb=1, kb=20, η=0.06, δ=0.5, α=6, β=1, rv=3, and kv=100. (a) q1=0; (b) q1=0.15; (c) q1=0.35; (d) q1=0.5.
(a)
(b)
(c)
(d)The sliding mode phenomena in the three subcases of Case A are discussed as follows.Case A.1: subsystemS2 has only one unstable equilibrium point and stable limit cycles.Figure2(a) shows the sliding bifurcation of the Filippov system when the value of ET varies in the range [3.8, 5.8]. V1 and V2 are two endpoints of the sliding segment. Moreover, sliding limit cycles appear. Figures 2(b)–2(d) present the three cases of Figure 2(a). In Figure 2(b), if B11>B21>ET (ET=3.9), Filippov system (7) has limit cycles and stabilizes to the larger limit cycle and virtual equilibria E11V of subsystem S1. In Figure 2(c), if B11>ET>B21 (ET=4.8), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment with EP. In Figure 2(d), if ET>B11>B21 (ET=5.9), there are limit cycles. Filippov system (7) stabilizes to the larger limit cycle and virtual equilibria E21V of subsystem S2.The results show that the Filippov system has limit cycles regardless of the initial values (Vt and Bt) and stabilizes to the larger limit cycle. In other words, when q1=0.2, the pest population density stabilizes in a relatively small range, i.e., no pest outbreaks will occur. Other interesting problems can be considered, for example, how to decide the time and choose strategies for controlling pests, how to determine the equilibria of the Filippov system, and how to find regions of pseudoequilibrium when there is sliding bifurcation. To solve these problems, sliding bifurcation diagrams with nullclines of the Filippov system for different situations are investigated.Case A.2: subsystemS2 is a bistable state.In Figure3, V1V2 is the sliding segment. In this section, for the Filippov system, red, black and magenta dashed lines, respectively, indicate nullclines fS1/fS2, gS1, and gS2. The points of intersection between the nullclines are equilibria of the Filippov system. In Figure 3(a), when the ET is set as different values in the range [2, 9], the Filippov system shows sliding bifurcation and limit cycles. The difference compared with Figure 2(a) is that there is only one larger limit cycle. Subsystem S1 has the equilibrium E11 while subsystem S2 has the three equilibria E21, E22, and E23. In particular, E21 and E23 are stable. Therefore, subsystem S2 is a bistable state. Figures 3(b)–3(d) present the three cases of Figure 3(a), i.e., if B11>B21>ET>B22>B23 (ET=3), regardless of the initial values, the Filippov system is stable at E21. E21 is regular and denoted by E21R (see Figure 3(b)). If B11>ET>B21>B22>B23 (ET=4.7), the Filippov system has a pseudoequilibrium EP but no sliding limit cycle, and it stabilizes on the V1V2 with EP (see Figure 3(c)). If ET>B11>B21>B22>B23 (ET=6.45), the Filippov system has only one sliding limit cycle and stabilizes to this limit cycle (see Figure 3(d)). Therefore, when q1=0.45, the pest population density will be stable in a larger range than in the case of q1=0.2 as shown in Figure 1.We conclude that the equilibria (E11, E21, E22, and E23) are the points of intersection of nullclines (i.e., fS1/fS2, gS1, and gS2) for the Filippov system. The endpoints (V1 and V2) of the sliding segment are located on two nullclines (i.e., gS1 and gS2). The pseudoequilibrium is the point of intersection of the sliding segment and nullclines of the Filippov system. In addition, the pseudoequilibrium exists on the sliding segment (E11E21). These findings are also confirmed in the following discussions.Case A.3: subsystemS2 has only one stable refuge equilibrium point.Figure4(a) presents the sliding bifurcation of the Filippov system for the different values of ET in the range [0.5, 7.8]. Subsystem S2 has only one stable equilibrium E21. Figures 4(b)–4(d) present the three cases of Figure 4(a). If B11>ET>B21 (ET=1.54), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP. There exists EP on the E11E21 (see Figure 4(b)). If B11>B21>ET (ET=0.5), the ET is sufficiently small for the Filippov system to be stable at a refuge equilibrium point (see Figure 4(c)). If ET>B11>B21 (ET=6.76), regardless of the initial values, there is a sliding limit cycle and the Filippov system stabilizes to the limit cycle as shown in Figure 4(d). Hence, when q1=0.6, the pest population density will be stable in a smaller range than in the case of q1=0.45 as shown in Figure 3.Figure 2
Sliding bifurcation for Case A.1 of Filippov system (7). Subsystem S2 has only one unstable equilibrium point, and stable limit cycles exist. q1=0.2, the other parameters are the same as those in Figure 1. (a) ET∈3.8,5.8; (b) ET=3.9; (c) ET=4.8; (d) ET=5.9.
(a)
(b)
(c)
(d)Figure 3
Sliding bifurcation for Case A.2 of Filippov system (7). Subsystem S2 is a bistable state. q1 = 0.45, the other parameters are the same as those in Figure 1. (a) ET∈2,9; (b) ET=3; (c) ET=4.7; (d) ET=6.45.
(a)
(b)
(c)
(d)Figure 4
Sliding bifurcation for Case A.3 of Filippov system (7). Subsystem S2 has only one stable refuge equilibrium point. q1 = 0.6, the other parameters are the same as those in Figure 1. (a) ET∈0.5,7.8; (b) ET=1.54; (c) ET=0.5; (d) ET=6.76.
(a)
(b)
(c)
(d)Therefore, whenq1 = 0.45, the range of pests that can be controlled is largest. At the same time, the strategy is the most suitable for controlling pests.


## 4.2. Case B

SubsystemS1 has a stable equilibrium point while there are three cases of subsystem S2.(i)
SubsystemS2 has only one stable equilibrium point(ii)
SubsystemS2 has two stable equilibrium points (i.e., Subsystem S2 is a bistable state)(iii)
SubsystemS2 has only one stable refuge equilibrium pointIn Figure5(a), only one stable equilibrium E11 emerges in subsystem S1. In Figure 5(b), subsystem S2 has only one stable equilibrium E21. In Figure 5(c), subsystem S2 has two stable equilibria E21 and E23, which is a bistable state. In Figure 5(d), subsystem S2 has a stable equilibrium E23 that becomes a stable refuge equilibrium point with relatively small number of pests.Figure 5
Existence of equilibria for Case B of Filippov system (7). Parameters are fixed as rb=0.5, kb=10, η=0.4, δ=0.15, α=5.7, β=1, rv=2, and kv=45. (a) q1=0; (b) q1=0.05; (c) q1=0.5; (d) q1=0.9.
(a)
(b)
(c)
(d)The sliding mode phenomena in the three subcases of Case B are discussed as follows:Case B.1: subsystemS2 has a stable equilibrium point.Figure6(a) shows that there is sliding bifurcation of Filippov system (7) when the ET is set as the different values in [2, 9.07]. There is only one stable equilibrium E11 of subsystem S1 and a stable equilibrium E21 of subsystem S2. The three cases of Figure 6(a) are presented in Figures 6(b)–6(d). If B11>B21>ET (ET=4), the Filippov system stabilizes at the equilibrium E21, which is regular and denoted by E21R (see Figure 6(b)). If ET>B11>B21 (ET=8), the Filippov system stabilizes at E11. E11 is virtual and defined by E11R (see Figure 6(c)). If B11>ET>B21 (ET=5), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP (see Figure 6(d)). In particular, there exists a pseudoequilibrium on the E11E21. While there is no limit cycle in this case, there are no pest outbreaks and the system stabilizes within a relatively small range.Case B.2: subsystemS2 is a bistable state.Figure7(a) shows the sliding bifurcation of the Filippov system for the value of ET varying in the range [4.2, 11.8]. Subsystem S1 has only one stable equilibrium E11, subsystem S2 is a bistable state where E21 and E23 are stable. Figures 7(b)–7(d) presents the three cases of Figure 7(a). In Figure 7(b), if B11>B21>ET>B22>B23 (ET=5.29), the Filippov system always stabilizes at E21, which is real and denoted by E21R. In Figure 7(c), if B11>ET>B21>B22>B23 (ET=8.54), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP. There is a pseudoequilibrium on the E11E21. In Figure 7(d), if ET>B11>B21>B22>B23 (ET=11.8), Filippov system (7) is stable at E11 with different initial values. E11 is regular and denoted by E11R. In this case, when q1=0.5, the pest population density will stabilize within a relatively large range.Case B.3: subsystemS2 has only one stable refuge equilibrium point.Figure8(a) presents the sliding mode phenomena for the different values of ET (ET∈0.1,12). Subsystem S1 has only one stable equilibrium E11 while subsystem S2 has only one stable equilibrium E21, which is a stable refuge equilibrium point. Figures 8(b)–8(d) present the three cases of Figure 8(a). In Figure 8(b), if B11>B21>ET (ET=0.1), no matter how large the initial values are, the Filippov system stabilizes at E21 (i.e., a stable refuge equilibrium point). In Figure 8(c), if B11>ET>B21 (ET=6.9), the Filippov system has a pseudoequilibrium EP and stabilizes on the V1V2 (i.e., the sliding segment) with EP. Moreover, EP emerges on the E11E21. In Figure 8(d), if ET=B11>B21 (ET=10.6), the equilibrium E11 becomes a stable refuge equilibrium point. The Filippov system stabilizes on the V1V2 with E11. E11 is virtual and denoted by E11V. When q1=0.9, the pest population density stabilizes within a smaller range than in the case of q1=0.5 as shown in Figure 7. This means that the proper strategy can better control pests.Figure 6
Sliding bifurcation for Case B.1 of Filippov system (7). Both subsystem S1 and subsystem S2 have one stable equilibrium point. q1=0.6, the other parameters are rb=0.6, kb=10, η=0.06, δ=0.5, α=3, β=1, rv=3, and kv=150. (a) ET∈2,9.07; (b) ET=4; (c) ET=8; (d) ET=5.
(a)
(b)
(c)
(d)Figure 7
Sliding bifurcation for Case B.2 of Filippov system (7). Subsystem S1 has one stable equilibrium point and subsystem S2 is a bistable state. q1=0.5 and the other parameters are fixed as rb=0.5, kb=10, η=0.4, δ=0.15, α=5.7, β=1, rv=2, and kv=45. (a) ET∈4.2,11.8; (b) ET=5.29; (c) ET=8.54; (d) ET=11.8.
(a)
(b)
(c)
(d)Figure 8
Sliding bifurcation for Case B.3 of Filippov system (7). Subsystem S1 has a stable equilibrium point. Subsystem S2 has only one stable refuge equilibrium point. q1=0.9, the other parameters are the same as those in Figure 7. (a) ET∈0.1,12; (b) ET=0.1; (c) ET=6.9; (d) ET=10.6.
(a)
(b)
(c)
(d)


## 4.3. Case C

SubsystemS1 has three equilibrium points, two of which are stable. There are two subcases for subsystem S2.(i)
SubsystemS2 is a bistable state(ii)
SubsystemS2 has a stable refuge equilibrium pointThe sliding mode phenomena in the two subcases of CaseC are discussed as follows:Case C.1: SubsystemS2 is bistable.Figure9(a) presents the sliding bifurcation of the Filippov system for the different values of ET in the range [6.4, 13.6]. Both subsystem S1 and subsystem S2 have three equilibrium points, namely E11, E12, E13, E21, E22 and E23. E11, E13, E21 and E23 are stable. Subsystems S1 and S2 are bistable states. The three cases of Figure 9(a) are presented in Figures 9(b)–9(d). If ET>B11>B21>B22>B12>B13>B23 (ET=15), the Filippov system stabilizes at E11, which is called a regular equilibrium of subsystem S1 and denoted by E11R (see Figure 9(b)). If B11>ET>B21>B22>B12>B13>B23 (ET=10.5), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP (see Figure 9(c)). If B11>B21>ET>B22>B13>B23 (ET=6.4), the Filippov system stabilizes at E21, which is called a regular equilibrium of subsystem S2 and denoted by E21R (see Figure 9(d)). Therefore, in this case, if the density of the pest population is high, the pest population can be well controlled; if the initial values is relatively small, the Filippov system stabilize at a refuge equilibrium point.Case C.2: SubsystemS2 has only one stable refuge equilibrium point.Figure10(a) shows the sliding bifurcation of the Filippov system under the different values of ET in the range [5.3, 13.8]. Subsystem S1 has three equilibrium points, namely E11, E12 and E13. Subsystem S2 has an equilibrium E21. There is a pseudoequilibrium on the E11E12. Figure 10(b) shows that a pseudoequilibrium EP emerges if B11>ET>B12>B13>B21 (ET=6.5). When the value of B is large, the Filippov system is stable on the V1V2 with EP. When the value of B is small, the Filippov system stabilizes at a refuge equilibrium point. Therefore, when q1=0.6, the pest population density can be controlled in a larger range than in the case of q1=0.4 as shown in Figure 9.Figure 9
Sliding bifurcation for Case C.1 of Filippov system (7). Both subsystem S1 and subsystem S2 are bistable states. q1=0.4, parameters are rb=0.8, kb=15, η=0.4, δ=0.1, α=6, β=1, rv=1.5, and kv=40. (a) ET∈6.4,11.6; (b) ET=15; (c) ET=10.5; (d) ET=6.4.
(a)
(b)
(c)
(d)Figure 10
Sliding bifurcation for Case C.2 of Filippov system (7). Subsystem S2 has only one refuge equilibrium point. q1=0.6, the other parameters are the same as those in Figure 9. (a) ET∈5.3,13.8; (b) ET=6.5.
(a)
(b)


## 4.4. Case D

SubsystemS1 and subsystem S2 have three equilibrium points. Only one is a stable refuge equilibrium point.In Figure11(a), subsystem S1 has the three equilibria E11, E12, and E13. E11 and E12 are unstable and colored black. Only the red E13 is stable. In Figure 11(b), subsystem S1 has the three equilibria E21, E22 and E23. The black E21 and E22 are unstable, and only the smallest red dot E23 is stable.Figure 11
Existence of equilibria for Case D of Filippov system (7). Parameters are fixed as rb=0.1, kb=10, η=0.15, σ=0.2, α=6, β=1, rv=2, and kv=100. (a) q1=0; (b) q1=0.4.
(a)
(b)The sliding mode phenomena of Case D are discussed as follows.Figure12(a) shows the sliding bifurcation of the Filippov system for the ET in the range [3.6, 16.4]. Both subsystem S1 and subsystem S2 have three equilibrium points. Only one is a stable refuge equilibrium point. Figures 12(b)–12(d) present the three cases of Figure 12(a). If B11>ET>B21>B22>B12>B13>B23 (ET=9.1), the Filippov system has a pseudoequilibrium EP and stabilizes on the sliding segment V1V2 with EP (see Figure 12(b)). If ET>B11>B21>B22>B12>B13>B23 (ET=12.7), Filippov system (7) has a stable sliding limit cycle and stabilizes on it (see Figure 12(c)). Figure 12(d) has no limit cycle in contrast with Figure 12(c). Therefore, regardless of their initial values, the Filippov system stabilizes at a stable refuge equilibrium point as shown in Figures 12(b)–12(d). In this case, when q1=0.4, the pest population remains stable within a relatively large range.Figure 12
Sliding bifurcation for Case D of Filippov system (7). q1=0.4, the other parameters are the same as those in Figure 11. (a) ET∈3.6,16.4; (b) ET=5.7; (c) ET=9.1; (d) ET=12.7.
(a)
(b)
(c)
(d)Overall, in Cases A–D, the value ofq1 is large and the number of beetles is controlled within a certain range that is better. Taking spraying pesticides as an example, from the perspective of economic costs, there is no desire to use more pesticides. In addition, from a biomathematical point of view, the spraying of pesticides (i.e., a chemical measure) is performed frequently to kill beetles, which creates pollution. Therefore, the results show that it is crucial to choose a optimal strategy (or the dosage of pesticide) if the pest density reaches the ET.


## 5. Conclusion

A novel Filippov forest-pest system with TPC concerning IPM was proposed. The system comprises two subsystemsS1 and S2. Our aim is to find the best time and strategy for successful pest control. The conditions for the existence and stability of equilibria of subsystems were addressed. Moreover, the sliding segment was determined. Several types of equilibria of the proposed system were defined, i.e., the regular equilibrium, virtual equilibrium, pseudoequilibrium, boundary equilibrium, and a special point called the tangent point. Meanwhile, nullclines of the Filippov system were also defined, which were related to the equilibria of these two susystems and used to determine their existence. That is, the points of intersection between nullclines were these equilibrium points of the system, and the two endpoints of the sliding segment V1V2 were on the nullclines.Four cases of the sliding bifurcations of the Filippov system with respect to different types of equilibrium points of subsystems were discussed. Details are given in Table1. In Case A.1, both subsystem S1 and subsystem S2 had only an unstable equilibrium point (see Figures 1(a) and 1(b)), the system had limit cycles and stabilized to the larger limit cycle regardless of the initial values (V and B) (see Figure 2).SubsystemS2 was a bistable state in Cases A.2, B.2, and C.1 (see Figures 3, 7, and 9). For instance, in Cases A.2 and B.2, subsystem S1 had an unstable equilibrium point (see Figure 1(a)) and subsystem S1 had a stable equilibrium point (see Figure 5(a)), subsystem S2 had three equilibria (i.e., E21, E22, and E23) and equilibria (i.e., E21 and E23) were stable (see Figures 1(c) and 5(c)). In Case C.1, both subsystem S1 and subsystem S2 are bistable states.SubsystemS2 had only one stable refuge equilibrium point in Cases A.3, B.3, C.2, and D. For example, in Case A.3, subsystem S1 had only an unstable equilibrium point. In Case B.3, subsystem S1 had only a stable equilibrium point. When the number of pests and forests was relatively small, subsystem S2 had only one stable refuge equilibrium point (see Figures 1(d) and 5(d)), the system was stable at it (see Figures 4 and 8). In Case C.2, subsystem S1 was the bistable state and only one stable refuge equilibrium point existed in subsystem S2. Moreover, the system stabilized on the V1V2 with pseudoequilibrium EP (see Figure 10). In Case D, both subsystems S1 and S2 had three equilibrium points (see Figure 11) and only one was a stable refuge equilibrium point. The Filippov system stabilized at a stable refuge equilibrium point with ET in the range [3.6, 16.4] (see Figure 12). A stable equilibrium point appeared in subsystem S1 and subsystem S2 (see Figures 5(a) and 5(b)). Especially, in Case A, the stable limit cycles existed in subsystem S1.In addition, it was verified that the pseudoequilibrium was the point of intersection of the sliding segment and nullclines of the Filippov system, and the pseudoequilibrium existed on theV1V2. In Case A, when q1 = 0.45, it was the most suitable for controlling pests. In Case B, the results revealed that the proper use of pesticides can effectively control pests. In Case C, when q1 = 0.6, the effects of pest control were the best. Therefore, to some extent, the more the pesticides were used, the better the number of beetles will be controlled within a certain range so that the aim of IPM strategies was achieved.In summary, the Filippov system had sliding limit cycles, a bistable state, and a stable refuge equilibrium point by means of sliding bifurcation analysis. From the perspective of practical biological significance, when the pest density reached the ET, the optimal strategy should be used, namely, the proper number of pests was captured, transferred, or killed by using the cultural and chemical strategies (i.e., the right amount of pesticide needed to be sprayed). From an economic point of view, it was beneficial to reduce the use of manpower, material resources, and financial resources and protected the environment. In the future, different control policies in which the threshold is also a linear or nonlinear function about forests and pests deserve investigation.



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*Source: 2371838-2019-11-11.xml*