# Filippov Ratio-Dependent Prey-Predator Model with Threshold Policy Control

Authors: Xianghong Zhang; Sanyi Tang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The Filippov ratio-dependent prey-predator model with economic threshold is proposed and studied. In particular, the sliding mode domain, sliding mode dynamics, and the existence of four types of equilibria and tangent points are investigated firstly. Further, the stability of pseudoequilibrium is addressed by using theoretical and numerical methods, and also the local sliding bifurcations including regular/virtual equilibrium bifurcations and boundary node bifurcations are studied. Finally, some global sliding bifurcations are addressed numerically. The globally stable touching cycle indicates that the density of pest population can be successfully maintained below the economic threshold level by designing suitable threshold policy strategies.


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## Body

## 1. Introduction

Ordinary differential equation models (ODE models) are widely used to describe the dynamics between predators and their prey, which has long been and will continue to be one of significant fields in mathematical ecology owing to its universal existence and importance [1, 2]. The simplest prey-predator dynamic model is the Lotka-Volterra model [3], which has been modified in many ways since its original and realism formulation in the 1920s.One important component of the prey-predator relation is predator's functional response which refers to the change in the density of prey attached per unit time per predator as the prey density changes and makes the prey-predator system more realistic. There are several famous functional response types in previous work, which are monotonically increasing and uniformly bounded functions in the first quadrant. Another functional response is the Michaelis-Menten (or Holling-type II) functional response, which is the most common type of functional response among arthropod predators [4, 5]. It takes the form p(x,y)=ax/(b+x), where a and b are positive constants that stand for capturing rate, and half capturing saturation constant, respectively.Considering predators having to search for food, a more suitable general prey-predator theory based on the so-called ratio-dependent theory is involved. It can be roughly comprehended as the per capita predator growth rate should be a function of the ratio of prey to predator abundance. And it is also strongly supported by numerous fields, laboratory experiments, and observations [6–8].Therefore, we can write the ratio-dependent prey-predator model with Michaelis-Menten functional response as follows:(1)x˙(t)=rx(1-xk)-axyby+x,y˙(t)=-δy+βaxyby+x,
where x and y represent the density of prey (pest) and predator (natural enemy), respectively. The prey is assumed to grow logistically and k is the carrying capacity of prey. The positive constants r and δ stand for intrinsic growth rate of prey and mortality rate of predator, respectively. β denotes the conversion rate of prey captured by predator.In population, both ecologist and mathematicians are interested in the ratio-dependent prey-predator model with Michaelis-Menten functional response [2, 7–9]. Hsu et al. [7] resolved a complete classification of the asymptotic behavior of the solutions of ratio-dependent model with the Michaelis-Menten functional response. They also studied the global stability of all equilibria in various cases and reconsidered the uniqueness of limit cycle.It is well known that pests have been one of the principal threats to crops, important plants, animals, and humans all over the world. Therefore, it is necessary to apply acceptable and effective strategies to control pest outbreak. In practice, it is impossible to eradicate the pests completely, nor is it biologically or economically desirable. Integrated pest management (IPM) is a long term management strategy [10–12], which uses a combination of biological, cultural, and chemical tactics so as to lower cost to the growers, minimize effect on the environment, and maintain pest population below the economic injury level (EIL). On the basis of IPM, biological strategy is useful and effective to suppress pest population, such as releasing beneficial natural enemies; culture strategy makes the environment less favorable to pests, such as catching or harvesting artificially. In most cropping systems, when the above two tactics are unable to keep pest population below the ET, chemical strategy (i.e., insecticide) is still a principal means to control pests and prevent economic loss. Thus, in order to control pest outbreak, we should carry out control strategies when the number of pests reaches or exceeds the ET which is lower than the EIL, and the control strategies should be suspended once the density of pest population falls below the ET, which is the so-called threshold policy control (TPC). Considering IPM strategies, either fixed moment or state-dependent impulsive models with the ratio-dependent or Michaelis-Menten-type response function of prey-predator model have been studied in [13–16].However, Zhao et al. [17] have stated some disadvantages of the impulsive differential equation models mentioned above. First, in the fixed moment impulsive model, without consideration whether the density of pest reaches the ET or not, control strategies are invariably implemented, which leads to consumption of vast resources. Second, in reality, all kinds of control strategies need some time and cannot be finished instantaneously, but in the state-dependent impulsive differential models, control strategies are carried out instantaneously, which is not reasonable.Therefore, we use Filippov system which is a vector differential equation with discontinuous right-hand side to describe prey-predator model with both noninstantaneous interventions and the threshold policy. Recently, although Filippov systems have been widely utilized in science and engineering, including harvesting thresholds, oil well drilling, and liquid-gas reaction [18–23]. However, very little is involved that they are used to investigate the ratio-dependent-type predator-prey model with Michaelis-Menten-type functional response. We assume that a proportion of preys are caught or transferred (culture strategy) or killed (chemical strategy), denoted by q1; a proportion of predators are released (biological strategy), denoted by q2. So we have the following control model for x>ET:
(2)x˙(t)=rx(1-xk)-axyby+x-q1x,y˙(t)=-δy+βaxyby+x+q2y.In this paper, we aim to give a detailed analysis of Filippov ratio-dependent prey-predator model with threshold policy control, which describes that control measures are implemented only when the density of pest in a population exceeds the ET. We investigate the sliding mode domain, sliding mode dynamics, the existence of four types of equilibria and tangent point of Filippov system, regular/virtual equilibrium bifurcation, and boundary node bifurcations. In addition, the local stability of pseudoequilibrium implies global stability in our numerical simulations. Globally touching bifurcation especially indicates that the density of pest can be successfully maintained below the ET by designing suitable threshold policy strategy. Therefore our control objective can be achieved in the above two cases, which are desired situations in crop, livestock sectors and forestry.The organization of this paper is as follows: in Section2, we give some basic results and preliminaries for ODE system and Filippov system. In Section 3, the existence of sliding segments and sliding mode dynamics for the Filippov system (3) are addressed. In Section 4, we give the null-isoclines and equilibria. Based on those results, in Section 5, we consider the bifurcation sets of equilibria and sliding bifurcation analyses. Then, in the last section, we give some discussions.


## 2. The ODE System and Filippov System

### 2.1. The Basic Preliminaries and Results for ODE System

The ODE model (1) has been well studied in [2, 7], and a complete classification of the asymptotic behavior of the solutions of the ratio-dependent model with Michaelis-Menten functional response has been proposed. In the following, we present some primary results in the following Lemma which are useful for this study.Lemma 1.
System (1) includes three equilibria (0,0), (k,0), and a unique positive equilibrium E*(x*,y*) if and only if the following two conditions are true: βa-δ>0 and r>(βa-δ)/βb, where E*=(x*,y*)=(k(βbr+δ-βa)/βbr,k(βa-δ)(βbr+δ-βa)/βb2rδ). If βa-δ>0 and βa-δ<r≤(a-bδ)/b, (0,0) is globally asymptotically stable; if βa-δ<0 and r≥(a-bδ)/b, (k,0) is globally asymptotically stable; if βa-δ>0 and (βa-δ)/βb<r≤βa-δ or βa-δ>0, r>βa-δ, and r≥(a-bδ)/b, the positive equilibrium of E*=(x*,y*) is globally asymptotically stable. However if the positive equilibrium is locally asymptotically stable, then the system (1) has no nontrivial positive periodic solutions. If βa-δ>0 and max{βa-δ,(βa-δ)/βb}<r<(a-bδ)/b hold true, then the system (1) has at most one stable limit cycle.


### 2.2. Filippov Ratio-Dependent Prey-Predator Model and Preliminaries

By now, based on IPM strategies and TPC, the models (1) and (2) can be incorporated and rewritten in the following form:
(3)x˙(t)=rx(1-xk)-axyby+x-ɛq1x,y˙(t)=-δy+βaxyby+x+ɛq2y,
with
(4)ɛ={0,x<ET,1,x>ET.We first introduce some useful properties and definitions on Filippov system according to [24, 25], so that we can investigate the model (3) in more detail. Let H(Z)=x-ET with vector Z=(x,y)T, and
(5)FS1(Z)=(rx(1-xk)-axyby+x,-δy+βaxyby+x)T,FS2(Z)=(rx(1-xk)-axyby+x-q1x,-δy+βaxyby+x+q2y)T.
Then the system (3) can be rewritten as the following Filippov system:
(6)Z˙(t)={FS1(Z),Z∈S1,FS2(Z),Z∈S2.In addition, we define the discontinuity boundary set (or the switching line)Σ={Z∈R+2∣H(Z)=0}, which divides R+2 into two regions S1 and S2, where
(7)S1={Z∈R+2∣H(Z)<0},S2={Z∈R+2∣H(Z)>0}.From now on, Filippov system (3) in different regions S1 or S2 is named as system S1 (i.e., system (1)) or system S2 (i.e., system (2)) correspondingly.Denote(8)σ(Z)=〈HZ(Z),FS1(Z)〉〈HZ(Z),FS2(Z)〉,
where HZ is a nonvanishing gradient of the smooth scale function H on Σ, and 〈·〉 denotes the standard scalar product; then the sliding mode domain can be defined as
(9)ΣS={Z∈Σ∣σ(Z)≤0}.We distinguish the following regions onΣ:(i)
Σ1∈Σ is the escaping region if 〈HZ(Z),FS1(Z)〉<0 and 〈HZ(Z),FS2(Z)〉>0 on Σ1;(ii)
Σ2∈Σ is the sliding region if 〈HZ(Z),FS1(Z)〉>0 and 〈HZ(Z),FS2(Z)〉<0 on Σ2;(iii)
Σ3∈Σ is the sewing region if 〈HZ(Z),FS1(Z)〉〈HZ(Z),FS2(Z)〉>0 on Σ3.The following definitions about all types of equilibria for Filippov system are necessary throughout the paper, so we list them as follows.Definition 2.
A pointZ* is called a regular equilibrium of system (3) if FS1(Z*)=0, with H(Z*)<0, or FS2(Z*)=0, with H(Z*)>0. A point Z* is called a virtual equilibrium of system (3), if FS1(Z*)=0, with H(Z*)>0, or FS2(Z*)=0, with H(Z*)<0.Definition 3.
A pointZ* is called a pseudoequilibrium if it is an equilibrium of the sliding mode of system (3); that is, (1-λ)FS1(Z*)+λFS2(Z*)=0, H(Z*)=0, and 0<λ<1, where
(10)λ=〈HZ(Z),FS1(Z)〉〈HZ(Z),FS1(Z)-FS2(Z)〉.Definition 4.
A pointZ* is called a boundary equilibrium of system (3) if FS1(Z*)=0, with H(Z*)=0, or FS2(Z*)=0, with H(Z*)=0.Definition 5.
A pointZ* is called a tangent point of system (3) if Z*∈ΣS and 〈HZ(Z*),FS1(Z*)〉=0, or 〈HZ(Z*),FS2(Z*)〉=0.The details and knowledge about the Filippov system, such as the concepts of Filippov solution, sliding mode solution, and bifurcation can be found in reference [24].


## 2.1. The Basic Preliminaries and Results for ODE System

The ODE model (1) has been well studied in [2, 7], and a complete classification of the asymptotic behavior of the solutions of the ratio-dependent model with Michaelis-Menten functional response has been proposed. In the following, we present some primary results in the following Lemma which are useful for this study.Lemma 1.
System (1) includes three equilibria (0,0), (k,0), and a unique positive equilibrium E*(x*,y*) if and only if the following two conditions are true: βa-δ>0 and r>(βa-δ)/βb, where E*=(x*,y*)=(k(βbr+δ-βa)/βbr,k(βa-δ)(βbr+δ-βa)/βb2rδ). If βa-δ>0 and βa-δ<r≤(a-bδ)/b, (0,0) is globally asymptotically stable; if βa-δ<0 and r≥(a-bδ)/b, (k,0) is globally asymptotically stable; if βa-δ>0 and (βa-δ)/βb<r≤βa-δ or βa-δ>0, r>βa-δ, and r≥(a-bδ)/b, the positive equilibrium of E*=(x*,y*) is globally asymptotically stable. However if the positive equilibrium is locally asymptotically stable, then the system (1) has no nontrivial positive periodic solutions. If βa-δ>0 and max{βa-δ,(βa-δ)/βb}<r<(a-bδ)/b hold true, then the system (1) has at most one stable limit cycle.


## 2.2. Filippov Ratio-Dependent Prey-Predator Model and Preliminaries

By now, based on IPM strategies and TPC, the models (1) and (2) can be incorporated and rewritten in the following form:
(3)x˙(t)=rx(1-xk)-axyby+x-ɛq1x,y˙(t)=-δy+βaxyby+x+ɛq2y,
with
(4)ɛ={0,x<ET,1,x>ET.We first introduce some useful properties and definitions on Filippov system according to [24, 25], so that we can investigate the model (3) in more detail. Let H(Z)=x-ET with vector Z=(x,y)T, and
(5)FS1(Z)=(rx(1-xk)-axyby+x,-δy+βaxyby+x)T,FS2(Z)=(rx(1-xk)-axyby+x-q1x,-δy+βaxyby+x+q2y)T.
Then the system (3) can be rewritten as the following Filippov system:
(6)Z˙(t)={FS1(Z),Z∈S1,FS2(Z),Z∈S2.In addition, we define the discontinuity boundary set (or the switching line)Σ={Z∈R+2∣H(Z)=0}, which divides R+2 into two regions S1 and S2, where
(7)S1={Z∈R+2∣H(Z)<0},S2={Z∈R+2∣H(Z)>0}.From now on, Filippov system (3) in different regions S1 or S2 is named as system S1 (i.e., system (1)) or system S2 (i.e., system (2)) correspondingly.Denote(8)σ(Z)=〈HZ(Z),FS1(Z)〉〈HZ(Z),FS2(Z)〉,
where HZ is a nonvanishing gradient of the smooth scale function H on Σ, and 〈·〉 denotes the standard scalar product; then the sliding mode domain can be defined as
(9)ΣS={Z∈Σ∣σ(Z)≤0}.We distinguish the following regions onΣ:(i)
Σ1∈Σ is the escaping region if 〈HZ(Z),FS1(Z)〉<0 and 〈HZ(Z),FS2(Z)〉>0 on Σ1;(ii)
Σ2∈Σ is the sliding region if 〈HZ(Z),FS1(Z)〉>0 and 〈HZ(Z),FS2(Z)〉<0 on Σ2;(iii)
Σ3∈Σ is the sewing region if 〈HZ(Z),FS1(Z)〉〈HZ(Z),FS2(Z)〉>0 on Σ3.The following definitions about all types of equilibria for Filippov system are necessary throughout the paper, so we list them as follows.Definition 2.
A pointZ* is called a regular equilibrium of system (3) if FS1(Z*)=0, with H(Z*)<0, or FS2(Z*)=0, with H(Z*)>0. A point Z* is called a virtual equilibrium of system (3), if FS1(Z*)=0, with H(Z*)>0, or FS2(Z*)=0, with H(Z*)<0.Definition 3.
A pointZ* is called a pseudoequilibrium if it is an equilibrium of the sliding mode of system (3); that is, (1-λ)FS1(Z*)+λFS2(Z*)=0, H(Z*)=0, and 0<λ<1, where
(10)λ=〈HZ(Z),FS1(Z)〉〈HZ(Z),FS1(Z)-FS2(Z)〉.Definition 4.
A pointZ* is called a boundary equilibrium of system (3) if FS1(Z*)=0, with H(Z*)=0, or FS2(Z*)=0, with H(Z*)=0.Definition 5.
A pointZ* is called a tangent point of system (3) if Z*∈ΣS and 〈HZ(Z*),FS1(Z*)〉=0, or 〈HZ(Z*),FS2(Z*)〉=0.The details and knowledge about the Filippov system, such as the concepts of Filippov solution, sliding mode solution, and bifurcation can be found in reference [24].


## 3. Sliding Region and Sliding Mode Dynamics

A sliding mode exists if there are regions in the discontinuity boundaryΣ, where the vectors for the two subsystems of the system (3) are directed towards each other. It is well known that two basic methods the so-called Filippov convex method [24] and Utkin equivalent control method [18], are developed for the sliding mode and its domains, which are shown in the appendix.


### 3.1. Sliding Segment and Region

Based on the appendix, we have(11)λ(Z)=r(1-ET/k)-ay/(by+ET)q1,
because the sliding mode regions can be determined by solving the inequalities; that is, λ(Z)≥0 and λ(Z)≤1. In order to solve the above two inequalities with respect to y, we need to consider the following two algebraic equations:
(12)r(1-ETk)-ayby+ET=0,r(1-ETk)-ayby+ET-q1=0.Solving the above two algebraic equations with respect toy yields two roots, denoted by
(13)y1=rET(k-ET)ak-br(k-ET),y2=(r(k-ET)-q1k)ETak-b(r(k-ET)-q1k).Based on the relations betweeny1 and y2, there exist two cases for the existence of sliding segments of Filippov system (3). By simply calculating and arranging, we have the following results. (i)
Whenr<ak/b(k-ET), the sliding segment can be described as
(14)ΣS1={(x,y)∣max{0,y2}≤y≤y1,x=ET}.(ii)
Whenmax{q1k/(k-ET),ak/b(k-ET)}<r<min{k(a+bq1)/b(k-ET),a(k+bq1)/b(k-ET)}, the sliding segment can be described as
(15)ΣS2={(x,y)∣0≤y≤y2,x=ET}.


### 3.2.  Sliding Mode Dynamics

Filippov system (3) only has one piece of sliding segment, and the solutions defined in it can be obtained from the sliding mode dynamics, which can be determined by employing the Utkin equivalent control method (see the appendix).FromH=0, we get that
(16)HZ=x′=rx(1-xk)-axyby+x-ɛq1x=0,x=ET.
And solving the above equations with respect to ɛ yields
(17)ɛ=r(1-ET/k)-ay/(by+ET)q1.Hence, the dynamics on the sliding modeΣS can be determined by the following scalar differential equation:
(18)y˙(t)=-δy+βayETby+ET+r(1-ET/k)-ay/(by+ET)q1q2y,=P1(y)(P2y+P3)≜ϕ(y),
where y∈ΣS1 or ΣS2,  P1(y)=y/q1(by+ET), P2=-δbq1+rbq2(1-ET/k)-aq2, and P3=-δq1ET+βaq1ET+rq2(1-ET/k)ET.


## 3.1. Sliding Segment and Region

Based on the appendix, we have(11)λ(Z)=r(1-ET/k)-ay/(by+ET)q1,
because the sliding mode regions can be determined by solving the inequalities; that is, λ(Z)≥0 and λ(Z)≤1. In order to solve the above two inequalities with respect to y, we need to consider the following two algebraic equations:
(12)r(1-ETk)-ayby+ET=0,r(1-ETk)-ayby+ET-q1=0.Solving the above two algebraic equations with respect toy yields two roots, denoted by
(13)y1=rET(k-ET)ak-br(k-ET),y2=(r(k-ET)-q1k)ETak-b(r(k-ET)-q1k).Based on the relations betweeny1 and y2, there exist two cases for the existence of sliding segments of Filippov system (3). By simply calculating and arranging, we have the following results. (i)
Whenr<ak/b(k-ET), the sliding segment can be described as
(14)ΣS1={(x,y)∣max{0,y2}≤y≤y1,x=ET}.(ii)
Whenmax{q1k/(k-ET),ak/b(k-ET)}<r<min{k(a+bq1)/b(k-ET),a(k+bq1)/b(k-ET)}, the sliding segment can be described as
(15)ΣS2={(x,y)∣0≤y≤y2,x=ET}.


## 3.2.  Sliding Mode Dynamics

Filippov system (3) only has one piece of sliding segment, and the solutions defined in it can be obtained from the sliding mode dynamics, which can be determined by employing the Utkin equivalent control method (see the appendix).FromH=0, we get that
(16)HZ=x′=rx(1-xk)-axyby+x-ɛq1x=0,x=ET.
And solving the above equations with respect to ɛ yields
(17)ɛ=r(1-ET/k)-ay/(by+ET)q1.Hence, the dynamics on the sliding modeΣS can be determined by the following scalar differential equation:
(18)y˙(t)=-δy+βayETby+ET+r(1-ET/k)-ay/(by+ET)q1q2y,=P1(y)(P2y+P3)≜ϕ(y),
where y∈ΣS1 or ΣS2,  P1(y)=y/q1(by+ET), P2=-δbq1+rbq2(1-ET/k)-aq2, and P3=-δq1ET+βaq1ET+rq2(1-ET/k)ET.


## 4.  Null-Isoclines and Equilibria

### 4.1. The Null-Isoclines of Filippov System (3)

Null-isoclines of both systemsS1 and S2 are related to the existence of equilibria and are useful for analysis of sliding dynamic.Null-isoclinesx˙(t)=0 and y˙(t)=0 for both systems S1 and S2 can be determined as follows. For the system S1, solving the equation of the null-isocline x˙(t)=0, yeilds
(19)fS1(x)=rx(1-x/k)a-br(1-x/k),
and null-isocline y˙(t)=0 gives
(20)gS1(x)=(βa-δ)xbδ.For the systemS2, solving the equation of the null-isocline x˙(t)=0, yields
(21)fS2(x)=(rx(1-x/k)-q1)xa+bq1-br(1-x/k),
and null-isocline y˙(t)=0 gives
(22)gS2(x)=(βa-(δ-q2))xb(δ-q2).


### 4.2. The Equilibria of Filippov System (3)

Because the solutions of Filippov system (3) are composed of connecting standard solutions in subsystems S1, S2 and sliding mode solutions on ΣS1 or ΣS2. The definitions of all types of equilibria of Filippov system have been provided in Section 2, which are important to bifurcation analysis. There may be several types of equilibria for Filippov system (3) which include regular equilibrium (denoted by ER), virtual equilibrium (denoted by EV), pseudoequilibrium (denoted by EP), boundary equilibrium (denoted by EB), and one type of special point named as tangent point (denoted by ET). Detailed definitions of these equilibria and the tangent point can be found in the literature [24].From Lemma1, we know that the subsystem S1 with x<ET has three possible equilibria: (0,0), (k,0), and interior equilibrium,
(23)ES1*=(xS1*,yS1*)=(k(βbr+δ-βa)βbr,k(βa-δ)(βbr+δ-βa)βb2rδ).Ifβa-δ>0 and r>(βa-δ)/βb, then there is a positive interior equilibrium for the system S1.Equilibria for the subsystemS2 with x>ET satisfy
(24)rx(1-xk)-axyby+x-q1x=0,-δy+βaxyby+x+q2y=0.Solving the above equations with respect tox and y yields three possible equilibria, that is, (0,0), (k(r-q1)/r,0), and interior equilibrium,
(25)ES2*=(xS2*,yS2*)=(k(βb(r-q1)+δ-q2-βa)βbr,k(βa-δ+q2)(βb(r-q1)+δ-q2-βa)βb2r(δ-q2)),
which indicates that, if r>(βa-δ+q2)/βb+q1 and q2<δ<βa+q2, then there is a positive interior equilibrium for the system S2.


#### 4.2.1. Regular Equilibria

For the subsystemS1 with x<ET, (0,0) is a regular equilibrium, while (k,0) is a virtual equilibrium. In addition, according to the coordinate of equilibrium ES1*, we have the following results. If r>(βa-δ)/βb>0 and
(26)k(βbr+δ-βa)βbr<ET,
then it is a regular equilibrium for the system S1, denoted by ER1. If r>(βa-δ)/βb>0 and
(27)k(βbr+δ-βa)βbr>ET,
then the equilibrium ES1* becomes a virtual equilibrium, denoted by EV1.About regular equilibria for the subsystemS2 with x>ET,  (0,0) is a virtual equilibrium, while (k(r-p1)/r,0) may be a regular or virtual equilibrium which depends on the parameter space. Moreover, according to the coordinate of equilibrium ES2*, we have the following conclusions. If r>(βa-δ+q2)/βb+q1, q2<δ<βa+q2 and
(28)k(βb(r-q1)+δ-q2-βa)βbr>ET,
then it is a regular equilibrium for the system S2, denoted by ER2. Note that if r>(βa-δ+q2)/βb+q1, q2<δ<βa+q2 and
(29)k(βb(r-q1)+δ-q2-βa)βbr<ET,
then the equilibrium ES2* becomes a virtual equilibrium, denoted by EV2. Based on the above analysis, if
(30)k(βa-δ)βb(k-ET)<r<k(βa-δ+q2+βbq1)βb(k-ET),
then the two virtual equilibria EV1 and EV2 can coexist.


#### 4.2.2. Pseudoequilibrium

For the existence of pseudoequilibriumEP=(ET,yP), yP component of the pseudoequilibrium of sliding flow satisfies the following equation:
(31)P1(y)(P2y+P3)=0,
where y∈ΣS1 or ΣS2. Solving the above equation with respect to y yields two possible roots denoted by yP1=0 and
(32)yP2=(δq1-βaq1-rq2(1-ET/k))ET-δbq1+rbq2(1-ET/k)-aq2.Further, if(33)kq1(δ-βa)q2(k-ET)<r<k(aq2+δbq1)bq2(k-ET)
holds true, then yP2 is a positive root. Note that if EP1=(ET,yP1) or EP2=(ET,yP2) lies in the sliding region ΣS1 or ΣS2, then the model has pseudoequilibrium. To do this, we consider the following two cases.Case 1 (sliding segment defined byΣS1).
If the inequalityr<q1k/(k-ET) holds true, then EP1=(ET,yP1)∈ΣS1 is a pseudoequilibrium of the Filippov system (3). Note that -δbq1+rbq2(1-ET/k)-aq2<0 (i.e., r<k(aq2+δbq1)/bq2(k-ET)) is well defined in this case. Therefore if the inequalities
(34)max{kq1(δ-βa)q2(k-ET),k(βa-δ)βb(k-ET)}<r<min{k(βa-δ+q2+βbq1)βb(k-ET),k(aq2+δbq1)bq2(k-ET)}
hold true, then EP2=(ET,yP2)∈ΣS1 is a positive pseudoequilibrium of the Filippov system (3).Case 2 (sliding segment defined byΣS2).
In this case,EP1=(ET,yP1)∈ΣS2 is a pseudoequilibrium of the Filippov system (3). If the inequalities
(35)max{k(βa-δ+q2+βbq1)βb(k-ET),kq1(δ-βa)q2(k-ET)}≤r<k(aq2+δbq1)bq2(k-ET)
hold true, EP2=(ET,yP2)∈ΣS2 is a positive pseudoequilibrium of the Filippov system (3).


#### 4.2.3. Boundary Equilibrium

The boundary equilibria of Filippov system (3) satisfy
(36)rx(1-xk)-axyby+x-ɛq1x=0,-δy+βaxyby+x+ɛq2y=0,x=ET,
with ɛ=0 or 1, which indicate that, if
(37)k(βbr+δ-βa)βbr=ETork(βb(r-q1)+δ-q2-βa)βbr=ET,
then we have the boundary equilibria
(38)EB1=(ET,(βa-δ)ETbδ),EB2=(ET,(βa-δ+q2)ETb(δ-q2)).


#### 4.2.4. Tangent Point

According to the definition of tangent point, we can see that the tangent pointET=(ET,yT) on sliding segment ΣS satisfies
(39)rx(1-xk)-axyby+x-ɛq1x=0,x=ET.Solving the above equations with respect tox and y yields two tangent points, including
(40)ET1=(ET,rET(1-ET/k)a-br(1-ET/k)),ET2=(ET(r(1-ET/k)-q1)a+q1b-br(1-ET/k)).If we fix all parameters, then the relations among null-isoclines, regular/virtual equilibria, pseudoequilibrium, and sliding segment are provided in Figure1. Note that virtual equilibria of both systems S1 and S2 imply the existence of pseudoequilibrium, and we will prove this general result later.Figure 1
Notations for null-isoclines, regular/virtual equilibria, pseudoequilibrium, and sliding segment. The parameter values are fixed as follows:a=0.7, b=0.6, r=1.2, k=12, β=0.5, δ=0.25, q1=0.25, q2=0.1, and ET=5.


### 4.3.  The Stability of Pseudoequilibrium

In the process of pest control, we should apply all kinds of control strategies so as to prevent multiple pest outbreaks or make sure that the total density of the pest stabilizes at a desired level of ET. If the unique positive equilibrium of systemS1 and system S2 is virtual simultaneously, then the sliding flow has a unique pseudoequilibrium. In order to realize this goal, we can choose a set of parameters such that all the equilibria of subsystems S1 and S2 are virtual equilibria, and the pseudoequilibria are globally stable, which have been widely used in pest control [26–28]. For example, if we fixed all parameter values as those in Figure 2, then both virtual equilibria coexist. Therefore, we address the stability of pseudoequilibrium EP2 in the following which is important to control pest.Figure 2
Regular/virtual equilibrium. The parameter values are fixed as follows:a=0.7, b=0.6, k=12, β=0.5, δ=0.25, q1=0.25, and q2=0.1.Theorem 6.
Either the inequalities (34) or (35) hold true or the two virtual equilibria EV1 and EV2 can coexist; then Filippov system (3) contains a positive pseudoequilibrium EP2. Regardless of which cases would occur, the pseudoequilibrium EP2 is locally stable with respect to sliding mode domain.Proof.
According to the conditions of Theorem6 we see that the inequalities (30) hold true, which implies that the inequalities (34) are true. Based on the discussions about the existence of pseudoequilibrium in Section 4.2, we have concluded that if inequalities (34) or (35) hold true, then the system (3) contains a positive pseudoequilibrium EP2.
It follows from (18) that we have(41)dϕ(y)dy|y=yP2=(-δb2kq1+rb2q2(k-ET)-abkq2)yP22kq1(byP2+ET)2+(-2δbkq1ET-2akq2ET+2rbq2ET(k-ET))yP2kq1(byP2+ET)2+-δkq1ET2+aβkq1ET2+rkq2ET2-rq2ET3kq1(byP2+ET)2=(kq1(δ-βa)-rq2(k-ET))(δbkq1-rbq2(k-ET)+akq2)aq1k2(bβq1+q2).Therefore, if the inequalities (33) hold true, then dϕ(y)/dy<0 and yP2 is a positive root of (31). Note that the inequalities (34) and (35) indicate the inequalities (33). Thus, the positive pseudoequilibrium EP2 is locally stale if it exists.It is difficult to directly prove the global stability of pseudoequilibriumEP2 in this case. Because we cannot employ the classical Bendixson-Dulac theorem due to the discontinuity of vector fields. However, if there is not crossing cycle surrounding sliding segment, then pseudoequilibrium is globally stable [29]. From the analysis of global bifurcation for the system (3) in the following section, the system just has touching bifurcation and there is nonexistence of a sliding cycle which surrounds the EP2. By using the similar methods, we have that pseudoequilibrium EP2 is globally stable. That is to say, the local stability of pseudoequilibrium EP2 with respect to sliding mode domain indicates its global stability in the first quadrant (shown in Figures 3(a), 3(b), 4(b), and 4(c)). In practice, in order to control pest outbreak, we should choose the desirable ET at first, so that all equilibria of each system such as system S1 and system S2 become virtual; then pseudoequilibrium not only exists but also is globally stable. In other words, the density of pest can be stable at the ET. When the density of pest reaches or exceeds the ET, we should carry out control strategies (e.g., releasing natural enemy, etc.), until it falls below the ET. In this way, our control goal can be realized fully.Boundary node bifurcation for Filippov system (3). The parameter values are fixed as follows: a=0.7, b=0.6, r=1.5, k=12, β=0.5, δ=0.25, q1=0.25, and q2=0.1.
(a)
ET
=
8.3
(b)
ET
=
9.3333
(c)
ET
=
9.8Boundary node bifurcation for Filippov system (3). The parameter values are fixed as follows: a=0.7, b=0.6, r=1.5, k=12, β=0.5, δ=0.25, q1=0.25, and q2=0.1.
(a)
ET
=
4
(b)
ET
=
4.6667
(c)
ET
=
5.8


## 4.1. The Null-Isoclines of Filippov System (3)

Null-isoclines of both systemsS1 and S2 are related to the existence of equilibria and are useful for analysis of sliding dynamic.Null-isoclinesx˙(t)=0 and y˙(t)=0 for both systems S1 and S2 can be determined as follows. For the system S1, solving the equation of the null-isocline x˙(t)=0, yeilds
(19)fS1(x)=rx(1-x/k)a-br(1-x/k),
and null-isocline y˙(t)=0 gives
(20)gS1(x)=(βa-δ)xbδ.For the systemS2, solving the equation of the null-isocline x˙(t)=0, yields
(21)fS2(x)=(rx(1-x/k)-q1)xa+bq1-br(1-x/k),
and null-isocline y˙(t)=0 gives
(22)gS2(x)=(βa-(δ-q2))xb(δ-q2).


## 4.2. The Equilibria of Filippov System (3)

Because the solutions of Filippov system (3) are composed of connecting standard solutions in subsystems S1, S2 and sliding mode solutions on ΣS1 or ΣS2. The definitions of all types of equilibria of Filippov system have been provided in Section 2, which are important to bifurcation analysis. There may be several types of equilibria for Filippov system (3) which include regular equilibrium (denoted by ER), virtual equilibrium (denoted by EV), pseudoequilibrium (denoted by EP), boundary equilibrium (denoted by EB), and one type of special point named as tangent point (denoted by ET). Detailed definitions of these equilibria and the tangent point can be found in the literature [24].From Lemma1, we know that the subsystem S1 with x<ET has three possible equilibria: (0,0), (k,0), and interior equilibrium,
(23)ES1*=(xS1*,yS1*)=(k(βbr+δ-βa)βbr,k(βa-δ)(βbr+δ-βa)βb2rδ).Ifβa-δ>0 and r>(βa-δ)/βb, then there is a positive interior equilibrium for the system S1.Equilibria for the subsystemS2 with x>ET satisfy
(24)rx(1-xk)-axyby+x-q1x=0,-δy+βaxyby+x+q2y=0.Solving the above equations with respect tox and y yields three possible equilibria, that is, (0,0), (k(r-q1)/r,0), and interior equilibrium,
(25)ES2*=(xS2*,yS2*)=(k(βb(r-q1)+δ-q2-βa)βbr,k(βa-δ+q2)(βb(r-q1)+δ-q2-βa)βb2r(δ-q2)),
which indicates that, if r>(βa-δ+q2)/βb+q1 and q2<δ<βa+q2, then there is a positive interior equilibrium for the system S2.


### 4.2.1. Regular Equilibria

For the subsystemS1 with x<ET, (0,0) is a regular equilibrium, while (k,0) is a virtual equilibrium. In addition, according to the coordinate of equilibrium ES1*, we have the following results. If r>(βa-δ)/βb>0 and
(26)k(βbr+δ-βa)βbr<ET,
then it is a regular equilibrium for the system S1, denoted by ER1. If r>(βa-δ)/βb>0 and
(27)k(βbr+δ-βa)βbr>ET,
then the equilibrium ES1* becomes a virtual equilibrium, denoted by EV1.About regular equilibria for the subsystemS2 with x>ET,  (0,0) is a virtual equilibrium, while (k(r-p1)/r,0) may be a regular or virtual equilibrium which depends on the parameter space. Moreover, according to the coordinate of equilibrium ES2*, we have the following conclusions. If r>(βa-δ+q2)/βb+q1, q2<δ<βa+q2 and
(28)k(βb(r-q1)+δ-q2-βa)βbr>ET,
then it is a regular equilibrium for the system S2, denoted by ER2. Note that if r>(βa-δ+q2)/βb+q1, q2<δ<βa+q2 and
(29)k(βb(r-q1)+δ-q2-βa)βbr<ET,
then the equilibrium ES2* becomes a virtual equilibrium, denoted by EV2. Based on the above analysis, if
(30)k(βa-δ)βb(k-ET)<r<k(βa-δ+q2+βbq1)βb(k-ET),
then the two virtual equilibria EV1 and EV2 can coexist.


### 4.2.2. Pseudoequilibrium

For the existence of pseudoequilibriumEP=(ET,yP), yP component of the pseudoequilibrium of sliding flow satisfies the following equation:
(31)P1(y)(P2y+P3)=0,
where y∈ΣS1 or ΣS2. Solving the above equation with respect to y yields two possible roots denoted by yP1=0 and
(32)yP2=(δq1-βaq1-rq2(1-ET/k))ET-δbq1+rbq2(1-ET/k)-aq2.Further, if(33)kq1(δ-βa)q2(k-ET)<r<k(aq2+δbq1)bq2(k-ET)
holds true, then yP2 is a positive root. Note that if EP1=(ET,yP1) or EP2=(ET,yP2) lies in the sliding region ΣS1 or ΣS2, then the model has pseudoequilibrium. To do this, we consider the following two cases.Case 1 (sliding segment defined byΣS1).
If the inequalityr<q1k/(k-ET) holds true, then EP1=(ET,yP1)∈ΣS1 is a pseudoequilibrium of the Filippov system (3). Note that -δbq1+rbq2(1-ET/k)-aq2<0 (i.e., r<k(aq2+δbq1)/bq2(k-ET)) is well defined in this case. Therefore if the inequalities
(34)max{kq1(δ-βa)q2(k-ET),k(βa-δ)βb(k-ET)}<r<min{k(βa-δ+q2+βbq1)βb(k-ET),k(aq2+δbq1)bq2(k-ET)}
hold true, then EP2=(ET,yP2)∈ΣS1 is a positive pseudoequilibrium of the Filippov system (3).Case 2 (sliding segment defined byΣS2).
In this case,EP1=(ET,yP1)∈ΣS2 is a pseudoequilibrium of the Filippov system (3). If the inequalities
(35)max{k(βa-δ+q2+βbq1)βb(k-ET),kq1(δ-βa)q2(k-ET)}≤r<k(aq2+δbq1)bq2(k-ET)
hold true, EP2=(ET,yP2)∈ΣS2 is a positive pseudoequilibrium of the Filippov system (3).


### 4.2.3. Boundary Equilibrium

The boundary equilibria of Filippov system (3) satisfy
(36)rx(1-xk)-axyby+x-ɛq1x=0,-δy+βaxyby+x+ɛq2y=0,x=ET,
with ɛ=0 or 1, which indicate that, if
(37)k(βbr+δ-βa)βbr=ETork(βb(r-q1)+δ-q2-βa)βbr=ET,
then we have the boundary equilibria
(38)EB1=(ET,(βa-δ)ETbδ),EB2=(ET,(βa-δ+q2)ETb(δ-q2)).


### 4.2.4. Tangent Point

According to the definition of tangent point, we can see that the tangent pointET=(ET,yT) on sliding segment ΣS satisfies
(39)rx(1-xk)-axyby+x-ɛq1x=0,x=ET.Solving the above equations with respect tox and y yields two tangent points, including
(40)ET1=(ET,rET(1-ET/k)a-br(1-ET/k)),ET2=(ET(r(1-ET/k)-q1)a+q1b-br(1-ET/k)).If we fix all parameters, then the relations among null-isoclines, regular/virtual equilibria, pseudoequilibrium, and sliding segment are provided in Figure1. Note that virtual equilibria of both systems S1 and S2 imply the existence of pseudoequilibrium, and we will prove this general result later.Figure 1
Notations for null-isoclines, regular/virtual equilibria, pseudoequilibrium, and sliding segment. The parameter values are fixed as follows:a=0.7, b=0.6, r=1.2, k=12, β=0.5, δ=0.25, q1=0.25, q2=0.1, and ET=5.


## 4.2.1. Regular Equilibria

For the subsystemS1 with x<ET, (0,0) is a regular equilibrium, while (k,0) is a virtual equilibrium. In addition, according to the coordinate of equilibrium ES1*, we have the following results. If r>(βa-δ)/βb>0 and
(26)k(βbr+δ-βa)βbr<ET,
then it is a regular equilibrium for the system S1, denoted by ER1. If r>(βa-δ)/βb>0 and
(27)k(βbr+δ-βa)βbr>ET,
then the equilibrium ES1* becomes a virtual equilibrium, denoted by EV1.About regular equilibria for the subsystemS2 with x>ET,  (0,0) is a virtual equilibrium, while (k(r-p1)/r,0) may be a regular or virtual equilibrium which depends on the parameter space. Moreover, according to the coordinate of equilibrium ES2*, we have the following conclusions. If r>(βa-δ+q2)/βb+q1, q2<δ<βa+q2 and
(28)k(βb(r-q1)+δ-q2-βa)βbr>ET,
then it is a regular equilibrium for the system S2, denoted by ER2. Note that if r>(βa-δ+q2)/βb+q1, q2<δ<βa+q2 and
(29)k(βb(r-q1)+δ-q2-βa)βbr<ET,
then the equilibrium ES2* becomes a virtual equilibrium, denoted by EV2. Based on the above analysis, if
(30)k(βa-δ)βb(k-ET)<r<k(βa-δ+q2+βbq1)βb(k-ET),
then the two virtual equilibria EV1 and EV2 can coexist.


## 4.2.2. Pseudoequilibrium

For the existence of pseudoequilibriumEP=(ET,yP), yP component of the pseudoequilibrium of sliding flow satisfies the following equation:
(31)P1(y)(P2y+P3)=0,
where y∈ΣS1 or ΣS2. Solving the above equation with respect to y yields two possible roots denoted by yP1=0 and
(32)yP2=(δq1-βaq1-rq2(1-ET/k))ET-δbq1+rbq2(1-ET/k)-aq2.Further, if(33)kq1(δ-βa)q2(k-ET)<r<k(aq2+δbq1)bq2(k-ET)
holds true, then yP2 is a positive root. Note that if EP1=(ET,yP1) or EP2=(ET,yP2) lies in the sliding region ΣS1 or ΣS2, then the model has pseudoequilibrium. To do this, we consider the following two cases.Case 1 (sliding segment defined byΣS1).
If the inequalityr<q1k/(k-ET) holds true, then EP1=(ET,yP1)∈ΣS1 is a pseudoequilibrium of the Filippov system (3). Note that -δbq1+rbq2(1-ET/k)-aq2<0 (i.e., r<k(aq2+δbq1)/bq2(k-ET)) is well defined in this case. Therefore if the inequalities
(34)max{kq1(δ-βa)q2(k-ET),k(βa-δ)βb(k-ET)}<r<min{k(βa-δ+q2+βbq1)βb(k-ET),k(aq2+δbq1)bq2(k-ET)}
hold true, then EP2=(ET,yP2)∈ΣS1 is a positive pseudoequilibrium of the Filippov system (3).Case 2 (sliding segment defined byΣS2).
In this case,EP1=(ET,yP1)∈ΣS2 is a pseudoequilibrium of the Filippov system (3). If the inequalities
(35)max{k(βa-δ+q2+βbq1)βb(k-ET),kq1(δ-βa)q2(k-ET)}≤r<k(aq2+δbq1)bq2(k-ET)
hold true, EP2=(ET,yP2)∈ΣS2 is a positive pseudoequilibrium of the Filippov system (3).


## 4.2.3. Boundary Equilibrium

The boundary equilibria of Filippov system (3) satisfy
(36)rx(1-xk)-axyby+x-ɛq1x=0,-δy+βaxyby+x+ɛq2y=0,x=ET,
with ɛ=0 or 1, which indicate that, if
(37)k(βbr+δ-βa)βbr=ETork(βb(r-q1)+δ-q2-βa)βbr=ET,
then we have the boundary equilibria
(38)EB1=(ET,(βa-δ)ETbδ),EB2=(ET,(βa-δ+q2)ETb(δ-q2)).


## 4.2.4. Tangent Point

According to the definition of tangent point, we can see that the tangent pointET=(ET,yT) on sliding segment ΣS satisfies
(39)rx(1-xk)-axyby+x-ɛq1x=0,x=ET.Solving the above equations with respect tox and y yields two tangent points, including
(40)ET1=(ET,rET(1-ET/k)a-br(1-ET/k)),ET2=(ET(r(1-ET/k)-q1)a+q1b-br(1-ET/k)).If we fix all parameters, then the relations among null-isoclines, regular/virtual equilibria, pseudoequilibrium, and sliding segment are provided in Figure1. Note that virtual equilibria of both systems S1 and S2 imply the existence of pseudoequilibrium, and we will prove this general result later.Figure 1
Notations for null-isoclines, regular/virtual equilibria, pseudoequilibrium, and sliding segment. The parameter values are fixed as follows:a=0.7, b=0.6, r=1.2, k=12, β=0.5, δ=0.25, q1=0.25, q2=0.1, and ET=5.


## 4.3.  The Stability of Pseudoequilibrium

In the process of pest control, we should apply all kinds of control strategies so as to prevent multiple pest outbreaks or make sure that the total density of the pest stabilizes at a desired level of ET. If the unique positive equilibrium of systemS1 and system S2 is virtual simultaneously, then the sliding flow has a unique pseudoequilibrium. In order to realize this goal, we can choose a set of parameters such that all the equilibria of subsystems S1 and S2 are virtual equilibria, and the pseudoequilibria are globally stable, which have been widely used in pest control [26–28]. For example, if we fixed all parameter values as those in Figure 2, then both virtual equilibria coexist. Therefore, we address the stability of pseudoequilibrium EP2 in the following which is important to control pest.Figure 2
Regular/virtual equilibrium. The parameter values are fixed as follows:a=0.7, b=0.6, k=12, β=0.5, δ=0.25, q1=0.25, and q2=0.1.Theorem 6.
Either the inequalities (34) or (35) hold true or the two virtual equilibria EV1 and EV2 can coexist; then Filippov system (3) contains a positive pseudoequilibrium EP2. Regardless of which cases would occur, the pseudoequilibrium EP2 is locally stable with respect to sliding mode domain.Proof.
According to the conditions of Theorem6 we see that the inequalities (30) hold true, which implies that the inequalities (34) are true. Based on the discussions about the existence of pseudoequilibrium in Section 4.2, we have concluded that if inequalities (34) or (35) hold true, then the system (3) contains a positive pseudoequilibrium EP2.
It follows from (18) that we have(41)dϕ(y)dy|y=yP2=(-δb2kq1+rb2q2(k-ET)-abkq2)yP22kq1(byP2+ET)2+(-2δbkq1ET-2akq2ET+2rbq2ET(k-ET))yP2kq1(byP2+ET)2+-δkq1ET2+aβkq1ET2+rkq2ET2-rq2ET3kq1(byP2+ET)2=(kq1(δ-βa)-rq2(k-ET))(δbkq1-rbq2(k-ET)+akq2)aq1k2(bβq1+q2).Therefore, if the inequalities (33) hold true, then dϕ(y)/dy<0 and yP2 is a positive root of (31). Note that the inequalities (34) and (35) indicate the inequalities (33). Thus, the positive pseudoequilibrium EP2 is locally stale if it exists.It is difficult to directly prove the global stability of pseudoequilibriumEP2 in this case. Because we cannot employ the classical Bendixson-Dulac theorem due to the discontinuity of vector fields. However, if there is not crossing cycle surrounding sliding segment, then pseudoequilibrium is globally stable [29]. From the analysis of global bifurcation for the system (3) in the following section, the system just has touching bifurcation and there is nonexistence of a sliding cycle which surrounds the EP2. By using the similar methods, we have that pseudoequilibrium EP2 is globally stable. That is to say, the local stability of pseudoequilibrium EP2 with respect to sliding mode domain indicates its global stability in the first quadrant (shown in Figures 3(a), 3(b), 4(b), and 4(c)). In practice, in order to control pest outbreak, we should choose the desirable ET at first, so that all equilibria of each system such as system S1 and system S2 become virtual; then pseudoequilibrium not only exists but also is globally stable. In other words, the density of pest can be stable at the ET. When the density of pest reaches or exceeds the ET, we should carry out control strategies (e.g., releasing natural enemy, etc.), until it falls below the ET. In this way, our control goal can be realized fully.Boundary node bifurcation for Filippov system (3). The parameter values are fixed as follows: a=0.7, b=0.6, r=1.5, k=12, β=0.5, δ=0.25, q1=0.25, and q2=0.1.
(a)
ET
=
8.3
(b)
ET
=
9.3333
(c)
ET
=
9.8Boundary node bifurcation for Filippov system (3). The parameter values are fixed as follows: a=0.7, b=0.6, r=1.5, k=12, β=0.5, δ=0.25, q1=0.25, and q2=0.1.
(a)
ET
=
4
(b)
ET
=
4.6667
(c)
ET
=
5.8


## 5. Equilibria and Sliding Bifurcation Set

### 5.1. Regular/Virtual Equilibrium Bifurcation

According to the above discussions, it is obvious thatr and ET are primary factors in determining the existence of the above different types of equilibria of the system (3). So we define four curves about parameters r and ET as follows:
(42)L1={(r,ET)∣r=βa-δβb},L2={(r,ET)∣r=βa-δ+q2βb+q1},L3={(r,ET)∣ET=k(βbr+δ-βa)βbr},L4={(r,ET)∣ET=k(βb(r-q1)+δ-q2-βa)βbr}.The four curves (i.e.,L1, L2, L3, and L4) divide the r and ET parameter space into six regions, and the existence or coexistence of regular or virtual equilibria is indicated in each region. The boundary equilibria EB1 and EB2 can appear on the lines L3 and L4 accordingly. In particular, it follows from Figure 2 that the two virtual equilibria EV1 and EV2 can coexist, which is very important to pest control. However, the two regular equilibria ER1 and ER2 cannot coexist.


### 5.2.  Boundary Node Bifurcations

This type of bifurcation may occur for Filippov system (3) once equilibria EP, ET, and ER or ET and ER collide together simultaneously when ET passes through a critical value. In this part, we choose ET as bifurcation parameter, and all other parameters are fixed as those in Figure 3. Note that once the parameter ET passes through the first critical value ETc1=9.3333, the regular equilibrium ER1, tangent point ET, and pseudoequilibrium EP collide together (see Figure 3(b)), where ETc1 is determined by
(43)ETc1=k(βbr+δ-βa)βbr.A virtual equilibriumEV1, an invisible tangent point ET1 and a pseudoequilibrium EP coexist, as shown in Figure 3(a), when ET<ETc1. They collide at ET=ETc1 and are substituted by a visible tangent point ET1, as shown in Figure 3(c), when ET>ETc1.Similarly, another boundary node bifurcation of Filippov system (3) occurs at ETc2=4.6667 (see Figure 4(b)), where ETc2 is determined by
(44)ETc2=k(βb(r-q1)+δ-q2-βa)βbr.A stable regular equilibriumER2 and a visible tangent point ET2 coexist as shown in Figure 4(a) when ET<ETc2. They collide with a pseudoequilibrium EP at ET=ETc2 and are substituted by an invisible tangent point ET2 as shown in Figure 4(c) when ET>ETc2.


### 5.3. Global Sliding Bifurcation

Global sliding bifurcations involve nonvanishing cycles, which include sliding disconnection, touching (or grazing) bifurcation, buckling bifurcation, crossing bifurcation, bifurcation of a sliding homoclinic orbit, and heteroclinic orbit [22]. Touching (or grazing) bifurcation implies that a positive period solution can collide with the sliding segments. From the work of Kuang and Beretta and Hsu et al. [2, 7], they concluded that when the value of the parameter a passed slightly through the bifurcation value, a stable limit cycle bifurcates from the unstable positive interior equilibrium for the system (1). According to numerical simulations, the system (3) just has touching bifurcation. For example, if we choose ET as bifurcation parameter and fix all other parameters as shown in Figure 5, when the value of parameter ET varies, touching bifurcation occurs at the critical value ETc3=2.59 for the system (3). Note that extensive numerical simulations indicate that, no matter what the value of ET is in touching bifurcation, the whole periodic solution lies in the region S1 (shown in Figure 5). This implies that the density of pest can be successfully maintained below ET by designing suitable threshold policy strategies. So our control objective can be fully realized, which is a desired situation in crop, livestock sectors and forestry.Globally touching bifurcation for Filippov system (3). The parameter values are fixed as follows: a=0.51, b=0.4, r=1.2, k=6, β=0.6863, δ=0.05, q1=0.4, and q2=0.1.
(a)
ET
=
1.5
(b)
ET
=
2.59
(c)
ET
=
3.5


## 5.1. Regular/Virtual Equilibrium Bifurcation

According to the above discussions, it is obvious thatr and ET are primary factors in determining the existence of the above different types of equilibria of the system (3). So we define four curves about parameters r and ET as follows:
(42)L1={(r,ET)∣r=βa-δβb},L2={(r,ET)∣r=βa-δ+q2βb+q1},L3={(r,ET)∣ET=k(βbr+δ-βa)βbr},L4={(r,ET)∣ET=k(βb(r-q1)+δ-q2-βa)βbr}.The four curves (i.e.,L1, L2, L3, and L4) divide the r and ET parameter space into six regions, and the existence or coexistence of regular or virtual equilibria is indicated in each region. The boundary equilibria EB1 and EB2 can appear on the lines L3 and L4 accordingly. In particular, it follows from Figure 2 that the two virtual equilibria EV1 and EV2 can coexist, which is very important to pest control. However, the two regular equilibria ER1 and ER2 cannot coexist.


## 5.2.  Boundary Node Bifurcations

This type of bifurcation may occur for Filippov system (3) once equilibria EP, ET, and ER or ET and ER collide together simultaneously when ET passes through a critical value. In this part, we choose ET as bifurcation parameter, and all other parameters are fixed as those in Figure 3. Note that once the parameter ET passes through the first critical value ETc1=9.3333, the regular equilibrium ER1, tangent point ET, and pseudoequilibrium EP collide together (see Figure 3(b)), where ETc1 is determined by
(43)ETc1=k(βbr+δ-βa)βbr.A virtual equilibriumEV1, an invisible tangent point ET1 and a pseudoequilibrium EP coexist, as shown in Figure 3(a), when ET<ETc1. They collide at ET=ETc1 and are substituted by a visible tangent point ET1, as shown in Figure 3(c), when ET>ETc1.Similarly, another boundary node bifurcation of Filippov system (3) occurs at ETc2=4.6667 (see Figure 4(b)), where ETc2 is determined by
(44)ETc2=k(βb(r-q1)+δ-q2-βa)βbr.A stable regular equilibriumER2 and a visible tangent point ET2 coexist as shown in Figure 4(a) when ET<ETc2. They collide with a pseudoequilibrium EP at ET=ETc2 and are substituted by an invisible tangent point ET2 as shown in Figure 4(c) when ET>ETc2.


## 5.3. Global Sliding Bifurcation

Global sliding bifurcations involve nonvanishing cycles, which include sliding disconnection, touching (or grazing) bifurcation, buckling bifurcation, crossing bifurcation, bifurcation of a sliding homoclinic orbit, and heteroclinic orbit [22]. Touching (or grazing) bifurcation implies that a positive period solution can collide with the sliding segments. From the work of Kuang and Beretta and Hsu et al. [2, 7], they concluded that when the value of the parameter a passed slightly through the bifurcation value, a stable limit cycle bifurcates from the unstable positive interior equilibrium for the system (1). According to numerical simulations, the system (3) just has touching bifurcation. For example, if we choose ET as bifurcation parameter and fix all other parameters as shown in Figure 5, when the value of parameter ET varies, touching bifurcation occurs at the critical value ETc3=2.59 for the system (3). Note that extensive numerical simulations indicate that, no matter what the value of ET is in touching bifurcation, the whole periodic solution lies in the region S1 (shown in Figure 5). This implies that the density of pest can be successfully maintained below ET by designing suitable threshold policy strategies. So our control objective can be fully realized, which is a desired situation in crop, livestock sectors and forestry.Globally touching bifurcation for Filippov system (3). The parameter values are fixed as follows: a=0.51, b=0.4, r=1.2, k=6, β=0.6863, δ=0.05, q1=0.4, and q2=0.1.
(a)
ET
=
1.5
(b)
ET
=
2.59
(c)
ET
=
3.5


## 6. Discussion

Recently, the threshold policy and IPM strategies have attracted great attention in agriculture, forestry, animal husbandry, and so on [10–12]. In the process of pest control, IPM strategies would be used only when the density of pest reaches or exceeds the ET. In addition, Filippov system provides a natural and rational framework for those real world problems, so it has been widely used in different fields such as in science and engineering [19–21, 23]. In this paper, we employ the sliding analysis of Filippov system to describe and investigate the long term dynamical behavior of the ratio-dependent-type prey-predator model with Michaelis-Menten-type functional response, so that we can use the Filippov system to model intervention of pest control policy. Firstly, we investigate the sliding mode domain, sliding mode dynamics. Secondly, the null-isoclines and the existence of four types of equilibria for Filippov system, including regular, virtual, boundary, and pseudoequilibrium and the tangent points are discussed in detail. Moreover, the stability of pseudoequilibrium is also studied. Thirdly, we have investigated the local sliding bifurcations including regular/virtual equilibrium bifurcation, and boundary node bifurcations. Further, global touching bifurcation is also studied by numerical techniques.In the process of pest control, we should apply all kinds of control strategies so as to prevent multiple pest outbreaks or make sure that the density of pest stabilizes at a desired level of ET. In order to realize these goals, on the one hand, we can determine a set of parameters such that not only all the equilibria of subsystemsS1 and S2 are virtual equilibria, which ensures that the pseudoequilibria exist, but also the pseudoequilibrium is globally stable. From Theorem 6, we provided the conditions of the existence of pseudoequilibrium EP2 for the system (3) and showed that it is locally stable if it exists. Further, we showed that the local stability of pseudoequilibrium EP2 implies global stability by numerical simulation. On the other hand, globally touching bifurcation indicates that the density of pest can be successfully maintained below the ET by designing suitable threshold policy strategies (shown in Figure 5). Therefore our control objective can be achieved fully in the above two cases, which can be used to pest control in crop, livestock sectors and forestry.Although impulsive prey-predator models with the ratio-dependent- or Michaelis-Menten-type response function have been studied in [13–16], the Filippov systems have many advantages in describing interventions including spraying pesticides and releasing natural enemies compared with impulsive models. In this work, the number of natural enemy to be released is proportional to its number. It is interesting that releasing number of natural enemy can be described by constant, independently of the existing numbers of pest and natural enemy. Moreover, in practice, considering environmental energy resource finiteness, we should choose the total number of both populations as a guide to switch the system, which is called the weighted escapement policy (WEP) [19, 30–32]. Therefore, in the future work, we will focus on the above two cases, which could result in richer dynamics.



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*Source: 280945-2013-10-23.xml*