# Poincaré Map Approach to Global Dynamics of the Integrated Pest Management Prey-Predator Model

Authors: Zhenzhen Shi; Qingjian Li; Weiming Li; Huidong Cheng
Journal: Complexity
 (2020)

---

## Abstract

An integrated pest management prey-predator model with ratio-dependent and impulsive feedback control is investigated in this paper. Firstly, we determine the Poincaré map which is defined on the phase set and discuss its main properties including monotonicity, continuity, and discontinuity. Secondly, the existence and stability of the boundary order-one periodic solution are proved by the method of Poincaré map. According to the Poincaré map and related differential equation theory, the conditions of the existence and global stability of the order-one periodic solution are obtained whenΦyA<yA, and we prove the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution when ΦyA>yA. Furthermore, we prove the existence of the order-k k≥2 periodic solution under certain conditions. Finally, we verify the main results by numerical simulation.


---

## Body

## 1. Introduction

Differential equations can be widely used in many ways, such as chemostat cultures in ecological systems [1–4], protection of biological resources and management of pests and diseases [5–8], viral power system (HIV) [9–12], and infectious disease research [13–17].In recent decades, differential equations with a predator-prey model have attracted the attention of many experts. Some researchers have done lots of research studies on several famous predator models including Holling I type [18], Holling II type [19], Holling III type [20], Beddington–DeAngelis type [21], ratio-dependent type [22], and Lotka–Volterra type [23], where the ratio-dependent prey-predator model has an important impact on the interaction between populations in recent years.(1)x′t=rxt1−xtK−mxtytαyt+xt,y′t=yt−d+βxtαyt+xt,where xt are the densities of the pest, yt are the densities of the natural enemy, r represents the intrinsic birth rate of pests, K is the carrying capacity for the pest when natural enemy y=0, and β denotes the ratio of biomass conversion. The death rate of the natural enemy is denoted as d. mxt/αyt+xt represents the ratio-dependent functional response. And r,β,K,d,m,andα are positive constants.The most common methods of pest management in agriculture are biological control and chemical control. There are usually three methods for biological control using natural enemies to prey on pests, virus control, and releasing bacteria or fungi. Among them, it is common to use natural enemies to prey on pests [24–27]. In addition, chemical control can also be called pesticide control. The main method is to spray insecticides when the pests are flooding [28–33]. But, both methods have drawbacks, for instance, biological control is only suitable for pests with relatively low density, while chemical control is highly toxic and pollutes the environment. Therefore, the researchers proposed an integrated pest management (IPM) method, which is a combination of pesticide control and biological control [34–38]. The IPM method does not make pests become extinct but keep pests below economic thresholds. In recent years, some scholars have studied many IPM models, such as Sun et al. proposed an IPM model and researched its dynamics analysis and control optimization in [39]. Thus, combined with system (1), we shall get system (2) as follows:(2)x′t=rxt1−xtK−mxtytαyt+xty′t=yt−d+βxtαyt+xtx<h,Δxt=−axtΔyt=−byt+cx=h,where h denotes the threshold of the pest, that is, integrated control strategy is adopted when x=h. 0<a<1 and 0<b<1 denote the proportion of prey and predator killed by spraying pesticides when x=h, respectively. c represents the number of predator released.In fact, many researchers have studied models with ratio-dependent over the past few decades. Berezovskaya et al. investigated parametric analysis on a prey-predator model with ratio-dependent [40]. Bandyopadhyay et al. obtained the condition that the ratio-dependent deterministic model enters the Hopf bifurcation and studied the stochastic stability of the system [41]. Nie et al. studied existence and local asymptotic stability of the positive periodic solution of the system [42]. However, research studies on the existence and global asymptotic stability of the order-k (k≥1) periodic solution seem to be rare. Therefore, this paper studies the global dynamics of the order-k (k≥1) periodic solution by using the related differential equation theory and properties of Poincaré map.This consists of the following parts. We study qualitative analysis of system (1) in the next section. In Section 3, some properties of Poincaré map and its expression are given. Moreover, the conditions of the existence and stability of order-k (k≥1) periodic solutions of system (2) are obtained. The main results are verified by numerical simulation in Section 4. In Section 5, we give the final conclusion.


## 2. Qualitative Analysis for System (1)

Model (1) possesses two boundary equilibrium pointsO0,0 and E1K,0 and an internal equilibrium point E∗x∗,y∗, where x∗=Kαβr−Kmβ−d/αβr and y∗=β−d/αdx∗.Lemma 1 (see [43]).
H1: if m≤αr and β≤d, then model (1) has no positive equilibrium and E1K,0 is globally asymptotically stableH2: O0,0 is always a saddle point for model (1)H3: if md2+αrβ2/2mdβ+mβ2≥1 and m≤αr, then E∗x∗,y∗ is a globally asymptotically stable node or focus (see Figure 1)
In fact, if the corresponding control strategy is not adopted, the pest population may reach the carrying capacityK. Therefore, we only discuss the condition where condition H1 is established in this paper, that is, m≤αr and β≤d hold in the remaining part of the paper.Figure 1
Phase diagram for system (1). (a) Boundary equilibria. (b) Internal equilibrium is a node. (c) Internal equilibrium is a focus.
(a)
(b)
(c)


## 3. Poincaré Map

In order to facilitate the expression, we make the following regulations.Two isoclines of system (1) are defined as L1:y=rx1−x/K/m−αr1−x/K and L2:y=β−dx/αd. Define two straight lines as L3:x=1−ah and L4:x=h. The intersection point of L1 and L3 is expressed as A1−ah,yA with yA=r1−ahK−1−ah/K/m−αrK−1−ah/K, while L1 intersects L4 at point Bh,yB with yB=rhK−h/K/m−αrK−h/K. Ω=x,y0<x<h,y>0⊂R+2 is defined as the open set in R+2. We define set x,yx=h,0≤y≤yB as impulsive set M and set x+,y+x+=1−ah,c≤y+≤1−byB+c as phase set N. I is defined as the continuous function which satisfies N=IM. Assume that the initial point x0+,y0+ is on the phase set N.


### 3.1. Poincaré Map and Its Properties

We define the following two sets:(3)Sah=x,yx=1−ah,y≥0,Sh=x,yx=h,y≥0.We choose to defineSah as Poincaré map. Assuming point Qk+1−ah,yk+ lies in set Sah, the trajectory Γt,t0,1−ah,yk+=xt,t0,1−ah,yk+,yt,t0,1−ah,yk+ starting from Qk+ reaches at Sh in a limited time t1 (xt1,t0,1−ah,yk+=h), and the intersection point is expressed as Qk+1=h,yk+1. This means that yk+1 is determined by yk+. Thus we shall get yk+1=yt1,t0,1−ah,yk+=Ψyk+. For convenience, denote that y1−ah,yk+=yt1,t0,1−ah,yk+ throughout this paper. Point Qk+1 jumps to point Qk+1+1−ah,yk+1+ on Sah after impulsive effect, where yk+1+=1−byk+1+c. Thus, the Poincare′ map is expressed as:(4)yk+1+=1−bΨyk++c=1−by1−ah,yk++c=Φyk+.For convenience, we assume(5)Px,y=rx1−xK−mxyαy+x,Qx,y=y−d+βxαy+x.We can convert model (2) to the following form:(6)dydx=y−d+βx/αy+xrx1−x/K−mxy/αy+x=gx,y,y1−ah=y0+.For model (6), the region Ω′=x,yx>0,0<y<r/m1−x/Kx+αy is studied.We setx0+,y0+=1−ah,Z, where Z<yB and Z∈N, and then x0+,y0+∈Ω′. Therefore, we obtain(7)yx=yx;1−ah,Z≜yx,Z.Moreover, according to model (6), we have that(8)yx,Z=Z+∫1−ahxgz,yz,Zdz.Therefore, Poincaré mapΦ takes the form in the region Ω′:(9)ΦZ=1−by1−ah,Z+c.Theorem 1.
If conditionH1 holds, then Poincare′ map Φ exists, and it has the following properties, see Figures 2 and 3:(I)
The domain and range ofΦ are 0,+∞ and c,1−by1−ah,yA+c, respectively. It increases on 0,yA and decreases on yA,+∞(II)
Φ is continuously differentiable(III)
Ifd>βx/αy+x and c>0 hold, then Φ always possesses a unique fixed point(IV)
Φ exists, and it has a horizontal asymptote y=c as yk+⟶∞Figure 2
The relationship betweenΦ and yk+ with r=1.5,K=100,m=0.9,α=0.1,d=0.5,β=0.2,a=0.25,b=0.1,andh=40, (a) c=10, and (b) c=21.
(a)
(b)Figure 3
The relationship betweenΦ and yk+ with r=1,K=1,m=0.8,α=0.2,d=0.5,β=0.2,a=0.4,b=0.1,andc=0.3.Proof.
(I)
For∀Z∈0,+∞, the trajectory Γt,t0,1−ah,Z crossing point C1−ah,Z intersects Sh at point C′h,ΨZ. Therefore, the domain of Φ is 0,+∞. We arbitrarily select two points U1−ah,yU+ and V1−ah,yV+ with yU+,yV+∈0,yA and assume yU+<yV+. According to the uniqueness of the solution of system (1), y1−ah,yU+<y1−ah,yV+ can be obtained. After one time pulse, then we have(10)ΦyV+−ΦyU+=1−by1−ah,yV++c−1−by1−ah,yU+−c=1−by1−ah,yV+−y1−ah,yU+>0.ForyU+,yV+∈yA,+∞ with yV+>yU+, the orbits that start at these two points will first cross the line L3 and then hit the line L4. Due to the vector field of model (1), y1−ah,yU+>y1−ah,yV+ is obtained. After one pulse, we have(11)ΦyV+−ΦyU+=1−by1−ah,yV++c−1−by1−ah,yU+−c=1−by1−ah,yV+−y1−ah,yU+<0.Therefore,Φ increases on 0,yA and decreases on yA,+∞, and the range of Φ is c,1−by1−ah,yA+c.(II)
It is easy to see thatPx,y and Qx,y are continuous and differentiable in the first quadrant. Φ is also continuously differentiable according to Cauchy–Lipschitz theorem with parameters.(III)
BecauseΦ decreases on yA,+∞, there is y^∈yA,+∞ that satisfies Φy^<y^. Furthermore, we know Φ0=c≥0. Therefore, Φ has y∗∈0,y^ satisfying Φy∗=y∗, i.e., there is a fixed point on 0,y^ for Φ. If c>0 and ΦyA<yA, then y∗∈0,yA. Because Φ decreases on yA,y^, then Φyk+<ΦyA<yA for yk+∈yA,y^ which means no fixed point exists for Φ on yA,+∞. In the case c>0 and ΦyA>yA, no fixed point exists for Φ on 0,yA because Φ increases on 0,yA and Φc>0. Since Φy^<y^ and ΦyA>yA, Φ has at least a fixed point on yA,y^.It is assumed that system (2) admits two fixed points y˜1 and y˜2, where y˜1,y˜2∈0,y^ and y˜1>y˜2 such that Φy˜1=y˜1 and Φy˜2=y˜2. Then,(12)dy˜1y˜2′x=y˜1′−y˜2′=y˜1−d+βx/αy˜1+xrx1−x/K−mxy˜1/αy˜1+x−y˜2−d+βx/αy˜2+xrx1−x/K−mxy˜2/αy˜2+x=g′ξy˜1−y˜2,wheregy=y−d+βx/αy+x/rx1−x/K−mxy/αy+x with(13)g′y=rx1−x/K−mxy/αy+x−d+βx/αy+x−αβxy/αy+x2+mx2y/αy+x2−d+βx/αy+xrx1−x/K−mxy/αy+x2.Sinced>βx/αy+x, we have g′y<0. Thus, dy˜1y˜2′x<0 for x∈1−ah,h, which indicates that dy˜1y˜2′x is decreasing on 0,y^ and dy˜1y˜2h<dy˜1y˜21−ah. Then,(14)ch=y˜1−1−bΨy˜1=y˜2+dy˜1y˜21−ah−1−bΨy˜2+dy˜1y˜2h>y˜2−1−bΨy˜2,which leads to a contradiction. Therefore, if conditionsd>βx/αy+x and c>0 hold, Φ has a unique fixed point on 0,y^.(IV)
The closure ofΩ is denoted as(15)Ω¯=x,y:x≥0,0≤y≤r1−x/Kxm−αr1−x/K.The setΩ¯ is an invariant set of system (1) provided d>β. Assume(16)L=y−r1−x/Kxm−αr1−x/KandΩ¯ is an invariant set of system (1) if the vector field will flow into the boundary Ω¯. This is true if(17)Px,y,Qx,y⋅αr/Krx1−x/K−r1−2x/Km−αr1−x/Km−αr1−x/K2,1L=0≤0,where⋅ represents the scalar product of two vectors; then, we have(18)HxL=0=y−d+βxαy+x−rx1−x/K−mxyαy+xr1−2x/Km−αr1−x/K−αr/Krx1−x/Km−αr1−x/K2=y−d+βxαy+x<0.In addition, we choosex,y arbitrarily which satisfies x,y∈Ω, and then x′t>0 and y′t<0 are obtained. Thus, we shall get Ψ+∞=0 with 1−ah,+∞∈N, that is, Φ+∞=c. Furthermore, there is y1>0 which satisfies Ψ+∞=y1 with B1h,y1∈M. For any point B2h,y2 with 0<y2<y1, the backward orbit initiating B2 reaches a point B0+1−ah,y0+∈N with y0+>+∞ by the uniqueness of the solution of system (2) and the invariance of the set Ω¯, which is a contradiction. Therefore, we shall get Ψ+∞=0 and Φ+∞=c. Therefore, there exists a horizontal asymptote y=c for Φ (Figure 3).


### 3.2. Stability of Boundary Order-One Periodic Solution Whenc=0

We can get the following equation when predatory populations tend to die and predator release is stopped:(19)x′t=rxt1−xtK,x<h,Δx=−axt,x=h,where x0+=1−ah, and by calculation, we have(20)xt=K1+K/1−ah−1exp−rt.Ifxt reaches the straight line L4 at time T, then(21)h=K1+K/1−ah−1exp−rT.Then, we shall get(22)T=1rlnK−h1−a1−aK−h.Thus, system (19) possesses a periodic solution, expressed by xTt, and(23)xTt=K1+K/1−ah−1exp−rt,with period T, i.e., there exists a boundary order-one periodic solution xTt,0.Theorem 2.
Ifc=0, then the boundary order-one periodic solution xTt,0 is orbitally asymptotically stable provided(24)g2≜−d+βrlnK−1−ah1−aK−h<0.Proof.
ϱ
x
,
y
=
x
−
h, νx,y=−ax, σx,y=−by+c, xTT,yTT=h,0, and xTT+,yTT+=1−ah,0.(25)∂P∂x=r−2rxK−αmy2αy+x2,∂Q∂y=−d+βx2αy+x2,∂ν∂x=−a,∂ν∂y=0,∂σ∂x=0,∂σ∂y=−b,∂ϱ∂x=1,∂ϱ∂y=0,Δ1=P+1−ah,0−b+1Ph,0=1−a1−bK−1−ahK−h,∫0T∂P∂xxTt,yTt+∂Q∂yxTt,yTtdt=∫0Tr−2rxTtK−d+βdt=∫0Tr−2rertert+K/1−ah−1−d+βdt=lnK−h1−aK−1−ah+−d+βrlnK−1−ah1−aK−h,μ2=1−a1−bK−1−ahK−h⋅explnK−h1−aK−1−ah+−d+βrlnK−1−ah1−aK−h=1−bexp−d+βrlnK−1−ah1−aK−h=1−bexpg2.
By a simple calculation, we shall getμ2<1; therefore, the conclusion of Theorem 2 is proved.


### 3.3. Order-k Periodic Solution When c>0

We first give a generalized result of the stability of the order-one periodic solutionζt,δt and assume that the period of the order-one periodic solution is T. Then, we shall get(26)ζT,δT=h,y¯,ζT+,δT+=1−ah,1−by¯+c.Therefore,(27)μ2=Δ1exp∫0T∂P∂xxTt,yTt+∂Q∂yxTt,yTtdt=P+1−ah,1−by¯+c−b+1Ph,y¯exp∫0Tr−2rζtK−αmδt2+βζt2αδt+ζt2−ddt=P+1−ah,1−by¯+c−b+1Ph,y¯exp∫0Tχtdt.Thus, according to [44], Theorem 2.3, the following theorem is obtained.Theorem 3.
Ifμ2<1, then the order-one periodic solution ζt,δt of system (2) is locally asymptotically stable.Theorem 4.
IfΦyA<yA, then the order-one periodic solution of system (2) is globally asymptotically stable.Proof.
IfΦyA<yA, according to Theorem 1, Φ possesses a unique fixed y∗, where y∗∈0,yA. Then, it can be seen from Theorem 1 that model (2) has a unique local asymptotically stable order-one periodic solution.
For any trajectoryΓt,t0,1−ah,y0+, if y0+∈[0,y∗, then y0+<Φy0+<y∗ by Theorem 1. After n time pulses, Φny0+ monotonically increases; thus, limn⟶+∞=y∗.
Fory0+>y∗, there are two cases which are discussed: (a) for all n, Φny0+>y∗. When n increases and limn⟶∞Φny0+=y∗, the series of Φy0+ decreases monotonically by Φy0+<y0+. (b) Φny0+>y∗ is not applicable to all n, and we have a minimum positive integer i that satisfies Φiy0+<y∗ according to Theorem 1. Thus, by using a method similar to case (a), we shall get that Φi+jy0+ increases monotonically when j increases and limj⟶+∞Φi+jy0+=y∗. Therefore, the result in Theorem 4 is proved.Remark 1.
Similarly, ifΦyA=yA, then Φ has a unique fixed point yA. Therefore, model (2) possesses a unique globally asymptotically stable order-one periodic solution.Theorem 5.
IfΦyA>yA and Φ2yA≥yA hold, then model (2) possesses a stable order-one periodic solution or order-two periodic solution.Proof.
There exists an integerξ satisfying yξ+=Φξy0+∈yA,ΦyA for 1−ah,y0+∈N with y0+∈0,+∞. By Theorem 1, for y0+∈0,yA, we shall get that there is no fixed point for Φ on interval 0,yA, and Φ increases monotonically on 0,yA. Thus, there exists an integer ξ satisfying yξ−1+<yA and yξ+≥yA. It follows that yξ+=Φyξ−1+≤ΦA, which means that yξ+∈yA,ΦyA. If y0+∈yA,+∞, there exists an integer ξ satisfying yξ+∈yA,ΦyA because Φy decreases monotonically on yA,+∞, y+=Φy0+≤ΦyA.
SinceΦ is monotonically decreasing on yA,ΦyA and Φ2 is monotonically increasing on yA,ΦyA, thus,(28)ΦyA,ΦyA=Φ2yA,ΦyA⊂yA,ΦyA.
By the above analysis, for anyy0+∈yA,ΦyA, we chose that y1+=Φy0+≠y0+ and y2+=Φ2y0+≠y0+ with yi+=Φiy0+, that is, trajectory starting from point 1−ah,y0+ is not order-one periodic solution or order-two periodic solution. Therefore, there are four cases as follows:(I)
IfΦyA≥y1+>y2+>y0+>yA, then Φy1+=y2+<y3+=Φy2+<Φy0+=y1+ and Φy2+=y3+>y4+=y3+>Φy1+=y2+, that is, y1+>y3+>y4+>y2+>y0+ (see Figure 4(a)). By induction, we can obtain(29)ΦyA≥y1+>⋯>y2i−1+>y2i+1+>⋯>y2i+2+>y2i+>⋯>y2+>y0+≥yA.(II)
IfΦyA>y1+>y0+>y2+>yA, then y3+=Φy2+>Φy0+=y1+ and Φy3+>Φy1+<y2+, Therefore, y3+>y1+>y0+>y2+>y4+ (see Figure 4(b)). By induction, we have(30)ΦyA≥⋯>y2i+1+>y2i−1+>⋯>y1+>y0+>y2+>⋯>y2i+>y2i+2+>⋯≥yA.(III)
IfyA<y1+<y0+<y2+≤ΦyA (see Figure 4(c)), then(31)yA≤⋯<y2i+1+<y2i−1+<⋯<y1+<y0+<y2+<⋯<y2i+<y2i+2+<⋯<ΦyA.(IV)
IfyA≤y1+<y2+<y0+≤ΦyA (see Figure 4(d)), then(32)yA≤y1+<⋯<y2i−1+<y2i+1+<⋯<y2i+2+<y2i+<⋯<y2+<y0+≤ΦyA.
By the above analysis, for cases (I) and (IV), there is uniquey∗∈yA,ΦyA satisfying limi⟶∞y2i+1=limi⟶∞y2i=y∗. For (II) and (III), there are two different values y1∗ and y2∗ with y1∗≠y2∗ such that limn⟶∞y2i+1=y1∗ and limi⟶∞y2i=y2∗. Therefore, the above conclusion is proved.Figure 4
(a)ΦyA≥y1+>y2+>y0+>yA. (b) ΦyA>y1+>y0+>y2+>yA. (c) yA<y1+<y0+<y2+≤ΦyA. (d) yA≤y1+<y2+<y0+≤ΦyA.
(a)
(b)
(c)
(d)Theorem 6.
WhenΦyA>yA, then the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution of system (2) are as follows: for any y0+∈yA,y∗ satisfies Φ2y0+>y0+, where Φy∗=y∗.Proof.
We get that model (2) possesses a stable order-one periodic solution when ΦyA>yA by Theorem 5. Then, for system (2), there exists y∗∈yA,ΦyA which satisfies Φy∗=y∗. For any y0+∈yA,y∗, make yn+=Φny0+,n=1,2,…, and according to the monotonicity of Poincare′ map Φ, we can get yA<y0+<y2+<y∗<y1+<ΦyA. Furthermore, yA<y2n+<y∗<y2n+1+<ΦyA is obtained. From the monotonicity of the sequence, we know limn⟶∞y2n+=limn⟶∞y2n+1+=y∗. Therefore, the order-one periodic solution of system (2) is globally asymptotically stable.
We assume the order-one periodic solution is globally stable for anyy+∈yA,y∗, Φ2y+>y+; if not, there exists minimum y0∈yA,y∗ which makes Φ2y0≤y0. From the theorem, we get that, for any small ε>0, there exists y1∈y∗−ε,y∗+ε which makes Φ2y1>y1. We know from the continuity of Φ that there exists y∗∗∈y0,y1 which makes Φ2y∗∗=y∗∗, that is, system (2) possesses a order-two periodic solution, which is a contradiction with the assumption.Theorem 7.
WhenΦyA>yA and Φ2yA<ym+, where Φym+=yA, then system (2) has the order-three periodic solution.Proof.
SinceΦyA>yA, according to Theorem 1, it is easy to know that there is unique y∗∈yA,ΦyA, which makes Φy∗=y∗. Since Poincare′ map is continuous in closed intervals 0,y∗ and Φ0=c>0 and Φy∗=y∗, according to the intermediate value theorem, there is ym+∈0,y∗ which makes Φym+=yA; furthermore, Φ3ym+=Φ2yA<ym+, that is, Φ3ym+−ym+<0.
On the contrary, according to the expression of Poincare′ map, we shall get that Φ30=τ>0. So, there is a number y⌣∈0,ym+ which makes Φ3y⌣=y⌣. This means that model (2) has the order-three periodic solution represented by the initial point 1−ah,y⌣.
In the same way, we can prove that system (2) has the order-k periodic solution provided Φk−1yA<ym+ and Φym+=yA.


## 3.1. Poincaré Map and Its Properties

We define the following two sets:(3)Sah=x,yx=1−ah,y≥0,Sh=x,yx=h,y≥0.We choose to defineSah as Poincaré map. Assuming point Qk+1−ah,yk+ lies in set Sah, the trajectory Γt,t0,1−ah,yk+=xt,t0,1−ah,yk+,yt,t0,1−ah,yk+ starting from Qk+ reaches at Sh in a limited time t1 (xt1,t0,1−ah,yk+=h), and the intersection point is expressed as Qk+1=h,yk+1. This means that yk+1 is determined by yk+. Thus we shall get yk+1=yt1,t0,1−ah,yk+=Ψyk+. For convenience, denote that y1−ah,yk+=yt1,t0,1−ah,yk+ throughout this paper. Point Qk+1 jumps to point Qk+1+1−ah,yk+1+ on Sah after impulsive effect, where yk+1+=1−byk+1+c. Thus, the Poincare′ map is expressed as:(4)yk+1+=1−bΨyk++c=1−by1−ah,yk++c=Φyk+.For convenience, we assume(5)Px,y=rx1−xK−mxyαy+x,Qx,y=y−d+βxαy+x.We can convert model (2) to the following form:(6)dydx=y−d+βx/αy+xrx1−x/K−mxy/αy+x=gx,y,y1−ah=y0+.For model (6), the region Ω′=x,yx>0,0<y<r/m1−x/Kx+αy is studied.We setx0+,y0+=1−ah,Z, where Z<yB and Z∈N, and then x0+,y0+∈Ω′. Therefore, we obtain(7)yx=yx;1−ah,Z≜yx,Z.Moreover, according to model (6), we have that(8)yx,Z=Z+∫1−ahxgz,yz,Zdz.Therefore, Poincaré mapΦ takes the form in the region Ω′:(9)ΦZ=1−by1−ah,Z+c.Theorem 1.
If conditionH1 holds, then Poincare′ map Φ exists, and it has the following properties, see Figures 2 and 3:(I)
The domain and range ofΦ are 0,+∞ and c,1−by1−ah,yA+c, respectively. It increases on 0,yA and decreases on yA,+∞(II)
Φ is continuously differentiable(III)
Ifd>βx/αy+x and c>0 hold, then Φ always possesses a unique fixed point(IV)
Φ exists, and it has a horizontal asymptote y=c as yk+⟶∞Figure 2
The relationship betweenΦ and yk+ with r=1.5,K=100,m=0.9,α=0.1,d=0.5,β=0.2,a=0.25,b=0.1,andh=40, (a) c=10, and (b) c=21.
(a)
(b)Figure 3
The relationship betweenΦ and yk+ with r=1,K=1,m=0.8,α=0.2,d=0.5,β=0.2,a=0.4,b=0.1,andc=0.3.Proof.
(I)
For∀Z∈0,+∞, the trajectory Γt,t0,1−ah,Z crossing point C1−ah,Z intersects Sh at point C′h,ΨZ. Therefore, the domain of Φ is 0,+∞. We arbitrarily select two points U1−ah,yU+ and V1−ah,yV+ with yU+,yV+∈0,yA and assume yU+<yV+. According to the uniqueness of the solution of system (1), y1−ah,yU+<y1−ah,yV+ can be obtained. After one time pulse, then we have(10)ΦyV+−ΦyU+=1−by1−ah,yV++c−1−by1−ah,yU+−c=1−by1−ah,yV+−y1−ah,yU+>0.ForyU+,yV+∈yA,+∞ with yV+>yU+, the orbits that start at these two points will first cross the line L3 and then hit the line L4. Due to the vector field of model (1), y1−ah,yU+>y1−ah,yV+ is obtained. After one pulse, we have(11)ΦyV+−ΦyU+=1−by1−ah,yV++c−1−by1−ah,yU+−c=1−by1−ah,yV+−y1−ah,yU+<0.Therefore,Φ increases on 0,yA and decreases on yA,+∞, and the range of Φ is c,1−by1−ah,yA+c.(II)
It is easy to see thatPx,y and Qx,y are continuous and differentiable in the first quadrant. Φ is also continuously differentiable according to Cauchy–Lipschitz theorem with parameters.(III)
BecauseΦ decreases on yA,+∞, there is y^∈yA,+∞ that satisfies Φy^<y^. Furthermore, we know Φ0=c≥0. Therefore, Φ has y∗∈0,y^ satisfying Φy∗=y∗, i.e., there is a fixed point on 0,y^ for Φ. If c>0 and ΦyA<yA, then y∗∈0,yA. Because Φ decreases on yA,y^, then Φyk+<ΦyA<yA for yk+∈yA,y^ which means no fixed point exists for Φ on yA,+∞. In the case c>0 and ΦyA>yA, no fixed point exists for Φ on 0,yA because Φ increases on 0,yA and Φc>0. Since Φy^<y^ and ΦyA>yA, Φ has at least a fixed point on yA,y^.It is assumed that system (2) admits two fixed points y˜1 and y˜2, where y˜1,y˜2∈0,y^ and y˜1>y˜2 such that Φy˜1=y˜1 and Φy˜2=y˜2. Then,(12)dy˜1y˜2′x=y˜1′−y˜2′=y˜1−d+βx/αy˜1+xrx1−x/K−mxy˜1/αy˜1+x−y˜2−d+βx/αy˜2+xrx1−x/K−mxy˜2/αy˜2+x=g′ξy˜1−y˜2,wheregy=y−d+βx/αy+x/rx1−x/K−mxy/αy+x with(13)g′y=rx1−x/K−mxy/αy+x−d+βx/αy+x−αβxy/αy+x2+mx2y/αy+x2−d+βx/αy+xrx1−x/K−mxy/αy+x2.Sinced>βx/αy+x, we have g′y<0. Thus, dy˜1y˜2′x<0 for x∈1−ah,h, which indicates that dy˜1y˜2′x is decreasing on 0,y^ and dy˜1y˜2h<dy˜1y˜21−ah. Then,(14)ch=y˜1−1−bΨy˜1=y˜2+dy˜1y˜21−ah−1−bΨy˜2+dy˜1y˜2h>y˜2−1−bΨy˜2,which leads to a contradiction. Therefore, if conditionsd>βx/αy+x and c>0 hold, Φ has a unique fixed point on 0,y^.(IV)
The closure ofΩ is denoted as(15)Ω¯=x,y:x≥0,0≤y≤r1−x/Kxm−αr1−x/K.The setΩ¯ is an invariant set of system (1) provided d>β. Assume(16)L=y−r1−x/Kxm−αr1−x/KandΩ¯ is an invariant set of system (1) if the vector field will flow into the boundary Ω¯. This is true if(17)Px,y,Qx,y⋅αr/Krx1−x/K−r1−2x/Km−αr1−x/Km−αr1−x/K2,1L=0≤0,where⋅ represents the scalar product of two vectors; then, we have(18)HxL=0=y−d+βxαy+x−rx1−x/K−mxyαy+xr1−2x/Km−αr1−x/K−αr/Krx1−x/Km−αr1−x/K2=y−d+βxαy+x<0.In addition, we choosex,y arbitrarily which satisfies x,y∈Ω, and then x′t>0 and y′t<0 are obtained. Thus, we shall get Ψ+∞=0 with 1−ah,+∞∈N, that is, Φ+∞=c. Furthermore, there is y1>0 which satisfies Ψ+∞=y1 with B1h,y1∈M. For any point B2h,y2 with 0<y2<y1, the backward orbit initiating B2 reaches a point B0+1−ah,y0+∈N with y0+>+∞ by the uniqueness of the solution of system (2) and the invariance of the set Ω¯, which is a contradiction. Therefore, we shall get Ψ+∞=0 and Φ+∞=c. Therefore, there exists a horizontal asymptote y=c for Φ (Figure 3).


## 3.2. Stability of Boundary Order-One Periodic Solution Whenc=0

We can get the following equation when predatory populations tend to die and predator release is stopped:(19)x′t=rxt1−xtK,x<h,Δx=−axt,x=h,where x0+=1−ah, and by calculation, we have(20)xt=K1+K/1−ah−1exp−rt.Ifxt reaches the straight line L4 at time T, then(21)h=K1+K/1−ah−1exp−rT.Then, we shall get(22)T=1rlnK−h1−a1−aK−h.Thus, system (19) possesses a periodic solution, expressed by xTt, and(23)xTt=K1+K/1−ah−1exp−rt,with period T, i.e., there exists a boundary order-one periodic solution xTt,0.Theorem 2.
Ifc=0, then the boundary order-one periodic solution xTt,0 is orbitally asymptotically stable provided(24)g2≜−d+βrlnK−1−ah1−aK−h<0.Proof.
ϱ
x
,
y
=
x
−
h, νx,y=−ax, σx,y=−by+c, xTT,yTT=h,0, and xTT+,yTT+=1−ah,0.(25)∂P∂x=r−2rxK−αmy2αy+x2,∂Q∂y=−d+βx2αy+x2,∂ν∂x=−a,∂ν∂y=0,∂σ∂x=0,∂σ∂y=−b,∂ϱ∂x=1,∂ϱ∂y=0,Δ1=P+1−ah,0−b+1Ph,0=1−a1−bK−1−ahK−h,∫0T∂P∂xxTt,yTt+∂Q∂yxTt,yTtdt=∫0Tr−2rxTtK−d+βdt=∫0Tr−2rertert+K/1−ah−1−d+βdt=lnK−h1−aK−1−ah+−d+βrlnK−1−ah1−aK−h,μ2=1−a1−bK−1−ahK−h⋅explnK−h1−aK−1−ah+−d+βrlnK−1−ah1−aK−h=1−bexp−d+βrlnK−1−ah1−aK−h=1−bexpg2.
By a simple calculation, we shall getμ2<1; therefore, the conclusion of Theorem 2 is proved.


## 3.3. Order-k Periodic Solution When c>0

We first give a generalized result of the stability of the order-one periodic solutionζt,δt and assume that the period of the order-one periodic solution is T. Then, we shall get(26)ζT,δT=h,y¯,ζT+,δT+=1−ah,1−by¯+c.Therefore,(27)μ2=Δ1exp∫0T∂P∂xxTt,yTt+∂Q∂yxTt,yTtdt=P+1−ah,1−by¯+c−b+1Ph,y¯exp∫0Tr−2rζtK−αmδt2+βζt2αδt+ζt2−ddt=P+1−ah,1−by¯+c−b+1Ph,y¯exp∫0Tχtdt.Thus, according to [44], Theorem 2.3, the following theorem is obtained.Theorem 3.
Ifμ2<1, then the order-one periodic solution ζt,δt of system (2) is locally asymptotically stable.Theorem 4.
IfΦyA<yA, then the order-one periodic solution of system (2) is globally asymptotically stable.Proof.
IfΦyA<yA, according to Theorem 1, Φ possesses a unique fixed y∗, where y∗∈0,yA. Then, it can be seen from Theorem 1 that model (2) has a unique local asymptotically stable order-one periodic solution.
For any trajectoryΓt,t0,1−ah,y0+, if y0+∈[0,y∗, then y0+<Φy0+<y∗ by Theorem 1. After n time pulses, Φny0+ monotonically increases; thus, limn⟶+∞=y∗.
Fory0+>y∗, there are two cases which are discussed: (a) for all n, Φny0+>y∗. When n increases and limn⟶∞Φny0+=y∗, the series of Φy0+ decreases monotonically by Φy0+<y0+. (b) Φny0+>y∗ is not applicable to all n, and we have a minimum positive integer i that satisfies Φiy0+<y∗ according to Theorem 1. Thus, by using a method similar to case (a), we shall get that Φi+jy0+ increases monotonically when j increases and limj⟶+∞Φi+jy0+=y∗. Therefore, the result in Theorem 4 is proved.Remark 1.
Similarly, ifΦyA=yA, then Φ has a unique fixed point yA. Therefore, model (2) possesses a unique globally asymptotically stable order-one periodic solution.Theorem 5.
IfΦyA>yA and Φ2yA≥yA hold, then model (2) possesses a stable order-one periodic solution or order-two periodic solution.Proof.
There exists an integerξ satisfying yξ+=Φξy0+∈yA,ΦyA for 1−ah,y0+∈N with y0+∈0,+∞. By Theorem 1, for y0+∈0,yA, we shall get that there is no fixed point for Φ on interval 0,yA, and Φ increases monotonically on 0,yA. Thus, there exists an integer ξ satisfying yξ−1+<yA and yξ+≥yA. It follows that yξ+=Φyξ−1+≤ΦA, which means that yξ+∈yA,ΦyA. If y0+∈yA,+∞, there exists an integer ξ satisfying yξ+∈yA,ΦyA because Φy decreases monotonically on yA,+∞, y+=Φy0+≤ΦyA.
SinceΦ is monotonically decreasing on yA,ΦyA and Φ2 is monotonically increasing on yA,ΦyA, thus,(28)ΦyA,ΦyA=Φ2yA,ΦyA⊂yA,ΦyA.
By the above analysis, for anyy0+∈yA,ΦyA, we chose that y1+=Φy0+≠y0+ and y2+=Φ2y0+≠y0+ with yi+=Φiy0+, that is, trajectory starting from point 1−ah,y0+ is not order-one periodic solution or order-two periodic solution. Therefore, there are four cases as follows:(I)
IfΦyA≥y1+>y2+>y0+>yA, then Φy1+=y2+<y3+=Φy2+<Φy0+=y1+ and Φy2+=y3+>y4+=y3+>Φy1+=y2+, that is, y1+>y3+>y4+>y2+>y0+ (see Figure 4(a)). By induction, we can obtain(29)ΦyA≥y1+>⋯>y2i−1+>y2i+1+>⋯>y2i+2+>y2i+>⋯>y2+>y0+≥yA.(II)
IfΦyA>y1+>y0+>y2+>yA, then y3+=Φy2+>Φy0+=y1+ and Φy3+>Φy1+<y2+, Therefore, y3+>y1+>y0+>y2+>y4+ (see Figure 4(b)). By induction, we have(30)ΦyA≥⋯>y2i+1+>y2i−1+>⋯>y1+>y0+>y2+>⋯>y2i+>y2i+2+>⋯≥yA.(III)
IfyA<y1+<y0+<y2+≤ΦyA (see Figure 4(c)), then(31)yA≤⋯<y2i+1+<y2i−1+<⋯<y1+<y0+<y2+<⋯<y2i+<y2i+2+<⋯<ΦyA.(IV)
IfyA≤y1+<y2+<y0+≤ΦyA (see Figure 4(d)), then(32)yA≤y1+<⋯<y2i−1+<y2i+1+<⋯<y2i+2+<y2i+<⋯<y2+<y0+≤ΦyA.
By the above analysis, for cases (I) and (IV), there is uniquey∗∈yA,ΦyA satisfying limi⟶∞y2i+1=limi⟶∞y2i=y∗. For (II) and (III), there are two different values y1∗ and y2∗ with y1∗≠y2∗ such that limn⟶∞y2i+1=y1∗ and limi⟶∞y2i=y2∗. Therefore, the above conclusion is proved.Figure 4
(a)ΦyA≥y1+>y2+>y0+>yA. (b) ΦyA>y1+>y0+>y2+>yA. (c) yA<y1+<y0+<y2+≤ΦyA. (d) yA≤y1+<y2+<y0+≤ΦyA.
(a)
(b)
(c)
(d)Theorem 6.
WhenΦyA>yA, then the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution of system (2) are as follows: for any y0+∈yA,y∗ satisfies Φ2y0+>y0+, where Φy∗=y∗.Proof.
We get that model (2) possesses a stable order-one periodic solution when ΦyA>yA by Theorem 5. Then, for system (2), there exists y∗∈yA,ΦyA which satisfies Φy∗=y∗. For any y0+∈yA,y∗, make yn+=Φny0+,n=1,2,…, and according to the monotonicity of Poincare′ map Φ, we can get yA<y0+<y2+<y∗<y1+<ΦyA. Furthermore, yA<y2n+<y∗<y2n+1+<ΦyA is obtained. From the monotonicity of the sequence, we know limn⟶∞y2n+=limn⟶∞y2n+1+=y∗. Therefore, the order-one periodic solution of system (2) is globally asymptotically stable.
We assume the order-one periodic solution is globally stable for anyy+∈yA,y∗, Φ2y+>y+; if not, there exists minimum y0∈yA,y∗ which makes Φ2y0≤y0. From the theorem, we get that, for any small ε>0, there exists y1∈y∗−ε,y∗+ε which makes Φ2y1>y1. We know from the continuity of Φ that there exists y∗∗∈y0,y1 which makes Φ2y∗∗=y∗∗, that is, system (2) possesses a order-two periodic solution, which is a contradiction with the assumption.Theorem 7.
WhenΦyA>yA and Φ2yA<ym+, where Φym+=yA, then system (2) has the order-three periodic solution.Proof.
SinceΦyA>yA, according to Theorem 1, it is easy to know that there is unique y∗∈yA,ΦyA, which makes Φy∗=y∗. Since Poincare′ map is continuous in closed intervals 0,y∗ and Φ0=c>0 and Φy∗=y∗, according to the intermediate value theorem, there is ym+∈0,y∗ which makes Φym+=yA; furthermore, Φ3ym+=Φ2yA<ym+, that is, Φ3ym+−ym+<0.
On the contrary, according to the expression of Poincare′ map, we shall get that Φ30=τ>0. So, there is a number y⌣∈0,ym+ which makes Φ3y⌣=y⌣. This means that model (2) has the order-three periodic solution represented by the initial point 1−ah,y⌣.
In the same way, we can prove that system (2) has the order-k periodic solution provided Φk−1yA<ym+ and Φym+=yA.


## 4. Numerical Simulation

In this paper, we establish an IPM prey-predator model. We give some special examples to verify the previous theoretical results in this section.Firstly, we setr=1.5,β=0.2,K=100,d=0.3,m=0.9,α=0.5,a=0.3,b=0.1,h=60,andc=0, and then we obtain Figure 5 through numerical simulation. We can know that predator population decreases and becomes extinct eventually provided yield of releases of the predator c=0, while the prey population will periodically oscillate at a relatively high frequency. Secondly, parameter values are assumed as r=1.5,β=0.2,K=100,d=0.5,m=0.9,α=0.2,a=0.45,b=0.1,h=80,andc=35, respectively, and obtain Figure 6 that shows model (2) has an order-one periodic solution if ΦyA<yA. We assume the value of handc as 90and95, respectively, and other values are unchanged. Therefore, we get that model (2) possesses an order-one periodic solution provided ΦyA>yA (see Figure 7). It can be seen from Figures 6 and 7 that the trajectories of different initial points eventually tend to be order-one periodic solution when t⟶∞, which means system (2) has a globally asymptotically stable order-one periodic solution, and the changes of natural enemies and pests are periodic. Therefore, pests are effectively controlled. Meanwhile, we simulate the solution of free system (1), and the results show that natural enemies without impulse will soon become extinct. Finally, we set r=1.5,K=100,m=0.9,α=0.1,d=0.5,β=0.2,a=0.25,b=0.1,h=40,c=20,andτ=20. Then, we get the order-2 periodic solution for system (2) (see Figure 8). Therefore, we verify the existence of the order-2 periodic solution for system (2). The existence of the order-2 periodic solution indicates that natural enemies and pests have periodic changes after two controls. Furthermore, pests are effectively controlled.Figure 5
Numerical simulations whenc=0. (a) Time series of x(t). (b) Time series of y(t). (c) Phase portrait of x(t) and y(t).
(a)
(b)
(c)Figure 6
Numerical simulations whenΦyA<yA. (a) Phase portrait of x(t) and y(t). (b) Time series of x(t). (c) Time series of y(t). The solution of system (2) is presented in full line, and the solution of free system (1) is represented in dotted lines.
(a)
(b)
(c)Figure 7
Numerical simulations whenΦyA>yA. (a) Phase portrait of x(t) and y(t). (b) Time series of x(t). (c) Time series of y(t). The solution of system (2) is presented in full line, and the solution of free system (1) is represented in dotted lines.
(a)
(b)
(c)Figure 8
Numerical simulation of the order-2 periodic solution. (a) Phase portrait ofx(t) and y(t). (b) Time series of y(t). (c) Time series of x(t).
(a)
(b)
(c)


## 5. Conclusion

We construct an IPM model with ratio-dependent in this paper. We take integrated control of biological control and chemical control to keep pest levels below the economic threshold.Firstly, we conduct a qualitative analysis for model (1) and obtain the conditions for stability of the internal equilibrium point and boundary equilibrium points, respectively. Secondly, we prove some properties of Poincare′ map. By using the Poincare′ map method, the conditions of stability of the order-one periodic solution are obtained when c=0 and c>0, respectively. Furthermore, we know that system (2) possesses order-k k≥1 periodic solution provided Φk−1yA<ym+ and Φym+=yA. Finally, the main results are verified by numerical simulation.In recent years, many scholars have established optimization problems to minimize the cost of pest management [45, 46]. In the future, we will put pest optimization issues into our research work.



---

*Source: 2376374-2020-04-17.xml*