# Dynamical Analysis of a Pest Management Model with Saturated Growth Rate and State Dependent Impulsive Effects

Authors: Wencai Zhao; Tongqian Zhang; Xinzhu Meng; Yang Yang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A new pest management mathematical model with saturated growth is proposed. The integrated pest management (IPM) strategy by introducing two state dependent pulses into the model is considered. Firstly, we analyze singular points of the model qualitatively and get the condition for focus point. Secondly, by using geometry theory of impulsive differential equation, the existence and stability of periodic solution of the system are discussed. Lastly, some examples and numerical simulations are given to illustrate our results.


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## Body

## 1. Introduction

Forest diseases and insect pests are the most destructive natural disasters to forestry; they are always an important constraint to forestry production worldwide and cause major economic losses in forestry. For example, spruce budworm, mountain pine beetle, and gypsy moth have led to significant losses in Canadian forests [1]. Therefore, how to prevent the occurrence and spreading of forest diseases and insect pests and reduce the economic loss is always an important subject in theory and practice. Mathematical models have been established to study change rule of the number of forest pests; these models include ordinary differential equation (ODE) system, difference equation system, partial differential equation (PDE) system, and so forth. In order to control pests, integrated pest management (IPM) [2–4] including two measures will be taken: one is chemical control (spraying pesticide) and the other is biological control (releasing natural enemy). In the process of pest management, if the amount of pests is small, biological control will be implemented, and if the amount of pests is large, integrated pest control will be applied. Usually, system with sudden disturbance always leads to impulsive system. In the forest pest management system, human’s interference can cause a sudden change of the number or density of pests. So impulsive differential equation (IDE) can be established to explain this phenomenon. Many biological models based on time impulse [5–13] or one state impulse [14, 15] have been studied by many authors. The model with two state impulses is very rare. In [16, 17], two models with two state impulses are established to study the existence and the stability of periodic solution of the models, respectively, by Poincaré map and the Lambert M function. In [18], a model with two state impulses is built up and the geometrical analysis of the model is made by Bendixson theorem of impulsive differential equations and the successor function. Motivated by [15, 17, 18], we get the following reasonable model by introducing two state dependent impulses:
(1)x′(t)=rx1+ax-bxy,y′(t)=cxy-dy,x≠h1,h2,orx=h1,y>y*,Δx=0,Δy=α,x=h1,y≤y*,Δx=-px,Δy=-qy+τ,x=h2,
where Δx(t)=x(t+)-x(t), Δy(t)=y(t+)-y(t),x(t) is the density of pests at time t, and y(t) is the density of natural enemies. rx/(1+ax) is saturated growth rate, b is predation rate of natural enemies, and c is the growth rate of natural enemies, after digestion and absorption. d is the death rate of natural enemies. h1 and h2 are the threshold with slight damage and serious damage to the forest, respectively. y* is the ordinate of intersection of the line x=h1 and (r/(1+ax))-by=0. α,τ are the amount of natural enemies y(t) released one time. p and q are death rates of pests and natural enemies due to pesticide, respectively. According to the biological significance, we only discuss R+2={(x,y)∣x≥0,y≥0}.The paper is organized as follows. In Section2, some definitions and theorems are given for the later use. In Section 3, the stability of system (1) without impulsive effect is investigated qualitatively. In Section 4, the existence and stability of periodic solution to system (1) are investigated using the successor function, Bendixson theorem, and stability theorem of periodic solution of impulsive differential equations. In Section 5, an example and some simulations are exerted to prove the results.


## 2. Preliminaries

We consider the state impulsive differential equation(2)x′(t)=P(x,y),y′(t)=Q(x,y),(x,y)∉M{x,y},Δx=α(x,y),Δy=β(x,y),(x,y)∈M{x,y},
where M is one-dimensional line or curve in R2. M is called impulsive set, and the function I is continuous mapping, I(M)=N. I is called the impulse function, and N is called the corresponding image set.Definition 1.
If there exists a pointP in the image set such that f(P,T)=Q∈M{x,y}, and the impulsive mapping I(Q)=I(f(P,T))=P∈N, then one calls f(P,T) the periodic solution of system (2).Definition 2.
Function(3)f(x)=f(x+)-f(x)
is called a successor function of the point x (see Figure 1).Figure 1
The successor function of system (2).Theorem 3 (Bendixson theorem of impulsive differential equations).
AssumeG is a Bendixson region of system (2); if G does not contain any critical points of system (2), then system (2) has a closed orbit in G.Theorem 4.
Assume that, in continuous dynamic system(X,Π), there exist two points x1, x2 in the pulse phase concentration such that the successor function f(x1)>0 and f(x2)<0; then there exists a point P between the two points x1, x2 that makes f(P)=0; that is, the system has periodic solution.Theorem 5 (see [19, 20]).
Consider the state impulsive differential equation(4)x′(t)=f(x,y),y′(t)=g(x,y),Φ(x,y)≠0,Δx=α(x,y),Δy=β(x,y),Φ(x,y)=0,
where the function Φ(x,y)=0 is corresponding to the impulsive set M. The T-periodic solution x=ϕ(t), y=φ(t) of the model is orbitally asymptotically stable if the multiplier μ2<1 is calculated by the following formula:
(5)μ2=∏k=1q‍Δkexp[∫0T‍(∂f∂x(ϕ(t),φ(t))+∂g∂y(ϕ(t),φ(t)))dt],
where
(6)Δk=(f+(∂β∂y·∂Φ∂x-∂β∂x·∂Φ∂y+∂Φ∂x)+g+(∂α∂x·∂Φ∂y-∂α∂y·∂Φ∂x+∂Φ∂y))×(f∂Φ∂x+g∂Φ∂y)-1
and f, g, (∂α/∂x), (∂α/∂y), (∂β/∂x), (∂β/∂y), (∂Φ/∂x), (∂Φ/∂y) are calculated at the point (ϕ(tk),φ(tk)) and f+=f(ϕ(tk+),φ(tk+)), g+=g(ϕ(tk+),φ(tk+)).


## 3. The Stability of System (1) without Impulsive Effect

In this section, we consider system (1) without impulsive effect. Let α,τ,p,q=0; then we get the following system:
(7)x′(t)=rx1+ax-bxy,y′(t)=y(-d+cx).
The system has two equilibrium points  E0(0,0)  and E(xℵ,yℵ), where xℵ=d/c and yℵ=rc/(b(ad+c)). Assume that the variational matrix of the system (7) is as follows:
(8)J=(r(1+ax)2-by-bxcycx-d).
At E0(0,0),
(9)J(E0)=(r00-d).
Obviously, E0(0,0) is saddle point. At E(xℵ,yℵ),
(10)J(E(xℵ,yℵ))=(-racd(c+ad)2-bdcrc2b(c+ad)0).
The characteristic equation of J(E) satisfies f(λ)=λ2+pλ+q=0, where p=racd/(c+ad)2, q=(rcd/(ad+c))>0 and Δ=p2-4q=rcd[ra2cd-4(ad+c)3]/(ad+c)4. Analyse Δ; we have the following conclusion: (i)
ifΔ>0, that is, ra2cd>4(ad+c)3, then E(xℵ,yℵ) is stable node;(ii)
ifΔ=0, that is, ra2cd=4(ad+c)3, then E(xℵ,yℵ) is stable critical node;(iii)
ifΔ<0, that is, ra2cd<4(ad+c)3, then E(xℵ,yℵ) is stable focus.Denote (H1): ra2cd<4(ad+c)3; we have the following theorem.Theorem 6.
The positive equilibrium point of system (7) is stable focus provided (H1) is true.Theorem 7.
The solution of system (7) is bounded.Proof.
Let(x(t),y(t)) be solution of system (7) satisfied initial conditions x(t0)=x0>0, y(t0)=y0>0. From the second equation of system (7), it follows that dy/dt>0 for x>d/c. And there exists T0>t0, such that orbit (x(t),y(t)) intersects with isoclinic line y=r/(b(1+ax)) at point (x(T0),y(T0)). Thus, we have x(t)<x(T0) for t>T0. Let K>x(T0); we have x(t)<K for t>T0.
On the other hand, making three lines in the first quadrant,Γ1: x-K=0, Γ2: y+(c/b)x-M=0 and Γ3: y-M+(d/b)=0, we can acquire that
(11)dΓ2dt=[y(cx-d)+cbx(r1+ax-by)]Γ2=[crxb(1+ax)+dcxb]-Md.
Then we have dΓ2/dt<0 for d/c<x<K and M large enough. Similarly,
(12)dΓ3dt=y(cx-d).
Then we have dΓ3/dt<0 for 0<x<d/c. According to the previous discussion, there exists an area Ω which is composed of x=0, y=0, Γ1, Γ2, Γ3. For all initial points (x(t0),y(t0)), ∃T>0, for t>T, we can acquire (x(t),y(t))∈Ω. Thus, the solution of system (7) satisfied initial conditions (x(t0),y(t0)) is bounded.Theorem 8.
The positive equilibrium point of system (7) is globally asymptotically stable provided condition (H1) is true.Proof.
Firstly, we claim that the positive equilibrium point of system (7) is stable focus point and around it there does not exist closed rail line provided condition (H1) is true. In fact, let Dulac function be B=1/xy; in regard to the system (7), it follows that
(13)D=∂(PB)∂x+(QB)∂y=-ray(1+ax)2<0.
Due to the Bendixson-Dulac theorem, around E, there does not exist closed rail line. Synthesize Theorems 6 and 7; therefore we have that the positive equilibrium point of system (7) is globally asymptotically stable provided condition (H1) is true.Vector graph of system (7) with a stable focus can be seen in Figure 2.Figure 2
Vector graph of system (7) with a stable focus, where r=0.6, a=1, b=0.5, c=0.8, and d=0.4.


## 4. The Analysis of System (1) with State Dependent on Pulse

### 4.1. The Existence of Periodic Solutions of the System

In this section, we will discuss the existence of periodic solution of system (1) by using the successor function defined in this paper and qualitative analysis. Let x˙=0, y˙=0; we know the curve L′: y=r/(b(1+ax)) and y-axis are X-isoline and the line L: x=d/c and x-axis are Y-isoline. From the previous discussion, we know E(xℵ,yℵ) is a stable focus provided (H1) is true. Denote the first impulsive set as M1={(x,y)∣x=h1,0≤y≤y*}, the second impulsive set as M2={(x,y)∣x=h2,y≥0}, the image set corresponding to set M1 as N1={(x,y)∣x=h1,0<y≤y*+α}, and the image set corresponding to set M2 as N2={(x,y)∣x=(1-p)h2,y≥τ}. Let points P, Q, and R be the intersection of line L′ with lines N1, N2, and M2, respectively. Due to the practical significance, in this paper we assume the set always lies in the left side of stable focus E; that is, h1<(1-p)h2<h2<d/c. We will start our discussion about the existence of periodic solution for (1) with the initial point C0(xc0,yc0). The structure graph of system (1) can be seen in Figure 3.Figure 3
The structure graph of system (1).


#### 4.1.1. The Orbit Starting from the PointC0 on N1 above P

Starting fromC0, the orbit intersects with M1 at point C1(xc1,yc1) and then produces pulse to the point C1′(xc1′,yc1′) on M1. According to (1), the following is got: yC1′=yC1+α. After n time impulse, the point C1+(xc1+,yc1+) should satisfy y*<yC1+≤y*+α. Thus, we only need to consider case y*<yC0≤y*+α, and three cases should be discussed.Case 1.
If the impulsive pointC1+ corresponding to C1 is exactly C0, thus, the curve C0C1C1+ shall constitute a periodic orbit of (1) (see Figure 4).Figure 4
The orbit starting from the pointC0 above P (Case 1 in Section 4.1.1).Case 2.
If the impulsive pointC1+ corresponding to C1 is below C0 on N1, obviously, the successor function of C0 satisfies f(C0)=yC1+-yC0<0. On the other hand, choose a point D0 next to P on N1 (i.e., |D0P|<ε). The orbit of (1) starting from point D0 intersects with M1 at point D1 and then jumps to point D1+ in the line N1. Because D0 is next to P, D1 is next to P, and yD1+=yD1+α, thus, the successor function of D0 satisfies f(D0)=yD1+-yD0>0. By Theorem 4, there exists a period solution (see Figure 5).Figure 5
The orbit starting from the pointC0 (Case 2 in Section 4.1.1).Case 3.
If the impulsive pointC1+ corresponding to C1 is above C0 on N1, obviously f(C0)>0. On the other hand, the orbit starting from point A(h1,y*+α) intersects with M1 at point A1 and then jumps to point A1+(h1,yA1+) in the line N1. According to system (1), the following is got: y*<yA1+<y*+α; then the successor function of A satisfies f(A)<0. Based on the previous discussion, according to Theorem 4, there exists a periodic orbit of (1) (see Figure 6).Figure 6
The orbit starting from the pointC0 (Case 3 in Section 4.1.1).


#### 4.1.2. The Orbit Starting from the PointQ

Starting fromQ, the orbit intersects with M2 at point C1 and then produces pulse to the point C1+ on N2. According to (1), the following is got:
(14)xC1+=(1-p)h2,yC1+=(1-q)yC1+τ.
By selecting τ, three cases should be discussed.Case 1.
If the impulsive pointC1+ corresponding to C1 is exactly Q, thus, the curve QC1C1+ shall constitute a periodic orbit of (1) (see Figure 7).Figure 7
The orbit starting from the pointQ (Case 1 in Section 4.1.2).Case 2.
If the impulsive pointC1+ corresponding to C1 is above Q on N2, thus, f(Q)=yC1+-yQ>0. Choose a point D0 above C1+ in the straight line N2. Only when it intersects with N1 at P, can the orbit of (1) starting from point D0 be vertical. After intersecting with M2 at point D2 through N2 from left to right, it jumps to point D2+ in the line N2. According to the existence and uniqueness of impulsive differential equations, D2 must be definitely below C1 on M2, and D2+ must be below C1+ on N2; therefore the successor function of D0 is f(D0)=yD2+-yD0<0. Based on the previous discussion, we can see that the region κ surrounded by the closed curve D0PD1D2C1C1+D0 is a positive invariant set of (1) and it contains no equilibrium point. According to Theorem 3, there exists a periodic orbit of (1) in κ (see Figure 8).Figure 8
The orbit starting from the pointQ (Case 2 in Section 4.1.2).Case 3.
If the impulsive pointC1+ corresponding to C1 is below Q on N2, we shall have the successor function f(Q)=yC1+-yQ<0.
In the meantime, take a pointD0 next to B((1-p)h2,0) on N2 (|D0B|≪ε.) Starting from D0, the orbit intersects with M2 at point D1 and then produces pulse to the point D1+ on N2. According to (1), the following is got:
(15)xD1+=(1-p)h2,yD1+=(1-q)yD1+τ.
Thus, D1+ must be above D0, and the successor function of D0 is f(D0)=yD1+-yD0>0.
As a result, the regionκ surrounded by the closed curve QD0D1C1Q involves a periodic solution of (1) (see Figure 9).Figure 9
The orbit starting from the pointQ (Case 3 in Section 4.1.2).


#### 4.1.3. The Orbit Starting from the PointC0 between the Second Impulsive Set M2 and Its Image Set N2

The orbit starting from pointC0, between line N2 and line M2, intersects with line M2 at C1 and jumps onto point C1+ on N2. There are the following cases with changing τ.Case 1.
IfC1+ on N2 is exactly Q, the orbit from point C1+ moves to the point C2 on M2. For C2, there are the following cases.(a)
IfC2 is exactly C1, then the curve C1C1+C2 forms a periodic orbit of (1) (see Figure 10).(b)
IfC2 on M2 is below C1, the successor function of C1 is f(C1)=yC2-yC1<0.In the meantime, take a pointD0 between N2 and M2 next to x-axis. After starting from D0, the orbit hits the point D1 on M2, then jumps onto the point D1+ on N2, and then moves to the point D2 on M2. Obviously, D2 is above D1; thus, the successor function of D1 is f(D1)=yD2-yD1>0.As a result, we get a periodic orbit in the regionκ encircled by the closed curve C1C1+D1+D1C1 (see Figure 11).(c)
IfC2 on M2 is above C1, which does not exist periodic orbit, otherwise the curve C1+C2 and C0C1 must be crossed, which leads to the conflict with the existence and uniqueness of solution to impulsive differential equations.Figure 10
The orbit starting from the pointC0 (Case 1(a) in Section 4.1.3).Figure 11
The orbit starting from the pointC0 (Case 1(b) in Section 4.1.3).Case 2.
IfC1+ on N2 is below Q, the same conclusion can be got in the similar discussion previously mentioned (see Figures 12 and 13).Figure 12
The orbit starting from the pointC0 (Case 2 in Section 4.1.3).Figure 13
The orbit starting from the pointC0 (Case 2 in Section 4.1.3).Case 3.
IfC1+ on N2 is above Q, for the orbit starting from C1+, there exist the following cases.(a)
If the orbit starting fromC1+ crosses N1 from the right to the left and becomes vertical only when it goes through the line L′, the point C2 is the intersection of the orbit and N1. The same conclusion can be got as the discussion in Section 4.1.1 (see Figure 14).(b)
The orbit starting fromC1+ becomes vertical only when it crosses PQ, and then it moves on automatically to C2 on M2, which is exactly C1, or below C1, or above C1.IfC2 on M2 is exactly C1, then the curve C1C1+C2 forms a periodic orbit (see Figure 15).IfC2 on M2 is below C1, then the successor function of C1 is f(C1)=yC2-yC1<0.On the other hand, take a pointD0 between N2 and M2 near to x-axis, and the orbit starting from D0 hits the point D1 on M2, then jumps onto the point D1+ on N2, and then returns to the point D2 on M2. Obviously, D2 on M2 is above D1. Thus, the successor function of D1 is f(D1)=yD2-yD1>0.Based on the previous discussion above, there exists a periodic orbit in the regionκ encircled D1C1C1+C2 and C1+D1+D2C2 (see Figure 16).(c)
IfC2 on M2 is above C1, which does not exist periodic orbit, the reason is the same as in Case 1(c) in Section 4.1.3.Figure 14
The orbit starting from the pointC0 (Case 3(a) in Section 4.1.3).Figure 15
The orbit starting from the pointC0 (Case 3(b) in Section 4.1.3).Figure 16
The orbit starting from the pointC0 (Case 3(b) in Section 4.1.3).


### 4.2. The Stability Analysis of Periodic Solutions of the System (1)

#### 4.2.1. The Stability of Periodic Solution about Impulsive SetM1

Theorem 9.
Letx=ϕ(t), y=φ(t) be the periodic solution of system (1), and ϕ(0)=h1, φ(0)=φ0; if φ0<(r/b(1+ah1))+α, the periodic solution is stable.Proof.
Let(16)f(x,y)=rx1+ax-bxy,g(x,y)=y(-d+cx),α(x,y)=0,β(x,y)=α,Φ(x,y)=x-h1,
and (ϕ(0),φ(0))=(h1,φ0), (ϕ(T),φ(T))=(x1,y1), (ϕ(T+),φ(T+))=(x1,y1+α)=(h1,φ0); one gets
(17)∂f∂x=r(1+ax)2-by,∂g∂y=cx-d,∂α∂x=0,∂α∂y=0,∂β∂x=0,∂β∂y=0,∂Φ∂x=1,∂Φ∂y=0.
According to Theorem 5, we have
(18)Δ1=f(ϕ(T+),φ(T+))f(ϕ(T),φ(T))=f(h1,φ0)f(h1,φ0-α)=(rh1/(1+ah1))-bh1φ0+bh1α-bh1α(rh1/(1+ah1))-bh1(φ0-α)=1-bα(r/(1+ah1))-b(φ0-α),μ2=Δ1exp∫0T‍[∂f∂x(ϕ(t),φ(t))+∂g∂y(ϕ(t),φ(t))]dt=Δ1exp∫0T‍[r(1+aϕ)2-bφ+cϕ-d]dt<Δ1exp∫0T‍[r1+aϕ-bφ+cϕ-d]dt=Δ1exp[∫0T‍1ϕ(t)dϕ(t)+∫0T‍1φ(t)dφ(t)]=Δ1φ0-αφ0=(1-bα(r/(1+ah1))-b(φ0-α))φ0-αφ0.
Obviously, if (r/(1+ah1))-b(φ0-α)>0, that is, φ0<(r/b(1+ah1))+α, we have μ2<1. Thus the periodic solution to the system (1) is stable. This completes the proof.


#### 4.2.2. The Stability of Periodic Solution about Impulsive SetM2

Theorem 10.
Letx=ϕ(t), y=φ(t) be the periodic solution of system (1), and (ϕ(0),φ(0))=((1-p)h2,φ0); if φ0<min{(r(1-q)/b(1+ah2))+τ,(1/bq)(bτ-(ap(1-q)rh2/(1+ah2)(1+a(1-p)h2)))} or φ0>max{(r(1-q)/b(1+ah2))+τ,(1/bq)(bτ-(ap(1-q)rh2/(1+ah2)(1+a(1-p)h2)))}, the periodic solution to the system (1) is stable.Proof.
Letx=ϕ(t), y=φ(t) be the T periodic solution of system (1), and
(19)f(x,y)=rx1+ax-bxy,g(x,y)=y(-d+cx),α(x,y)=-px,β(x,y)=-qy+τ,Φ(x,y)=x-h2,(ϕ(T+),φ(T+))=((1-p)ϕ(T),(1-q)φ(T)+τ),(ϕ(T),φ(T))=(h2,(φ0-τ)/(1-q)); one gets
(20)∂f∂x=r(1+ax)2-by,∂g∂y=cx-d,∂α∂x=-p,∂α∂y=0,∂β∂x=0,∂β∂y=-q,∂Φ∂x=1,∂Φ∂y=0.
Thus,
(21)Δ1=(1-q)f(ϕ(T+),φ(T+))f(ϕ(T),φ(T))=((1-q)[rϕ(T+)1+aϕ(T+)-bϕ(T+)φ(T+)])×(rϕ(T)1+aϕ(T)-bϕ(T)φ(T))-1=((1-q)[r(1-p)h21+a(1-p)h2-b(1-p)h2φ0])×(rh21+ah2-bh2φ0-τ1-q)-1=((1-p)(1-q)[r1+a(1-p)h2-bφ0])×(r1+ah2-bφ0-τ1-q)-1,μ2=Δ1exp∫0T‍(∂f∂x(ϕ(t),φ(t))+∂g∂y(ϕ(t),φ(t)))dt=Δ1exp∫0T‍(r(1+aϕ)2-bφ+cϕ-d)dt<Δ1exp∫0T‍(r1+aϕ-bφ+cϕ-d)dt=Δ1exp(∫0T‍1ϕ(t)dϕ(t)+∫0T‍1φ(t)dφ(t))=Δ11(1-q)(1-p)φ0-τφ0=(1-p)(1-q)[(r/(1+a(1-p)h2))-bφ0](r/(1+ah2))-b((φ0-τ)/(1-q))×1(1-q)(1-p)φ0-τφ0≤(r/(1+a(1-p)h2))-bφ0(r/(1+ah2))-b((φ0-τ)/(1-q)).
If
(22)φ0<min{r(1-q)b(1+ah2)+τ,1bq(bτ-ap(1-q)rh2(1+ah2)(1+a(1-p)h2))}
or
(23)φ0>max{r(1-q)b(1+ah2)+τ,1bq(bτ-ap(1-q)rh2(1+ah2)(1+a(1-p)h2))},
we have μ2<1; then the periodic solution to the system (1) is stable. This completes the proof.


## 4.1. The Existence of Periodic Solutions of the System

In this section, we will discuss the existence of periodic solution of system (1) by using the successor function defined in this paper and qualitative analysis. Let x˙=0, y˙=0; we know the curve L′: y=r/(b(1+ax)) and y-axis are X-isoline and the line L: x=d/c and x-axis are Y-isoline. From the previous discussion, we know E(xℵ,yℵ) is a stable focus provided (H1) is true. Denote the first impulsive set as M1={(x,y)∣x=h1,0≤y≤y*}, the second impulsive set as M2={(x,y)∣x=h2,y≥0}, the image set corresponding to set M1 as N1={(x,y)∣x=h1,0<y≤y*+α}, and the image set corresponding to set M2 as N2={(x,y)∣x=(1-p)h2,y≥τ}. Let points P, Q, and R be the intersection of line L′ with lines N1, N2, and M2, respectively. Due to the practical significance, in this paper we assume the set always lies in the left side of stable focus E; that is, h1<(1-p)h2<h2<d/c. We will start our discussion about the existence of periodic solution for (1) with the initial point C0(xc0,yc0). The structure graph of system (1) can be seen in Figure 3.Figure 3
The structure graph of system (1).


### 4.1.1. The Orbit Starting from the PointC0 on N1 above P

Starting fromC0, the orbit intersects with M1 at point C1(xc1,yc1) and then produces pulse to the point C1′(xc1′,yc1′) on M1. According to (1), the following is got: yC1′=yC1+α. After n time impulse, the point C1+(xc1+,yc1+) should satisfy y*<yC1+≤y*+α. Thus, we only need to consider case y*<yC0≤y*+α, and three cases should be discussed.Case 1.
If the impulsive pointC1+ corresponding to C1 is exactly C0, thus, the curve C0C1C1+ shall constitute a periodic orbit of (1) (see Figure 4).Figure 4
The orbit starting from the pointC0 above P (Case 1 in Section 4.1.1).Case 2.
If the impulsive pointC1+ corresponding to C1 is below C0 on N1, obviously, the successor function of C0 satisfies f(C0)=yC1+-yC0<0. On the other hand, choose a point D0 next to P on N1 (i.e., |D0P|<ε). The orbit of (1) starting from point D0 intersects with M1 at point D1 and then jumps to point D1+ in the line N1. Because D0 is next to P, D1 is next to P, and yD1+=yD1+α, thus, the successor function of D0 satisfies f(D0)=yD1+-yD0>0. By Theorem 4, there exists a period solution (see Figure 5).Figure 5
The orbit starting from the pointC0 (Case 2 in Section 4.1.1).Case 3.
If the impulsive pointC1+ corresponding to C1 is above C0 on N1, obviously f(C0)>0. On the other hand, the orbit starting from point A(h1,y*+α) intersects with M1 at point A1 and then jumps to point A1+(h1,yA1+) in the line N1. According to system (1), the following is got: y*<yA1+<y*+α; then the successor function of A satisfies f(A)<0. Based on the previous discussion, according to Theorem 4, there exists a periodic orbit of (1) (see Figure 6).Figure 6
The orbit starting from the pointC0 (Case 3 in Section 4.1.1).


### 4.1.2. The Orbit Starting from the PointQ

Starting fromQ, the orbit intersects with M2 at point C1 and then produces pulse to the point C1+ on N2. According to (1), the following is got:
(14)xC1+=(1-p)h2,yC1+=(1-q)yC1+τ.
By selecting τ, three cases should be discussed.Case 1.
If the impulsive pointC1+ corresponding to C1 is exactly Q, thus, the curve QC1C1+ shall constitute a periodic orbit of (1) (see Figure 7).Figure 7
The orbit starting from the pointQ (Case 1 in Section 4.1.2).Case 2.
If the impulsive pointC1+ corresponding to C1 is above Q on N2, thus, f(Q)=yC1+-yQ>0. Choose a point D0 above C1+ in the straight line N2. Only when it intersects with N1 at P, can the orbit of (1) starting from point D0 be vertical. After intersecting with M2 at point D2 through N2 from left to right, it jumps to point D2+ in the line N2. According to the existence and uniqueness of impulsive differential equations, D2 must be definitely below C1 on M2, and D2+ must be below C1+ on N2; therefore the successor function of D0 is f(D0)=yD2+-yD0<0. Based on the previous discussion, we can see that the region κ surrounded by the closed curve D0PD1D2C1C1+D0 is a positive invariant set of (1) and it contains no equilibrium point. According to Theorem 3, there exists a periodic orbit of (1) in κ (see Figure 8).Figure 8
The orbit starting from the pointQ (Case 2 in Section 4.1.2).Case 3.
If the impulsive pointC1+ corresponding to C1 is below Q on N2, we shall have the successor function f(Q)=yC1+-yQ<0.
In the meantime, take a pointD0 next to B((1-p)h2,0) on N2 (|D0B|≪ε.) Starting from D0, the orbit intersects with M2 at point D1 and then produces pulse to the point D1+ on N2. According to (1), the following is got:
(15)xD1+=(1-p)h2,yD1+=(1-q)yD1+τ.
Thus, D1+ must be above D0, and the successor function of D0 is f(D0)=yD1+-yD0>0.
As a result, the regionκ surrounded by the closed curve QD0D1C1Q involves a periodic solution of (1) (see Figure 9).Figure 9
The orbit starting from the pointQ (Case 3 in Section 4.1.2).


### 4.1.3. The Orbit Starting from the PointC0 between the Second Impulsive Set M2 and Its Image Set N2

The orbit starting from pointC0, between line N2 and line M2, intersects with line M2 at C1 and jumps onto point C1+ on N2. There are the following cases with changing τ.Case 1.
IfC1+ on N2 is exactly Q, the orbit from point C1+ moves to the point C2 on M2. For C2, there are the following cases.(a)
IfC2 is exactly C1, then the curve C1C1+C2 forms a periodic orbit of (1) (see Figure 10).(b)
IfC2 on M2 is below C1, the successor function of C1 is f(C1)=yC2-yC1<0.In the meantime, take a pointD0 between N2 and M2 next to x-axis. After starting from D0, the orbit hits the point D1 on M2, then jumps onto the point D1+ on N2, and then moves to the point D2 on M2. Obviously, D2 is above D1; thus, the successor function of D1 is f(D1)=yD2-yD1>0.As a result, we get a periodic orbit in the regionκ encircled by the closed curve C1C1+D1+D1C1 (see Figure 11).(c)
IfC2 on M2 is above C1, which does not exist periodic orbit, otherwise the curve C1+C2 and C0C1 must be crossed, which leads to the conflict with the existence and uniqueness of solution to impulsive differential equations.Figure 10
The orbit starting from the pointC0 (Case 1(a) in Section 4.1.3).Figure 11
The orbit starting from the pointC0 (Case 1(b) in Section 4.1.3).Case 2.
IfC1+ on N2 is below Q, the same conclusion can be got in the similar discussion previously mentioned (see Figures 12 and 13).Figure 12
The orbit starting from the pointC0 (Case 2 in Section 4.1.3).Figure 13
The orbit starting from the pointC0 (Case 2 in Section 4.1.3).Case 3.
IfC1+ on N2 is above Q, for the orbit starting from C1+, there exist the following cases.(a)
If the orbit starting fromC1+ crosses N1 from the right to the left and becomes vertical only when it goes through the line L′, the point C2 is the intersection of the orbit and N1. The same conclusion can be got as the discussion in Section 4.1.1 (see Figure 14).(b)
The orbit starting fromC1+ becomes vertical only when it crosses PQ, and then it moves on automatically to C2 on M2, which is exactly C1, or below C1, or above C1.IfC2 on M2 is exactly C1, then the curve C1C1+C2 forms a periodic orbit (see Figure 15).IfC2 on M2 is below C1, then the successor function of C1 is f(C1)=yC2-yC1<0.On the other hand, take a pointD0 between N2 and M2 near to x-axis, and the orbit starting from D0 hits the point D1 on M2, then jumps onto the point D1+ on N2, and then returns to the point D2 on M2. Obviously, D2 on M2 is above D1. Thus, the successor function of D1 is f(D1)=yD2-yD1>0.Based on the previous discussion above, there exists a periodic orbit in the regionκ encircled D1C1C1+C2 and C1+D1+D2C2 (see Figure 16).(c)
IfC2 on M2 is above C1, which does not exist periodic orbit, the reason is the same as in Case 1(c) in Section 4.1.3.Figure 14
The orbit starting from the pointC0 (Case 3(a) in Section 4.1.3).Figure 15
The orbit starting from the pointC0 (Case 3(b) in Section 4.1.3).Figure 16
The orbit starting from the pointC0 (Case 3(b) in Section 4.1.3).


## 4.1.1. The Orbit Starting from the PointC0 on N1 above P

Starting fromC0, the orbit intersects with M1 at point C1(xc1,yc1) and then produces pulse to the point C1′(xc1′,yc1′) on M1. According to (1), the following is got: yC1′=yC1+α. After n time impulse, the point C1+(xc1+,yc1+) should satisfy y*<yC1+≤y*+α. Thus, we only need to consider case y*<yC0≤y*+α, and three cases should be discussed.Case 1.
If the impulsive pointC1+ corresponding to C1 is exactly C0, thus, the curve C0C1C1+ shall constitute a periodic orbit of (1) (see Figure 4).Figure 4
The orbit starting from the pointC0 above P (Case 1 in Section 4.1.1).Case 2.
If the impulsive pointC1+ corresponding to C1 is below C0 on N1, obviously, the successor function of C0 satisfies f(C0)=yC1+-yC0<0. On the other hand, choose a point D0 next to P on N1 (i.e., |D0P|<ε). The orbit of (1) starting from point D0 intersects with M1 at point D1 and then jumps to point D1+ in the line N1. Because D0 is next to P, D1 is next to P, and yD1+=yD1+α, thus, the successor function of D0 satisfies f(D0)=yD1+-yD0>0. By Theorem 4, there exists a period solution (see Figure 5).Figure 5
The orbit starting from the pointC0 (Case 2 in Section 4.1.1).Case 3.
If the impulsive pointC1+ corresponding to C1 is above C0 on N1, obviously f(C0)>0. On the other hand, the orbit starting from point A(h1,y*+α) intersects with M1 at point A1 and then jumps to point A1+(h1,yA1+) in the line N1. According to system (1), the following is got: y*<yA1+<y*+α; then the successor function of A satisfies f(A)<0. Based on the previous discussion, according to Theorem 4, there exists a periodic orbit of (1) (see Figure 6).Figure 6
The orbit starting from the pointC0 (Case 3 in Section 4.1.1).


## 4.1.2. The Orbit Starting from the PointQ

Starting fromQ, the orbit intersects with M2 at point C1 and then produces pulse to the point C1+ on N2. According to (1), the following is got:
(14)xC1+=(1-p)h2,yC1+=(1-q)yC1+τ.
By selecting τ, three cases should be discussed.Case 1.
If the impulsive pointC1+ corresponding to C1 is exactly Q, thus, the curve QC1C1+ shall constitute a periodic orbit of (1) (see Figure 7).Figure 7
The orbit starting from the pointQ (Case 1 in Section 4.1.2).Case 2.
If the impulsive pointC1+ corresponding to C1 is above Q on N2, thus, f(Q)=yC1+-yQ>0. Choose a point D0 above C1+ in the straight line N2. Only when it intersects with N1 at P, can the orbit of (1) starting from point D0 be vertical. After intersecting with M2 at point D2 through N2 from left to right, it jumps to point D2+ in the line N2. According to the existence and uniqueness of impulsive differential equations, D2 must be definitely below C1 on M2, and D2+ must be below C1+ on N2; therefore the successor function of D0 is f(D0)=yD2+-yD0<0. Based on the previous discussion, we can see that the region κ surrounded by the closed curve D0PD1D2C1C1+D0 is a positive invariant set of (1) and it contains no equilibrium point. According to Theorem 3, there exists a periodic orbit of (1) in κ (see Figure 8).Figure 8
The orbit starting from the pointQ (Case 2 in Section 4.1.2).Case 3.
If the impulsive pointC1+ corresponding to C1 is below Q on N2, we shall have the successor function f(Q)=yC1+-yQ<0.
In the meantime, take a pointD0 next to B((1-p)h2,0) on N2 (|D0B|≪ε.) Starting from D0, the orbit intersects with M2 at point D1 and then produces pulse to the point D1+ on N2. According to (1), the following is got:
(15)xD1+=(1-p)h2,yD1+=(1-q)yD1+τ.
Thus, D1+ must be above D0, and the successor function of D0 is f(D0)=yD1+-yD0>0.
As a result, the regionκ surrounded by the closed curve QD0D1C1Q involves a periodic solution of (1) (see Figure 9).Figure 9
The orbit starting from the pointQ (Case 3 in Section 4.1.2).


## 4.1.3. The Orbit Starting from the PointC0 between the Second Impulsive Set M2 and Its Image Set N2

The orbit starting from pointC0, between line N2 and line M2, intersects with line M2 at C1 and jumps onto point C1+ on N2. There are the following cases with changing τ.Case 1.
IfC1+ on N2 is exactly Q, the orbit from point C1+ moves to the point C2 on M2. For C2, there are the following cases.(a)
IfC2 is exactly C1, then the curve C1C1+C2 forms a periodic orbit of (1) (see Figure 10).(b)
IfC2 on M2 is below C1, the successor function of C1 is f(C1)=yC2-yC1<0.In the meantime, take a pointD0 between N2 and M2 next to x-axis. After starting from D0, the orbit hits the point D1 on M2, then jumps onto the point D1+ on N2, and then moves to the point D2 on M2. Obviously, D2 is above D1; thus, the successor function of D1 is f(D1)=yD2-yD1>0.As a result, we get a periodic orbit in the regionκ encircled by the closed curve C1C1+D1+D1C1 (see Figure 11).(c)
IfC2 on M2 is above C1, which does not exist periodic orbit, otherwise the curve C1+C2 and C0C1 must be crossed, which leads to the conflict with the existence and uniqueness of solution to impulsive differential equations.Figure 10
The orbit starting from the pointC0 (Case 1(a) in Section 4.1.3).Figure 11
The orbit starting from the pointC0 (Case 1(b) in Section 4.1.3).Case 2.
IfC1+ on N2 is below Q, the same conclusion can be got in the similar discussion previously mentioned (see Figures 12 and 13).Figure 12
The orbit starting from the pointC0 (Case 2 in Section 4.1.3).Figure 13
The orbit starting from the pointC0 (Case 2 in Section 4.1.3).Case 3.
IfC1+ on N2 is above Q, for the orbit starting from C1+, there exist the following cases.(a)
If the orbit starting fromC1+ crosses N1 from the right to the left and becomes vertical only when it goes through the line L′, the point C2 is the intersection of the orbit and N1. The same conclusion can be got as the discussion in Section 4.1.1 (see Figure 14).(b)
The orbit starting fromC1+ becomes vertical only when it crosses PQ, and then it moves on automatically to C2 on M2, which is exactly C1, or below C1, or above C1.IfC2 on M2 is exactly C1, then the curve C1C1+C2 forms a periodic orbit (see Figure 15).IfC2 on M2 is below C1, then the successor function of C1 is f(C1)=yC2-yC1<0.On the other hand, take a pointD0 between N2 and M2 near to x-axis, and the orbit starting from D0 hits the point D1 on M2, then jumps onto the point D1+ on N2, and then returns to the point D2 on M2. Obviously, D2 on M2 is above D1. Thus, the successor function of D1 is f(D1)=yD2-yD1>0.Based on the previous discussion above, there exists a periodic orbit in the regionκ encircled D1C1C1+C2 and C1+D1+D2C2 (see Figure 16).(c)
IfC2 on M2 is above C1, which does not exist periodic orbit, the reason is the same as in Case 1(c) in Section 4.1.3.Figure 14
The orbit starting from the pointC0 (Case 3(a) in Section 4.1.3).Figure 15
The orbit starting from the pointC0 (Case 3(b) in Section 4.1.3).Figure 16
The orbit starting from the pointC0 (Case 3(b) in Section 4.1.3).


## 4.2. The Stability Analysis of Periodic Solutions of the System (1)

### 4.2.1. The Stability of Periodic Solution about Impulsive SetM1

Theorem 9.
Letx=ϕ(t), y=φ(t) be the periodic solution of system (1), and ϕ(0)=h1, φ(0)=φ0; if φ0<(r/b(1+ah1))+α, the periodic solution is stable.Proof.
Let(16)f(x,y)=rx1+ax-bxy,g(x,y)=y(-d+cx),α(x,y)=0,β(x,y)=α,Φ(x,y)=x-h1,
and (ϕ(0),φ(0))=(h1,φ0), (ϕ(T),φ(T))=(x1,y1), (ϕ(T+),φ(T+))=(x1,y1+α)=(h1,φ0); one gets
(17)∂f∂x=r(1+ax)2-by,∂g∂y=cx-d,∂α∂x=0,∂α∂y=0,∂β∂x=0,∂β∂y=0,∂Φ∂x=1,∂Φ∂y=0.
According to Theorem 5, we have
(18)Δ1=f(ϕ(T+),φ(T+))f(ϕ(T),φ(T))=f(h1,φ0)f(h1,φ0-α)=(rh1/(1+ah1))-bh1φ0+bh1α-bh1α(rh1/(1+ah1))-bh1(φ0-α)=1-bα(r/(1+ah1))-b(φ0-α),μ2=Δ1exp∫0T‍[∂f∂x(ϕ(t),φ(t))+∂g∂y(ϕ(t),φ(t))]dt=Δ1exp∫0T‍[r(1+aϕ)2-bφ+cϕ-d]dt<Δ1exp∫0T‍[r1+aϕ-bφ+cϕ-d]dt=Δ1exp[∫0T‍1ϕ(t)dϕ(t)+∫0T‍1φ(t)dφ(t)]=Δ1φ0-αφ0=(1-bα(r/(1+ah1))-b(φ0-α))φ0-αφ0.
Obviously, if (r/(1+ah1))-b(φ0-α)>0, that is, φ0<(r/b(1+ah1))+α, we have μ2<1. Thus the periodic solution to the system (1) is stable. This completes the proof.


### 4.2.2. The Stability of Periodic Solution about Impulsive SetM2

Theorem 10.
Letx=ϕ(t), y=φ(t) be the periodic solution of system (1), and (ϕ(0),φ(0))=((1-p)h2,φ0); if φ0<min{(r(1-q)/b(1+ah2))+τ,(1/bq)(bτ-(ap(1-q)rh2/(1+ah2)(1+a(1-p)h2)))} or φ0>max{(r(1-q)/b(1+ah2))+τ,(1/bq)(bτ-(ap(1-q)rh2/(1+ah2)(1+a(1-p)h2)))}, the periodic solution to the system (1) is stable.Proof.
Letx=ϕ(t), y=φ(t) be the T periodic solution of system (1), and
(19)f(x,y)=rx1+ax-bxy,g(x,y)=y(-d+cx),α(x,y)=-px,β(x,y)=-qy+τ,Φ(x,y)=x-h2,(ϕ(T+),φ(T+))=((1-p)ϕ(T),(1-q)φ(T)+τ),(ϕ(T),φ(T))=(h2,(φ0-τ)/(1-q)); one gets
(20)∂f∂x=r(1+ax)2-by,∂g∂y=cx-d,∂α∂x=-p,∂α∂y=0,∂β∂x=0,∂β∂y=-q,∂Φ∂x=1,∂Φ∂y=0.
Thus,
(21)Δ1=(1-q)f(ϕ(T+),φ(T+))f(ϕ(T),φ(T))=((1-q)[rϕ(T+)1+aϕ(T+)-bϕ(T+)φ(T+)])×(rϕ(T)1+aϕ(T)-bϕ(T)φ(T))-1=((1-q)[r(1-p)h21+a(1-p)h2-b(1-p)h2φ0])×(rh21+ah2-bh2φ0-τ1-q)-1=((1-p)(1-q)[r1+a(1-p)h2-bφ0])×(r1+ah2-bφ0-τ1-q)-1,μ2=Δ1exp∫0T‍(∂f∂x(ϕ(t),φ(t))+∂g∂y(ϕ(t),φ(t)))dt=Δ1exp∫0T‍(r(1+aϕ)2-bφ+cϕ-d)dt<Δ1exp∫0T‍(r1+aϕ-bφ+cϕ-d)dt=Δ1exp(∫0T‍1ϕ(t)dϕ(t)+∫0T‍1φ(t)dφ(t))=Δ11(1-q)(1-p)φ0-τφ0=(1-p)(1-q)[(r/(1+a(1-p)h2))-bφ0](r/(1+ah2))-b((φ0-τ)/(1-q))×1(1-q)(1-p)φ0-τφ0≤(r/(1+a(1-p)h2))-bφ0(r/(1+ah2))-b((φ0-τ)/(1-q)).
If
(22)φ0<min{r(1-q)b(1+ah2)+τ,1bq(bτ-ap(1-q)rh2(1+ah2)(1+a(1-p)h2))}
or
(23)φ0>max{r(1-q)b(1+ah2)+τ,1bq(bτ-ap(1-q)rh2(1+ah2)(1+a(1-p)h2))},
we have μ2<1; then the periodic solution to the system (1) is stable. This completes the proof.


## 4.2.1. The Stability of Periodic Solution about Impulsive SetM1

Theorem 9.
Letx=ϕ(t), y=φ(t) be the periodic solution of system (1), and ϕ(0)=h1, φ(0)=φ0; if φ0<(r/b(1+ah1))+α, the periodic solution is stable.Proof.
Let(16)f(x,y)=rx1+ax-bxy,g(x,y)=y(-d+cx),α(x,y)=0,β(x,y)=α,Φ(x,y)=x-h1,
and (ϕ(0),φ(0))=(h1,φ0), (ϕ(T),φ(T))=(x1,y1), (ϕ(T+),φ(T+))=(x1,y1+α)=(h1,φ0); one gets
(17)∂f∂x=r(1+ax)2-by,∂g∂y=cx-d,∂α∂x=0,∂α∂y=0,∂β∂x=0,∂β∂y=0,∂Φ∂x=1,∂Φ∂y=0.
According to Theorem 5, we have
(18)Δ1=f(ϕ(T+),φ(T+))f(ϕ(T),φ(T))=f(h1,φ0)f(h1,φ0-α)=(rh1/(1+ah1))-bh1φ0+bh1α-bh1α(rh1/(1+ah1))-bh1(φ0-α)=1-bα(r/(1+ah1))-b(φ0-α),μ2=Δ1exp∫0T‍[∂f∂x(ϕ(t),φ(t))+∂g∂y(ϕ(t),φ(t))]dt=Δ1exp∫0T‍[r(1+aϕ)2-bφ+cϕ-d]dt<Δ1exp∫0T‍[r1+aϕ-bφ+cϕ-d]dt=Δ1exp[∫0T‍1ϕ(t)dϕ(t)+∫0T‍1φ(t)dφ(t)]=Δ1φ0-αφ0=(1-bα(r/(1+ah1))-b(φ0-α))φ0-αφ0.
Obviously, if (r/(1+ah1))-b(φ0-α)>0, that is, φ0<(r/b(1+ah1))+α, we have μ2<1. Thus the periodic solution to the system (1) is stable. This completes the proof.


## 4.2.2. The Stability of Periodic Solution about Impulsive SetM2

Theorem 10.
Letx=ϕ(t), y=φ(t) be the periodic solution of system (1), and (ϕ(0),φ(0))=((1-p)h2,φ0); if φ0<min{(r(1-q)/b(1+ah2))+τ,(1/bq)(bτ-(ap(1-q)rh2/(1+ah2)(1+a(1-p)h2)))} or φ0>max{(r(1-q)/b(1+ah2))+τ,(1/bq)(bτ-(ap(1-q)rh2/(1+ah2)(1+a(1-p)h2)))}, the periodic solution to the system (1) is stable.Proof.
Letx=ϕ(t), y=φ(t) be the T periodic solution of system (1), and
(19)f(x,y)=rx1+ax-bxy,g(x,y)=y(-d+cx),α(x,y)=-px,β(x,y)=-qy+τ,Φ(x,y)=x-h2,(ϕ(T+),φ(T+))=((1-p)ϕ(T),(1-q)φ(T)+τ),(ϕ(T),φ(T))=(h2,(φ0-τ)/(1-q)); one gets
(20)∂f∂x=r(1+ax)2-by,∂g∂y=cx-d,∂α∂x=-p,∂α∂y=0,∂β∂x=0,∂β∂y=-q,∂Φ∂x=1,∂Φ∂y=0.
Thus,
(21)Δ1=(1-q)f(ϕ(T+),φ(T+))f(ϕ(T),φ(T))=((1-q)[rϕ(T+)1+aϕ(T+)-bϕ(T+)φ(T+)])×(rϕ(T)1+aϕ(T)-bϕ(T)φ(T))-1=((1-q)[r(1-p)h21+a(1-p)h2-b(1-p)h2φ0])×(rh21+ah2-bh2φ0-τ1-q)-1=((1-p)(1-q)[r1+a(1-p)h2-bφ0])×(r1+ah2-bφ0-τ1-q)-1,μ2=Δ1exp∫0T‍(∂f∂x(ϕ(t),φ(t))+∂g∂y(ϕ(t),φ(t)))dt=Δ1exp∫0T‍(r(1+aϕ)2-bφ+cϕ-d)dt<Δ1exp∫0T‍(r1+aϕ-bφ+cϕ-d)dt=Δ1exp(∫0T‍1ϕ(t)dϕ(t)+∫0T‍1φ(t)dφ(t))=Δ11(1-q)(1-p)φ0-τφ0=(1-p)(1-q)[(r/(1+a(1-p)h2))-bφ0](r/(1+ah2))-b((φ0-τ)/(1-q))×1(1-q)(1-p)φ0-τφ0≤(r/(1+a(1-p)h2))-bφ0(r/(1+ah2))-b((φ0-τ)/(1-q)).
If
(22)φ0<min{r(1-q)b(1+ah2)+τ,1bq(bτ-ap(1-q)rh2(1+ah2)(1+a(1-p)h2))}
or
(23)φ0>max{r(1-q)b(1+ah2)+τ,1bq(bτ-ap(1-q)rh2(1+ah2)(1+a(1-p)h2))},
we have μ2<1; then the periodic solution to the system (1) is stable. This completes the proof.


## 5. Example and Numerical Simulation

To verify the validity of our results, we give an example. Letr=0.6, a=1, b=0.5, c=0.8, d=0.4, p=0.5, q=0.2, h1=0.2, and h2=0.45. By calculation, we obtain y*=1, P(0.2,1), Q(0.225,0.9796), and E(xℵ,yℵ)=(0.5,0.8). Then we have
(24)x′(t)=0.6x1+x-0.5xy,y′(t)=0.8xy-0.4y,x≠0.2,0.45,orx=0.2,y>1,Δx=0,Δy=α,x=0.2,y≤1,Δx=-0.5x,Δy=-0.2y+τ,x=0.45.
Numerical analysis of system (24) is being done using Maple 14.0. We have the following cases.Case 1.
Initial pointC0 is in the line N1, where α=0.5.(a)
LetC0 be (0.2,1.2) above P(0.2,1) (see Figure 17).(b)
LetC0 be (0.2,0.6) below P(0.2,1) (see Figure 18).Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)Case 2.
Initial pointC0 is in the line N2. Let C0 be Q(0.225,0.9796).(a)
The impulsive pointC1+ corresponding to C1 is exactly Q, where τ=0.62 (see Figure 19).(b)
The impulsive pointC1+ corresponding to C1 is above Q on N2, where τ=0.8 (see Figure 20).(c)
The impulsive pointC1+ corresponding to C1 is below Q on N2, where τ=0.5 (see Figure 21).Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)Case 3.
Initial pointC0 is between the second impulsive set M2 and its image set N2. Let C0 be (0.25,0.9).(a)
The impulsive pointC1+ corresponding to C1 is exactly Q, where τ=0.62 (see Figure 22).(b)
The impulsive pointC1+ corresponding to C1 is below Q on N2, where τ=0.4 (see Figure 23).(c)
(1) The impulsive point C1+ corresponding to C1 is above Q on N2, where τ=0.8 (see Figure 24).(2) The impulsive point C1+ corresponding to C1 is above Q on N2, where τ=1 and α=0.5 (see Figure 25).Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)Illustration of basic behavior of system (24).
(a)
Phase diagram of system  (24)
(b)
Time series of system  (24)
(c)
Time series of system  (24)All the simulations show agreement with the results in this paper.


## 6. Conclusion

Certainly, the actual pest management will be very complicated; such biological models may be described by more state dependent pulses and time impulse, but they are difficult to study by the use of successor functions. In this paper, a pest management model with saturated growth rate is constructed. We proposed two state dependent impulsive effects in the model according to quantity of pests and natural enemies. Our primary results are to explain the effects of human interference in the actual pest management on dynamics of the system. If the quantity of pests is less thanh1, but the quantity of natural enemies is enough relatively (>y*), human interference is unnecessary. If the quantity of pests reaches a certain amount h1, but the quantity of natural enemies is less than or  equal to y*, only biological control (releasing natural enemy) is implemented, and if the quantity of pests is large (reach h2), integrated pest control (spraying pesticide and releasing natural enemy) is applied. We think our results will offer help to the actual pest management.



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*Source: 204642-2013-06-09.xml*