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

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.

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: 𝑥^′(𝑡)=𝑟𝑥𝑦1+𝑎𝑥−𝑏𝑥𝑦,_′(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, (1) 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 𝑅²₊={(𝑥,𝑦)∣𝑥≥0,𝑦≥0}.

The paper is organized as follows. In Section 2, 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 𝑥^′(𝑡),𝑦=𝑃𝑥,𝑦_′(𝑡),,,,,=𝑄𝑥,𝑦𝑥,𝑦∉𝑀𝑥,𝑦Δ𝑥=𝛼𝑥,𝑦Δ𝑦=𝛽𝑥,𝑦𝑥,𝑦∈𝑀𝑥,𝑦 (2) 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 point 𝑃 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 𝑓(𝑥)=𝑓𝑥⁺−𝑓(𝑥) (3) 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). Assume 𝐺 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 𝑥^′(𝑡),𝑦=𝑓𝑥,𝑦_′(𝑡),Φ,,Φ=𝑔𝑥,𝑦𝑥,𝑦≠0,Δ𝑥=𝛼𝑥,𝑦Δ𝑦=𝛽𝑥,𝑦𝑥,𝑦=0, (4) 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: 𝜇₂=^𝑞_𝑘=1Δ_𝑘exp^𝑇₀𝜕𝑓𝜙(𝑡)(𝑡)+𝜕𝑥,𝜑𝜕𝑔𝜙(𝑡)(𝑡,𝜕𝑦,𝜑)𝑑𝑡 (5) where Δ_𝑘=𝑓₊𝜕𝛽⋅𝜕𝑦𝜕Φ−𝜕𝑥𝜕𝛽⋅𝜕𝑥𝜕Φ+𝜕𝑦𝜕Φ𝜕𝑥+𝑔₊𝜕𝛼⋅𝜕𝑥𝜕Φ−𝜕𝑦𝜕𝛼⋅𝜕𝑦𝜕Φ+𝜕𝑥𝜕Φ×𝑓𝜕𝑦𝜕Φ𝜕𝑥+𝑔𝜕Φ𝜕𝑦⁻¹ (6) and 𝑓, 𝑔, (𝜕𝛼/𝜕𝑥), (𝜕𝛼/𝜕𝑦), (𝜕𝛽/𝜕𝑥), (𝜕𝛽/𝜕𝑦), (𝜕Φ/𝜕𝑥), (𝜕Φ/𝜕𝑦) are calculated at the point (𝜙(𝑡_𝑘),𝜑(𝑡_𝑘)) and 𝑓₊=𝑓(𝜙(𝑡⁺_𝑘),𝜑(𝑡⁺_𝑘)), 𝑔₊=𝑔(𝜙(𝑡⁺_𝑘),𝜑(𝑡⁺_𝑘)).

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: 𝑥^′(𝑡)=𝑟𝑥𝑦1+𝑎𝑥−𝑏𝑥𝑦,_′(𝑡)=𝑦(−𝑑+𝑐𝑥). (7) The system has two equilibrium points 𝐸₀(0,0) and 𝐸(𝑥^ℵ,𝑦^ℵ), where 𝑥^ℵ=𝑑/𝑐 and 𝑦^ℵ=𝑟𝑐/(𝑏(𝑎𝑑+𝑐)). Assume that the variational matrix of the system (7) is as follows: 𝑟𝐽=(1+𝑎𝑥)²−𝑏𝑦−𝑏𝑥𝑐𝑦𝑐𝑥−𝑑. (8) At 𝐸₀(0,0), 𝐽𝐸₀=𝑟00−𝑑. (9) Obviously, 𝐸₀(0,0) is saddle point. At 𝐸(𝑥^ℵ,𝑦^ℵ), 𝐽𝐸𝑥^ℵ,𝑦^ℵ=−𝑟𝑎𝑐𝑑(𝑐+𝑎𝑑)²−𝑏𝑑𝑐𝑟𝑐²𝑏()0𝑐+𝑎𝑑. (10) 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)if Δ>0, that is, 𝑟𝑎²𝑐𝑑>4(𝑎𝑑+𝑐)³, then 𝐸(𝑥^ℵ,𝑦^ℵ) is stable node; (ii)if Δ=0, that is, 𝑟𝑎²𝑐𝑑=4(𝑎𝑑+𝑐)³, then 𝐸(𝑥^ℵ,𝑦^ℵ) is stable critical node; (iii)if Δ<0, that is, 𝑟𝑎²𝑐𝑑<4(𝑎𝑑+𝑐)³, then 𝐸(𝑥^ℵ,𝑦^ℵ) 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 𝑑Γ₂=𝑦()+𝑐𝑑𝑡𝑐𝑥−𝑑𝑏𝑥𝑟1+𝑎𝑥−𝑏𝑦_Γ2=𝑐𝑟𝑥𝑏()+1+𝑎𝑥𝑑𝑐𝑥𝑏−𝑀𝑑. (11) Then we have 𝑑Γ₂/𝑑𝑡<0 for 𝑑/𝑐<𝑥<𝐾 and 𝑀 large enough. Similarly, 𝑑Γ₃().𝑑𝑡=𝑦𝑐𝑥−𝑑 (12) 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 𝜕()𝐷=𝑃𝐵+()𝜕𝑥𝑄𝐵=𝜕𝑦−𝑟𝑎𝑦(1+𝑎𝑥)²<0. (13) 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 𝑥̇=0, 𝑦̇=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(𝑥^ℵ,𝑦^ℵ) 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(𝑥_𝑐0,𝑦_𝑐0). 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 Point 𝐶₀ on 𝑁₁ above 𝑃

Starting from 𝐶₀, the orbit intersects with 𝑀₁ at point 𝐶₁(𝑥_𝑐1,𝑦_𝑐1) and then produces pulse to the point 𝐶^′₁(𝑥_𝑐′1,𝑦_𝑐′1) on 𝑀₁. According to (1), the following is got: 𝑦_𝐶′1=𝑦_𝐶1+𝛼. After 𝑛 time impulse, the point 𝐶⁺₁(𝑥_𝑐⁺₁,𝑦_𝑐+1) should satisfy 𝑦^∗<𝑦_𝐶+1≤𝑦^∗+𝛼. Thus, we only need to consider case 𝑦^∗<𝑦_𝐶0≤𝑦^∗+𝛼, and three cases should be discussed.

Case 1. If the impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is exactly 𝐶₀, thus, the curve 𝐶₀𝐶₁𝐶⁺₁ shall constitute a periodic orbit of (1) (see Figure 4).

Figure 4 

The orbit starting from the point 𝐶0 above 𝑃 (Case 1 in Section 4.1.1).

Case 2. If the impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is below 𝐶₀ on 𝑁₁, obviously, the successor function of 𝐶₀ satisfies 𝑓(𝐶₀)=𝑦_𝐶+1−𝑦_𝐶0<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 𝐷⁺₁ in the line 𝑁₁. Because 𝐷₀ is next to 𝑃, 𝐷₁ is next to 𝑃, and 𝑦_𝐷+1=𝑦_𝐷1+𝛼, thus, the successor function of 𝐷₀ satisfies 𝑓(𝐷₀)=𝑦_𝐷+1−𝑦_𝐷0>0. By Theorem 4, there exists a period solution (see Figure 5).

Figure 5 

The orbit starting from the point 𝐶0 (Case 2 in Section 4.1.1).

Case 3. If the impulsive point 𝐶⁺₁ 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 𝐴⁺₁(ℎ₁,𝑦_𝐴+1) in the line 𝑁₁. According to system (1), the following is got: 𝑦^∗<𝑦_𝐴+1<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 point 𝐶0 (Case 3 in Section 4.1.1).

4.1.2. The Orbit Starting from the Point 𝑄

Starting from 𝑄, the orbit intersects with 𝑀₂ at point 𝐶₁ and then produces pulse to the point 𝐶⁺₁ on 𝑁₂. According to (1), the following is got: 𝑥_𝐶⁺₁=1−𝑝ℎ₂,𝑦_𝐶+1=1−𝑞𝑦_𝐶1+𝜏. (14) By selecting 𝜏, three cases should be discussed.

Case 1. If the impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is exactly 𝑄, thus, the curve 𝑄𝐶₁𝐶⁺₁ shall constitute a periodic orbit of (1) (see Figure 7).

Figure 7 

The orbit starting from the point 𝑄 (Case 1 in Section 4.1.2).

Case 2. If the impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is above 𝑄 on 𝑁₂, thus, 𝑓(𝑄)=𝑦_𝐶+1−𝑦_𝑄>0. Choose a point 𝐷₀ above 𝐶⁺₁ 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 𝐷⁺₂ in the line 𝑁₂. According to the existence and uniqueness of impulsive differential equations, 𝐷₂ must be definitely below 𝐶₁ on 𝑀₂, and 𝐷⁺₂ must be below 𝐶⁺₁ on 𝑁₂; therefore the successor function of 𝐷₀ is 𝑓(𝐷₀)=𝑦_𝐷+2−𝑦_𝐷0<0. Based on the previous discussion, we can see that the region 𝜅 surrounded by the closed curve 𝐷₀𝑃𝐷₁𝐷₂𝐶₁𝐶⁺₁𝐷₀ 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 point 𝑄 (Case 2 in Section 4.1.2).

Case 3. If the impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is below 𝑄 on 𝑁₂, we shall have the successor function 𝑓(𝑄)=𝑦_𝐶+1−𝑦_𝑄<0.
In the meantime, take a point 𝐷₀ 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 𝐷⁺₁ on 𝑁₂. According to (1), the following is got: 𝑥_𝐷⁺₁=1−𝑝ℎ₂,𝑦_𝐷+1=1−𝑞𝑦_𝐷1+𝜏. (15) Thus, 𝐷⁺₁ must be above 𝐷₀, and the successor function of 𝐷₀ is 𝑓(𝐷₀)=𝑦_𝐷+1−𝑦_𝐷0>0.
As a result, the region 𝜅 surrounded by the closed curve 𝑄𝐷₀𝐷₁𝐶₁𝑄 involves a periodic solution of (1) (see Figure 9).

Figure 9 

The orbit starting from the point 𝑄 (Case 3 in Section 4.1.2).

4.1.3. The Orbit Starting from the Point 𝐶₀ between the Second Impulsive Set 𝑀₂ and Its Image Set 𝑁₂

The orbit starting from point 𝐶₀, between line 𝑁₂ and line 𝑀₂, intersects with line 𝑀₂ at 𝐶₁ and jumps onto point 𝐶⁺₁ on 𝑁₂. There are the following cases with changing 𝜏.

Case 1. If 𝐶⁺₁ on 𝑁₂ is exactly 𝑄, the orbit from point 𝐶⁺₁ moves to the point 𝐶₂ on 𝑀₂. For 𝐶₂, there are the following cases.(a) If 𝐶₂ is exactly 𝐶₁, then the curve 𝐶₁𝐶⁺₁𝐶₂ forms a periodic orbit of (1) (see Figure 10). (b) If 𝐶₂ on 𝑀₂ is below 𝐶₁, the successor function of 𝐶₁ is 𝑓(𝐶₁)=𝑦_𝐶2−𝑦_𝐶1<0.?In the meantime, take a point 𝐷₀ 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 𝐷⁺₁ 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)=𝑦_𝐷2−𝑦_𝐷1>0.?As a result, we get a periodic orbit in the region 𝜅 encircled by the closed curve 𝐶₁𝐶⁺₁𝐷⁺₁𝐷₁𝐶₁ (see Figure 11).(c) If 𝐶₂ on 𝑀₂ is above 𝐶₁, which does not exist periodic orbit, otherwise the curve 𝐶⁺₁𝐶₂ 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 point 𝐶0 (Case 1(a) in Section 4.1.3).

Figure 11 

The orbit starting from the point 𝐶0 (Case 1(b) in Section 4.1.3).

Case 2. If 𝐶⁺₁ 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 point 𝐶0 (Case 2 in Section 4.1.3).

Figure 13 

The orbit starting from the point 𝐶0 (Case 2 in Section 4.1.3).

Case 3. If 𝐶⁺₁ on 𝑁₂ is above 𝑄, for the orbit starting from 𝐶⁺₁, there exist the following cases.(a) If the orbit starting from 𝐶⁺₁ 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 from 𝐶⁺₁ 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.?If 𝐶₂ on 𝑀₂ is exactly 𝐶₁, then the curve 𝐶₁𝐶⁺₁𝐶₂ forms a periodic orbit (see Figure 15). ?If 𝐶₂ on 𝑀₂ is below 𝐶₁, then the successor function of 𝐶₁ is 𝑓(𝐶₁)=𝑦_𝐶2−𝑦_𝐶1<0.?On the other hand, take a point 𝐷₀ 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 𝐷⁺₁ 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)=𝑦_𝐷2−𝑦_𝐷1>0.?Based on the previous discussion above, there exists a periodic orbit in the region 𝜅 encircled 𝐷₁𝐶₁𝐶⁺₁𝐶₂ and 𝐶⁺₁𝐷⁺₁𝐷₂𝐶₂ (see Figure 16).(c) If 𝐶₂ 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 point 𝐶0 (Case 3(a) in Section 4.1.3).

Figure 15 

The orbit starting from the point 𝐶0 (Case 3(b) in Section 4.1.3).

Figure 16 

The orbit starting from the point 𝐶0 (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 Set 𝑀₁

Theorem 9. Let 𝑥=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 𝑓𝑥,𝑦=𝑟𝑥𝑔(),𝛼Φ1+𝑎𝑥−𝑏𝑥𝑦,𝑥,𝑦=𝑦−𝑑+𝑐𝑥𝑥,𝑦=0,𝛽𝑥,𝑦=𝛼,𝑥,𝑦=𝑥−ℎ₁, (16) and (𝜙(0),𝜑(0))=(ℎ₁,𝜑₀), (𝜙(𝑇),𝜑(𝑇))=(𝑥₁,𝑦₁), (𝜙(𝑇⁺),𝜑(𝑇⁺))=(𝑥₁,𝑦₁+𝛼)=(ℎ₁,𝜑₀); one gets 𝜕𝑓=𝑟𝜕𝑥(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. (17) According to Theorem 5, we have Δ₁=𝑓𝜙𝑇⁺,𝜑𝑇⁺𝑓𝜙(𝑇)(𝑇)=𝑓ℎ,𝜑¹,𝜑₀𝑓ℎ₁,𝜑₀=−𝛼𝑟ℎ₁/1+𝑎ℎ₁−𝑏ℎ₁𝜑₀+𝑏ℎ₁𝛼−𝑏ℎ₁𝛼𝑟ℎ₁/1+𝑎ℎ₁−𝑏ℎ₁𝜑₀−𝛼=1−𝑏𝛼𝑟/1+𝑎ℎ₁−𝑏𝜑₀,𝜇−𝛼₂=Δ₁exp^𝑇₀𝜕𝑓𝜙(𝑡)(𝑡+𝜕𝑥,𝜑)𝜕𝑔𝜙(𝑡)(𝑡𝜕𝑦,𝜑)𝑑𝑡=Δ₁exp^𝑇₀𝑟1+𝑎𝜙²−𝑏𝜑+𝑐𝜙−𝑑𝑑𝑡<Δ₁exp^𝑇₀𝑟1+𝑎𝜙−𝑏𝜑+𝑐𝜙−𝑑𝑑𝑡=Δ₁exp^𝑇₀1𝜙(𝑡)𝑑𝜙(𝑡)+^𝑇₀1𝜑(𝑡)𝑑𝜑(𝑡)=Δ₁𝜑₀−𝛼𝜑₀=1−𝑏𝛼𝑟/1+𝑎ℎ₁−𝑏𝜑₀𝜑−𝛼⁰−𝛼𝜑₀. (18) 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 Set 𝑀₂

Theorem 10. Let 𝑥=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. Let 𝑥=𝜙(𝑡), 𝑦=𝜑(𝑡) be the 𝑇 periodic solution of system (1), and 𝑓𝑥,𝑦=𝑟𝑥𝑔(),𝛼Φ1+𝑎𝑥−𝑏𝑥𝑦,𝑥,𝑦=𝑦−𝑑+𝑐𝑥𝑥,𝑦=−𝑝𝑥,𝛽𝑥,𝑦=−𝑞𝑦+𝜏,𝑥,𝑦=𝑥−ℎ₂, (19)(𝜙(𝑇⁺),𝜑(𝑇⁺))=((1−𝑝)𝜙(𝑇),(1−𝑞)𝜑(𝑇)+𝜏),(𝜙(𝑇),𝜑(𝑇))=(ℎ₂,(𝜑₀−𝜏)/(1−𝑞)); one gets 𝜕𝑓=𝑟𝜕𝑥(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. (20) Thus, Δ₁=1−𝑞𝑓𝜙𝑇⁺,𝜑𝑇⁺𝑓𝜙(𝑇)(𝑇)=𝑇,𝜑1−𝑞𝑟𝜙⁺1+𝑎𝜙𝑇⁺−𝑏𝜙𝑇⁺𝜑𝑇⁺×𝑟𝜙(𝑇)(𝑇)(𝑇)𝜑(𝑇)1+𝑎𝜙−𝑏𝜙⁻¹=𝑟ℎ1−𝑞1−𝑝₂1+𝑎1−𝑝ℎ₂−𝑏1−𝑝ℎ₂𝜑₀×𝑟ℎ₂1+𝑎ℎ₂−𝑏ℎ₂𝜑₀−𝜏1−𝑞⁻¹=𝑟1−𝑝1−𝑞1+𝑎1−𝑝ℎ₂−𝑏𝜑₀×𝑟1+𝑎ℎ₂𝜑−𝑏⁰−𝜏1−𝑞⁻¹,𝜇₂=Δ₁exp^𝑇₀𝜕𝑓𝜙(𝑡)(𝑡)+𝜕𝑥,𝜑𝜕𝑔𝜙(𝑡)(𝑡)𝜕𝑦,𝜑𝑑𝑡=Δ₁exp^𝑇₀𝑟1+𝑎𝜙²−𝑏𝜑+𝑐𝜙−𝑑𝑑𝑡<Δ₁exp^𝑇₀𝑟1+𝑎𝜙−𝑏𝜑+𝑐𝜙−𝑑𝑑𝑡=Δ₁exp^𝑇₀1𝜙(𝑡)𝑑𝜙(𝑡)+^𝑇₀1𝜑(𝑡)𝑑𝜑(𝑡)=Δ₁1𝜑1−𝑞1−𝑝⁰−𝜏𝜑₀=1−𝑝1−𝑞𝑟/1+𝑎1−𝑝ℎ₂−𝑏𝜑₀𝑟/1+𝑎ℎ₂−𝑏𝜑₀/×1−𝜏1−𝑞𝜑1−𝑞1−𝑝⁰−𝜏𝜑₀≤𝑟/1+𝑎1−𝑝ℎ₂−𝑏𝜑₀𝑟/1+𝑎ℎ₂−𝑏𝜑₀/.−𝜏1−𝑞 (21) If 𝜑₀𝑟<min1−𝑞𝑏1+𝑎ℎ₂1+𝜏,𝑏𝑞𝑏𝜏−𝑎𝑝1−𝑞𝑟ℎ₂1+𝑎ℎ₂1+𝑎1−𝑝ℎ₂ (22) or 𝜑₀𝑟>max1−𝑞𝑏1+𝑎ℎ₂1+𝜏,𝑏𝑞𝑏𝜏−𝑎𝑝1−𝑞𝑟ℎ₂1+𝑎ℎ₂1+𝑎1−𝑝ℎ₂, (23) 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. Let 𝑟=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(𝑥^ℵ,𝑦^ℵ)=(0.5,0.8). Then we have 𝑥^′(𝑡)=0.6𝑥𝑦1+𝑥−0.5𝑥𝑦,_′(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. (24) Numerical analysis of system (24) is being done using Maple 14.0. We have the following cases.

Case 1. Initial point 𝐶₀ is in the line 𝑁₁, where 𝛼=0.5.(a) Let 𝐶₀ be (0.2,1.2) above 𝑃(0.2,1) (see Figure 17). (b)Let 𝐶₀ be (0.2,0.6) below 𝑃(0.2,1) (see Figure 18).

(a)  Phase diagram of system  (24)

(b) Time series of system  (24)

(c) Time series of system  (24)

Figure 17 

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)

Figure 18 

Illustration of basic behavior of system (24).

Case 2. Initial point 𝐶₀ is in the line 𝑁₂. Let 𝐶₀ be 𝑄(0.225,0.9796).(a)The impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is exactly 𝑄, where 𝜏=0.62 (see Figure 19). (b)The impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is above 𝑄 on 𝑁₂, where 𝜏=0.8 (see Figure 20). (c)The impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is below 𝑄 on 𝑁₂, where 𝜏=0.5 (see Figure 21).

(a) Phase diagram of system  (24)

(b) Time series of system  (24)

(c) Time series of system  (24)

Figure 19 

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)

Figure 20 

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)

Figure 21 

Illustration of basic behavior of system (24).

Case 3. Initial point 𝐶₀ is between the second impulsive set 𝑀₂ and its image set 𝑁₂. Let 𝐶₀ be (0.25,0.9).(a)The impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is exactly 𝑄, where 𝜏=0.62 (see Figure 22).(b)The impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is below 𝑄 on 𝑁₂, where 𝜏=0.4 (see Figure 23).(c)(1) The impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is above 𝑄 on 𝑁₂, where 𝜏=0.8 (see Figure 24).?(2) The impulsive point 𝐶⁺₁ corresponding to 𝐶₁ is above 𝑄 on 𝑁₂, where 𝜏=1 and 𝛼=0.5 (see Figure 25).

(a)  Phase diagram of system  (24)

(b) Time series of system  (24)

(c) Time series of system  (24)

Figure 22 

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)

Figure 23 

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)

Figure 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)

Figure 25 

Illustration of basic behavior 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 than ℎ₁, 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.

Acknowledgments

This work is supported by the Shandong Provincial Natural Sciences Funds of China (Grant no. ZR2012AM012) and a Project of Shandong Province Higher Educational Science and Technology Program of China (Grant no. J13LI05).