The Mathematical Study of Pest Management Strategy

Abstract

The theory of impulsive state feedback control is used to establish a mathematical model in the pest management strategy. Then, the qualitative analysis of the mathematical model was provided. Here, a successor function in the geometry theory of differential equations is used to prove the sufficient conditions for uniqueness of the 1-periodic solution. It proved the orbital asymptotic stability of the periodic solution. In addition, numerical analysis is used to discuss the application significance of the mathematical model in the pest management strategy.

1. Introduction

Impulse is an interference in the thing at a short time in the course of its development. It is a method of external control. This kind of method is widely used in biological control, prevention of epidemic, cancer cells of chemotherapeutics, and so on. We use impulsive differential equation to reflect the method of external control. We can use impulsive differential equation to describe some biological phenomena in population ecology. There are mainly two kinds of impulsive differential equation. One kind is fixed times impulsive differential equation, and the other kind is differential system with state impulses. In the recent thirty years, many authors have studied the impulsive differential equation [1–5]. They obtained some theories of impulsive differential equation; particularly the theory of fixed times impulsive differential equation is widely used in population ecology. Many authors have studied the dynamics of predator-prey models with impulsive control strategies [6–12].

Pest management is a focus which people are concerned with. Because the technological revolutions have recently hit the industrial world and the experience and lessons are accumulated, the ideology and strategy of pest management have changed a lot. Pest management changes from chemical control to integrated control. It is fully integrated into the development of agriculture and forestry sustainability.

The study of pest management strategy has good application value and significant agriculture production. In the past few decades, many authors have made a lot of research and discussion it [13]. There are two major methods of pest management. The first is chemical control. It means that the main method to control the amount of pests is spraying insecticide. But its drawback is that it will cause pollution to the environment. In addition, spray insecticide will kill natural enemies and other beneficial organisms. Although this can control pest, it had a negative impact. The second is biological control, which means that the method to control the amount of pests is culturing the natural enemies of pests. Because the biological control can avoid the environmental pollution, many scholars studied biological control. Some people put forward the integrated control method (IPM) by combining chemical control and biological control. Thus we not only can use the fast speed of chemical control, but also can use biological control to avoid the environmental pollution. In the process of pest management, we see culturing the natural enemies of pests or insecticide spraying as an instant action, and this action is not regular. This action is decided by the number of pests; when the amount of pests reached a critical value, we spray insecticide or release the natural enemies of pests at the instant of that time; here, the critical value is called economic threshold or ET. Here, the instant action of culturing the natural enemies of pests or insecticide spraying is impulsive control as we said before, so we use differential system with state impulses to describe integrated control method (IPM) in pest management.

In recent years, the application of differential system with state impulses in integrated pest management has been greatly developed. Tang used differential system with state impulses in pest management [14, 15]. They established a system with state impulses: 𝑑𝑥𝑑𝑡=𝑥𝑎−𝑏𝑦𝑑𝑦()𝑑𝑡=𝑦𝑐𝑥−𝑑𝑥<𝑥₁Δ𝑥=−𝑝𝑥,Δ𝑦=ℎ,𝑥=𝑥₁. (1.1) Here 𝑥 is the densities of the pest, 𝑦 is the densities of natural enemies of the pest, 𝑥₁ is the critical value of economic, 𝑎 is the growth rate of the pest, 𝑏 is the trapping rate of natural enemies of the pest, 𝑐 is the absorption rate of natural enemies of the pest, 𝑑 is the death rate of natural enemies of the pest, 0<𝑝<1 is the rate of killed pest by spraying insecticide, and ℎ is the amount of natural enemies of the pest that we released; they are all positive numbers. This system is a spatial model; we can get the explicit solution of it. For system (1.1), the stability and existence of 1-periodic solution and the existence of 2-periodic solution all can be gotten by using comparison principle to transform the system into difference equation.

System (1.1) considered a two-species predator-prey model (Lotka-Volterra) that there is not density dependence for the continuous process of pulse points; this disagrees with practical significance. In order to be closer to the actual, Zeng at [16, 17] made the system (1.1) to be that there is density dependence for the continuous process of pulse points. This can reflect the practical situation; the model is as follows: 𝑑𝑥𝑑𝑡=𝑥𝑎−𝑟𝑥−𝑏𝑦𝑑𝑦()𝑑𝑡=𝑦𝑐𝑥−𝑑𝑥<𝑥₁Δ𝑥=−𝑝𝑥,Δ𝑦=ℎ,𝑥=𝑥₁. (1.2) Here 𝑟 is the density-dependent coefficient of the pest. They use tectonic Lambert-𝑊 function and comparison principle to get the condition of the existence of 1-periodic solution.

For system (1.2), it did not consider the influence of spraying insecticide on natural enemies. In order to reflect the actual more accurately, we introduced the rate of killed natural enemies by spraying insecticide (0<𝑞<1) at the foundation on system (1.2); then we get the model: 𝑑𝑥𝑑𝑡=𝑥𝑎−𝑟𝑥−𝑏𝑦𝑑𝑦()𝑑𝑡=𝑦𝑐𝑥−𝑑𝑥<𝑥₁,Δ𝑥=−𝑝𝑥Δ𝑦=−𝑞𝑦+ℎ𝑥=𝑥₁. (1.3) The significance of parameters is the same as the aforementioned.

The remainder of this paper is organized as follows. In Section 3 we use the successor function about geometry theory of semicontinuous dynamical systems to get the condition of existence and stability of 1-periodic solution for system (1.3). Section 4 combined with numerical simulations gives the application for system (1.3) in pest management.

2. Preliminaries

Definition 2.1. For the state impulse differential equation 𝑑𝑥,𝑑𝑡=𝑓𝑥,𝑦𝑑𝑦,,,.𝑑𝑡=𝑔𝑥,𝑦𝑥,𝑦∉𝑀𝑥,𝑦Δ𝑥=𝛼𝑥,𝑦,Δ𝑦=𝛽𝑥,𝑦𝑥,𝑦∈𝑀𝑥,𝑦 (2.1) Here 𝑀{𝑥,𝑦}, 𝑁{𝑥,𝑦}, and 𝑅₂(𝑥,𝑦) are lines or curves on the plane, 𝑀{𝑥,𝑦} is the pulse set, and 𝑁{𝑥,𝑦} is the phase set. We describe a dynamical system made by the solution maps of system (2.1) as a semicontinuous dynamical system, which is denoted as (Ω,𝑓,𝜑,𝑀). The initial mapping point 𝑝 is not in the pulse set, 𝑃∈Ω=𝑅₂−𝑀{𝑥,𝑦},  𝜑 is a continuous mapping, 𝜑(𝑀)=𝑁, and  𝜑 is known as pulse mapping.

Definition 2.2. 𝑓(𝑃,𝑡) is the semicontinuous dynamical system mapping described by system (2.1) at Ω→Ω; 𝑓(𝑃,𝑡) is a mapping in itself. It includes two parts:(1) differential equation 𝑑𝑥,𝑑𝑡=𝑓𝑥,𝑦𝑑𝑦.𝑑𝑡=𝑔𝑥,𝑦 (2.2) The Poincare mapping 𝜋(𝑃,𝑡) is the mapping of (2.2) at the initial mapping point 𝑃; if 𝑓(𝑃,𝑡)∩𝑀(𝑥,𝑦)=0, then the semicontinuous dynamical system mapping at the initial mapping point 𝑃 is 𝑓(𝑃,𝑡)=𝜋(𝑃,𝑡).(2) If there is a 𝑇₁, then 𝑓(𝑃,𝑇₁)=𝑄₁∈𝑀{𝑥,𝑦}; pulse mapping is 𝜑𝑄₁=𝜑𝑓𝑃,𝑇₁=𝑃₁∈𝑁, (2.3) and if 𝑓(𝑃₁,𝑡)∩𝑀{𝑥,𝑦}=0, then the semicontinuous dynamical system mapping at the initial mapping point 𝑃 is 𝑓(𝑃,𝑡)=𝜋(𝑃,𝑇₁)+𝜋(𝑃₁,𝑡−𝑇₁).(3) At the situation of (2), if 𝑓(𝑃₁,𝑡)∩𝑀{𝑥,𝑦}≠0, and having a 𝑇₂ made 𝑓(𝑃₁,𝑇₂)=𝑄₂∈𝑀{𝑥,𝑦}, then 𝑓(𝑃,𝑡)=𝜋𝑃,𝑇₁+𝑓𝑃₁,𝑡−𝑇₁=𝜋𝑃,𝑇₁+𝜋𝑃₁,𝑇₂+𝑓𝑃₂,𝑡−𝑇₁−𝑇₂. (2.4)(4) For repeated superior surface, 𝑓(𝑃₁,𝑡)∩𝑀{𝑥,𝑦}≠0; then we have 𝑓(𝑃₁,𝑇₁)=∑_𝑘=1𝜋(𝑃_𝑘,𝑇_𝑘)+𝑓(𝑃_𝑘,𝑡)𝑓𝑃₁=𝑓𝑃,𝑡¹,𝑡0≤𝑡<𝑇₁𝜋𝑃₁,𝑇₁+𝑓𝑃₂,𝑡−𝑇₁𝑇₁≤𝑡<𝑇₁+𝑇₂𝜋𝑃₁,𝑇₁+𝜋𝑃₂,𝑇₂+𝑓𝑃₃,𝑡−𝑇₁−𝑇₂𝑇₁+𝑇₂≤𝑡<𝑇₁+𝑇₂+𝑇₃⋮_𝑛𝑘∑₌₁𝜋𝑃_𝑘,𝑇_𝑘+𝑓𝑃_𝑘+1,𝑡−∑^𝑛_𝑘=1𝑇∑_𝑘^𝑛_𝑘=1𝑇_𝑘≤𝑡≤∑^𝑛+1_𝑘=1𝑇_𝑘⋮ (2.5)

Property 1. The mapping of the semi-continuous dynamical system:
(1) 𝑓(𝑃,0)=𝑃; (2) 𝑓(𝑓(𝑃,𝑡₁),𝑡₂)=𝑓(𝑃,𝑡₁+𝑡₂); (3) 𝑓(𝑃,𝑡) is continuously at initial mapping point 𝑃.

Definition 2.3. If the periodic solution Γ₀ of system (2.1) does not intersect with pulse set 𝑀{𝑥,𝑦}, then Γ₀ is also the periodic solution for system (2.1).

Definition 2.4. When there is a point 𝑃 at phase set 𝑁 and a 𝑇₁, make 𝑓(𝑃,𝑇₁)=𝑄₁∈𝑀{𝑥,𝑦}; it also has 𝜑(𝑄₁)=𝜑(𝑓(𝑃,𝑇₁))=𝑃∈𝑁; then 𝑓(𝑃,𝑇₁) is said to be 1-periodic solution.

Definition 2.5. Successor function: let 𝐿 be a coordinate axis defined at 𝑁, the origin point is intersection point of line 𝑥=(1−𝑝)𝑥₁ with 𝑥 axis, and the positive direction consistent with the positive direction of 𝑦 axis, an arbitrary point 𝑥∈𝑁, 𝑙(𝑥) is the coordinate of 𝑥 at 𝑁, 𝑙(𝑥)∈𝐶⁰; if there exists a 𝑡₁∈𝑅⁺, making 𝜋(𝑥,𝑡₁)∈𝑀,𝑥⁺=𝜑(𝜋(𝑥,𝑡₁))∈𝑁, then 𝐹(𝑥) is the successor function of point 𝑥; here 𝐹(𝑥)=𝑙(𝑥⁺)−𝑙(𝑥).

Lemma 2.6. The successor function 𝐹(𝑥) is continuous.
In fact, successor function  𝐹(𝑥) is that continuous solution 𝜋(𝑥,𝑡₁) of differential equation compound with continuous functions 𝐼(𝑥) and 𝐹(𝑥) is a complex function of two continuous functions, so 𝐹(𝑥) is continuous.

Lemma 2.7. Let continuous dynamical system be as (𝑋,Π); if there are two points 𝑥₁ and 𝑥₂ at phase set, making 𝐹(𝑥₁)⋅𝐹(𝑥₂)<0, then there must exist a point 𝑃 between 𝑥₁ and 𝑥₂ such that 𝐹(𝑃)=0; thus there must exist 1-periodic solution by point 𝑃.

Lemma 2.8 (Poincaré’s criterion). The 𝑇-periodic solution 𝑥=𝜙(𝑡),𝑦=𝜑(𝑡) of system 𝑑𝑥,𝑑𝑡=𝑓𝑥,𝑦𝑑𝑦𝑑𝑡=𝑔𝑥,𝑦,Φ𝑥,𝑦≠0,Δ𝑥=𝛼𝑥,𝑦,Δ𝑦=𝛽𝑥,𝑦,Φ𝑥,𝑦=0 (2.6) is orbitally asymptotically stable if the multiplier 𝜇₂ satisfies the condition |𝜇₂|<1, where 𝑢₂=^𝑞_𝑘=1Δ_𝑘exp^𝑇₀𝜕𝑓𝜙(𝑡)(𝑡)+𝜕𝑥,𝜑𝜕𝑔𝜙(𝑡)(𝑡),Δ𝜕𝑦,𝜑𝑑𝑡_𝑘=𝑓₊𝜕𝛽⋅𝜕𝑦𝜕Φ−𝜕𝑥𝜕𝛽⋅𝜕𝑥𝜕Φ𝜕𝑦+𝜕Φ/𝜕𝑥+𝑔₊𝜕𝛼⋅𝜕𝑥𝜕Φ−𝜕𝑦𝜕𝛼⋅𝜕𝑦𝜕Φ×𝑓𝜕𝑥+𝜕Φ/𝜕𝑦𝜕Φ𝜕𝑥+𝑔𝜕Φ𝜕𝑦⁻¹. (2.7)(2.8) Here 𝑓, 𝑔, 𝜕𝛼/𝜕𝑥, 𝜕𝛼/𝜕𝑦, 𝜕𝛽/𝜕𝑥, 𝜕𝛽/𝜕𝑦, 𝜕Φ/𝜕𝑥, 𝜕Φ/𝜕𝑦 are calculated for the point (𝜙(𝜏_𝑘),𝜑(𝜏_𝑘)), 𝑓₊=𝑓𝜙𝜏⁺_𝑘,𝜑𝜏⁺_𝑘,𝑔₊=𝑔𝜙𝜏⁺_𝑘,𝜑𝜏⁺_𝑘. (2.9)

3. The Stability and Existence of 1-Periodic Solution to Pest Management Model with Impulsive State Control

Statement 3.1. At system (1.3), if 𝑝=𝑞=ℎ=0, then we get Lotka-Volterra predator-prey model: 𝑑𝑥,𝑑𝑡=𝑥𝑎−𝑟𝑥−𝑏𝑦𝑑𝑦().𝑑𝑡=𝑦𝑐𝑥−𝑑 (3.1) When 0<𝑎𝑐−𝑟𝑑<𝑟²𝑑/4𝑐, system has stable focus 𝐸(𝑑/𝑐,(𝑎𝑐−𝑟𝑑)/𝑏𝑐). This stable focus is asymptotically stable. When 𝑝>0, 𝑞>0, ℎ>0, we get system (1.3).

Statement 3.2. The intersection point of pulse set 𝑥=𝑥₁ and isoclines 𝑎−𝑟𝑥−𝑏𝑦=0 is denoted by 𝐻(𝐻_𝑥,𝐻_𝑦); then there exists a trajectory of system Γ₁ that tangency with 𝑥=𝑥₁ to 𝐻(𝐻_𝑥,𝐻_𝑦), and the phase point of 𝐻 at phase set 𝑥=(1−𝑝)𝑥₁ is denoted by 𝐻₁(𝐻_1𝑥,𝐻_1𝑦).

Theorem 3.3. When pulse set is 𝑥₁=𝑑/𝑐, then there exists a point 𝑀 at phase set  𝑥=(1−𝑝)(𝑑/𝑐); make 𝐹(𝑀)=0, and then system (1.3) has a 1-periodic solution.

Proof. Let pulse set be 𝑥₁=𝑑/𝑐, phase set is 𝑥=(1−𝑝)𝑥₁, the intersection point of pulse set 𝑥=(1−𝑝)𝑥₁ and isoclines 𝑎−𝑟𝑥−𝑏𝑦=0 is denoted by 𝐴(𝐴_𝑥,𝐴_𝑦)=𝐴((1−𝑝)(𝑑/𝑐),(𝑎𝑐−𝑟𝑑(1−𝑝))/𝑏𝑐), 𝐴_𝑥 is the 𝑥 coordinate of 𝐴, 𝐴_𝑦 is the 𝑦 coordinate of 𝐴, there exists a trajectory 𝐿₁ at initial point 𝐴 of system, its tangency with 𝑥=(1−𝑝)𝑥₁ at 𝐴 and intersects with 𝑥=𝑥₁ at 𝐶₁(𝑑/𝑐,𝐶_1𝑦), that pulse to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐴₁(𝐴_1𝑥,𝐴_1𝑦)=𝐴((1−𝑝)(𝑑/𝑐),(1−𝑞)𝐶_1𝑦+ℎ), it is called the successor point of 𝐴. From Definition 2.5, we get the successor faction is 𝐹(𝐴)=𝐴_1𝑦−𝐴_𝑦 of 𝐴.(i)If 𝐹(𝐴)=𝐴_1𝑦−𝐴_𝑦<0, (see Figure 1), there exists a point 𝑆∶𝑆_𝑦≫0 at 𝑥=(1−𝑝)𝑥₁, the trajectory 𝐿₂ over 𝑆 intersects with 𝑥=(1−𝑝)𝑥₁, the intersection point is denoted by 𝐵, let 0<𝐵_𝑦≪1, trajectory 𝐿₂ intersects with 𝑥=𝑥₁ at point 𝐶₂, that is pulsed to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐵₁(𝐵_1𝑥,𝐵_1𝑦), so 𝐹(𝐵)=𝐵_1𝑦−𝐵_𝑦>0, so there must exist a point 𝑀 at phase set 𝑥=(1−𝑝)𝑥₁, it satisfies 𝐵_𝑦<𝑀_𝑦<𝐴_𝑦, that make 𝐹(𝑀)=0, from Lemma 2.7, we get that system (1.3) has a 1-periodic solution.(ii)If 𝐹(𝐴)=𝐴_1𝑦−𝐴_𝑦>0, (see Figure 2), then there exists a trajectory 𝐿₃ which can sufficiently approach trajectory 𝐿₁ to make the intersection point 𝐴^′∶0<𝐴^′_𝑦−𝐴_𝑦≪1 of trajectory 𝐿₃ and 𝑥=(1−𝑝)𝑥₁. It means that point 𝐴^′ can sufficiently approach point 𝐴. The trajectory 𝐿₃ over 𝑆 intersects with 𝑥=𝑥₁ at point 𝐶₃, that pulse to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐴^′₁(𝐴^′_1𝑥,𝐴^′_1𝑦), and point 𝐵₁ satisfies 0<𝐴_1𝑦−𝐴^′_1𝑦≪1. That means 𝐴^′₁ can sufficiently approach point 𝐴₁; then 𝐹(𝐴^′)=𝐴^′_1𝑦−𝐴^′_𝑦>0. At the same time, there exists a point 𝑆∶𝑆_𝑦≫0 at 𝑥=(1−𝑝)𝑥₁, the trajectory 𝐿₂ over 𝑆 intersects with 𝑥=𝑥₁, the intersection point is denoted by 𝐶₂, that is pulsed to 𝑥=(1−𝑝)𝑥₁, phase point is 𝑆₁(𝑆_1𝑥,𝑆_1𝑦), then 𝐹(𝑆)=𝑆_1𝑦−𝑆_𝑦<0. So, there must exist a point 𝑀 at phase set 𝑥=(1−𝑝)𝑥₁, it satisfy 𝐴^′_𝑦<𝑀_𝑦<𝑆_𝑦, that makes 𝐹(𝑀)=0, from Lemma 2.7, we get that system (1.3) has a 1-periodic solution. This completes the proof.

Figure 1 

Figure 2 

Theorem 3.4. When pulse set is 0<𝑥₁<𝑑/𝑐, there exists a point 𝑀 at phase set 𝑥=(1−𝑝)𝑥₁; make 𝐹(𝑀)=0, and then system (1.3) has a 1-periodic solution.

Proof. Let pulse set is 0<𝑥₁<𝑑/𝑐, phase set be 𝑥=(1−𝑝)𝑥₁, the intersection point of pulse set 𝑥=(1−𝑝)𝑥₁ and isoclines 𝑎−𝑟𝑥−𝑏𝑦=0 is denoted by 𝐴(𝐴_𝑥,𝐴_𝑦), and there exists a trajectory 𝐿₁ at initial point 𝐴 of system, its tangency with 𝑥=(1−𝑝)𝑥₁ at 𝐴 and intersects with 𝑥=𝑥₁ at 𝐶₁(𝐶_1𝑥,𝐶_1𝑦), that is pulsed to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐴₁(𝐴_1𝑥,𝐴_1𝑦), it is called the successor point of 𝐴. From Definition 2.5, we get the successor function 𝐹(𝐴)=𝐴_1𝑦−𝐴_𝑦 of 𝐴. Here, we make the discussion similar to the proof of Theorem 3.3; then we get that system (1.3) has a 1-periodic solution whether at 𝐹(𝐴)=𝐴_1𝑦−𝐴_𝑦<0 (see Figure 3) or at 𝐹(𝐴)=𝐴_1𝑦−𝐴_𝑦>0, (see Figure 4). This completes the proof.

Figure 3 

Figure 4 

Statement 3.5. Let pulse set be 𝑥=𝑥₁(𝑑/𝑐<𝑥₁<𝑎/𝑟), and phase set is 𝑥=(1−𝑝)𝑥₁; we know that trajectory 𝐿₀ of system tangency with 𝑥=𝑥₁ at 𝐻(𝐻_𝑥,𝐻_𝑦); the negative semiorbits of point 𝐻 are denoted by 𝑅(𝐻,𝑡); here 𝑡≤0 the phase point of 𝐻 at phase set 𝑥=(1−𝑝)𝑥₁ is denoted by 𝐴₁(𝐴_1𝑥,𝐴_1𝑦).

Theorem 3.6. When the pulse set is 𝑑/𝑐<𝑥₁<𝑎/𝑟 and phase set is 0<(1−𝑝)𝑥₁<𝑑/𝑐, (1)if the negative semiorbits of point 𝐻 are 𝑅(𝐻,𝑡)∩𝑁=𝜙, then there exists a point 𝑀 at 𝑥=(1−𝑝)𝑥₁ making 𝐹(𝑀)=0. It means that system (1.3) has a 1-periodic solution.(2)The negative semiorbits of point 𝐻 are 𝑅(𝐻,𝑡). If for the first time it intersects with phase set at 𝐴(𝐴_𝑥,𝐴_𝑦), the second time it intersects with phase set at 𝐵(𝐵_𝑥,𝐵_𝑦), here 𝐴_𝑦>𝐵_𝑦, when 𝐴_1𝑦−𝐴_𝑦>0 or 𝐴_1𝑦−𝐵_𝑦<0, there exists a point 𝑀 at phase set 𝑥=(1−𝑝)𝑥₁, making 𝐹(𝑀)=0; it means that system (1.3) has a 1-periodic solution.

Proof. (1) If the negative semiorbits of point 𝐻 are 𝑅(𝐻,𝑡)∩𝑁=𝜙, there exists a trajectory 𝐿₁ at initial point 𝐴 of system, that tangency with 𝑥=(1−𝑝)𝑥₁ at 𝐴 and intersects with 𝑥=𝑥₁ at 𝐶₁(𝐶_1𝑥,𝐶_1𝑦), that is pulsed to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐴₁(𝐴_1𝑥,𝐴_1𝑦), it is called the successor point of 𝐴, the successor faction is 𝐹(𝐴)=𝐴_𝑦−𝐴_1𝑦.(i)If 𝑓(𝐴)=𝐴_1𝑦−𝐴_𝑦<0 (see Figure 5), there exists a point 𝑆∶𝑆_𝑦≫0, the trajectory of system 𝐿₂ which cross the point 𝑆 intersect with 𝑥=(1−𝑝)𝑥₁ at 𝐵, it make 0<𝐵_𝑦≪1, the trajectory 𝐿₂ of system intersect with 𝑥=𝑥₁ at point 𝐶₂, that is pulse to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐵₁(𝐵_1𝑥,𝐵_1𝑦), then 𝐹(𝐵)=𝐵_1𝑦−𝐵_𝑦>0, so there must exist a point 𝑀 at phase set 𝑥=(1−𝑝)𝑥₁, it satisfies 𝐵_𝑦<𝑀_𝑦<𝐴_𝑦, it makes 𝐹(𝑀)=0. From Lemma 2.7, we get that system (1.3) has a 1-periodic solution.(ii)If 𝐹(𝐴)=𝐴_1𝑦−𝐴_𝑦>0 (see Figure 6), there exists a trajectory 𝐿₃ which can sufficiently approach trajectory 𝐿₁ to make the intersection point 𝐴^′∶0<𝐴^′_𝑦−𝐴_𝑦≪1 of trajectory 𝐿₃ and 𝑥=(1−𝑝)𝑥₁; it means that point 𝐴^′ can sufficiently approach point 𝐴, the trajectory 𝐿₃ over 𝑆 intersects with 𝑥=𝑥₁ at point 𝐶₃, that is pulsed to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐴^′₁(𝐴^′_1𝑥,𝐴^′_1𝑦), point 𝐴^′₁ satisfies 0<𝐴_1𝑦−𝐴^′_1𝑦≪1, and that means 𝐴^′₁ can sufficiently approach point 𝐴₁; then 𝐹(𝐴^′)=𝐴^′_1𝑦−𝐴^′_𝑦>0. At the same time, there exists a point 𝑆∶𝑆_𝑦≫0 at 𝑥=(1−𝑝)𝑥₁, the trajectory 𝐿₂ over 𝑆 intersects with 𝑥=𝑥₁, the intersection point is denoted by 𝐶₂, that pulse to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐹₁(𝐹_1𝑥,𝐹_1𝑦), and then 𝐹(𝑆)=𝑆_1𝑦−𝑆_𝑦<0, so there must exist a point 𝑀 at phase set 𝑥=(1−𝑝)𝑥₁; it satisfies 𝐴^′_𝑦<𝑀_𝑦<𝑆_𝑦, making 𝐹(𝑀)=0. From Lemma 2.7, we get that system (1.3) has a 1-periodic solution.
(2) The negative semiorbits of point 𝐻 are (𝐻,𝑡). If for the first time it intersects with phase set at 𝐴^′′(𝐴_𝑥^′′,𝐴_𝑦^′′), the second time it intersects with phase set at 𝐵^′′(𝐵_𝑥^′′,𝐵_𝑦^′′); here 𝐴_𝑦^′′>𝐵_𝑦^′′; here we consider the phase point 𝐴₁^′′(𝐴^′′_1𝑥,𝐴^′′_1𝑦) of 𝐻 at 𝑥=(1−𝑝)𝑥₁.(i)If 𝐹(𝐴^′′)=𝐴^′′_1𝑦−𝐴^′′>0 (see Figure 7), then there exists a trajectory 𝐿₃ which can sufficiently approach trajectory 𝐿₁ to make the intersection point 𝐷∶0<𝐷_𝑦−𝐴_𝑦≪1 of trajectory 𝐿₃ and 𝑥=(1−𝑝)𝑥₁. It means that point 𝐷 can sufficiently approach point 𝐴, the trajectory 𝐿₃ intersect with 𝑥=𝑥₁ at point 𝐶₃, that pulse to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐷₁(𝐷_1𝑥,𝐷_1𝑦), it satisfies 𝐷₁∶0<𝐴_1𝑦−𝐷_1𝑦≪1; that means 𝐷₁ can sufficiently approach point 𝐴₁; then we have 𝑓(𝐷)=𝐷_1𝑦−𝐷_𝑦>0, there exists a point 𝑆∶𝑆_𝑦≫0 at 𝑥=(1−𝑝)𝑥₁, the trajectory 𝐿₂ over 𝑆 intersects with 𝑥=𝑥₁, the intersection point is denoted by 𝐶₂, that pulse to 𝑥=(1−𝑝)𝑥₁, phase point is 𝐹₁(𝐹_1𝑥,𝐹_1𝑦), and then 𝐹(𝑆)=𝑆_1𝑦−𝑆_𝑦<0, so there must exist a point 𝑀 at phase set 𝑥=(1−𝑝)𝑥₁; it satisfies 𝐷_𝑦<𝑀_𝑦<𝑆_𝑦, making 𝐹(𝑀)=0. From Lemma 2.7, we get that system (1.3) has a 1-periodic solution.(ii)If 𝐹(𝐵^′′)=𝐴^′′_1𝑦−𝐵_𝑦^′′<0 (see Figure 8), there exists a point 𝑆∶𝑆_𝑦≫0, the trajectory of system 𝐿₂ which crosses the point 𝑆 intersects with 𝑥=(1−𝑝)𝑥₁ at 𝐺, it makes 0<𝐺_𝑦≪1, the trajectory of system 𝐿₂ intersects with 𝑥=𝑥₁ at point 𝐶₂, that pulse to 𝑥=(1−𝑝)𝑥₁; phase point is 𝐺₁(𝐺_1𝑥,𝐺_1𝑦), and then 𝑓(𝐺)=𝐺_1𝑦−𝐺_𝑦>0, so there must exist a point 𝑀 at phase set 𝑥=(1−𝑝)𝑥₁; it satisfies 𝐺_𝑦<𝑀_𝑦<𝐵_𝑦^′′, making 𝐹(𝑀)=0. From Lemma 2.7, we get that system (1.3) has a 1-periodic solution.

Figure 5 

Figure 6 

Figure 7 

Figure 8 

Statement 3.7. If 𝐵_𝑦<ℎ<𝐴_1𝑦<𝐴_𝑦, then system (1.3) has no 1-periodic solution.
If 𝑑/𝑐≤(1−𝑝)𝑥₁<𝑥₁, the periodic solution at the right of stable focus under the influence of impulsive control, then it does not have practical significance, so we did not discuss it.

Theorem 3.8. If the condition 0<(𝑏ℎ−𝑟𝑝𝑥₁−𝑏𝑞𝜙₀)/(𝑎−𝑟𝑥₁−𝑏𝜙₀)<2 holds, then the 1-periodic solution Γ₀ which crosses the point (𝑥₁,𝜑₀) of system (1.3) is orbitally asymptotically stable.

Proof. From system (1.3), we have 𝜕𝑓𝜕𝑥=𝑎−2𝑟𝑥−𝑏𝑦,𝜕𝑔𝜕𝑦=𝑐𝑥−𝑑,𝜕𝛼𝜕𝑥=−𝑝,𝜕𝛽Φ𝜕𝑦=−𝑞,𝑥,𝑦=𝑥−𝑥₁,𝜕Φ𝜕𝑥=1,𝜕Φ𝜙(𝑇)(𝑇)=𝑥𝜕𝑦=0,,𝜑₁,𝜑₀,𝜕𝛼𝜕𝑦=0,𝜕𝛽𝜙𝑇𝜕𝑥=0,₊,𝜑𝑇⁺=1−𝑝𝑥₁,1−𝑞𝜑₀,Δ+ℎ₂=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=𝑓₊⋅1−𝑞𝑔=1−𝑞⋅𝑓₊𝜙𝑇⁺⋅𝜑𝑇⁺𝑓𝜙(𝑇)(𝑇)=𝑥⋅𝜑1−𝑝₁𝑎−𝑟1−𝑝𝑥₁−𝑏1−𝑞𝜑₀+ℎ1−𝑞𝑥₁𝑎−𝑟𝑥₁−𝑏𝜑₀=1−𝑝𝑎−𝑟1−𝑝𝑥₁−𝑏1−𝑞𝜑₀+ℎ1−𝑞𝑎−𝑟𝑥₁−𝑏𝜑₀,𝜇₂=Δ₂⋅exp^𝑇₀𝜕𝑓+𝜕𝑥𝜙,𝜑𝜕𝑔𝜕𝑦𝜙,𝜑𝑑𝑡=Δ₂⋅exp^𝑇₀𝑎−𝑟𝜙−𝑏𝜑+𝑐𝜙−𝑑−𝑟𝜙𝑑𝑡=Δ₂⋅exp^𝑥1_{𝑥1−𝑝1}1𝜙(𝑡)𝑑𝜙(𝑡)+^𝜑0_{𝜑1−𝑞0+ℎ}1𝜑(𝑡)𝑑𝜑(𝑡)−^𝑇₀𝑟𝜙(𝑡)𝑑𝑡=Δ₂1⋅expln11−𝑝+ln1−𝑞𝜑₀+ℎ−𝑟^𝑇₀𝜙(𝑡)𝑑𝑡=Δ₂⋅1⋅𝜑1−𝑝1−𝑞⁰𝜑₀+ℎ⋅exp−𝑟^𝑇₀𝜙(𝑡)𝑑𝑡, (3.2) then 𝑢₂=1−𝑝𝑎−𝑟1−𝑝𝑥₁−𝑏1−𝑞𝜑₀+ℎ1−𝑞𝑎−𝑟𝑥₁−𝑏𝜑₀⋅1⋅𝜑1−𝑝₀1−𝑞𝜑₀+ℎ⋅exp−𝑟^𝑇₀𝜙(𝑡)=𝑥𝑑𝑡𝑎−𝑟1−𝑝₁−𝑏1−𝑞𝜑₀+ℎ1−𝑞𝑎−𝑟𝑥₁−𝑏𝜑₀⋅𝜑₀1−𝑞𝜑₀+ℎ⋅exp−𝑟^𝑇₀𝜙(𝑡)≤𝑥𝑑𝑡𝑎−𝑟1−𝑝₁−𝑏1−𝑞𝜑₀+ℎ1−𝑞𝑎−𝑟𝑥₁−𝑏𝜑₀⋅𝜑₀1−𝑞𝜑₀⋅exp−𝑟^𝑇₀𝜙(𝑡)=𝑥𝑑𝑡𝑎−𝑟1−𝑝₁−𝑏1−𝑞𝜑₀+ℎ𝑎−𝑟𝑥₁−𝑏𝜑₀⋅exp−𝑟^𝑇₀𝜙(𝑡)≤𝑥𝑑𝑡𝑎−𝑟1−𝑝₁−𝑏1−𝑞𝜑₀+ℎ𝑎−𝑟𝑥₁−𝑏𝜑₀=1−𝑏ℎ−𝑟𝑝𝑥₁−𝑏𝑞𝜑₀𝑎−𝑟𝑥₁−𝑏𝜑₀. (3.3) When 0<(𝑏ℎ−𝑟𝑝𝑥₁−𝑏𝑞𝜙₀)/(𝑎−𝑟𝑥₁−𝑏𝜙₀)<2, then |𝑢₂|<1, so the 1-periodic solution Γ₀ is orbitally asymptotically stable.

Statement 3.9. If 𝑝=𝑞=0, ℎ>0, we get the pest management model of nonpollution; when the economic threshold of system (1.3) is 𝑥=𝑥₁, then 𝑦₁=(𝑎−𝑟𝑥₁)/𝑏, and it is called the key point of the natural enemies of pests. At the time that the number of the natural enemies of pests is less than 𝑦₁, we release the natural enemies of pests, so we have the following model: 𝑑𝑥𝑑𝑡=𝑥𝑎−𝑟𝑥−𝑏𝑦𝑑𝑦()𝑑𝑡=𝑦𝑐𝑥−𝑑𝑦>𝑦₁,Δ𝑦=ℎ,𝑦=𝑦₁. (3.4) here 0<ℎ<(𝑟/𝑏)𝑥₁ or 0<ℎ<((𝑎/𝑏)−𝑦₁) means the number of the natural enemies we released at one time; the significance of other parameters is the same as the aforementioned.

Theorem 3.10. (1) If 𝑥₁≤𝑑/𝑐, then the system (3.4) has a 1-periodic solution.
(2) If 𝑥₁>𝑑/𝑐, then the system (3.4) has a 1-periodic solution or it has 𝑥(𝑡)≤𝑥₁ for any 𝑡.

Theorem 3.11. The 1-periodic solution of system (3.4) is orbitally asymptotically stable.

4. Numerical Analysis and Biological Significance

In this part, we use numerical simulation to analyse the dynamical behavior and ecological significance of system (1.3). We fixed the coefficients of the system, then we get system (2.1): 𝑑𝑥𝑑𝑡=𝑥16−𝑥−2𝑦𝑑𝑦()𝑑𝑡=𝑦2𝑥−10𝑥<𝑥₁,Δ𝑥=−𝑝𝑥,Δ𝑦=−𝑞𝑦+ℎ,𝑦=𝑦₁. (4.1)

Statement 4.1. If there is no impulse, then system has the unique positive equilibrium (5, 4.5) which is globally asymptotically stable.
Next, we consider the existence of 1-periodic solution for system (4.1) under the different values of pulse set 𝑥₁, the parameters 𝑝,𝑞,ℎ, and initial point (𝑥₀,𝑦₀).
Case 1. 𝑝=0.6,𝑞=0.4,ℎ=1,(𝑥₀,𝑦₀)=(3.77,1.42), pulse set is 𝑥₁=4.5<5, see Figure 9, and the system has a 1-periodic solution.
Case 2. 𝑝=0.6,𝑞=0.4,ℎ=5,(𝑥₀,𝑦₀)=(3.77,1.42), pulse set is 𝑥₁=4.5<5, see Figure 10, and the system has a 1-periodic solution.
Case 3. 𝑝=0.6,𝑞=0.4,ℎ=1,(𝑥₀,𝑦₀)=(3.77,1.42), pulse set is 𝑥₁=5, see Figure 11, and the system has a 1-periodic solution.
Case 4. 𝑝=0.6,𝑞=0.4,ℎ=5,(𝑥₀,𝑦₀)=(3.77,1.42), pulse set is 𝑥₁=5, see Figure 12, and the system has a 1-periodic solution.
Case 5. 𝑝=0.6,𝑞=0.4,ℎ=1,(𝑥₀,𝑦₀)=(3.77,1.42), pulse set is 𝑥₁=6>5, phase set is (1−𝑝)𝑥₁=2.4<5, see Figure 13, and the system has a 1-periodic solution.
Case 6. 𝑝=0.6,𝑞=0.4,ℎ=5,(𝑥₀,𝑦₀)=(3.77,1.42), pulse set is 𝑥₁=6>5, phase set is (1−𝑝)𝑥₁=2.4<5, see Figure 14, and the system has a 1-periodic solution.
Case 7. 𝑝=0.2,𝑞=0.5,ℎ=1.1,(𝑥₀,𝑦₀)=(6.44,0.435), pulse set is 𝑥₁=7>5, phase set is (1−𝑝)𝑥₁=5.6>5, see Figure 15, and the system has a 1-periodic solution.
Case 8. 𝑝=0.2,𝑞=0.5,ℎ=12,(𝑥₀,𝑦₀)=(6.44,0.435), pulse set is 𝑥₁=7>5, phase set is (1−𝑝)𝑥₁=5.6>5, see Figure 16, and the system has a 1-periodic solution.
Case 9. 𝑝=0.2,𝑞=0.5,ℎ=1.2,(𝑥₀,𝑦₀)=(6.44,0.435), pulse set is 𝑥₁=7>5, phase set is (1−𝑝)𝑥₁=5.6>5, see Figure 17, and the system has no 1-periodic solution.
Case 10. 𝑝=0.2,𝑞=0.5,ℎ=11,(𝑥₀,𝑦₀)=(6.44,0.435), pulse set is 𝑥₁=7>5, phase set is (1−𝑝)𝑥₁=5.6>5, see Figure 18, and the system has no 1-periodic solution.

Figure 9 

Figure 10 

Figure 11 

Figure 12 

Figure 13 

Figure 14 

Figure 15 

Figure 16 

Figure 17 

Figure 18 

From the numerical analyses, we know that it is better to use comprehensive control including chemistry control and biological technique according to different values of economic threshold to pest. When the pulse set is below or equal to the number of the pests of the system at the equilibrium state without pulse, the system has a 1-periodic solution, which is consistent with the proof of the theorem. When the pulse set is more than the number of the pests of the system at the equilibrium state without pulse and the phase set is less than the number of the pests of the system at the equilibrium state without pulse, the system has a 1-periodic solution, which is consistent with the proof of the theorem. When the pulse set is more than the number of the pests of the system at the equilibrium state without pulse and the phase set is more than the number of the pests of the system at the equilibrium state without pulse, the 1-periodic solution of system may not necessarily exist; we must consider different kinds of the number of the natural enemies we released; then the 1-periodic solution exists and has different periods, which is consistent with the proof of the theorem. So we take different release strategies according to different growth periods of the crop. In order to decide how to control the number of the natural enemies we released, the control strategy with impulsive state needs observing and recording the number of the pests and the natural enemies. In theory, we can predict the cycle time without repeated measurements, which can save a lot of manpower and material resources. The model in this paper is closer to the reality than the model that there is no density dependence for the continuous process of pulse points; it is also closer to the reality than the model that did not consider the influence of natural enemies of spraying insecticide.

Acknowledgments

This work is supported by Natural Science Foundation of Fujian Education Department (JB09078), Minnan Science and Technology Institute and the Young Core Instructor (mkq201006).