Multi-State Dependent Impulsive Control for Pest Management

Abstract

According to the integrated pest management strategies, we propose a model for pest control which adopts different control methods at different thresholds. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution of such system, and further, the attractiveness of the order one periodic solution by sequence convergence rules and qualitative analysis. Numerical simulations are carried out to illustrate the feasibility of our main results. Our results show that our method used in this paper is more efficient and easier than the existing ones for proving the existence of order one periodic solution.

1. Introduction

It is of great value to study pest management method applied in agricultural production; entomologists and the whole society have been paying close attention to how to control pests effectively and to save manpower and material resources. In agricultural production, pesticides-spraying (chemical control) and release of natural enemies (biological control) are the ways commonly used for pest control. But if we implement chemical control as soon as pests appear, many problems are caused: the first is environmental pollution; the second is increase of costs including human and material resources and time; the third is killing natural enemies, such as parasitic wasp; the last is pests’ resistance to pesticides, which brings great negative effects instead of working as well as had been expected [1–3]. The second way, which controls pests with the help of the increasing natural enemies, can avoid problems caused by chemical control and gets more and more attention. So many scholars have been studying and discussing it [4–8]. Considering the effectiveness of the chemical control and nonpollution and limitations of the biological one, people have proposed the method of integrated pest management (IPM), which is a pest management system integrating all appropriate ways and technologies to control economic injury level (EIL) caused by pest populations in view of population dynamics and its relevant environment. In the process of practical application, people usually implement the following two schemes for the integrated pest management: one is to implement control at a fixed time to eradicate pests [9, 10]; the other is to implement measures only when the amount of pests reaches a critical level, which is to make the amount less than certain economic impairment level, not to wipe out pests [11–13]. Salazar conducted an experiment of broad bean being damaged by bean sprouts worm in 1976 and found “crops’ compensation to damage of pests”, that is, yields of crops which had been damaged a little by pests in the early growth are actually higher than those without damage. In other words, we do not want to wipe out pests but to control them to a certain economic injury level (EIL). So, the second is used most in the process of agricultural industry. Tang and Cheke [14] first proposed the “Volterra” model in the from of a state-dependent impulsive model: 𝑥^′(𝑡)(𝑡)(𝑡),𝑦=𝑥𝑎−𝑏𝑦_′(𝑡)(𝑡(𝑡,(𝑡)(𝑡),(𝑡)=𝑦)(−𝑑+𝑐𝑥))𝑥≠𝐸𝑇,Δ𝑥=−𝛼𝑥Δ𝑦=𝑞,𝑥=𝐸𝑇, (1.1) and they applied this model to pest management and proved existence and stability of periodic solution of first and second order. Then Tang and Cheke [14] also proposed bait-dependent digestive model with state pulse: 𝑥^′(𝑡)(𝑡)(𝑡),𝑦=𝑥𝑎−𝑏𝑦_′(𝑡)(𝑡)(𝑡)=𝑦𝜆𝑏𝑥(𝑡),1+𝑏ℎ𝑥−𝑑𝑥≠ℎ_max,(𝑡)(𝑡),(𝑡)Δ𝑥=−𝛼𝑥Δ𝑦=𝑞,𝑥=ℎ_max, (1.2) they had the existence of positive periodic solution and stability of orbit. Recently Jiang and Lu et al. [15–17] have proposed pest management model with state pulse and phase structure and several predator-prey models with state pulse and had the existence of semi-trivial periodic solution and positive periodic solution and stability of orbit.

It is worth mentioning that the vast majority of research on population dynamics system with state pulse considers single state pulse, which is to say, only when the amount of population reaches the same economic threshold can measures be taken (e.g., chemical control and biological control); but this single state-pulse control does not confirm to reality. In fact, we often need to use different control methods under different states in real life. For example, in the process of pest management, when the amount of pests is small, biological control is implemented; when the amount is large, combination control is applied. Tang et al. [18] have investigated and developed a mathematical model with hybrid impulsive model: 𝑥^′(𝑡)(𝑡(𝑡(𝑡)𝑦(𝑡)𝑦=𝑟𝑥)(1−𝛿𝑥))−𝑏𝑥,𝑥<𝐸𝑇,_′(𝑡)=𝑦(𝑡)(𝑐𝑥(𝑡)−𝑎),𝑡=𝜆_𝑚,𝑥𝑡⁺=1−𝑝₁𝑥(𝑡),𝑦𝑡⁺=1−𝑝₂𝑦(𝑡),𝑥(𝑡)𝑦𝜆=𝐸𝑇,_+𝑚=1+𝑝₃𝑦𝜆₊+𝑞,𝑡=𝜆_𝑚. (1.3) Motivated by Tang, on the basis of the above analysis, we set up the following predicator-prey system with different control methods in different thresholds: 𝑥^′(𝑡)(𝑡)(𝑡),𝑦=𝑥𝑎−𝑏𝑦_′(𝑡)(𝑡)(𝑡)=𝑦𝜆𝑏𝑥(𝑡),1+𝑏ℎ𝑥−𝑑𝑥≠ℎ₁,ℎ₂or𝑥=ℎ₁,𝑦>𝑦^∗,(𝑡)(𝑡)Δ𝑥=0,Δ𝑦=𝛿,𝑥=ℎ₁,𝑦⩽𝑦^∗,(𝑡)(𝑡),(𝑡)(𝑡)Δ𝑥=−𝛼𝑥Δ𝑦=−𝛽𝑦+𝑞,𝑥=ℎ₂, (1.4) where 𝑥(𝑡) and 𝑦(𝑡) represent, respectively, the prey and the predator population densities at time 𝑡; 𝑎, 𝑏, 𝜆, ℎ₁, ℎ₂ and 𝑑 are all positive constants and ℎ₁<ℎ₂; 𝑦^∗=𝑎/𝑏. 𝛼,𝛽∈(0,1) represent the fraction of pest and predator, respectively, which die due to the pesticide when the amount of prey reaches economic threshold ℎ₂ and 𝑞 is the release amount of predator. 𝜆𝑏𝑥(𝑡)/(1+𝑏ℎ𝑥(𝑡)) is the per capita functional response of the predator. When the amount of the prey reaches the threshold ℎ₁ at time 𝑡_ℎ1, controlling measures are taken (releasing natural enemies) and the amount of predator abruptly turns to 𝑦(𝑡_ℎ1)+𝛿. When the amount of the prey reaches the threshold ℎ₂ at time 𝑡_ℎ2, spraying pesticide, and releasing natural enemies and the amount of prey and predator abruptly turn to (1−𝛼)𝑥(𝑡_ℎ2) and (1−𝛽)𝑦(𝑡_ℎ2)+𝑞, respectively. Refer to [17] Liu et al. for details.

2. Preliminaries

First, we give some basic definitions and lemmas.

Definition 2.1. A triple (𝑋,Π,𝑅⁺) is said to be a semidynamical system if 𝑋 is a metric space, 𝑅⁺ is the set of all nonnegative real, and Π(𝑃,𝑡)∶𝑋×𝑅⁺→𝑋 is a continuous map such that:(i)Π(𝑃,0)=𝑃 for all 𝑃∈𝑋;(ii)Π(𝑃,𝑡) is continuous for 𝑡 and 𝑠;(iii)Π(Π(𝑃,𝑡))=Π(𝑃,𝑡+𝑠) for all 𝑃∈𝑋 and 𝑡,𝑠∈𝑅⁺. Sometimes a semi-dynamical system (𝑋,Π,𝑅⁺) is denoted by (𝑋,Π).

Definition 2.2. Assuming that(i)(𝑋,Π) is a semi-dynamical system;(ii)𝑀 is a nonempty subset of 𝑋;(iii)function 𝐼∶𝑀→𝑋 is continuous and for any 𝑃∈𝑀, there exists a 𝜀>0 such that for any 0<|𝑡|<𝜀, Π(𝑃,𝑡)∉𝑀.
Then, (𝑋,Π,𝑀,𝐼) is called an impulsive semi-dynamical system.
For any 𝑃, the function Π_𝑃∶𝑅⁺→𝑋 defined as Π_𝑃(𝑡)=Π(𝑃,𝑡) is continuous, and we call Π_𝑃(𝑡) the trajectory passing through point 𝑃. The set 𝐶⁺(𝑃)={Π(𝑃,𝑡)/0≤𝑡<+∞} is called positive semitrajectory of point 𝑃. The set 𝐶⁻(𝑃)={Π(𝑃,𝑡)/−∞<𝑡≤0} is called the negative semi-trajectory of point 𝑃.

Definition 2.3. One considers state-dependent impulsive differential equations: 𝑥^′(𝑡),𝑦=𝑃𝑥,𝑦_′(𝑡),,(𝑡),(𝑡),,=𝑄𝑥,𝑦𝑥,𝑦∉𝑀𝑥,𝑦Δ𝑥=𝛼𝑥,𝑦Δ𝑦=𝛽𝑥,𝑦𝑥,𝑦∈𝑀𝑥,𝑦 (2.1) where 𝑀(𝑥,𝑦) and 𝑁(𝑥,𝑦) represent the straight line or curve line on the plane, 𝑀(𝑥,𝑦) is called impulsive set. The function 𝐼 is continuous mapping, 𝐼(𝑀)=𝑁, 𝐼 is called the impulse function. 𝑁(𝑥,𝑦) is called the phase set. We define “dynamic system” constituted by the definition of solution of state impulsive differential equation (2.1) as “semicontinuous dynamic systems”, which is denoted as (Ω,𝑓,𝐼,𝑀).

Definition 2.4. Suppose that the impulse set 𝑀 and the phase set 𝑁 are both lines, as shown in Figure 1. Define the coordinate in the phase set 𝑁 as follows: denote the point of intersection 𝑄 between 𝑁 and 𝑥-axis as 𝑂, then the coordinate of any point 𝐴 in 𝑁 is defined as the distance between 𝐴 and 𝑄 and is denoted by 𝑦_𝐴. Let 𝐶 denote the point of intersection between the trajectory starting from 𝐴 and the impulse set 𝑀, and let 𝐵 denote the phase point of 𝐶 after impulse with coordinate 𝑦_𝐵. Then, we define 𝐵 as the successor point of 𝐴, and then the successor function of point 𝐴 is that 𝑓(𝐴)=𝑦_𝐵−𝑦_𝐴.

Figure 1 

Successor function defined.

Definition 2.5. A trajectory Π(𝑃₀,𝑡) is called order one periodic solution with period 𝑇 if there exists a point 𝑃₀∈𝑁 and 𝑇>0 such that 𝑃=Π(𝑃₀,𝑡)∈𝑀 and 𝑃⁺=𝐼(𝑃)=𝑃₀.

We get these lemmas from the continuity of composite function and the property of continuous function.

Lemma 2.6. Successor function defined in Definition 2.1 is continuous.

Lemma 2.7. In system (1.4), if there exist 𝐴∈𝑁, 𝐵∈𝑁 satisfying successor function 𝑓(𝐴)𝑓(𝐵)<0, then there must exist a point 𝑃 (𝑃∈𝑁) satisfying 𝑓(𝑃)=0 the function between the point of 𝐴 and the point of 𝐵, thus there is an order one periodic solution in system (1.4).
Next, we consider the model (1.4) without impulse effects: 𝑥^′(𝑡)(𝑡)(𝑡),𝑦=𝑥𝑎−𝑏𝑦_′(𝑡)(𝑡)(𝑡)=𝑦𝜆𝑏𝑥(𝑡).1+𝑏ℎ𝑥−𝑑 (2.2) It is well known that the system (2.2) possesses(I)two steady states 𝑂(0,0)-saddle point, and 𝑅(𝑑/𝑏(𝜆−𝑑ℎ),𝑎/𝑏)=𝑅(𝑥^∗,𝑦^∗)(𝜆>𝑑ℎ)-stable centre;(II)a unique closed trajectory through any point in the first quadrant contained inside the point 𝑅.

In this paper, we assume that the condition 𝜆>𝑑ℎ holds. By the biological background of system (1.4), we only consider 𝐷={(𝑥,𝑦)∶𝑥≥0,𝑦≥0}. Vector graph of system (2.2) can be seen in Figure 2.

Figure 2 

Illustration of vector graph of system (2.2).

This paper is organized as follows. In the next section, we present some basic definitions and an important lemmas as preliminaries. In Section 3, we prove existence for an order one periodic solution of system (1.4). The sufficient conditions for the attractiveness of order one periodic solutions of system (1.4) are obtained in Section 4. At last, we state conclusion, and the main results are carried out to illustrate the feasibility by numerical simulations.

3. Existence of the Periodic Solution

In this section, we will investigate the existence of an order one periodic solution of system (1.4) by using the successor function defined in this paper and qualitative analysis. For this goal, we denote that 𝑀₁={(𝑥,𝑦)/𝑥=ℎ₁,0≤𝑦≤𝑎/𝑏}, and that 𝑀₂={(𝑥,𝑦)∣𝑥=ℎ₂,𝑦≥0}. Phase set of set 𝑀 is that 𝑁₁=𝐼(𝑀₁)={(𝑥,𝑦)∣𝑥=ℎ₁,𝑎/𝑏<𝑦≤(𝑎/𝑏)+𝛿} and that 𝑁₂=𝐼(𝑀₂)={(𝑥,𝑦)∣𝑥=(1−𝛼)ℎ₂,𝑦≥𝑞}. Isoclinic line is denoted, respectively, by lines: 𝐿₁={(𝑥,𝑦)∣𝑦=𝑎/𝑏,𝑥≥0} and 𝐿₂={(𝑥,𝑦)∣𝑥=𝑑/𝑏(𝜆−𝑑ℎ),𝑦≥0}.

For the convenience, if 𝑃∈Ω−𝑀, 𝐹(𝑃) is defined as the first point of intersection of 𝐶⁺(𝑃) and 𝑀, that is, there exists a 𝑡₁∈𝑅₊ such that 𝐹(𝑃)=Π(𝑃,𝑡₁)∈𝑀, and for 0<𝑡<𝑡₁,Π(𝑃,𝑡)∉𝑀; if 𝐵∈𝑁,𝑅(𝐵) is defined as the first point of intersection of 𝐶⁻(𝑃) and 𝑁, that is, there exists a 𝑡₂∈𝑅₊ such that 𝑅(𝐵)=Π(𝐵,−𝑡₂)∈𝑁, and for −𝑡<𝑡<0,Π(𝐵,𝑡)∉𝑁. For any point 𝑃, we denote 𝑦_𝑃 as its ordinate. If the point 𝑃(ℎ,𝑦_𝑃)∈𝑀, then pulse occurs at the point 𝑃, the impulsive function transfers the point 𝑃 into 𝑃⁺∈𝑁. Without loss of generality, unless otherwise specified we assume the initial point of the trajectory lies in phase set 𝑁.

Due to the practical significance, in this paper we assume the set always lies in the left side of stable centre 𝑅, that is, ℎ₁<𝑑/𝑏(𝜆−𝑑ℎ) and (1−𝛼)ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ).

In the light of the different position of the set 𝑁₁ and the set 𝑁₂, we consider the following three cases.

Case 1 (0<ℎ₁<𝑑/𝑏(𝜆−𝑑ℎ)). In this case, set 𝑀₁ and 𝑁₁ are both in the left side of stable center 𝑅 (as shown in Figure 3). Take a point 𝐵₁(ℎ₁,(𝑎/𝑏)+𝜀)∈𝑁₁ above 𝐴, where 𝜀>0 is small enough, then there must exist a trajectory passing through 𝐵₁ which intersects with 𝑀₁ at point 𝑃₁(ℎ₁,𝑦_𝑝1), we have 𝑦_𝑝1<𝑎/𝑏. Since 𝑝₁∈𝑀₁, pulse occurs at the point 𝑃₁, the impulsive function transfers the point 𝑃₁ into 𝑃⁺₁(ℎ₁,𝑦_𝑝1+𝛿) and 𝑃⁺₁ must lie above 𝐵₁, therefore inequation (𝑎/𝑏)+𝜀<𝑦_𝑝1+𝛿 holds, thus the successor function of 𝐵₁ is 𝑓(𝐵₁)=𝑦_𝑝1+𝛿−((𝑎/𝑏)+𝜀)>0.
On the other hand, the trajectory with the initial point 𝑃⁺₁ intersects 𝑀₁ at point 𝑃₂(ℎ₁,𝑦_𝑝2), in view of vector field and disjointness of any two trajectories, we know 𝑦_𝑝2<𝑦_𝑝1<𝑎/𝑏. Supposing the point 𝑃₂ is subject to impulsive effects to point 𝑃⁺₂(ℎ₁,𝑦_𝑝+2), where 𝑦_𝑝+2=𝑦_𝑝2+𝛿, the position of 𝑃⁺₂ has the following two cases.
Subcase 1.1 (𝑎/𝑏<𝑦_𝑝2+𝛿<𝑦_𝑝1+𝛿). In this case, the point 𝑃⁺₂ lies above the point 𝐴 and below 𝑃⁺₁, then we have 𝑓(𝑃⁺₁)=𝑦_𝑝2+𝛿−(𝑦_𝑝1+𝛿)<0.
By Lemma 2.7, there exists an order one periodic solution of system (1.4), whose initial point is between 𝐵₁ and 𝑃⁺₁ in set 𝑁₁.
Subcase 1.2 (𝑎/𝑏≥𝑦_𝑝2+𝛿 (as shown in Figure 4)). The point 𝑃⁺₂ lies below the point 𝐴, that is, 𝑃⁺₂∈𝑀₁, then pulse occurs at the point 𝑃⁺₂, the impulsive function transfers the point 𝑃⁺₂ into 𝑃₂⁺⁺(ℎ₁,𝑦_𝑝2+2𝛿). If 𝑎/𝑏<𝑦_𝑝2+2𝛿<𝑦_𝑝1+𝛿, like the analysis of Subcase 1.1, there exists an order one periodic solution of system (1.4). If 𝑎/𝑏>𝑦_𝑝2+2𝛿, that is, 𝑃₂⁺⁺∈𝑀₁, then we repent the above process until there exists 𝑘∈𝑍₊ such that 𝑃₂⁺⁺ jumps to 𝑃₂^𝑖+((ℎ₁,𝑦_𝑝2+(𝑘+2)𝛿) after 𝑘 times’ impulsive effects which satisfies 𝑎/𝑏<𝑦_𝑝2+(𝑘+2)𝛿<𝑦_𝑝1+𝛿. Like the analysis of Subcase 1.1, there exists an order one periodic solution of system (1.4).
Now, we can summarize the above results as the following theorem.

Figure 3 

0<ℎ1<(𝑑/𝑏(1−𝑑ℎ))(𝑎/𝑏)<𝑦𝑃2+𝛿+<𝑦𝑃1+𝛿.

Figure 4 

0<ℎ1<(𝑑/𝑏(1−𝑑ℎ))(𝑎/𝑏)>𝑦𝑃2+𝛿.

Theorem 3.1. If 𝜆>𝑑ℎ,0<ℎ₁<𝑑/𝑏(𝜆−𝑑ℎ), then there exists an order one periodic solution of the system (1.4).

Remark 3.2. It shows from the proved process of Theorem 3.1 that the number of natural enemies should be selected appropriately, which aims to reduce releasing impulsive times to save manpower and resources.

Case 2 (ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ)). In this case, set 𝑀₂ and 𝑁₂ are both in the left side of stable center 𝑅, in the light of the different position of the set 𝑁₂, we consider the following two cases.
Subcase 2.1 (0<ℎ₁<(1−𝛼)ℎ₂<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ)). In this case, the set 𝑁₂ is in the right side of 𝑀₁ (as shown in Figure 5). The trajectory passing through point 𝐴 which tangents to 𝑁₂ at point 𝐴 intersects with 𝑀₂ at point 𝑃₀(ℎ₂,𝑦_𝑝0). Since the point 𝑃₀∈𝑀₂, then impulse occurs at point 𝑃₀. Supposing the point 𝑃₀ is subject to impulsive effects to point 𝑃⁺₀((1−𝛼)ℎ₂,𝑦_𝑃+0), where 𝑦_𝑃+0=(1−𝛽)𝑦_𝑃0+𝑞, the position of 𝑃⁺₀ has the following three cases:
(1) ((1−𝛽)𝑦_𝑃0+𝑞>𝑎/𝑏). Take a point 𝐵₁((1−𝛼)ℎ₂,𝜀+𝑎/𝑏)∈𝑁₂ above 𝐴, where 𝜀>0 is small enough. Then there must exist a trajectory passing through the point 𝐵₁ which intersects with the set 𝑀₂ at point 𝑃₁(ℎ₂,𝑦_𝑃1). In view of continuous dependence of the solution on initial value and time, we know 𝑦_𝑃1<𝑦_𝑃0 and the point 𝑃₁ is close to 𝑃₀ enough, so we have the point 𝑃⁺₁ is close to 𝑃⁺₀ enough and 𝑦_𝑃+1<𝑦_𝑃+0, then we obtain 𝑓(𝐵₁)=𝑦_𝑃+1−𝑦_𝐵1>0.
On the other hand, the trajectory passing through point 𝐵 tangents to 𝑁₁ at point 𝐵. Set 𝐹(𝑆)=𝑃₂(ℎ₂,𝑦_𝑃2)∈𝑀₂. Denote the coordinates of impulsive point 𝑃⁺₂((1−𝛼)ℎ₂,𝑦_𝑃+2) corresponding to the point 𝑃₂(ℎ₂,𝑦_𝑃2). If 𝑦_𝑆≥𝑦_𝑃+0 then 𝑦_𝑃+2<𝑦_𝑃+0<𝑦_𝑆. So we obtain 𝑓(𝑆)=𝑦_𝑃+2−𝑦_𝑆<0. There exists an order one periodic solution of system (1.4), whose initial point is between the point 𝐵₁ and the point 𝑆 in set 𝑁₂ (Figure 5). If 𝑦_𝑆<𝑦_𝑃+0 and 𝑦_𝑃+2≤𝑦_𝑆, we have 𝑓(𝑆)=𝑦_𝑃+2−𝑦_𝑆≤0, we conclude that there exists an order one periodic solution of system (1.4). If 𝑦_𝑆<𝑦_𝑃+0 and 𝑦_𝑃+2>𝑦_𝑆, from the vector field of system (1.4), we know the trajectory of system (1.4) with any initiating point on the 𝑁₂ will ultimately stay in Ω₁={(𝑥,𝑦)/0≤𝑥≤ℎ₁,𝑦≥0} after one impulsive effect. Therefore there is no an order one periodic solution of system (1.4).
(2) ((1−𝛽)𝑦_𝑃0+𝑞<𝑎/𝑏 (as shown in Figure 6)). In this case, the point 𝑃⁺₀ lies below the point 𝐴, that is, (1−𝛽)𝑦_𝑃0+𝑞<𝑎/𝑏, thus the successor function of the point 𝐴 is 𝑓(𝐴)=(1−𝛽)𝑦_𝑃0+𝑞−𝑎/𝑏<0.
Take another point 𝐵₁((1−𝛼)ℎ₂,𝜀)∈𝑁₂, where 𝜀>0 is small enough. Then there must exist a trajectory passing through the point 𝐵₁ which intersects 𝑀₂ at point 𝑃₁(ℎ₂,𝑦_𝑃1)∈𝑀₂. Suppose the point 𝑃₁(ℎ₂,𝑦_𝑃1) is subject to impulsive effects to point 𝑃⁺₁((1−𝛼)ℎ₂,𝑦_𝑃+1), then we have 𝑦_𝑃+1>𝜀. So we have 𝑓(𝐵₁)=𝑦_𝑃+1−𝜀>0.
From Lemma 2.7, there exists an order one periodic solution of system (1.4), whose initial point is between 𝐵₁ and 𝐴 in set 𝑁₂.
(3) ((1−𝛽)𝑦_𝑃0+𝑞=𝑎/𝑏). 𝑃⁺₀ coincides with 𝐴, and the successor function of 𝐴 is that 𝑓(𝐴)=0, so there exists an order one periodic solution of system (1.4) which is just a part of the trajectory passing through the 𝐴.

Figure 5 

ℎ2<𝑑/𝑏(1−𝑑ℎ), ℎ1<(1−𝛼)ℎ2<ℎ2.

Figure 6 

ℎ2<𝑑/𝑏(1−𝑑ℎ), (1−𝛽)𝑦𝑃0+𝑞<𝑎/𝑏.

Now, we can summarize the above results as the following theorem.

Theorem 3.3. Assuming that 𝜆>𝑑ℎ and 0<ℎ₁<(1−𝛼)ℎ₂<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ). If 𝑦_𝑃+0≤𝑦_𝐴, there exists an order one periodic solutions of the system (1.4). If 𝑦_𝑃+0>𝑦_𝐴, if 𝑦_𝑆≥𝑦_𝑃+0 or 𝑦_𝑆<𝑦_𝑃+0, and 𝑦_𝑆>𝑦_𝑃+2, there exists an order one periodic solutions of the system (1.4). If 𝑦_𝑃+0>𝑦_𝐴, 𝑦_𝑆<𝑦_𝑃+0 and 𝑦_𝑆<𝑦_𝑃+2, there is no an order one periodic solutions of the system (1.4). The trajectory of system (1.4) with any initiating point on the 𝑁₂ will ultimately stay in Ω₁={(𝑥,𝑦)/0≤𝑥≤ℎ₁,𝑦≥0} after one impulsive effect.

Subcase 2.2 (0<(1−𝛼)ℎ₂<ℎ₁<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ)). In this case, the set 𝑁₂ is on the left side of 𝑁₁. Any trajectory from initial point (𝑥⁺₀,𝑦⁺₀)∈𝑁₂ will intersect with 𝑀₁ at some point with time increasing. By the analysis of Case 1, the trajectory from initial point (𝑥⁺₀,𝑦⁺₀)∈𝑁₂ on the set 𝑁₂ will stay in the region Ω₁={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₁}. Similarly, any trajectory from initial point (𝑥⁺₀,𝑦⁺₀)∈Ω₀={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₂} will stay in the region Ω₁={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₁} after one impulsive effect or free from impulsive effect.

Theorem 3.4. If 𝜆>𝑑ℎ, 0<(1−𝛼)ℎ₂<ℎ₁<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ), there is no an order one periodic solutions to the system (1.4), and the trajectory with initial point (𝑥⁺₀,𝑦⁺₀)∈Ω₀={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₂} will stay in the region Ω₁={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₁}.

Case 3 (𝑑/𝑏(𝜆−𝑑ℎ)<ℎ₂). In this case, the set 𝑀₂ is on the right side of stable center 𝑅. In the light of the different position of 𝑁₂, we consider the following two subcases.
Subcase 3.1 (ℎ₁<(1−𝛼)ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ)<ℎ₂). In this case, the set 𝑀₂ is in the right side of 𝑅. Then there exists a unique closed trajectory Γ₁ of system (1.4) which contains the point 𝑅 and is tangent to 𝑀₂ at the point 𝐴.
Since Γ₁ is closed trajectory, we take their the minimal value 𝛿_min of abscissas at the trajectory Γ₁, namely, 𝛿_min≤𝑥 holds for any abscissas of Γ₁.
(1) (ℎ₁<(1−𝛼)ℎ₂<𝛿_min<𝑑/𝑏(𝜆−𝑑ℎ)<ℎ₂). In this case, there is a trajectory, which contains the point 𝑅(𝑑/𝑏(𝜆−𝑑ℎ),𝑎/𝑏) and is tangent to the 𝑁₂ at the point 𝐵 intersects 𝑀₂ at a point 𝑃₁(ℎ₂,𝑦_𝑃1)∈𝑀₂. Suppose point 𝑃₁ is subject to impulsive effects to point 𝑃⁺₁((1−𝛼)ℎ₂,𝑦_𝑃+1), here 𝑦_𝑃+1=(1−𝛽)𝑦_𝑃1+𝑞. The position of 𝑃⁺₁ has the following three sub-cases. If (1−𝛽)𝑦_𝑃1+𝑞<𝑎/𝑏 (Figure 7), the point 𝑃⁺₁ lies below the point 𝐵. Like the analysis of Subcase 2.1(2), we can prove there exists an order one periodic solution to the system (1.4) in this case. If (1−𝛽)𝑦_𝑃1+𝑞>𝑎/𝑏, the point 𝑃⁺₁ lies above the point 𝐵; the trajectory from initiating point 𝑃⁺₁ intersects with the line 𝐿₁ at point 𝐶. If ℎ₁≤𝑦_𝐶 (Figure 8), we have 𝑦_𝑃1>𝑦_𝑃2 and 𝑦_𝑃+1>𝑦_𝑃+2, then the successor function of 𝑃⁺₁ is that 𝑓(𝑃⁺₁)=𝑦_𝑃+2−𝑦_𝑃+1<0. Then, we know that there exists an order one periodic solution of system (1.4), whose initial point is between the point 𝑃⁺₁ and 𝐵 in set 𝑁₂. If ℎ₁>𝑦_𝐶 (Figure 9), there is a trajectory which is tangent to the 𝑁₁ at a point 𝐷 intersects with 𝑀₂ at a point 𝑃₃(ℎ₂,𝑦_𝑃3)∈𝑀₂, 𝑃₃ jumps to 𝑃⁺₃ after the impulsive effects. If 𝑦_𝑃+3≤𝑦_𝐵1, we can easily know that there exists an order one periodic solution of system (1.4). If 𝑦_𝑃+3>𝑦_𝐵1, by the qualitative analysis of the system (1.4), we know that trajectory with any initiating point on the 𝑁₂ will ultimately stay in Γ₁ after a finite number of impulsive effects. If (1−𝛽)𝑦_𝑃1+𝑞=𝑎/𝑏, the point 𝑃⁺₁ coincides with the point 𝐵, and the successor function of the point 𝐵 is that 𝑓(𝐵)=0; then there exists an order one periodic solution which is just a part of the trajectory passing through the point 𝐵.

Figure 7 

ℎ1<(1−𝛼)ℎ2<𝛿min<𝑑/(𝜆−ℎ𝑑)(1−𝛽)𝑦𝑃1+𝑞<𝑎/𝑏.

Figure 8 

ℎ1<(1−𝛼)ℎ2<𝛿min<𝑑/(𝜆−ℎ𝑑)(1−𝛽)𝑦𝑃1+𝑞>𝑎/𝑏.

Figure 9 

ℎ1<(1−𝛼)ℎ2<𝛿min<𝑑/(𝜆−ℎ𝑑)<ℎ2(1−𝛽)𝑦𝑃1+𝑞>𝑎/𝑏, ℎ1>𝑦𝑐.

Now, we can summarize the above results as the following theorem.

Theorem 3.5. Assuming that 𝜆>𝑑ℎ and ℎ₁<(1−𝛼)ℎ₂<𝛿_min<𝑑/𝑏(𝜆−𝑑ℎ)<ℎ₂. If (1−𝛽)𝑦_𝑃1+𝑞≤𝑎/𝑏, there exists an order one periodic solution to the system (1.4). If (1−𝛽)𝑦_𝑃1+𝑞>𝑎/𝑏 and ℎ₁≤𝑦_𝐶, then there exists an order one periodic solution to the system (1.4). If (1−𝛽)𝑦_𝑃1+𝑞>𝑎/𝑏, ℎ₁>𝑦_𝐶 and 𝑦_𝑃+3≤𝑦_𝐵1, then there exists an order one periodic solution to the system (1.4).
(2)  (ℎ₁<𝛿_min<(1−𝛼)ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ)<ℎ₂). In this case, denote the closed trajectory Γ₁ of system (1.4) intersects with 𝑁₂ two points 𝐴₁((1−𝛼)ℎ₂,𝑦_𝐴1) and 𝐴₂((1−𝛼)ℎ₂,𝑦_𝐴2) (as shown in Figure 10). Since 𝐴∈𝑀₂, impulse occurs at the point 𝐴. Suppose point 𝐴 is subject to impulsive effects to point 𝑃⁺₀((1−𝛼)ℎ₂,𝑦_𝑃+0), here 𝑦_𝑃+0=(1−𝛽)(𝑎/𝑏)+𝑞. If (1−𝛽)(𝑎/𝑏)+𝑞=𝑦_𝐴1 or (1−𝛽)(𝑎/𝑏)+𝑞=𝑦_𝐴2, then 𝑃⁺₀ coincides with 𝐴₁ or 𝑃⁺₀ coincides with 𝐴₂, and the successor function of 𝐴₁ or 𝐴₂ is that 𝑓(𝐴₁)=0 or 𝑓(𝐴₂)=0. So, there exists an order one periodic solution of system (1.4) which is just a part of the trajectory Γ₁. If (1−𝛽)(𝑎/𝑏)+𝑞<𝑦_𝐴2, the point 𝑃⁺₀ lies below the point 𝐴₂. Like the analysis of Subcase 2.1(2), we can prove there exists an order one periodic solution to the system (1.4) in this case. If (1−𝛽)(𝑎/𝑏)+𝑞>𝑦_𝐴1 (as shown in Figure 11), the point 𝑃⁺₀ is above the point 𝐴₁. Like the analysis of Subcase 3.1(1), we obtain sufficient conditions of existence of order one periodic solution to the system (1.4).

Figure 10 

ℎ1<𝛿min<(1−𝛼)ℎ2<𝑑/(𝜆−ℎ𝑑)<ℎ2(1−𝛽)(𝑎/𝑏)+𝑞<𝑦𝐴2.

Figure 11 

ℎ1<𝛿min<(1−𝛼)ℎ2<𝑑/(𝜆−ℎ𝑑)<ℎ2.

Theorem 3.6. Assuming that 𝜆>𝑑ℎ,ℎ₁<(1−𝛼)ℎ₂<𝛿_min<𝑑/𝑏(𝜆−𝑑ℎ)<ℎ₂. If (1−𝛽)(𝑎/𝑏)+𝑞≤𝑦_𝐴2, there exists an order one periodic solution to the system (1.4). If (1−𝛽)(𝑎/𝑏)+𝑞>𝑦_𝐴1 and ℎ₁≤𝑦_𝐶, then there exists an order one periodic solution to the system (1.4). If (1−𝛽)(𝑎/𝑏)+𝑞>𝑦_𝐴1,(1−𝛽)𝑦_𝑃1+𝑞>𝑎/𝑏,ℎ₁>𝑦_𝐶, and 𝑦_𝑃+3≤𝑦_𝐵1, then there exists an order one periodic solution to the system (1.4).
(3)  (𝑦_𝐴2<(1−𝛽)(𝑎/𝑏)+𝑞<𝑦_𝐴1). In this case, we note that the point 𝑃⁺₀ must lie between the point 𝐴₁ and the point 𝐴₂ (As shown in Figure 12). Taking a point 𝐵₁∈𝑀₂ such that 𝐵₁ jumps to 𝐴₂ after the impulsive effect, denote 𝐴₂=𝐵⁺₁. Since 𝑦_𝑃+0>𝑦_𝐵+1, we have 𝑦_𝐴>𝑦_𝐵1. Let 𝑅(𝐵₁)=𝐵⁺₂∈𝑁₂, take a point 𝐵₂∈𝑀₂ such that 𝐵₂ jumps to 𝐵⁺₂ after the impulsive effects, then we have 𝑦_𝐵+1>𝑦_𝐵+2, 𝑦_𝐵1>𝑦_𝐵2. This process continues until there exists a 𝐵⁺_𝐾∈𝑁₂ (𝐾∈𝑍₊) satisfying 𝑦_𝐵+𝑘<𝑞. So we obtain a sequence {𝐵⁺_𝑘}_{𝑘=1,2,…,𝐾} of the set 𝑀₂ and a sequence {𝐵_𝑘}_{𝑘=1,2,…,𝐾} of set 𝑁₂ satisfying 𝑅(𝐵_𝑘−1)=𝐵⁺_𝑘∈𝑁₂,𝑦_{𝐵+𝑘−1}>𝑦_𝐵+𝑘. In the following, we will prove the trajectory of system (1.4) with any initiating point of set 𝑁₂ will ultimately stay in Γ₁.

Figure 12 

𝑦𝐴2<(1−𝛽)(𝑎/𝑏)+𝑞<𝑦𝐴1.

From the vector field of system (1.4), we know the trajectory of system (1.4) with any initiating point between the point 𝐴₁ and 𝐴₂ will be free from impulsive effect and ultimately will stay in Γ₁.

For any point below 𝐴₂, it must lie between 𝐵⁺_𝑘 and 𝐵⁺_𝑘−1, here 𝑘=2,3,…,𝐾+1 and 𝐴₂=𝐵⁺₁. After 𝑘 times’ impulsive effects, the trajectory with this initiating point will arrive at some point of the set 𝑁₂ which must be between 𝐴₁ and 𝐴₂, and then ultimately stay in Γ₁.

Denote the intersection of the trajectory passing through the point 𝐵 which tangents to 𝑁₁ at point 𝐵 with the set 𝑁₂ at 𝑆((1−𝛼)ℎ₂,𝑦_𝑆). With time increasing, the trajectory of system (1.4) from any initiating point on segment 𝐴₁𝑆 intersect with the set 𝑁₂ at some point which is below 𝐴₂; so just like the analysis above we obtain, it will ultimately stay in Γ₁. So for any point below 𝑆, will ultimately stay in region Γ₁ with time increasing.

Now, we can summarize the above results as the following theorem.

Theorem 3.7. Assuming that 𝜆>𝑑ℎ,ℎ₁<𝛿_min<(1−𝛼)ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ)<ℎ₂, and 𝑦_𝐴2<(1−𝛽)(𝑎/𝑏)+𝑞<𝑦_𝐴1, there is no periodic solution in system (1.4), and the trajectory with any initiating point on the set 𝑁₂ will stay in Γ₁ or in the region Ω₁={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₁}.

Subcase 3.2 (0<(1−𝛼)ℎ₂<ℎ₁<𝑑/𝑏(𝜆−𝑑ℎ)<ℎ₂). In this case, the set 𝑁₂ is on the left side of the set 𝑁₁ and 𝑀₂ in the right side of 𝑅. Like the analysis of Subcase 2.2, we can know that any trajectory with initial point (𝑥⁺₀,𝑦⁺₀)∈Ω₀={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₂} will stay in the region Ω₁={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₁} after one impulsive effect or free from impulsive effect.

4. Attractiveness of the Order One Periodic Solution

In this section, under the condition of existence of order one periodic solution to system (1.4) and the initial value of pest population 𝑥(0)≤ℎ₂, we discuss its attractiveness. We focus on Case 1 and Case 2; by similar method, we can obtain similar results about Case 3.

Theorem 4.1. Assuming that 𝜆>𝑑ℎ,ℎ₁<𝑑/(𝑏(𝜆−𝑑ℎ)) and 𝛿≥𝑎/𝑏.
If 𝑦_𝑃+0>𝑦_𝑃+2>𝑦_𝑃+ or 𝑦_𝑃+0<𝑦_𝑃+2<𝑦_𝑃+ (Figure 14), then(I)there exists a unique order one periodic solution of system (1.4),(II)if (1−𝛼)ℎ₂<ℎ₁, order one periodic solution of system (1.4) is attractive in the region Ω₀={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₂}.

Proof. By the derivation of Theorem 3.1, we know there exists an order one periodic solution of system (1.4). We assume trajectory 𝑃𝑃⁺ and segment 𝑃𝑃⁺ formulate an order one periodic solution of system (1.4), that is, there exists a 𝑃⁺∈𝑁₂ such that the successor function of 𝑃⁺ satisfies 𝑓(𝑃⁺)=0. First, we will prove the uniqueness of the order one periodic solution.
We take any two points 𝐶₁(ℎ₁,𝑦_𝐶1)∈𝑁₁,𝐶₂(ℎ₁,𝑦_𝐶2)∈𝑁₁ satisfying 𝑦_𝐶2>𝑦_𝐶1>𝑦_𝐴, then we obtain two trajectories whose initiate points are 𝐶₁ and 𝐶₂ intersects the set 𝑀₁ two points 𝐷₁(ℎ₁,𝑦_𝐷1) and 𝐷₂(ℎ₁,𝑦_𝐷2), respectively, (Figure 13). In view of the vector field of system (1.4) and the disjointness of any two trajectories without impulse, we know 𝑦_𝐷1>𝑦_𝐷2. Suppose the points 𝐷₁ and 𝐷₂ are subject to impulsive effect to points 𝐷⁺₁(ℎ₁,𝑦_𝐷+1) and 𝐷⁺₂(ℎ₂,𝑦_𝐷+2), respectively, then we have 𝑦_𝐷+1>𝑦_𝐷+2 and 𝑓(𝐶₁)=𝑦_𝐷+1−𝑦_𝐶1,𝑓(𝐶₂)=𝑦_𝐷+2−𝑦_𝐶2, so we get 𝑓(𝐶₁)−𝑓(𝐶₂)<0, thus we obtain the successor function 𝑓(𝑥) is decreasing monotonously of 𝑁₁, so there is a unique point 𝑃⁺∈𝑁₁ satisfying 𝑓(𝑃⁺)=0, and the trajectory 𝑃⁺𝑃𝑃⁺ is a unique order one periodic solution of system (1.4).
Next, we prove the attractiveness of the order one periodic solution 𝑃⁺𝑃𝑃⁺ in the region Ω₀={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₂}. We focus on the case 𝑦_𝑃+0>𝑦_𝑃+2>𝑦_𝑃+; by similar method, we can obtain similar results about case 𝑦_𝑃+0<𝑦_𝑃+2<𝑦_𝑃+ (Figure 14).
Take any point 𝑃⁺₀(ℎ₁,𝑦_𝑃+0)∈𝑁₁ above 𝑃⁺. Denote the first intersection point of the trajectory from initiating point 𝑃⁺₀(ℎ₁,𝑦_𝑃+0) with the set 𝑀₁ at 𝑃₁(ℎ₁,𝑦_𝑃1), and the corresponding consecutive points are 𝑃₂(ℎ₁,𝑦_𝑃2),𝑃₃(ℎ₁,𝑦_𝑃3),𝑃₄(ℎ₁,𝑦_𝑃4),…, respectively. Consequently, under the effect of impulsive function, the corresponding points after pulse are 𝑃⁺₁(ℎ₁,𝑦_𝑃+1),𝑃⁺₂(ℎ₁,𝑦_𝑃+2),𝑃⁺₃(ℎ₁,𝑦_𝑃+3),….
Due to conditions 𝑦_𝑃+0>𝑦_𝑃+2>𝑦_𝑃+, 𝑦_𝑃+𝑘=𝑦_𝑃𝑘+𝛿, 𝛿≥𝑎/𝑏 and disjointness of any two trajectories, then we get a sequence {𝑃⁺_𝑘}_𝑘=1,2,… of the set 𝑁₁ satisfying 𝑦_𝑃+1<𝑦_𝑃+3<⋯<𝑦_{𝑃+2𝑘−1}<𝑦_𝑃+2𝑘+1<⋯<𝑦_𝑃+<⋯<𝑦_𝑃+2𝑘<𝑦_{𝑃+2𝑘−2}<⋯<𝑦_𝑃+2<𝑦_𝑃+0. (4.1)
So the successor function 𝑓(𝑃⁺_2𝑘−1)=𝑦_𝑃+2𝑘−𝑦_{𝑃+2𝑘−1}>0 and 𝑓(𝑃⁺_2𝑘)=𝑦_𝑃+2𝑘+1−𝑦_𝑃+2𝑘<0 hold. Series {𝑦_{𝑃2𝑘−1}}_𝑘=1,2,… increases monotonously and has upper bound, so lim_𝑘→∞𝑦_{𝑃+2𝑘−1} exists. Next, we will prove lim_𝑘→∞𝑦_{𝑃+2𝑘−1}=𝑦_𝑃+. Set lim_𝑘→∞𝑃_2𝑘−1=𝐶⁺, we will prove 𝑃⁺=𝐶⁺. Otherwise 𝑃⁺≠𝐶⁺, then there is a trajectory passing through the point 𝐶⁺ which intersects the set 𝑀₁ at point 𝐶, then we have 𝑦_𝐶>𝑦_𝑃, 𝑦𝐶⁺>𝑦_𝑃+. Since 𝑓(𝐶⁺)≥0 and 𝑃⁺≠𝐶⁺, according to the uniqueness of the periodic solution, then we have 𝑓(𝐶⁺𝐶)=𝑦⁺−𝑦_𝐶+>0, thus 𝑦_𝐶+<𝑦_𝑃+𝐶<𝑦⁺ hold. Analogously, let trajectory passing through the point 𝐶⁺ which intersects the set 𝑀₁ at point 𝐶, and the corresponding consecutive points is 𝐶, then 𝑦_𝐶>𝑦𝐶>𝑦_𝑝>𝑦𝐶𝐶,𝑦⁺𝐶>𝑦⁺𝐶>𝑦⁺>𝑦_𝑝+>𝑦_𝐶+, then we have 𝑓(𝐶⁺𝐶)=𝑦⁺𝐶−𝑦⁺>0, this is, contradict to the fact that 𝐶⁺ is a limit of sequence {𝑃⁺_2𝑘−1}_𝑘=1,2,…, so we obtain 𝑃⁺=𝐶⁺. So, we obtain lim_𝑘→∞𝑦_{𝑃+2𝑘−1}=𝑦_𝑃+. Similarly, we can prove lim_𝑘→∞𝑦_𝑃+2𝑘=𝑦_𝑃+.
From above analysis, we know there exists a unique order one periodic solution in system (1.4), and the trajectory from initiating any point of the 𝑁₁ will ultimately tend to be order one periodic solution 𝑃⁺𝑃𝑃⁺.
Any trajectory from initial point (𝑥⁺₀,𝑦⁺₀)∈Ω₀={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥≤ℎ₂} will intersect with 𝑁₁ at some point with time increasing on the condition that (1−𝛼)ℎ₂<ℎ₁<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ); therefore the trajectory from initial point on 𝑁₁ ultimately tends to be order one periodic solution 𝑃⁺𝑃𝑃⁺. Therefore, order one periodic solution 𝑃⁺𝑃𝑃⁺ is attractive in the region Ω₀. This completes the proof.

Figure 13 

There is a unique order one periodic solution (Theorem 4.1).

Figure 14 

Order one periodic solution is attractive (Theorem 4.1).

Remark 4.2. Assuming that 𝜆>𝑑ℎ,ℎ₁<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ) and 𝛿≥𝑎/𝑏, if 𝑦_𝑃+<𝑦_𝑃+0<𝑦_𝑃+2 or 𝑦_𝑃+>𝑦_𝑃+0>𝑦_𝑃+2 then the order one periodic solution is unattractive.

Theorem 4.3. Assuming that 𝜆>𝑑ℎ,ℎ₁<(1−𝛼)ℎ₂<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ) and 𝑦_𝑃+0<𝑦_𝐴 (as shown in Figure 15), then(I)There exists an odd number of order one periodic solutions of system (1.4) with initial value between 𝐶⁺₁ and 𝐴 in set 𝑁₂.(II)If the periodic solution is unique, then the periodic solution is attractive in region Ω₂, here Ω₂ is open region which is constituted by trajectory 𝐺𝐵, segment 𝐵𝐻, segment 𝐻𝐸, and segment 𝐸𝐺.

Figure 15 

Attractneness of order one periodic solution (Theorem 4.3).

Proof. (I) According to the Subcase 2.1(2), 𝑓(𝐴)<0 and 𝑓(𝐶⁺₁)>0, and the continuous successor function 𝑓(𝑥), there exists an odd number of root satisfying 𝑓(𝑥)=0, then we can get there exists an odd number of order one periodic solutions of system (1.4) with initial value between 𝐶⁺₁ and 𝐴 in set 𝑁₂.
(II) By the derivation of Theorem 3.3, we know there exists an order one periodic solution of system (1.4) whose initial point is between 𝐶⁺₁ and 𝑃⁺₀ in the set 𝑁₂. Assume trajectory 𝑃⁺𝑃 and segment 𝑃𝑃⁺ formulate the unique order one periodic solution of system (1.4) with initial point 𝑃⁺∈𝑁₂.
On the one hand, take a point 𝐶⁺₁((1−𝛼)ℎ₂,𝑦_𝐶+1)∈𝑁₂ satisfying 𝑦_𝐶+1=𝜀<𝑞 and 𝑦_𝐶+1<𝑦_𝑃+. The trajectory passing through the point 𝐶⁺₁((1−𝛼)ℎ₂,𝜀) which intersects with the set 𝑀₂ at point 𝐶₂(ℎ₂,𝑦_𝐶2), that is, 𝐹(𝐶⁺₁)=𝐶₂∈𝑀₂, then we have 𝑦_𝐶2<𝑦_𝑃, thus 𝑦_𝐶+2<𝑦_𝑃+, since 𝑦_𝐶+2=(1−𝛽)𝑦_𝐶2+𝑞>𝜀. So, we obtain 𝑓(𝐶⁺₁)=𝑦_𝐶+2−𝑦_𝐶+1=𝑦_𝐶+2−𝜀>0; Set 𝐹(𝐶⁺₂)=𝐶₃∈𝑀₂, because 𝑦_𝐶+1<𝑦_𝐶+2<𝑦_𝑃+, we know 𝑦_𝐶2<𝑦_𝐶3<𝑦_𝑃, then we have 𝑦_𝐶+2<𝑦_𝐶+3<𝑦_𝑃+ and 𝑓(𝐶⁺₂)=𝑦_𝐶+3−𝑦_𝐶+2>0. This process is continuing, then we get a sequence {𝐶⁺_𝑘}_𝑘=1,2,… of the set 𝑁₂ satisfying 𝑦_𝐶+1<𝑦_𝐶+2<⋯<𝑦_𝐶+𝑘<⋯<𝑦_𝑃+ (4.2) and 𝑓(𝐶⁺_𝑘)=𝑦_𝐶+𝑘+1−𝑦_𝐶+𝑘>0. Series {𝑦_𝐶+𝑘}_𝑘=1,2,… increase monotonously and have upper bound, so lim_𝑘→∞𝑦_𝐶+𝑘 exists. Like the proof of Theorem 4.1, we can prove lim_𝑘→∞𝑦_𝐶+𝑘=𝑦_𝑃+.
On the other hand, set 𝐹(𝑃⁺₀)=𝐷₁∈𝑀₂, then 𝐷₁ jumps to 𝐷⁺₁∈𝑁₂ under the impulsive effects. Since 𝑦_𝑃+<𝑦_𝑃+0<𝑦_𝐴, we have 𝑦_𝑃<𝑦_𝐷1<𝑦_𝑃0, thus we obtain 𝑦_𝑃+<𝑦_𝐷+1<𝑦_𝑃+0,𝑓(𝑃⁺₀)=𝑦_𝐷+1−𝑦_𝑃+0<0. Set 𝐹(𝐷⁺₁)=𝐷₂∈𝑀₂, then 𝐷₂ jumps to 𝐷⁺₂∈𝑁₂ under the impulsive effects. We have 𝑦_𝑃+<𝑦_𝐷+2<𝑦_𝐷+1; this process is continuing, we can obtain a sequence {𝐷⁺_𝑘}_{𝑘=1,2,…,} of the set 𝑁₂ satisfying 𝑦_𝑃+0>𝑦_𝐷+1>𝑦_𝐷+2>⋯>𝑦_𝐷+𝑘>⋯>𝑦_𝑃+ (4.3) and 𝑓(𝐷⁺_𝑘)=𝑦_𝐷+𝑘+1−𝑦_𝐷+𝑘<0. Series {𝑦_𝐷+𝑘}_𝑘=1,2,… decreases monotonously and has lower bound, so lim_𝑘→∞𝑦_𝐷+𝑘 exists. Similarly, we can prove lim_𝑘→∞𝑦_𝐷+𝑘=𝑦_𝑃+.
Any point 𝑄∈𝑁₂ below 𝐴 must be in some interval [𝑦_𝐷+𝑘+1,𝑦_𝐷+𝑘)_𝑘=1,2,…, [𝑦_𝐷+1,𝑦_𝑃+0), [𝑦_𝑃+0,𝑦_𝐴), [𝑦_𝐶+𝑘,𝑦_𝐶+𝑘+1)_𝑘=1,2,…. Without loss of generality, we assume the point 𝑄∈[𝑦_𝐷+𝑘+1,𝑦_𝐷+𝑘). The trajectory with initiating point 𝑄 moves between trajectory 𝐷⁺_𝑘𝐷_𝑘+1 and 𝐷⁺_𝑘+1𝐷_𝑘+2 and intersects with 𝑀₂ at some point between 𝐷_𝑘+2 and 𝐷_𝑘+1; under the impulsive effects, it jumps to the point of 𝑁₂ which is between [𝑦_𝐷+𝑘+2,𝑦_𝐷+𝑘+1), then trajectory Π(𝑄,𝑡) continues to move between trajectory 𝐷⁺_𝑘+1𝐷_𝑘+2 and 𝐷⁺_𝑘+2𝐷_𝑘+3. This process can be continued unlimitedly. Since lim_𝑘→∞𝑦_𝐷+𝑘=𝑦_𝑃+, the intersection sequence of trajectory Π(𝑄,𝑡), and the set 𝑁₂ will ultimately tend to the point 𝑃⁺. Similarly, if 𝑄∈[𝑦_𝐶+𝑘,𝑦_𝐶+𝑘+1), we also can get the intersection sequence of trajectory Π(𝑄,𝑡) and the set 𝑁₂ will ultimately tend to point 𝑃⁺. Thus, the trajectory initiating any point below 𝐴 ultimately tend to the unique order one periodic solution 𝑃⁺𝑃𝑃⁺.
Denote the intersection of the trajectory passing through the point 𝐵 which tangents to 𝑁₁ at the point 𝐵 with the set 𝑁₂ at a point 𝑆((1−𝛼)ℎ₂,𝑦_𝑆). The trajectory from any initiating point on segment 𝐴𝑆 will intersect with the set 𝑁₂ at some point below 𝐴 with time increasing. So like the analysis above, we obtain the trajectory from any initiating point on segment 𝐴𝑆 will ultimately tend to be the unique order one periodic solution 𝑃⁺𝑃𝑃⁺.
Since the trajectory with any initiating point of the Ω₂ will certainly intersect with the set 𝑁₂, then from the above analysis, we know the trajectory with any initiating point on segment 𝐴𝑆 will ultimately tend to be order one periodic solution 𝑃⁺𝑃𝑃⁺. Therefore, the unique order one periodic solution 𝑃⁺𝑃𝑃⁺ is attractive in the region Ω₂. This completes the proof.

Remark 4.4. Assuming that 𝜆>𝑑ℎ, ℎ₁<(1−𝛼)ℎ₂<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ), and 𝑦_𝐶+1<𝑦_𝐴<𝑦_𝑃+0, then the order one periodic solution with initial point between 𝐴 and 𝑃⁺₀ is unattractive.

Theorem 4.5. Assuming that 𝜆>𝑑ℎ, ℎ₁<(1−𝛼)ℎ₂<ℎ₂<𝑑/𝑏(𝜆−𝑑ℎ), 𝑦_𝑃+0>𝑦_𝑃+1>𝑦_𝐴 (Figure 16) then, there exists a unique order one periodic solution of system (1.4) which is attractive in the region Ω₂, here Ω₂ is open region which enclosed by trajectory 𝐺𝐵, segment 𝐵𝐻, segment 𝐻𝐸 and segment 𝐸𝐺.

Figure 16 

Attractneness of order one periodic solution (Theorem 4.5).

Proof. By the derivation of Theorem 3.3, we know there exists an order one periodic solution of system (1.4). We assume trajectory 𝑃⁺𝑃 and segment 𝑃𝑃⁺ formulate an order one periodic solution of system (1.4), that is, 𝑃⁺∈𝑁₂ is its initial point satisfying 𝑓(𝑃⁺)=0. Like the proof of Theorem 4.1, we can prove the uniqueness of the order one periodic solution of system (1.4).
Next, we prove the attractiveness of the order one periodic solution 𝑃⁺𝑃𝑃⁺ in the region Ω₂.
Denote the first intersection point of the trajectory from initiating point 𝑃⁺₀ with the impulsive set 𝑀₂ at 𝑃₁(ℎ,𝑦_𝑃1), and the corresponding consecutive points are 𝑃₂(ℎ,𝑦_𝑃2),𝑃₃(ℎ,𝑦_𝑃3),𝑃₄(ℎ,𝑦_𝑃4)⋯ respectively. Consequently, under the effect of impulsive function 𝐼, the corresponding points after pulse are 𝑃⁺₁(ℎ,𝑦_𝑃+1),𝑃⁺₂(ℎ,𝑦_𝑃+2),𝑃⁺₃(ℎ,𝑦_𝑃+3), …. In view of 𝑦_𝑃+0>𝑦_𝑃+1>𝑦_𝐴 and disjointness of any two trajectories, we have 𝑦_𝑃+1<𝑦_𝑃+3<⋯<𝑦_{𝑃+2𝑘−1}<𝑦_𝑃+2𝑘+1<⋯𝑦_𝑃+2𝑘<𝑦_{𝑃+2𝑘−2}<𝑦_𝑃+2<𝑦_𝑃+0. (4.4) So 𝑓(𝑃⁺_2𝑘−1)=𝑦_𝑃+2𝑘−𝑦_{𝑃+2𝑘−1}>0 and 𝑓(𝑃⁺_2𝑘)=𝑦_𝑃+2𝑘+1−𝑦_𝑃+2𝑘<0 hold. Like the proof of Theorem 4.1, we can prove lim_𝑘→∞𝑦_{𝑃+2𝑘−1}=lim_𝑘→∞𝑦_𝑃+2𝑘=𝑦_𝑃+.
The trajectory from initiating point between 𝐵⁺₀ and 𝑃⁺₀ will intersect with impulsive set 𝑁₂ with time increasing, under the impulsive effects it arrives at a point of 𝑁₂ which is between [𝑦_{𝑃+2𝑘−1},𝑦_𝑃+2𝑘+1) or [𝑦_𝑃+2𝑘,𝑦_{𝑃+2𝑘−2}). Then like the analysis of Theorem 4.3, we know the trajectory from any initiating point between 𝐵⁺₀ and 𝑃⁺₀ will ultimately tend to be order one periodic solution 𝑃⁺𝑃𝑃⁺.
Denote the intersection of the trajectory passing through point 𝐵 which tangents to 𝑁₁ at point 𝐵 with the set 𝑁₂ at 𝑆. Since the trajectory from initiating any point below 𝑆 of the set 𝑁₂ will certain intersect with set 𝑁₂, next we only need to prove the trajectory with any initiating point below 𝑆 of the set 𝑁₂ will ultimately tend to be order one periodic solution 𝑃⁺𝑃𝑃⁺.
Assume a point 𝐵₀ of set 𝑀₂ jumps to 𝐵⁺₀ under the impulsive effect. Set 𝑅(𝐵₀)=𝐵⁺₁∈𝑁₂. Assume point 𝐵₁ of set 𝑁₂ jumps to 𝐵⁺₁ under the impulsive effect. Set 𝑅(𝐵₁)=𝐵⁺₂∈𝑁₂. This process is continuing until there exists a 𝐵_𝐾⁺₀∈𝑁 (𝐾⁺₀∈𝑁₂) satisfying 𝑦_𝐵𝐾+0<𝑞. So we obtain a sequence {𝐵_𝑘}_{𝑘=0,1,2,…,𝐾0} of set 𝑀₂ and a sequence {𝐵⁺_𝑘}_{𝑘=0,1,2,…,𝐾0} of set 𝑁₂ satisfying 𝑅(𝐵_𝑘−1)=𝐵⁺_𝑘,𝑦_𝐵+𝑘<𝑦_{𝐵+𝑘−1}. For any point of set 𝑁₂ below 𝐵⁺₀, it must lie between 𝐵⁺_𝑘+1 and 𝐵⁺_𝑘 here 𝑘=1,2,…,𝐾₀. After 𝐾₀+1 times’ impulsive effects, the trajectory with this initiating point will arrive at some point of the set 𝑁₂ which must be between 𝐵⁺₀ and 𝑃⁺₀, and then will ultimately tend to order one periodic solution 𝑃⁺𝑃𝑃⁺. There is no order one periodic solution with the initial point below 𝐵⁺₀.
The trajectory with any initiating point in segment 𝐴𝑆 will intersect with the set 𝑁₂ at some point below 𝐵⁺₀ with time increasing. Like the analysis above, we obtain the trajectory initiating any point on segment 𝐴𝑆 will ultimately tend to be the unique order one periodic solution 𝑃⁺𝑃𝑃⁺.
From above analysis, we know there exists a unique order one periodic solution in system (1.4), and the trajectory from any initiating point below 𝑆 will ultimately tend to be order one periodic solution 𝑃⁺𝑃𝑃⁺. Therefore, order one periodic solution 𝑃⁺𝑃𝑃⁺ is attractive in the region Ω₂. This completes the proof.

5. Conclusion

In this paper, a state-dependent impulsive dynamical model concerning different control methods at different thresholds is proposed, we find a new method to study existence and attractiveness of order one periodic solution of such system. We define semicontinuous dynamical system and successor function, demonstrate the sufficient conditions that system (1.4) exists order one periodic solution with differential geometry theory and successor function. Besides, we successfully prove the attractiveness of the order one periodic solution by sequence convergence rules and qualitative analysis.The method can be also extended to mechanical dynamical systems with impacts, for example [19, 20].

These results show that the state-dependent impulsive effects contribute significantly to the richness of the dynamics of the model. The conditions of existence of order one periodic solution in this paper have more extensively applicable scope than the conditions given in [14]. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. The methods of the theorems are proved to be new in this paper, and these methods are more efficient and easier to operate than the existing research methods which have been applied the models with impulsive state feedback control [16–18, 21], so they are deserved further promotion. In this paper, according to the integrated pest management strategies, we propose a model for pest control which adopts different control methods at different thresholds, the corresponding control is exerted, which leads to the two state impulses in model. Certainly, many biological systems will always be described by three or more state variables, which are the main work in the future.

In order to testify the validity of our results, we consider the following example. 𝑥^′(𝑡)(𝑡)(𝑡),𝑦=𝑥0.4−0.5𝑦_′(𝑡)(𝑡)(𝑡)=𝑦0.25𝑥(𝑡),1+0.1𝑥−0.6𝑥≠ℎ₁,ℎ₂or𝑥=ℎ₁,𝑦>𝑦^∗,(𝑡)(𝑡)Δ𝑥=0,Δ𝑦=𝛿,𝑥=ℎ₁,𝑦⩽𝑦^∗,(𝑡)(𝑡),(𝑡)(𝑡)Δ𝑥=−𝛼𝑥Δ𝑦=−𝛽𝑦+𝑞,𝑥=ℎ₂, (5.1) where 𝛼,𝛽∈(0,1), 𝛿>0, 𝑞>0, 0<ℎ₁<ℎ₂. Now, we consider the impulsive effects influences on the dynamics of system (5.1).

Example 5.1. Existence and attractiveness of order one periodic solution.
We set ℎ₁=1, 𝛼=0.6, 𝛽=0.8, 𝑞=0.8, ℎ₂=1.8, initiating points are (0.85,0.2) (red), (0.8,0.5) (green), and (0.75,0.11) (blue), respectively. Figure 17 shows that the conditions of Theorems 3.1 and 4.1 hold, system (5.1) exists order one periodic solution. The trajectory from different initiating must ultimately tend to be the order one periodic solution. Therefore, order one periodic solution is attractive.

(a)

(b)

(c)

Figure 17 

The time series and phase diagram for system (1.4) starting from initial value (0.85,0.2) (red), (0.8,0.5) (green), and (0.75,0.11) (blue),  𝛿=0.6, ℎ1=1<𝑥∗.

Example 5.2. Existence and attractiveness of positive periodic solution.
We set ℎ₁=0.7, 𝛼=0.6, 𝛽=0.8, 𝑞=0.8, ℎ₂=1.8, ℎ₁<(1−𝛼)ℎ₂<𝑥^∗, initiating points are (0.8,0.1) (red), (0.7,0.5) (green), and (0.75,0.3) (blue), respectively. Figure 18 shows that the conditions of Theorems 3.3 and 4.3 hold, there exists order one periodic solution of the system (5.1), and the trajectory from different initiating must ultimately tend to be the order one periodic solution. Therefore, order one periodic solution is attractive.

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Figure 18 

The time series and phase diagram for system (1.4) starting from initial value (0.8,0.1) (red), (0.7,0.5) (green), and (0.75,0.3) (blue) 𝛼=0.6, 𝛽=0.3, 𝑞=0.8, ℎ2=1.8, ℎ1<ℎ2<𝑥∗.

Example 5.3. Existence and attractive of positive periodic solutions.
We set ℎ₁=0.7, 𝛼=0.6, 𝛽=0.8, 𝑞=0.8, ℎ₂=3.5, ℎ₁<(1−𝛼)ℎ₂<𝑥^∗<ℎ₂, initiating points are (1,0.7) (red), (1.4,0.5) (green), and (1.2,1) (blue) as shown in Figure 19. Therefore, the conditions of Theorems 3.6 and 4.5 hold, then system (5.1) exists order one periodic solution, and it is attractive.

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Figure 19 

The time series and phase diagram for system (1.4) starting from initial value (1, 0.7) (red), (1.4, 0.5) (green), and  (1.2,1) (blue) 𝛼=0.6, 𝛽=0.3, 𝑞=0.8, ℎ2=3.5, ℎ1<𝑥∗<ℎ2.

Acknowledgment

This Project supported by the National Natural Science Foundation of China (no. 10872118).