Optimal Control of Agricultural Insects with a Stage-Structured Model

Abstract

A stage-structured pest control model with impulse effects by state feedback control is formulated, and a semicontinuous dynamic system and its successor functions are defined. The sufficient conditions of existence and attractiveness of order one periodic solution are obtained by the method of successor functions. The superiority of the state feedback control strategy in this paper is that we only need to monitor the sum of immature and mature pest populations. Moreover, our results show that our method used in this paper is more efficient and easier than the existing methods.

1. Introduction

Insects are among the most diverse group of animals on the planet and include more than a million described species and represent more than half of all known living organisms. The number of extant species is estimated to be at between six and ten million and potentially represents over 90 percent of the differing metazoan life forms on Earth. They may be found in nearly all environments, although only a small number of species occur in the oceans, a habitat dominated by another arthropod group, the crustaceans. Some insects damage crops by feeding on sap, leaves, or fruits, a few bite humans and livestock, alive and dead, to feed on blood and some are capable of transmitting diseases to humans, pets and livestock. So, humans regard certain insects as pests, and attempt to control them using a host of techniques. Pest control is very important because out of control pests can wreak havoc. For example, many countries in the world suffer deeply from plagues of locusts each year.

In pest management, we can control the pest population by many methods, such as spraying pesticides and releasing natural enemies. Among the methods, chemical pest control is still the predominant type of pest control today, although its long-term effects led to a renewed interest in traditional and biological pest controls towards the end of the 20th century. For most pests, such as aphid and silkworm, the survival rate has a relation with their age and development stage. In recent years, various stage-structure models have been proposed and studied for populations [1–12]. We suppose a stage-structure model as follows: 𝑑𝑥𝑑𝑡=−𝑎𝑥+𝑏𝑦,𝑑𝑦𝑑𝑡=𝑐𝑥−𝑑𝑦, (1) where 𝑥(𝑡) and 𝑦(𝑡) represent the densities of immature and mature pest populations, respectively; 𝑎, 𝑏, 𝑐, and 𝑑 are positive constants; 𝑎 denotes the sum of the rate of an immature population turning into a mature population and the death rate of the immature pest population; 𝑏 and 𝑑 denote the birth rate of the immature pest population and the death rate of the mature pest population, respectively; 𝑐 denotes the rate of an immature population turning into a mature population. The dynamics property of system (1) is very simple. The unique equilibrium point of system (1) is 𝑂(0,0). In the case of 𝑎𝑑−𝑏𝑐>0, 𝑂(0,0) is stable, so no pest control measure is needed. But in the case of 𝑎𝑑−𝑏𝑐<0, 𝑂(0,0) is a saddle with a separatrix Γ∶𝑘₁𝑥, where 𝑘₁=((𝑎−𝑑)+(𝑎−𝑑)²+4𝑏𝑐)/2𝑏. So the effective measures are necessary to control the pest population.

Apparently, when the pesticides are sprayed, it is natural to assume that both the mature and immature pest populations diminish abruptly; that is, impulse occurs, and we can use an impulsive differential equation to model the process of spraying pesticides. The theory of impulsive differential systems has been developed by numerous mathematicians [13–15]. There are three kinds of typical impulsive differential equations [13]: (1) systems with impulses at fixed time; (2) systems with impulses at variable time; (3) autonomous systems with impulses. In recent years, most of researches on IDEs concern systems of types 1 [16–23]. However, the drawbacks of type (1) lie in neglecting growth law of the pest and the cost of management. We must control the amount of pesticide and use them in the crucial phase in order to maximize the benefits of pesticides. Therefore we should do the insect survey well, establish the reasonable control index, reduce the times of spraying, and improve the economic and ecological benefits. Today, some researchers also propose the way of the state feedback strategy by IDEs systems of types 3, such as in [21] and the economic threshold (ET) is mentioned and defined in [22]. However, the shortcoming of state feedback control strategy in [21], is the that pesticides might be applied only when the density of the mature pests population reaches ET, which can lead to the sum of pests beyond the ET because of the overlook of immature pests. Considering the previous factors, we formulate the model as follows: 𝑑𝑥𝑑𝑡=−𝑎𝑥+𝑏𝑦,𝑑𝑦𝑑𝑡=𝑐𝑥−𝑑𝑦,𝑥+𝑦≠ℎ,Δ𝑥=−𝛼𝑥,Δ𝑦=−𝛽𝑦,𝑥+𝑦=ℎ, (2) where 𝑎, 𝑏, 𝑐, 𝑑, ℎ>0. Δ𝑥=𝑥(𝑡⁺)−𝑥(𝑡) and Δ𝑦=𝑦(𝑡⁺)−𝑦(𝑡); 𝛼 and 𝛽∈(0,1) represent the proportion of killed immature and mature pests by spraying pesticides, respectively; ℎ represents the ET of the pest totality. System (2) has been studied by Jiang et al. [23], and they obtained the sufficient conditions of existence and stability of periodic solutions by means of the sequence convergence rule and the analogue of the Poincare criterion. In this paper, we improve the demonstration method of [23] via the successor function.

The organization of this paper is as follows. In the next section, we give some important definitions and auxiliary lemmas. In Section 3, the existence of one order periodic solution is obtained by using successor function. In Section 4, we discuss the attractiveness of order one periodic solution under its existence conditions by means of sequence convergence rules and qualitative analysis. Finally, by numerical simulation, we can confirm our theoretical conclusions and give a brief discussion.

2. Definitions and Lemmas

Definition 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),𝑠)=Π(𝑃,𝑡+𝑠) for all 𝑃∈𝑋 and 𝑡, 𝑠∈𝑅₊.
Sometimes a semidynamical system (𝑋,Π,𝑅₊) is denoted by (𝑋,Π).

Definition 2. Assume that(i)(𝑋,Π) is a semidynamical system;(ii)𝑀 is a nonempty subset of 𝑋;(iii)function 𝐼∶𝑀→𝑋 is continuous and for any 𝑃∈𝑀, and 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 semidynamical system.

For any 𝑃∈𝑋, 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, for any subset 𝑀 of 𝑋; we define 𝑀⁺(𝑃)=𝐶⁺(𝑃)∩𝑀−{𝑃}. The set 𝐶⁻(𝑃)={Π(𝑃,𝑡)∣−∞<𝑡≤0} is called the negative semitrajectory of point 𝑃.

Let Π(𝑃,𝑡) be the trajectory of differential dynamical system 𝑑𝑥/𝑑𝑡=𝑓(𝑥) passing through the point 𝑃. Now we give the trajectory Π(𝑃,𝑡) of the following impulsive semidynamical system: 𝑑𝑥(𝑥)𝑥𝑑𝑡=𝑓,𝑥∉𝑀,₊=𝐼(𝑥),𝑥∈𝑀. (3)

Definition 3. Let (𝑋,Π,𝑀,𝐼) be an impulsive semidynamical system corresponding to the system (3), and, for any 𝑃∈𝑋−𝑀, the trajectory Π(𝑃,𝑡)∶(𝑋−𝑀)×𝑅₊→𝑋 passing through point 𝑃 is defined as follows: if 𝑀⁺(𝑝)=∅,Π(𝑃,𝑡)=Π(𝑃,𝑡), for 𝑡∈𝑅₊;   if 𝑀⁺(𝑝)≠∅, there exists a 𝑃₁∈𝑀⁺(𝑃), and 𝑇₀∈𝑅₊ such that Π(𝑃,𝑇₀)=𝑃₁∈𝑀 and 0<𝑡<𝑇₀, Π(𝑃,𝑡)∉𝑀. Let 𝐼(𝑃₁)=𝐼(Π(𝑃,𝑇₀))=𝑃⁺₁∈𝑁; then for 0≤𝑡≤𝑇₀, Π()=Π()𝑃,𝑡𝑃,𝑡,0≤𝑡<𝑇₀,𝑃⁺₁,𝑡=𝑇₀, (4) if 𝑀⁺(𝑃₁)=∅, Π(𝑃⁺₁,𝑡)=Π(𝑃⁺₁,𝑡−𝑇₀), 𝑡≥𝑇₀;   if 𝑀⁺(𝑃⁺₁)≠∅, there exists a 𝑃₂∈𝑀⁺(𝑃⁺₁), and 𝑇₁∈𝑅₊ such that Π(𝑃⁺₁,𝑇₁)=𝑃₂∈𝑀 and 𝑇₀<𝑡<𝑇₀+𝑇₁, Π(𝑃⁺₁,𝑡)∉𝑀. Let 𝐼(𝑃₂)=𝐼(Π(𝑃⁺₁),𝑇₁)=𝑃⁺₂∈𝑁; then for 𝑇₀≤𝑡≤𝑇₀+𝑇₁, Π()=Π𝑃𝑃,𝑡⁺¹,𝑡−𝑇₀,𝑇₀≤𝑡<𝑇₀+𝑇₁,𝑃⁺₂,𝑡=𝑇₀+𝑇₁. (5)
Repeat the above process; then there exists an 𝑁₀ such that 𝑀⁺(𝑃⁺_𝑁0)=∅ or for any 𝑛∈𝑍₊={1,2,…,}, 𝑀⁺(𝑃⁺_𝑛)≠∅. So, we can obtain a finite or infinite impulsive point sequence {𝑃_𝑛} in 𝑀 satisfying 𝑃_𝑛=𝐹(𝑃⁺_𝑛−1)=Π(𝑃⁺_𝑛−1,𝑇_𝑛−1); here 𝑃⁺_𝑛−1=𝐼(𝑃_𝑛)∈𝑁,𝑇_𝑛−1∈𝑅₊. Then we have Π()=Π()𝑃,𝑡𝑃,𝑡,0≤𝑡<𝑇₁,Π𝑃⁺₁,𝑡−𝑇₁,𝑇₁≤𝑡<𝑇₁+𝑇₂,⋮Π𝑃⁺_𝑛−1,𝑡−∑^𝑛−1_𝑘=1𝑇_𝑘,∑^𝑛−1_𝑘=1𝑇_𝑘≤𝑡<∑^𝑛_𝑘=1𝑇_𝑘,Π𝑃⁺_𝑁0,𝑡−∑^𝑁0_𝑘=1𝑇_𝑘,𝑡≥∑^𝑁0_𝑘=1𝑇_𝑘 (6) or Π()=Π()𝑃,𝑡𝑃,𝑡,0≤𝑡<𝑇₁,Π𝑃⁺₁,𝑡−𝑇₁,𝑇₁≤𝑡<𝑇₁+𝑇₂,⋮Π𝑃⁺_𝑛−1,𝑡−∑^𝑛−1_𝑘=1𝑇_𝑘,∑^𝑛−1_𝑘=1𝑇_𝑘≤𝑡<∑^𝑛_𝑘=1𝑇_𝑘.⋮ (7)
In system (2), 𝑀={(𝑥,𝑦)∈𝑅₊∣𝑥+𝑦=ℎ,𝑥≥0,𝑦≥0}, 𝑁=𝐼(𝑀)={(𝑥,𝑦)∈𝑅₊∣𝑥=(1−𝛼)ℎ, 𝑦=(1−𝛽)ℎ}. For the convenience, if 𝑃∈𝑋−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∈𝑋−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(𝑃,−𝑡₂)∈𝑁, and for −𝑡₂<𝑡<0, Π(𝐵,𝑡)∉𝑁. Furthermore, the isocline 𝑑𝑥/𝑑𝑡=0 is denoted by 𝐿₁; the isocline 𝑑𝑦/𝑑𝑡=0 is denoted by 𝐿₂. The saddle separatrix 𝑘₁𝑥 is denoted by Γ. Let 𝐵 be the intersection of the saddle separatrix Γ, and let 𝑁 and 𝐶 be the intersection of coordinate axis 𝑥 and set 𝑁. Next we will give the definition of the successor function of semidynamical system (2). First, we define a new number axis in set 𝑁. On straight line 𝑦−((1−𝛽)𝑥/(1−𝛼))=(1−𝛽)ℎ that corresponds to set 𝑁, take the origin at point 𝐵 and 𝐵𝐶 as positive direction and unit length to be consistent with coordinate axis 𝑥 or 𝑦; then we obtain a number axis 𝑙. For any 𝑥∈𝑙, 𝑙(𝑥)∈𝑁.

Definition 4. 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=𝐹(𝑃₀)=Π(𝑃₀,𝑇)∈𝑀 and 𝑃⁺=𝐼(𝑃)=𝑃₀.

Definition 5. Suppose 𝑔∶𝑁→𝑁 to be a map. For any 𝑥∈𝑁, there exists a 𝑡₁>0 such that 𝐹(𝑥)=Π(𝑥,𝑡₁)=𝑥₁∈𝑀, 𝑥⁺₁=𝐼(𝑥₁)∈𝑁; then 𝑔(𝑥)=𝑙(𝑥⁺₁)−𝑙(𝑥) is called the successor function of point 𝑥, and the point 𝑥⁺₁ is called the successor point of 𝑥.

Remark 6. If 𝑔(𝑥₀)=0, the trajectory Π(𝑥₀,𝑡) with initial point 𝑥₀ is an order one periodic solution of system (2).

According to the continuity of compound function, we have the following.

Lemma 7. The successor function 𝑔(𝑥) is continuous.

In system (2), if there exist two points 𝑥₁∈𝑁, 𝑥₂∈𝑁 satisfying 𝑔(𝑥₁)𝑔(𝑥₂)<0, by the zero point theorem of continuous function in the closed interval, we know that there must be a point 𝑥₀∈𝑁 which is between 𝑥₁ and 𝑥₂ such that 𝑔(𝑥₀)=0, so we have the following lemmas.

Lemma 8 (existence theorem of order one periodic solution). If there exist two points 𝑥₁∈𝑁, 𝑥₂∈𝑁, which are both in the left 𝐵 or right 𝐵, satisfying 𝑔(𝑥₁)𝑔(𝑥₂)<0, it has an order one periodic solution in system (2).

Lemma 9. Let 𝑙₁={(𝑥,𝑦)∣𝑦=𝑘𝑥,𝑥≥0,𝑦≥0}, 𝑙₂={(𝑥,𝑦)∣𝑥+𝑦=ℎ,𝑥≥0,𝑦≥0}=𝑀, 𝑙₃={(𝑥,𝑦)∣𝑦−((1−𝛽)𝑥/(1−𝛼))=(1−𝛽)ℎ,𝑥≥0,𝑦≥0}=𝑁. Denote 𝐹∈𝑀 as the intersect of 𝑙₁ and 𝑙₂ and 𝐺∈𝑁 as the intersect of 𝑙₁ and 𝑙₃; then,(i)if 𝛼>𝛽, 𝑙(𝐹⁺)<𝑙(𝐺);(ii)if 𝛼=𝛽, 𝑙(𝐹⁺)=𝑙(𝐺);(iii)if 𝛼<𝛽, 𝑙(𝐹⁺)>𝑙(𝐺).

Proof. We focus on proving case (i), by similar method we can prove case (ii) and (iii). By computing, we can easily obtain 𝐹(ℎ/(𝑘+1),𝑘ℎ/(𝑘+1)), 𝐺((1−𝛼)(1−𝛽)ℎ/(𝑘(1−𝛼)+(1−𝛽)), 𝑘(1−𝛼)(1−𝛽)ℎ/(𝑘(1−𝛼)+(1−𝛽))), and 𝐹⁺(ℎ(1−𝛼)/(𝑘+1), 𝑘(1−𝛽)ℎ/(𝑘+1)). If 𝛼>𝛽; then (1−𝛼)(1−𝛽)ℎ/(𝑘(1−𝛼)+(1−𝛽))>ℎ(1−𝛼)/(𝑘+1) and 𝑘(1−𝛼)(1−𝛽)ℎ/(𝑘(1−𝛼) + (1−𝛽))<𝑘(1−𝛽)ℎ/(𝑘+1). So, 𝑙(𝐹⁺)<𝑙(𝐺). This completes the proof.

3. Existence of Order One Periodic Solution of System (2)

In this section, we will investigate the existence of order one periodic solution of system (2) by using the successor function.

We note that system (2) is an impulsive semidynamical system, where 𝑀={(𝑥,𝑦)∣𝑥+𝑦=ℎ, (𝑥,𝑦)∈𝑅⁺₂} is the closed subset of 𝑅⁺₂ and 𝐼∶(𝑥,𝑦)∈𝑀→((1−𝛼)𝑥 and (1−𝛽)𝑦)∈𝑁 is a continuous function. For any point 𝐸, let 𝑥_𝑒 denote its abscissa, and let 𝑦_𝑒 be its ordinate. If 𝐸(𝑥_𝑒,𝑦_𝑒)∈𝑀, then pulse occurs at point 𝐸, and the impulsive function transfers the point 𝐸 into 𝐸⁺. Without loss of generality, unless otherwise specified we assume that the initial point of the trajectory lies in 𝑁.

Now we discuss the existence of order one periodic solution of system (2) from the following three cases.

Case I (𝛼>𝛽 (see Figure 1)). In this case, according to Lemma 9, 𝑙(𝐴⁺)<𝑙(𝐵) and set 𝐹(𝐴⁺)=𝐴₁(𝑥_𝐴1,𝑦_𝐴1)∈𝑀; then impulse occurs at point 𝐴₁. Suppose point 𝐴₁ is subject to impulsive effects to point 𝐴⁺₁((1−𝛼)𝑥_𝐴1,(1−𝛽)𝑦_𝐴1)∈𝑁. In view of vector field and disjointness of any two trajectories, we know 𝑥_𝐴>𝑥_𝐴1 and 𝑦_𝐴<𝑦_𝐴1, so we have 𝑥_𝐴⁺>𝑥_𝐴⁺₁ and 𝑦_𝐴+<𝑦_𝐴+1; then 𝑙(𝐴⁺₁)<𝑙(𝐴⁺), and the successor function of 𝐴⁺ is 𝑔(𝐴⁺)=𝑙(𝐴⁺₁)−𝑙(𝐴⁺)<0; there exists a trajectory Γ₁ intersecting to 𝑀 at point 𝐴₃ which is subject to impulsive effects to point 𝐴⁺₃∈𝑁. Set 𝐹(𝐴⁺₃)=𝐴₂∈𝑀; then impulse occurs at point 𝐴₂. Suppose point 𝐴₂ is subject to impulsive effects to point 𝐴⁺₂((1−𝛼)𝑥_𝐴2,(1−𝛽)𝑦_𝐴2)∈𝑁. We know 𝑥_𝐴2>𝑥_𝐴3 and 𝑦_𝐴2<𝑦_𝐴3, so we have 𝑥_𝐴⁺₂>𝑥_𝐴⁺₃ and 𝑦_𝐴+2<𝑦_𝐴+3; then 𝑙(𝐴⁺₂)>𝑙(𝐴⁺₃), and the successor function of 𝐴⁺₃ is 𝑔(𝐴⁺₃)=𝑙(𝐴⁺₂)−𝑙(𝐴⁺₃)>0.

Figure 1 

By Lemma 8, we know there exists an order one periodic solution of system (2), whose initial point is between 𝐴⁺₃ and 𝐴⁺ in set 𝑁.

Case II (𝛼=𝛽 (see Figure 2)). In this case, according to Lemma 9, the point 𝐴⁺ coincides with 𝐵, and the successor function of 𝐵 is 𝑔(𝐵)=𝑙(𝐵)−𝑙(𝐴⁺)=0, so the segment 𝐵𝐴 is an order one periodic solution of system (2).

Figure 2 

Case III (𝛼<𝛽 (see Figure 3)). In this case, according to Lemma 9, 𝑙(𝐴⁺)>𝑙(𝐵) and set 𝐹(𝐴⁺)=𝐴^′₁(𝑥_𝐴′1,𝑦_𝐴′1)∈𝑀; then impulse occurs at point 𝐴^′₁. Suppose point 𝐴^′₁ is subject to impulsive effects to point 𝐴₁^′+((1−𝛼)𝑥_𝐴′1,(1−𝛽)𝑦_𝐴′1)∈𝑁. We know 𝑥_𝐴<𝑥_𝐴′1 and 𝑦_𝐴>𝑦_𝐴′1, so we have 𝑥_𝐴⁺<𝑥_𝐴1′+ and 𝑦_𝐴+>𝑦_𝐴1′+; then 𝑙(𝐴₁^′+)>𝑙(𝐴⁺), and the successor function of 𝐴⁺ is 𝑔(𝐴⁺)=𝑙(𝐴₁^′+)−𝑙(𝐴⁺)>0; there exists a trajectory Γ₁ intersecting to 𝑀 at point 𝐴^′₃ which is subject to impulsive effects to point 𝐴₃^′+∈𝑁. Set 𝐹(𝐴₃^′+)=𝐴^′₂∈𝑀; then impulse occurs at point 𝐴^′₂. Suppose point 𝐴^′₂ is subject to impulsive effects to point 𝐴^′₂((1−𝛼)𝑥_𝐴′2,(1−𝛽)𝑦_𝐴′2)∈𝑁. We know 𝑥_𝐴′2<𝑥_𝐴′3 and 𝑦_𝐴′2>𝑦_𝐴′3, so we have 𝑥_𝐴2′+<𝑥_𝐴3′+ and 𝑦_𝐴2′+>𝑦_𝐴3′+; then 𝑙(𝐴₂^′+)<𝑙(𝐴₃^′+), and the successor function of 𝐴₃^′+ is 𝑔(𝐴₃^′+)=𝑙(𝐴₂^′+)−𝑙(𝐴₃^′+)<0.

Figure 3 

By Lemma 8 we know there exists an order one periodic solution of system (2), whose initial point is between 𝐴⁺ and 𝐴₃^′+ in set 𝑁.

Thus we have the following theorem.

Theorem 10. If 𝑎𝑑−𝑏𝑐<0, there exists an order one periodic solution in system (2).

4. Attractiveness of Order One Periodic Solution of System (2)

In this section, we discuss the attractiveness of system (2).

Theorem 11. Assume that conditions 𝛼>𝛽 and 𝑎𝑑−𝑏𝑐<0 hold; then,(a)there exist order one periodic solutions of system (2) in the region 𝐴𝐴₃𝐴⁺₃𝐴⁺.(b)If the periodic solution is unique, it is attractive, where the attractive region is in the region 𝐴𝐴₃𝐴⁺₃𝐴⁺ and the coordinate 𝐴(ℎ/(𝑘₁+1),𝑘₁ℎ/(𝑘₁+1)), 𝐴₃(𝑑ℎ/(𝑐+𝑑),𝑐ℎ/(𝑐+𝑑)), 𝐴⁺₃(𝑑ℎ(1−𝛼)/(𝑐+𝑑),𝑐ℎ(1−𝛽)/(𝑐+𝑑)), 𝐴⁺(ℎ(1−𝛼)/(𝑘+1),𝑘ℎ(1−𝛽)/(𝑘+1)) (see Figure 1).

Proof. (a) By the derivation of Theorem 10 and the conditions 𝛼>𝛽, we know 𝑔(𝐴⁺)=𝑙(𝐴⁺₁)−𝑙(𝐴⁺)<0, 𝑔(𝐴⁺₃)=𝑙(𝐴⁺₂)−𝑙(𝐴⁺₃)>0, so there exists an order one periodic solution at least of system (2) whose initial point is between 𝐴⁺₃ and 𝐴⁺ in set 𝑁. According to the continuity of successor function and the zero point theorem of continuous function, we can easily prove there exists odd-numbered order one periodic solution of system (2).
(b) If the periodic solution is unique, assume trajectory 𝑃₊𝑃 and segment 𝑃𝑃₊ are the periodic solution of system (2) with initial point 𝑃₊∈𝑁 (see Figure 4). Next, we will prove the periodic solution is attractive.
On the one hand, set 𝐹(𝐶)=𝐴₁; then owning to the disjointness of any two trajectories, we have 𝑥_𝐴1<𝑥_𝑃 and 𝑦_𝐴1>𝑦_𝑃; thus, 𝑥_𝐶<𝑥_𝐴⁺₁<𝑥_𝑃⁺ and 𝑦_𝐶>𝑦_𝐴+1>𝑦_𝑃+. So, we obtain 𝑙(𝐴⁺₁)>𝑙(𝐶) and 𝑔(𝐶)=𝑙(𝐴⁺₁)−𝑙(𝐶)>0; set 𝐹(𝐴⁺₁)=𝐴₂; we know 𝑥_𝐴1<𝑥_𝐴2<𝑥_𝑃 and 𝑦_𝐴1>𝑦_𝐴2>𝑦_𝑃; then we have 𝑥_𝐴⁺₁<𝑥_𝐴⁺₂<𝑥_𝑃⁺ and 𝑦_𝐴+1>𝑦_𝐴+2>𝑦_𝑃+. So 𝑙(𝐴⁺₂)>𝑙(𝐴⁺₁) and 𝑔(𝐴⁺₁)=𝑙(𝐴⁺₂)−𝑙(𝐴⁺₁)>0. This process is continuing; then we get a sequence {𝐴⁺_𝑘}_𝑘=1,2,… of set 𝑁 satisfying 𝑥_𝐴⁺₁<𝑥_𝐴⁺₂<⋯<𝑥_𝐴⁺_𝑘<⋯<𝑥_𝑃⁺ and 𝑦_𝐴+1>𝑦_𝐴+2>⋯>𝑦_𝐴+𝑘>⋯>𝑦_𝑃+, so 𝑙(𝐴⁺_𝑘+1)>𝑙(𝐴⁺_𝑘), and 𝑔(𝐴⁺_𝑘)=𝑙(𝐴⁺_𝑘+1)−𝑙(𝐴⁺_𝑘)>0. Series {𝑦_𝐴+𝑘}_𝑘=1,2,… decreases monotonously and has lower bound, and series {𝑥_𝐴⁺_𝑘}_𝑘=1,2,… increases monotonously and has upper bound, so lim_𝑘→+∞𝑥_𝐴⁺_𝑘 and lim_𝑘→+∞𝑦_𝐴+𝑘 exist. Next we will prove lim_𝑘→+∞𝑥_𝐴⁺_𝑘=𝑥_𝑃⁺ and lim_𝑘→+∞𝑦_𝐴+𝑘=𝑦_𝑃+. Set 𝐴⁺_∗=lim_𝑘→+∞𝐴⁺_𝑘; that is, we will prove 𝐴⁺_∗=𝑃⁺. Otherwise, 𝐴⁺_∗≠𝑃⁺, set 𝐹(𝐴⁺_∗)=𝐴; then we have 𝑥𝐴<𝑥_𝑃𝐴,𝑥⁺<𝑥_𝑃⁺ and 𝑦𝐴>𝑦_𝑃𝐴,𝑦⁺>𝑦_𝑃+. Since 𝑔(𝐴⁺_∗)≥0 and 𝐴⁺_∗≠𝑃⁺, we have 𝑔(𝐴⁺_∗)=𝑙(𝐴⁺)−𝑙(𝐴⁺_∗)>0; thus, we have 𝑥_𝐴⁺_∗𝐴<𝑥⁺<𝑥_𝑃 and 𝑦_𝐴+∗𝐴>𝑦⁺>𝑦_𝑃 holds. Set 𝐹(𝐴⁺)=𝐴; we know 𝑥𝐴𝐴<𝑥<𝑥_𝑃𝐴𝐴,𝑦>𝑦>𝑦_𝑃, and 𝑥𝐴⁺𝐴<𝑥⁺<𝑥_𝑃⁺, 𝑦𝐴⁺𝐴>𝑦⁺>𝑦_𝑃+; then we have 𝑔(𝐴⁺)=𝑙(𝐴⁺)−𝑙(𝐴⁺)>0, this is contradicted to the fact that 𝐴⁺_∗ is a limit of sequence {𝐴⁺_𝑘}_𝑘=1,2,…. So we obtain 𝐴⁺_∗=𝑃⁺ and lim_𝑘→+∞𝑥_𝐴⁺_𝑘=𝑥_𝑃⁺ and lim_𝑘→+∞𝑦_𝐴+𝑘=𝑦_𝑃+.
On the other hand, set 𝐹(𝐷)=𝐵₁; then owning to the disjointness of any two trajectories, we have 𝑥_𝐵1>𝑥_𝑃 and 𝑦_𝐵1<𝑦_𝑃; thus, 𝑥_𝐷>𝑥_𝐵⁺₁>𝑥_𝑃⁺ and 𝑦_𝐷<𝑦_𝐵+1<𝑦_𝑃+. So, 𝑔(𝐷)=𝑙(𝐵⁺₁)−𝑙(𝐷)>0; set 𝐹(𝐵⁺₁)=𝐵₂. Because 𝑥_𝐷>𝑥_𝐵⁺₁>𝑥_𝑃⁺ and 𝑦_𝐷<𝑦_𝐵+1<𝑦_𝑃+, we know 𝑥_𝐵1>𝑥_𝐵2>𝑥_𝑃 and 𝑦_𝐵1<𝑦_𝐵2<𝑦_𝑃; then we have 𝑥_𝐵⁺₁>𝑥_𝐵⁺₂>𝑥_𝑃⁺ and 𝑦_𝐵+1<𝑦_𝐵+2<𝑦_𝑃+. So 𝑙(𝐵⁺₂)<𝑙(𝐵⁺₁) and 𝑔(𝐵⁺₁)=𝑙(𝐵⁺₂)−𝑙(𝐵⁺₁)<0; this process is continuing; then we get a sequence {𝐵⁺_𝑘}_𝑘=1,2,… of set 𝑁 satisfying 𝑥_𝐵⁺₁>𝑥_𝐵⁺₂>⋯>𝑥_𝐵⁺_𝑘>⋯>𝑥_𝑃⁺ and 𝑦_𝐵+1<𝑦_𝐵+2<⋯<𝑦_𝐵+𝑘<⋯<𝑦_𝑃+ so 𝑙(𝐵⁺_𝑘+1)<𝑙(𝐵⁺_𝑘) and 𝑔(𝐵⁺_𝑘)=𝑙(𝐵⁺_𝑘+1)−𝑙(𝐵⁺_𝑘)<0. Series {𝑥_𝐵⁺_𝑘}_𝑘=1,2,… decreases monotonously and has lower bound and series {𝑦_𝐵+𝑘}_𝑘=1,2,… increases monotonously and has upper bound, so lim_𝑘→+∞𝑥_𝐵⁺_𝑘 and lim_𝑘→+∞𝑦_𝐵+𝑘 exist. Next we will prove lim_𝑘→+∞𝑥_𝐵⁺_𝑘=𝑥_𝑃⁺ and lim_𝑘→+∞𝑦_𝐵+𝑘=𝑦_𝑃+. Set 𝐵⁺_∗=lim_𝑘→+∞𝐵⁺_𝑘; that is, we will prove 𝐵⁺_∗=𝑃⁺. Otherwise, 𝐵⁺_∗≠𝑃⁺, set 𝐹(𝐵⁺_∗)=𝐵; then we have 𝑥𝐵>𝑥_𝑃𝐵,𝑥⁺>𝑥_𝑃⁺, and 𝑦𝐵<𝑦_𝑃𝐵,𝑦⁺<𝑦_𝑃+. Since 𝑔(𝐵⁺_∗)≥0 and 𝐵⁺_∗≤𝑃⁺; we have 𝑔(𝐵⁺_∗)=𝑙(𝐵⁺)−𝑙(𝐵⁺_∗)<0; thus, we have 𝑥_𝐵⁺_∗𝐵>𝑥⁺>𝑥_𝑃, and 𝑦_𝐵+∗𝐵<𝑦⁺<𝑦_𝑃 holds. Set 𝐹(𝐵⁺)=𝐵, we know 𝑥𝐵𝐵>𝑥>𝑥_𝑃, 𝑦𝐵𝐵<𝑦<𝑦_𝑃, and 𝑥𝐵⁺𝐵>𝑥⁺>𝑥_𝑃⁺, 𝑦𝐵⁺𝐵<𝑦⁺<𝑦_𝑃+; then we have 𝑔(𝐵⁺)=𝑙(𝐵⁺)−𝑙(𝐵⁺)<0, this is contradicted to the fact that 𝐵⁺_∗ is a limit of sequence {𝐵⁺_𝑘}_𝑘=1,2,…. So we obtain 𝐵⁺_∗=𝑃⁺ and lim_𝑘→+∞𝑥_𝐵⁺_𝑘=𝑥_𝑃⁺ and lim_𝑘→+∞𝑦_𝐵+𝑘=𝑦_𝑃+. Since the trajectory initiating any point of Ω₀={(𝑥,𝑦)∣𝑥≥0, 𝑦≥0, 𝑥(0)+𝑦(0)<ℎ} is certain to intersect with set 𝑁; next we only need to prove that the trajectory initiating any point of 𝑁 will ultimately tend to the unique order one periodic solution 𝑃⁺𝑃𝑃⁺.
Any point 𝑈∈𝑁 must be in some interval [𝑙(𝐵⁺_(𝑘+1)),𝑙(𝐵⁺_𝑘))_𝑘=2,3,…, [𝑙(𝐵⁺₁),𝑙(𝐶)),(𝑙(𝐶),𝑙(𝐴⁺₁)],(𝑙(𝐴⁺_𝑘),𝑙(𝐴⁺_𝑘+1)]_𝑘=2,3,…. Without loss of generality, we assume 𝑈∈[𝑙(𝐵⁺_𝑘+1),𝑙(𝐵⁺_𝑘)). The trajectory initiating point 𝑈 moves between trajectory 𝐵⁺_𝑘𝐵_𝑘+1 and 𝐵⁺_𝑘+1𝐵_𝑘+2 and intersects with 𝑀 at some point between 𝐵_𝑘+2 and 𝐵_𝑘+1; under the impulsive effects it jumps to the point of 𝑁 which is between [𝑙(𝐵⁺_𝑘+1),𝑙(𝐵⁺_𝑘+2)); then trajectory Π(𝑈,𝑡) continues to move between trajectory 𝐵⁺_𝑘+1𝐵_𝑘+2 and 𝐵⁺_𝑘+2𝐵_𝑘+3. This process can be continued unlimitedly. Since lim_𝑘→+∞𝑥_𝐵⁺_𝑘=𝑥_𝑃⁺ and lim_𝑘→+∞𝑦_𝐵+𝑘=𝑦_𝑃+, the intersection sequence of trajectory Π(𝑈,𝑡) and set 𝑁 will ultimately tend to point 𝑃⁺. Thus the trajectory initiating any point of 𝑁 will ultimately tend to the unique order one periodic solution 𝑃⁺𝑃𝑃⁺.
From the above analysis, we know the trajectory initiating any point of 𝑁 will ultimately tend to order one periodic solution 𝑃⁺𝑃𝑃⁺. Therefore, order one periodic solution 𝑃⁺𝑃𝑃⁺ is attractive in the region Ω₀={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥(0)+𝑦(0)<ℎ}. This completes the proof. Like the analysis of Theorem 11, we have the following theorem.

Figure 4 

Theorem 12. Assume that conditions 𝛼>𝛽 and 𝑎𝑑−𝑏𝑐<0 hold; then,(a)there exists an order one periodic solutions of system (2) in the region 𝐴^′₃𝐴𝐴⁺𝐴₃^′+.(b)If the periodic solution is unique, it is attractive, where the attractive region is in the region 𝐴^′₃𝐴𝐴⁺𝐴₃^′+ and the coordinate 𝐴^′₃(𝑎ℎ/(𝑎+𝑏),𝑏ℎ/(𝑎+𝑏)), 𝐴(ℎ/(𝑘₁+1),𝑘₁ℎ/(𝑘₁+1)), 𝐴⁺(ℎ(1−𝛼)/(𝑘+1),𝑘ℎ(1−𝛽)/(𝑘+1))𝐴₃^′+(𝑎ℎ(1−𝛼)/(𝑎+𝑏),𝑏ℎ(1−𝛽)/(𝑎+𝑏)) (see Figure 3).

Theorem 13. if 𝛼=𝛽 and 𝑎𝑑−𝑏𝑐<0 hold, in view of vector field and Lemma 9, the segment 𝐵𝐴 is a unique order one periodic solution of system (2) which is attractive in the region Ω₀={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑥(0)+𝑦(0)<ℎ}.

Proof. In view of vector field and Lemma 9, we can easily prove the theorem; thus, the proof is omitted.

5. Conclusion

To control pest, a stage-structure model with state-dependent impulsive control was investigated. We defined the successor functions of semicontinuous dynamic system and gave a new method to judge the existence of order one periodic solution of such system. By means of sequence convergence rules and qualitative analysis, we give sufficient conditions of the existence of order one periodic solution, as well as global attractiveness of order one periodic solution. To verify the theoretical results obtained in this paper, we will give some numerical simulations. Let 𝑎=0.6, 𝑏=0.8, 𝑐=0.5, 𝑑=0.2, ℎ=1. By comparing Figures 1, 2, and 3 with Figures 5(a), 5(b), and 5(c), it is obvious that our new method to judge the existence of order one periodic solution is precise. From Figures 6(a) and 6(b), using our spraying strategy, pest population can be controled under the economic threshold, whose method reduces insecticide application and protects environment. It is the first time to put forward the method. Our results have provided some fundamental theoretical conclusions that could be of applied importance.

(a)

(b)

(c)

Figure 5 

Phase portrait of system (2) to see the effect of the proportion of killed immature and mature 𝛼 and 𝛽 with 𝑎=0.6, 𝑏=0.8, 𝑐=0.5, 𝑑=0.2, and ℎ=1. (a) 𝛼>𝛽. (b) 𝛼=𝛽. (c) 𝛼<𝛽.

(a)

(b)

Figure 6 

Permanence of system (2) to see the effect of the the proportion of killed immature and mature 𝛼 and 𝛽. Other parameters are the same with those in Figure 5. (a) Time-series of the population 𝑥(𝑡). (b) Time-series of the population 𝑦(𝑡).

Remark 14. The attractiveness of order one periodic solution of system (2) can be shown by Figure 7. We take some different initial values, for example, (0.25,0.55), (0.25,0.15) and (0.4,0.15), respectively. By numerical simulations, we find that the trajectories initiating different values finally tend to a unique order one periodic solution of system (2). This indicates that the order one periodic solution of system (2) is globally attractive and we guess that it is unique order one periodic solution of system (2).

Figure 7 

The global attractiveness of system (2) with parameters 𝑎=0.6, 𝑏=0.8, 𝑐=0.5, 𝑑=0.2, ℎ=0.2, 𝛼=0.5, 𝛽=0.3. The initial values for trajectory 𝐿1, 𝐿2 and 𝐿3 are (0.25,0.55), (0.25,0.15) and (0.4,0.15), respectively.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (10971001) and the Excellent Talents Support Project of Universities and Colleges in Liaoning.