Periodic Solution of Prey-Predator Model with Beddington-DeAngelis Functional Response and Impulsive State Feedback Control

Abstract

A prey-predator model with Beddington-DeAngelis functional response and impulsive state feedback control is investigated. We obtain the sufficient conditions of the global asymptotical stability of the system without impulsive effects. By using the geometry theory of semicontinuous dynamic system and the method of successor function, we obtain the system with impulsive effects that has an order one periodic solution, and sufficient conditions for existence and stability of order one periodic solution are also obtained. Finally, numerical simulations are performed to illustrate our main results.

1. Introduction

The study of the dynamics of prey-predator system is one of the dominant subjects in both ecology and mathematical ecology due to the fact that predator-prey interaction is the fundamental structure in population dynamics. Many scholars have carried out the study of prey-predator system with various functional responses, such as Monod-type and Holling-type. It is well known that Beddington-DeAngelis functional response which was introduced by Beddington and DeAngelis et al. [1, 2] can avoid some of the singular behavior of ratio-dependent models at low densities and provide better description of predator feeding over a range of prey-predator abundances. Therefore, in this paper, we investigate prey-predator system with Beddington-DeAngelis functional response.

Impulsive differential equations have been widely used in various fields of applied sciences, for example, physics, ecology, and pest control. The majority of them just concern the system with impulses at fixed times [3–7]. However, in practical ecological system, the control measures (by poisoning or releasing the natural nenmy, etc.) are taken only when the amount of species reaches a threshold value, rather than the usual impulsive fixed-time control strategy. Impulsive state feedback equation is a powerful tool to manage these problems. Therefore, some researchers proposed the impulsive state feedback control models for population management [8–11].

Motivated by the above works, in this paper, we consider the following predator-prey system with Beddington-DeAngelis functional respose: 𝑥^′(𝜏)𝑥𝑥=𝑟1−𝐾−𝑥𝑦𝑦,𝛼+𝛽𝑥+𝛾𝑥<ℎ₁,𝑦^′(𝜏)=𝑘𝑥𝑦𝑦Δ𝛼+𝛽𝑥+𝛾−𝑑𝑦,𝑥=−𝑝𝑥,𝑥=ℎ₁,Δ𝑦=𝜄₁−𝑞𝑦, (1.1) where Δ𝑥=𝑥(𝜏⁺)−𝑥(𝜏), Δ𝑦=𝑦(𝜏⁺)−𝑦(𝜏), 𝑟, 𝐾, 𝛼, 𝛽, 𝛾, 𝑑, and  𝑘 are positive constants and 𝑥(𝜏),𝑦(𝜏) represent the population density of prey (pest) and predator (natural enemy) at time 𝜏, respectively. 𝑟 is the intrinsic growth rate and 𝐾 is the carrying capacity in the absence of predation. The predator consumes the prey with functional response of Beddington-DeAngelis type 𝑥𝑦/(𝛼+𝛽𝑥+𝛾𝑦) and contributes to its growth with rate 𝑘𝑥𝑦/(𝛼+𝛽𝑥+𝛾𝑦). 𝑑 is the death rate of predator. 0<𝑝<1,0<𝑞<1, when the amount of the prey reaches the threshold ℎ at time 𝜏_ℎ1, controlling measures are taken and the amount of the prey and predator abruptly turn to (1−𝑝)ℎ₁ and 𝜄₁+(1−𝑞)𝑦(𝜏_ℎ1), respectively. The functional response in system (1.1) is similar to the well-known Holling type II with an extra term 𝛾𝑦 in the denominator which models the mutual interference among predators. It also has some of the same qualitative behaviours as the classical ratio-dependent model (i.e., 𝛼=0), but is free from the singular behaviors of ratio-dependent model at low densities which is, in fact, the source of controversy [12–14]. For simplicity, let (𝛼+𝛽𝑥+𝛾𝑦)𝑑𝑡=𝑑𝜏, then system (1.1) becomes the form: 𝑥^′(𝑡)=𝑥𝑟𝑟𝛼+𝑟𝛽−𝐾𝑥+𝑟𝛾−1𝑦+𝑟𝛽𝐾𝑥²−𝑟𝛾𝐾𝑥𝑦,𝑥<ℎ₁,𝑦^′(𝑡)=𝑦𝑦,Δ−𝛼𝑑+𝑘−𝑑𝛽𝑥−𝛾𝑑𝑥=−𝑝𝑥,𝑥=ℎ₁.Δ𝑦=𝜄₁−𝑞𝑦, (1.2) We assume that 𝑘−𝑑𝛽>0, otherwise 𝑦^′(𝑡)<0, we have 𝑦→0 as 𝑡→∞, which means that the predator population will die out, therefore, in the following, we always assume that 𝑘−𝑑𝛽>0.

We nondimensionalize system (1.2) with the following scaling: 𝑥=(𝛼𝑑/(𝑘−𝑑𝛽))𝑥, 𝑦=(𝛼/𝛾)𝑦, and have 𝑥^′(𝑡)=𝑓𝑥1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄,𝑦𝑥𝑦𝑥<ℎ,_′(𝑡)=𝑒𝑦𝑥−𝑦−1,Δ𝑥=−𝑝𝑥,𝑥=ℎ,Δ𝑦=𝜄−𝑞𝑦, (1.3) where the nondimensional parameters are defined as 𝑎₁=(𝐾𝛽−𝛼)𝑑/(𝑘−𝑑𝛽)𝐾, 𝑎₂=1−(1/𝑟𝛾), 𝑓=𝑟𝛼, 𝑒=𝛼𝑑, 𝑎₃=𝛼𝛽𝑑²/𝐾(𝑘−𝑑𝛽)², 𝑎₄=𝛼𝑑/(𝑘−𝑑𝛽), ℎ=((𝑘−𝑑𝛽)/𝛼𝑑)ℎ₁, and 𝜄=(𝛾/𝛼)𝜄₁. Obviously that 𝑒>0, 𝑓>0, 𝑎₃>0, 𝑎₄>0.

In this paper, we mainly discuss the existence and stability of periodic solution of system (1.3) by using the geometry theory of semicontinuous dynamic system and the method of successor function which were introduced in the paper [15, 16], as far as we know, there are few papers to apply these methods to prove the existence of order one periodic solution, which makes the study simpler and clearer. What is worth saying is that modeling thoughts and mathematical methods used in this paper are of more important theoretical and practical value.

An outline of this paper is as follows: some definitions and theorems are given for the later use in the next section. The qualitative analysis of the system without impulsive effects is given in Section 3. In Section 4, the existence and stability of order one periodic solution of system (1.3) are investigated. Numerical simulations and some discussions are provided in Section 5.

2. Definitions and Lemmas

Definition 2.1 (see [15]). Differential equation with impulsive state feedback control 𝑑𝑥,𝑑𝑡=𝑃𝑥,𝑦𝑑𝑦,,,,𝑑𝑡=𝑄𝑥,𝑦𝑥,𝑦∉𝑀𝑥,𝑦Δ𝑥=𝛼𝑥,𝑦,Δ𝑦=𝛽𝑥,𝑦𝑥,𝑦∈𝑀𝑥,𝑦 (2.1) where 𝑀{𝑥,𝑦} and 𝑁{𝑥,𝑦} are lines or curves on the plane 𝑅²(𝑥,𝑦). 𝑀{𝑥,𝑦} is called pulse set and 𝑁{𝑥,𝑦} is called phase set. The dynamical system which is constituted by the solution map of system (2.1) is called semicontinuous dynamical system which is denoted by (Ω,𝑓,𝜑,𝑀). We assume that the map with the initial point 𝑝 is not in the pulse set 𝑀{𝑥,𝑦}, that is, 𝑝∈Ω=𝑅²−𝑀{𝑥,𝑦}, 𝜑 is a continuous map, 𝜑(𝑀)⊆𝑁, and 𝜑 is called pulse map.

In this paper, 𝑅⁺=[0,+∞), 𝑅²₊={(𝑥,𝑦)∣𝑥≥0,𝑦≥0}. 𝑀={(𝑥,𝑦)∈𝑅²₊∣𝑥=ℎ,𝑦≥0}, 𝜑∶(𝑥,𝑦)∈𝑀→((1−𝑝)𝑥,𝜄+(1−𝑞)𝑦)∈𝑅²₊, 𝑁=𝜑(𝑀)={(𝑥,𝑦)∈𝑅²₊∣𝑥=(1−𝑝)ℎ,𝑦≥0}, then system (1.3) constitutes a semicontinuous dynamical system (Ω,𝑓,𝜑,𝑀).

For any 𝑃∈Ω, the function 𝑓_𝑃∶𝑅⁺→Ω defined as 𝑓_𝑃(𝑡)=𝑓(𝑃,𝑡) is continuous and we call 𝑓_𝑃(𝑡) the trajectory passing through point 𝑃. The set 𝐶⁺(𝑃)={𝑓(𝑃,𝑡)∣0≤𝑡<+∞} is called positive semitrajectory of point 𝑃. The set 𝐶⁻(𝑃)={𝑓(𝑃,𝑡)∣−∞<𝑡≤0} is called the negative semitrajectory of point 𝑃. For the convenience, if 𝑃∈Ω, 𝐹(𝑃) is defined as the first intersection point of 𝐶⁺(𝑃) and 𝑀, that is, there exists a 𝑡₁∈𝑅⁺ such that 𝐹(𝑃)=𝑓(𝑃,𝑡₁)∈𝑀, and for 0<𝑡<𝑡₁,𝑓(𝑃,𝑡)∉𝑀.

Definition 2.2 (see [15]). A trajectory 𝑓(𝑃,𝑡) is called order one periodic solution with period 𝑇 if there exists a point 𝑃∈𝑁 and 𝑇>0 such that 𝑓(𝑃,𝑇)=𝑄∈𝑀 and 𝜑(𝑄)=𝜑(𝑓(𝑃,𝑇))=𝑃∈𝑁.

Next we will give the definition of the successor function of semicontinuous dynamical system (1.3). First, we define a new number axis in set 𝑁. On straight line 𝑥=(1−𝑝)ℎ, take the origin at point ((1−𝑝)ℎ,0) of coordinate axis 𝑥 and define positive direction and unit length to be consistent with coordinate axis 𝑦, then we obtain a number axis 𝑙. For any 𝑥∈𝑙, let 𝑙(𝑥) be coordinate of point 𝑥.

Definition 2.3. Suppose 𝑔∶𝑁→𝑁 be a map. For any 𝑥∈𝑁, there exists a 𝑡₁>0 such that 𝐹(𝑥)=𝑓(𝑥,𝑡₁)=𝑥₁∈𝑀, 𝑥⁺₁=𝜑(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 2.4. If 𝑔(𝑥₀)=0, the trajectory 𝑓(𝑥₀,𝑡) with initial point 𝑥₀ is an order one periodic solution of system (1.3).

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

Lemma 2.5. The successor function 𝑔(𝑥) is continuous.

In system (1.3), the isocline 𝑑𝑥/𝑑𝑡=0 is denoted by Γ₁, the isocline 𝑑𝑦/𝑑𝑡=0 is denoted by Γ₂. Let 𝐴((1−𝑝)ℎ,𝑦_𝐴) be the intersection point of isocline Γ₁ and set 𝑁. If there exist two points 𝑥₁∈𝑁, 𝑥₂∈𝑁, which are both below 𝐴 or above 𝐴, satisfying 𝑔(𝑥₁)𝑔(𝑥₂)<0, by the zero point theorem of continuous function in the closed interval, we know there exists a point 𝑥₀∈𝑁 which is between 𝑥₁ and 𝑥₂ such that 𝑔(𝑥₀)=0, so we have the following.

Lemma 2.6 (existence theorem of order one periodic solution). The system (1.3) exists an order one periodic solution if there exist two points 𝑥₁∈𝑁, 𝑥₂∈𝑁, which are both below 𝐴 or above 𝐴, satisfying 𝑔(𝑥₁)𝑔(𝑥₂)<0.

3. Qualitative Analysis of System (1.3) without Impulsive Effects

In this section, we will study the qualitative characteristic of system (1.3) without impulsive effects. If no impulsive effects is introduced, then system (1.3) is 𝑑𝑥𝑑𝑡=𝑓𝑥1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄𝑥𝑦,𝑑𝑦.𝑑𝑡=𝑒𝑦𝑥−𝑦−1 (3.1)

The equilibrium of system (3.1) satisfies 𝑓𝑥1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄𝑥𝑦=0,𝑒𝑦𝑥−𝑦−1=0. (3.2) It can be seen that system (3.1) exist boundary equilibrium 𝑂(0,0), 𝐴(𝑥₁,0), where 𝑥₁ satisfied 1+𝑎₁𝑥−𝑎₃𝑥²=0 and a positive equilibrium 𝐵(𝑥^∗,𝑦^∗) provided that 𝑎₃<1+𝑎₁,𝑎₂<0, where 𝑥^∗=(𝑎₁+𝑎₂+𝑎₄+(𝑎₁+𝑎₂+𝑎₄)²+4(𝑎₃+𝑎₄)(1−𝑎₂))/2(𝑎₃+𝑎₄), 𝑦^∗=𝑥^∗−1. In the following, we always suppose that 𝑎₁<0,𝑎₂<0.

Lemma 3.1. The system (3.1) is uniformly bounded.

Proof. Let the straight line 𝑙₁=𝑥−𝑥₁=0, we have 𝑑𝑙₁/𝑑𝑡=𝑑𝑥/𝑑𝑡|_𝑥=𝑥1=𝑓𝑥(𝑎₂𝑦−𝑎₄𝑥𝑦)<0, then the trajectory of system (3.1) from the right of 𝑙₁ through the 𝑙₁ into the left. Define a function 𝑉(𝑥,𝑦)=−(𝑒/𝑎₂)𝑥+𝑓𝑦−𝐾, where 0<𝑥<𝑥₁. The function 𝑉(𝑥,𝑦) intersects the line 𝑥=𝑥₁ and axis 𝑦 at the points 𝐻 and 𝐺, respectively. We have 𝑑𝑉/𝑑𝑡|_𝐻𝐺=−𝑒𝑓𝐾²[(1/𝑓²)−(𝑎₄𝑥²/𝐾𝑎₂𝑓)+(2𝑒𝑥/𝐾𝑎₂𝑓²)+(𝑛/𝑓)]+𝑒𝑓[−(𝑥/𝑎₂)−(𝑎₁𝑥²/𝑎₂)(𝑎₃𝑥³/𝑎₂)+(𝑎₄𝑒𝑥³/𝑎²₂𝑓)−(𝑒²𝑥²/𝑎²₂𝑓²)(𝑒𝑥/𝑎₂𝑓)]. we can choose 𝐾 enough large such that 𝑑𝑉/𝑑𝑡|_𝐻𝐺<0. Hence, the system (3.1) is uniformly bounded (see Figure 1). The proof is completed.

Figure 1 

The uniformly bounded region.

In the following, we will analysis the stability of equilibrium 𝑂(0,0),𝐴(𝑥₁,0) and 𝐵(𝑆^∗,𝑥^∗) of system (3.1). Clearly, 𝑂(0,0) is saddle point and if 𝑎₃>1+𝑎₁, 𝐴(𝑥₁,0) is stable, otherwise 𝐴(𝑥₁,0) is saddle point and the positive equilibrium point 𝐵(𝑆^∗,𝑥^∗) exists. For the positive equilibrium point, we have the following result.

Theorem 3.2. The positive equilibrium point 𝐵(𝑆^∗,𝑥^∗) is globally asymptotically stable if 𝑎₃<1+𝑎₁.

Proof. The Jacobian matrix 𝐽_𝐵=𝐽(𝑥^∗,𝑦^∗) of system (3.1) at 𝐵 takes the form of 𝐽_𝐵=𝑓𝑎₁𝑥^∗−2𝑎₃𝑥²−𝑎₄𝑥^∗𝑦^∗𝑓𝑥^∗𝑎₂−𝑎₄𝑥^∗𝑒𝑦^∗−𝑒𝑦^∗. (3.3) The eigenvalue problem for the 𝐽_𝐵=𝐽(𝑥^∗,𝑦^∗) provides the characteristic equation 𝜆²+𝑄₁𝜆+𝑄₂=0, (3.4) where the coefficients 𝑄₁, 𝑄₂ are 𝑄₁=𝑒𝑦^∗+𝑓−𝑎₁𝑥^∗+2𝑎₃𝑥^∗2+𝑎₄𝑥^∗𝑦^∗,𝑄₂=𝑒𝑓𝑦^∗2𝑎₃𝑥^∗2+𝑎₄𝑥^∗𝑦^∗−𝑎₁𝑥^∗−𝑎₂𝑥^∗+𝑎₄𝑥^∗2. (3.5) Note that 𝑄₁>0, 𝑄₂>0, then both of the eigenvalues have negative real part, we have that 𝐵 is locally asymptotically stable.
Let 𝑃(𝑥,𝑦)=𝑓𝑥(1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄𝑥𝑦), 𝑄(𝑥,𝑦)=𝑒𝑦(𝑥−𝑦−1) and 𝐵=𝑥^𝛼−1𝑦^𝛽−1, where 𝛼=−𝑎₁/(1+𝑎₁), 𝛽=𝛼𝑓/𝑒, then we have 𝜕()𝐷=𝐵𝑃+𝜕()𝜕𝑥𝐵𝑄𝜕𝑦=𝑥^𝛼−1𝑦^𝛽−1𝛼𝑓−𝑒𝛽+𝛼𝑓𝑎₁+𝑓𝑎₁+𝑒𝛽𝑥+𝛼𝑓𝑎₂𝑦−−𝑒𝛽−𝑒𝛼𝑓𝑎₃+2𝑎₃𝑓𝑥²−𝛼𝑓𝑎₄+𝑓𝑎₄𝑥𝑦=𝑥^{−((𝑎1/(1+𝑎1))−1)}𝑦^{−((𝑎1𝑓/((1+𝑎1)𝑒))−1)}𝛼𝑓𝑎₂𝑦−−𝑒𝛽−𝑒𝛼𝑓𝑎₃+2𝑎₃𝑓𝑥²−𝛼𝑓𝑎₄+𝑓𝑎₄𝑥𝑦<0, (3.6) the system (3.1) has no closed trajectory, therefore 𝐵(𝑆^∗,𝑥^∗) is globally asymptotically stable. This completes the proof.

4. Existence and Stability of Periodic Solutions of System (1.3)

4.1. Existence of Order One Periodic Solution

In this section, we will investigate the existence of order one periodic solution of system (1.3) by using the method of successor function. Note that system (1.3) is a semicontinuous dynamical system, for convenience, any point 𝐶, let 𝑥_𝐶 denote its abscissa and 𝑦_𝐶 be its ordinate. If 𝐶=(ℎ,𝑦_𝐶)∈𝑀, then impulse occurs at point 𝐶, the impulsive function transfers the point 𝐶 into 𝐶⁺. Without loss of generality, unless otherwise specified we assume the initial point of the trajectory lies in 𝑁. From discussion of Section 3, we know that 𝐵 is globally asymptotically stable when 𝑎₃<1+𝑎₁. If ℎ>𝑥^∗, (1−𝑝)ℎ>𝑥^∗, all the solutions of system (1.3) tend to the equilibrium 𝐵 after finite impulse, so we mainly pay attention to the case ℎ<𝑥^∗ or ℎ>𝑥^∗ and (1−𝑝)ℎ<𝑥^∗.

Case I (ℎ<𝑥^∗). In this case, sets 𝑀 and 𝑁 are both in the left side of point 𝐵. Trajectory passing through point 𝐴 tangents to 𝑁 at point 𝐴. Set 𝐹(𝐴)=𝐸(ℎ,𝑦_𝐸)∈𝑀, then impulse occurs at point 𝐸. Suppose point 𝐸 is subject to impulsive effects to point 𝐸⁺((1−𝑝)ℎ,𝑦_𝐸+)∈𝑁, here 𝑦_𝐸+=(1−𝑞)𝑦_𝐸+𝜄, the position of 𝐸⁺ has the following three cases.

Case 𝐼₁ (𝑦_𝐸+=𝑦_𝐴). 𝐸⁺ coincides with 𝐴, and the successor function of 𝐴 is 𝑔(𝐴)=𝑦_𝐸+−𝑦_𝐴=0, so trajectory 𝐴𝐸 and segment 𝐸𝐴 formulate an order one periodic solution of system (1.3).

Case 𝐼₂ (𝑦_𝐸+>𝑦_𝐴). In this case, 𝐸⁺ is above 𝐴. Set 𝐹(𝐸⁺)=𝐸₁∈𝑀, in view of vector field and disjointness of any two trajectories, we know 𝑦_𝐸1<𝑦_𝐸, so we have 𝑦_𝐸+1<𝑦_𝐸+, then the successor function of 𝐸⁺ is 𝑔(𝐸⁺)=𝑦_𝐸+1−𝑦_𝐸+<0.

Take another point 𝐷((1−𝑝)ℎ,𝑦_𝐴+𝜀) above A, where 𝜀>0 is small enough. Set 𝐹(𝐷)=𝐷₁∈𝑀, in view of continuous dependence of the solution on initial value and time, we know 𝑦_𝐷1<𝑦_𝐸 and the point 𝐷₁ is close to 𝐸 enough, so we have 𝑦_𝐷+1<𝑦_𝐸+ and the point 𝐷⁺₁ is close to 𝐸⁺ enough, since 𝑦_𝐸+>𝑦_𝐴, then we obtain 𝑔(𝐷)=𝑦_𝐷+1−𝑦_𝐷>0. By Lemma 2.6, we know there exists an order one periodic solution of system (1.3), whose initial point is between 𝐷 and 𝐸⁺ in set 𝑁 (see Figure 2).

Figure 2 

Illustration of system (1.3) for the case ℎ<𝑥∗, 𝑦𝐸+>𝑦𝐴.

Case 𝐼₃ (𝑦_𝐸+<𝑦_𝐴). In this case, 𝐸⁺ is below 𝐴. We have 𝑦_𝐸1<𝑦_𝐸, so 𝑦_𝐸+1<𝑦_𝐸+, then the successor function of 𝐸⁺ is 𝑔(𝐸⁺)=𝑦_𝐸+1−𝑦_𝐸+<0.

Take another point 𝐴₁((1−𝑝)ℎ,𝜀)∈𝑁 satisfying 0<𝜀<𝜄. Set 𝐹(𝐴₁)=𝐸₂∈𝑀, then 𝑦_𝐸+2=(1−𝑞)𝑦_𝐸2+𝜄, so we have 𝑔(𝐴₁)=𝑦_𝐸+2−𝑦_𝐴1>0. By Lemma 2.6, we know there exists an order one periodic solution of system (1.3), whose initial point is between 𝐸⁺ and 𝐴₁ in set 𝑁 (see Figure 3). Therefore, we have the following theorem.

Figure 3 

Illustration of system (1.3) for the case ℎ<𝑥∗, 𝑦𝐸+<𝑦𝐴.

Theorem 4.1. Suppose that 𝑎₃<1+𝑎₁, ℎ<𝑥^∗, then system (1.3) has an order one periodic solution.

Case II (ℎ>𝑥^∗,(1−𝑝)ℎ<𝑥^∗). In this case, set 𝑀 is in the right side of 𝐵 and set 𝑁 in the left of 𝐵. Denote the intersection point of isocline Γ₁(𝑑𝑥/𝑑𝑡=0) and set 𝑀 by 𝐴(ℎ,𝑦_𝐴). The trajectory passing through point 𝐴 tangents to 𝑀 at point 𝐴. 𝐺(𝑥_ℎ,𝑦_ℎ) is defined as the first intersection point of 𝐶⁻(𝐴) and Γ₁, that is, there exists a 𝑇₁>0 such that 𝑓(𝐴,−𝑇₁)=𝐺∈𝐶⁻(𝐴)⋂Γ₁, and for −𝑇₁<𝑡≤0, 𝑓(𝐴,𝑡)⋂Γ₁=∅.

Case 𝐼𝐼₁ (𝑥_𝐺<(1−𝑝)ℎ). There exist 𝑇₂,𝑇₃>0 such that 𝐶∈𝐶⁻(𝐴)⋂𝑁,𝐷∈𝐶⁻(𝐴)⋂𝑁, 𝑓(𝐴,−𝑇₂)=𝐷,𝑓(𝐴,−𝑇₃)=𝐶, and for −𝑇₃<𝑡<−𝑇₂,𝑓(𝐴,𝑡)⋂𝑁=∅. Suppose point 𝐴 is subject to impulsive effects to point 𝐴⁺((1−𝑝)ℎ,𝑦_𝐴+)∈𝑁.

If 𝐴⁺ coincides with 𝐶 or 𝐷, trajectory 𝐷𝐴 and segment 𝐷𝐴 or trajectory 𝐶𝐷𝐴 and segment 𝐴𝐶 formulate an order one periodic solution of system (1.3).

If 𝐴⁺ is below 𝐷, that is, 𝑦_𝐴+<𝑦_𝐷, the successor function of 𝐷 is 𝑔(𝐷)=𝑦_𝐴+−𝑦_𝐷<0. Take another point 𝐷₁((1−𝑝)ℎ,𝜀)∈𝑁 satisfying 0<𝜀<𝜄. Set 𝐹(𝐷₁)=𝐷₂∈𝑀, then 𝑦_𝐷+2=(1−𝑞)𝑦_𝐷2+𝜄>𝜀, then we have 𝑔(𝐷₁)=𝑦_𝐷+2−𝑦_𝐷1>0. We conclude that there exists an order one periodic solution of system (1.3), whose initial point is between 𝐷 and 𝐷₁ in set 𝑁 (see Figure 4).

Figure 4 

Illustration of system (1.3) for the case ℎ>𝑥∗, (1−𝑝)ℎ<𝑥∗, 𝑥𝐺<(1−𝑝)ℎ, 𝑦𝐴+<𝑦𝐷.

If 𝐴⁺ is above 𝐶, that is, 𝑦_𝐴+>𝑦_𝐶, the successor function of 𝐶 is 𝑔(𝐶)=𝑦_𝐴+−𝑦_𝐶>0. Set 𝐹(𝐴⁺)=𝐴₁∈𝑀, in view of the vector fields of system (1.3), we know 𝑦_𝐴1<𝑦_𝐴,𝑦_𝐴+1<𝑦_𝐴+, so we have 𝑔(𝐴⁺)=𝑦_𝐴+1−𝑦_𝐴+<0. We conclude that there exists an order one periodic solution of system (1.3), whose initial point is between 𝐶 and 𝐴⁺ in set 𝑁 (see Figure 5).

Figure 5 

Illustration of system (1.3) for the case ℎ>𝑥∗, (1−𝑝)ℎ<𝑥∗, 𝑥𝐺<(1−𝑝)ℎ, 𝑦𝐴+>𝑦𝐶.

If 𝐴⁺ is between 𝐶 and 𝐷, in view of vector fields, we know the trajectory of system (1.3) initiating any point between 𝐶 and 𝐷 of set 𝑁 will be free from impulsive effects and ultimately tend to stable point 𝐵, thus there does not exist order one periodic solution in system (1.3) in this case. So we have the following.

Theorem 4.2. Suppose that 𝑎₃<1+𝑎₁,ℎ>𝑥^∗, (1−𝑝)ℎ<𝑥^∗ and 𝑥_𝐺<(1−𝑝)ℎ, 𝑦_𝐴+≤𝑦_𝐷 or 𝑦_𝐴+≥𝑦_𝐶, then system (1.3) has an order one periodic solution.

Case 𝐼𝐼₂ (𝑥_𝐺>(1−𝑝)ℎ). Denote the intersection point of isocline Γ₁ and set 𝑁 by 𝐾((1−𝑝)ℎ,𝑦_𝐾). The trajectory passing through point 𝐾 tangents to 𝑁 at point 𝐾. Set 𝐹(𝐾)=𝑆(ℎ,𝑦_𝑆)∈𝑀, then impulse occurs at point 𝑆. Suppose point 𝑆 is subject to impulsive effects to point 𝑆⁺((1−𝑝)ℎ,𝑦_𝑆+)∈𝑁. Like the analysis of Case I, we can prove that there exists an order one periodic solution in system (1.3) in this case (see Figure 6).

Figure 6 

Illustration of system (1.3) for the case ℎ>𝑥∗, (1−𝑝)ℎ<𝑥∗, 𝑥𝐺>(1−𝑝)ℎ.

Theorem 4.3. Suppose that 𝑎₃<1+𝑎₁,ℎ>𝑥^∗,(1−𝑝)ℎ<𝑥^∗ and 𝑥_𝐺>(1−𝑝)ℎ, then system (1.3) exists an order one periodic solution.

In the following, we analyze the stability of order one periodic solution in system (1.3). Firstly, we give one lemma to discuss the stability of this periodic solution of system (1.3).

Lemma 4.4. The 𝑇-periodic solution 𝑥=𝜉(𝑡),𝑦=𝜂(𝑡) of the system, 𝑑𝑥,𝑑𝑡=𝑃𝑥,𝑦if𝜙𝑥,𝑦≠0,𝑑𝑦,,,𝑑𝑡=𝑄𝑥,𝑦Δ𝑥=𝛼𝑥,𝑦if𝜙𝑥,𝑦=0,Δ𝑦=𝛽𝑥,𝑦 (4.1) is orbitally asymptotically stable if the Floquet multiplier 𝜇₂ satisfies the condition |𝜇₂|<1, where 𝜇₂=Π^𝑞_𝑘=1Δ_𝑘exp^𝑇₀𝜕𝑃𝜉(𝑡)(𝑡)+𝜕𝑥,𝜂𝜕𝑄𝜉(𝑡)(𝑡)𝜕𝑦,𝜂𝑑𝑡, (4.2) with Δ_𝑘=𝑃₊𝜕𝛽/𝜕𝑦𝜕𝜙/𝜕𝑥−𝜕𝛽/𝜕𝑥𝜕𝜙/𝜕𝑦+𝜕𝜙/𝜕𝑥𝑃+𝑄𝜕𝜙/𝜕𝑥+𝑄𝜕𝜙/𝜕𝑦₊(𝜕𝛼/𝜕𝑥)𝜕𝜙/𝜕𝑦−𝜕𝛼/𝜕𝑦𝜕𝜙/𝜕𝑥+𝜕𝜙/𝜕𝑦𝑃,𝜕𝜙/𝜕𝑥+𝑄𝜕𝜙/𝜕𝑦 (4.3) and 𝑃, 𝑄, (𝜕𝛼/𝜕𝑥), (𝜕𝛼/𝜕𝑦), (𝜕𝛽/𝜕𝑥), (𝜕𝛽/𝜕𝑦), (𝜕𝜙/𝜕𝑥), and (𝜕𝜙/𝜕𝑦) are calculated at the point (𝜉(𝜏_𝑘),𝜂(𝜏_𝑘)), 𝑃₊=𝑃(𝜉(𝜏⁺_𝑘),𝜂(𝜏⁺_𝑘)), 𝑄₊=𝑄(𝜉(𝜏⁺_𝑘),𝜂(𝜏⁺_𝑘)). 𝜙(𝑥,𝑦) is a sufficiently smooth function with grad 𝜙(𝑥,𝑦)≠0, and 𝜏_𝑘(𝑘∈𝑁) is the time of the 𝑘th jump.

The proof of this lemma is referred to Simeonov and Baĭnov [17].

In the following, we suppose this periodic solution of system (1.3) with period 𝑇 passes through the points 𝐻⁺₁((1−𝑝)ℎ, 𝜄+(1−𝑞)𝜂₀)∈𝑁 and 𝐻₁(ℎ,𝜂₀)∈𝑀 (see Figures 2–6). As the expression and the period of this solution are unknown, we discuss the stability of this positive periodic solution by Lemma 4.4. In our case, 𝑃𝑥,𝑦=𝑓𝑥1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄,𝛼𝛽𝜉(𝑇)(𝑇)=𝑥𝑦,𝑄𝑥,𝑦=𝑒𝑦𝑥−𝑦−1𝑥,𝑦=−𝑝𝑥,𝑥,𝑦=𝜄−𝑞𝑦,𝜙𝑥,𝑦=𝑥−ℎ,,𝜂ℎ,𝜂₀,𝜉𝑇⁺,𝜂𝑇⁺=1−𝑝ℎ,𝜄+1−𝑞𝜂₀. (4.4)

Then 𝜕𝑃𝜕𝑥=𝑓1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄𝑥𝑦+𝑓𝑥𝑎₁−2𝑎₃𝑥−𝑎₄𝑦,𝜕𝑄𝜕𝑦=𝑒𝑥−𝑦−1−𝑒𝑦,𝜕𝛼𝜕𝑥=−𝑝,𝜕𝛼𝜕𝑦=0,𝜕𝛽𝜕𝑥=0,𝜕𝛽𝜕𝑦=−𝑞,𝜕𝜙𝜕𝑥=1,𝜕𝜙𝜕𝑦=0,_{Δ𝑘=𝑃+−+𝜕𝛽/𝜕𝑦𝜕𝜙/𝜕𝑥𝜕𝛽/𝜕𝑥𝜕𝜙/𝜕𝑦𝜕𝜙/𝜕𝑥𝑃+𝑄𝜕𝜙/𝜕𝑥+𝑄𝜕𝜙/𝜕𝑦+()−+𝜕𝛼/𝜕𝑥𝜕𝜙/𝜕𝑦𝜕𝛼/𝜕𝑦𝜕𝜙/𝜕𝑥𝜕𝜙/𝜕𝑦𝑃=𝑃𝜉𝑇𝜕𝜙/𝜕𝑥+𝑄𝜕𝜙/𝜕𝑦1−𝑞+𝑇,𝜂+𝑃𝜉(𝑇)(𝑇)=,𝜂1−𝑞1−𝑝1+𝑎11−𝑝ℎ+𝑎2𝜂𝜄+1−𝑞0−𝑎31−𝑝2ℎ2−𝑎4ℎ𝜂1−𝑝𝜄+1−𝑞01+𝑎1ℎ+𝑎2𝜂0−𝑎3ℎ2−𝑎4ℎ𝜂0.} (4.5)

Set 𝑁=^𝑇₀𝜕𝑃𝑥(𝑡)(𝑡)+𝜕𝑥,𝑦𝜕𝑄𝑥(𝑡)(𝑡)=𝜕𝑦,𝑦𝑑𝑡_𝑇0𝑓1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄𝑥𝑦+𝑓𝑥𝑎₁−2𝑎₃𝑥−𝑎₄𝑦=+𝑒𝑥−𝑦−1−𝑒𝑦𝑑𝑡_𝑇0𝑓1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄𝑥𝑦𝑑𝑡+^𝑇₀𝑒+𝑥−𝑦−1𝑑𝑡_𝑇0𝑓𝑥𝑎₁−2𝑎₃𝑥−𝑎₄𝑦−𝑒𝑦𝑑𝑡, (4.6) if (𝑥(𝑡),𝑦(𝑡)) are period function with 𝑇, then ^𝑇₀𝑓1+𝑎₁𝑥+𝑎₂𝑦−𝑎₃𝑥²−𝑎₄𝑥𝑦𝑑𝑡=^𝑇₀𝑑ln𝑥(𝑡)=0,_𝑇0𝑒𝑥−𝑦−1𝑑𝑡=^𝑇₀𝑑ln𝑦(𝑡)=0, (4.7) hence 𝑁=^𝑇₀𝑓𝑥𝑎₁−2𝑎₃𝑥−𝑎₄𝑦−𝑒𝑦𝑑𝑡<0. (4.8)

Suppose 𝐻(𝑡)=(𝜕𝑃/𝜕𝑥)(𝜉(𝑡),𝜂(𝑡))+(𝜕𝑄/𝜕𝑦)(𝜉(𝑡),𝜂(𝑡)), then 𝜇₂=Δ₁exp^𝑇₀𝜕𝑃𝜉(𝑡)(𝑡)+𝜕𝑥,𝜂𝜕𝑄𝜉(𝑡)(𝑡)=𝑟ℎℎ𝜂𝜕𝑦,𝜂𝑑𝑡1−𝑝𝑟−1−𝑝/𝐾−𝑎1−𝑝₀+𝜔−𝑏𝜂₀+𝜔𝑟−𝑟ℎ/𝐾−𝑎ℎ𝜂₀−𝑏𝜂₀exp^𝑇₀(𝐻(𝑡))𝑑𝑡. (4.9) Because (𝜉(𝑡),𝜂(𝑡)) is period solution of system (1.3), we have ∫^𝑇₀𝐻(𝑡)𝑑𝑡<0. Obviously, |𝜇₂|<1 if 1−𝑞1−𝑝1+𝑎₁1−𝑝ℎ+𝑎₂𝜄+1−𝑞𝜂₀−𝑎₃1−𝑝²ℎ²−𝑎₄1−𝑝ℎ𝜄+1−𝑞𝜂₀1+𝑎₁ℎ+𝑎₂𝜂₀−𝑎₃ℎ²−𝑎₄ℎ𝜂₀≤1. (4.10)

Theorem 4.5. If system (1.3) has an order one periodic solution, this order one periodic solution is orbitally asymptotically stable provided 1−𝑞1−𝑝1+𝑎₁1−𝑝ℎ+𝑎₂𝜄+1−𝑞𝜂₀−𝑎₃1−𝑝²ℎ²−𝑎₄1−𝑝ℎ𝜄+1−𝑞𝜂₀1+𝑎₁ℎ+𝑎₂𝜂₀−𝑎₃ℎ²−𝑎₄ℎ𝜂₀≤1. (4.11)

5. Numerical Analysis and Discussion

In order to verify the theoretical results in this paper, we consider the following example 𝑥^′(𝑡)=0.5𝑥1−0.35𝑥−0.4𝑦−0.01𝑥²,𝑦−0.8𝑥𝑦𝑥<ℎ,_′(𝑡)=0.4𝑦𝑥−𝑦−1,Δ𝑥=−𝑝𝑥,𝑥=ℎ.Δ𝑦=𝜄−𝑞𝑦, (5.1) In numerical simulation, let 𝑎₁=−0.35, 𝑎₂=−0.4, 𝑎₃=0.01, 𝑎₄=0.8, 𝑒=0.4, and 𝑓=0.5, by calculating we know 𝑎₃<1+𝑎₁, 𝑥^∗=1.34, 𝑦^∗=0.34. If ℎ=2.4>𝑥^∗, 𝑝=0.4, (1−𝑝)ℎ=1.44>𝑥^∗, 𝑞=0.5,𝜄=0.8, then the time series and phase portrait can be seen in Figure 7, by analysis of Section 4, we know that in this case, system (1.3) has no periodic solution and all the solutions tend to the equilibrium 𝐵 after finite impulses. If ℎ=1.2<𝑥^∗, by Theorem 4.1, we know that system (1.3) has an order one periodic solution which is shown in Figure 8. As shown in Figure 9, if ℎ=1.4>𝑥^∗, (1−𝑝)ℎ=0.84<𝑥^∗, by Theorems 4.2 and 4.3, we know system (1.3) also has an order one periodic solution. The results show that prey-predator model with Beddington-DeAnglis functional response and impulsive state feedback control tends to stable state or be periodic, which means that the amount of prey is under control.

(a)

(b)

(c)

Figure 7 

Time series and portrait phase of system (1.3) with 𝑝=0.4, 𝑞=0.5, 𝑥0=0.72, 𝑦0=0.18, ℎ=2.4>𝑥∗, and (1−𝑝)ℎ=1.44>𝑥∗.

(a)

(b)

(c)

Figure 8 

Time series and portrait phase of system (1.3) with 𝑝=0.4, 𝑞=0.5, 𝑥0=0.72, 𝑦0=0.18, and ℎ=1.2<𝑥∗.

(a)

(b)

(c)

Figure 9 

Time series and portrait phase of system (1.3) with 𝑝=0.4, 𝑞=0.5, 𝑥0=0.84, 𝑦0=0.18, and ℎ=1.4>𝑥∗.

Choosing 𝜄 as a control parameter, the phase portraits of system (1.3) are shown in Figure 10 with different values 𝜄, which not only indicate that the prey can be suppressed below certain level by using the impulsive state feedback strategy when the amount of the prey reaches the economic threshold ℎ, but also indicate that it is more easy to control the amount of the prey below certain economic threshold ℎ as 𝜄 increasing. On the other hand, if choosing ℎ as a control parameter, we can see that more numbers of impulses are needed to make the system tend to a stable periodic solution, that is, it is more difficult to control the amount of the prey below economic threshold ℎ as ℎ is decreasing (see Figure 11). Figure 12 shows that it is more easy to control the amount of the prey under certain economic threshold ℎ as 𝑞 is increasing or 𝑝 is increasing (see Figure 13).

(a)

(b)

(c)

Figure 10 

The phase portraits of system (1.3) with 𝑝=0.4, 𝑞=0.5, 𝑥0=0.72, 𝑦0=0.18, ℎ=1.2<𝑥∗, and different 𝜄: (a) 𝜄=0.75; (b) 𝜄=0.4; (c) 𝜄=0.14.

(a)

(b)

(c)

Figure 11 

The phase portraits of system (1.3) with 𝑝=0.4, 𝑞=0.4, 𝑥0=0.22, 𝑦0=0.114, 𝜄=0.6, and different ℎ: (a) ℎ=1.2; (b) ℎ=0.81; (c) ℎ=0.23.

(a)

(b)

(c)

Figure 12 

The phase portraits of system (1.3) with 𝑥0=0.72, 𝑦0=0.114, 𝑞=0.5, 𝜄=0.2, and different 𝑝: (a) 𝑝=0.79; (b) 𝑝=0.59; (c) 𝑝=0.39.

(a)

(b)

(c)

Figure 13 

The phase portraits of system (1.3) with 𝑥0=0.72, 𝑦0=0.114, 𝑝=0.4, 𝜄=0.62, and different 𝑞: (a) 𝑞=0.9; (b) 𝑞=0.6; (c) 𝑞=0.2.

According to the above analysis, it is obvious that the prey can be well suppressed below certain level by using impulsive state feedback strategy for the fact that the system has stable periodic solution under some conditions. The key to the system with impulsive state feedback control is to give the suitable feedback state (the value of ℎ) and the control parameters (𝑝, 𝑞 and 𝜄) according to practice.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11171284), the Fujian Provincial Natural Science Foundation of China (2012J01012), the Fujian Provincial Education Fundation (JA12198), and the Scientific Research Foundation of Jimei University of China (ZC2011003).