Dynamics of a Beddington-DeAngelis Type Predator-Prey Model with Impulsive Effect

Abstract

In view of the logical consistence, the model of a two-prey one-predator system with Beddington-DeAngelis functional response and impulsive control strategies is formulated and studied systematically. By using the Floquet theory of impulsive equation, small amplitude perturbation method, and comparison technique, we obtain the conditions which guarantee the global asymptotic stability of the two-prey eradication periodic solution. We also proved that the system is permanent under some conditions. Numerical simulations find that the system appears the phenomenon of competition exclusion.

1. Introduction

In an ecological system, understanding the dynamical relationships between predator and prey is one of the central goals. One important component of the predator-prey relationships is the predator's functional response. Functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes. In 1965, Holling [1] gave three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. The Beddington-DeAngelis functional response was introduced by Beddington [2] and DeAngelis et al. [3] in 1975. It is similar to the Holling type functional response but contains an extra term describing mutual interference by predators. Thus, a predator-prey model with Beddington-DeAngelis functional response is as the following form [2, 3]: 𝑑𝑥(𝑡)(𝑡)𝑥(𝑡)𝑑𝑡=𝑎𝑥1−𝐾−𝑒𝑥(𝑡)𝑦(𝑡)(𝑡)(𝑡),(𝑡)𝑏𝑦+𝑥+𝑐𝑑𝑦(𝑡)+(𝑡)𝑦(𝑡)𝑑𝑡=−𝐷𝑦𝑚𝑒𝑥(𝑡)(𝑡),𝑏𝑦+𝑥+𝑐 (1) where 𝑥(𝑡) and 𝑦(𝑡) represent the population density of the prey and the predator at time 𝑡, respectively. Usually, 𝐾 is called the carrying capacity of the prey. The constant 𝑎 is called intrinsic growth rate of the prey. 𝑚 is the conversion rate and 𝐷 is the death rate of the predator. The term 𝑏𝑦(𝑡) measures the mutual interference between predators. In 2004, Fan and Kuang [4] explored the dynamics of the nonautonomous, spatially homogeneous, and continuous time predator-prey system with the Beddington-DeAngelis functional response. The explorations involved the permanence, extinction, and global asymptotic stability (general nonautonomous case); the existence, uniqueness, and stability of a positive (almost) periodic solution; and a boundary (almost) periodic solution for the periodic (almost periodic) case.

It is straightforward to generalize the above two-species model to the situation when two prey species competing for the same resource but the two prey species are predated by one predator species. This results in the following three-species model: 𝑑𝑥(𝑡)𝑑𝑡=𝑎₁𝑥(𝑡)𝑥(𝑡)(𝑡)1−+𝑦𝐾−𝑒₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑧(𝑡)+𝑥(𝑡)+𝑐₁,𝑑𝑦(𝑡)𝑑𝑡=𝑎₂𝑦(𝑡𝑥(𝑡)(𝑡))1−+𝑦𝐾−𝑒₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑧(𝑡)+𝑦(𝑡)+𝑐₂,𝑑𝑧(𝑡)(𝑡)+𝑚𝑑𝑡=−𝐷𝑧¹𝑒₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑧(𝑡)+𝑥(𝑡)+𝑐₁+𝑚₂𝑒₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑧(𝑡)+𝑦(𝑡)+𝑐₂. (2)

If 𝑥 and 𝑦 are identical species, then 𝑎₁=𝑎₂=𝑎, 𝑒₁=𝑒₂=𝑒, 𝑏₁=𝑏₂=𝑏, 𝑐₁=𝑐₂=𝑐, and 𝑚₁=𝑚₂=𝑚. Let 𝑁=𝑥+𝑦; then, the model that is consistent with (2) should take the form 𝑑𝑁(𝑡)(𝑡)𝑁(𝑡)𝑑𝑡=𝑎𝑁1−𝐾−𝑒𝑁(𝑡)𝑧(𝑡)(𝑡)(𝑡),(𝑡)𝑏𝑧+𝑁+𝑐𝑑𝑧(𝑡)+(𝑡)𝑧(𝑡)𝑑𝑡=−𝐷𝑧𝑚𝑒𝑁(𝑡)(𝑡).𝑏𝑧+𝑁+𝑐 (3)

If model (2) is logically sound one, it should reduce itself to (3) in this situation. However, the result is not the case; instead, it becomes 𝑑𝑁(𝑡)(𝑡)𝑁(𝑡)𝑑𝑡=𝑎𝑁1−𝐾−𝑒𝑥(𝑡)𝑧(𝑡)(𝑡)(𝑡)−(𝑡)𝑧(𝑡)𝑏𝑧+𝑥+𝑐𝑒𝑦(𝑡)(𝑡),(𝑡)𝑏𝑧+𝑦+𝑐𝑑𝑧(𝑡)+(𝑡)𝑧(𝑡)𝑑𝑡=−𝐷𝑧𝑚𝑒𝑥(𝑡)(𝑡)+(𝑡)𝑧(𝑡)𝑏𝑧+𝑥+𝑐𝑚𝑒𝑦(𝑡)(𝑡).𝑏𝑧+𝑦+𝑐 (4)

According to the logical consistence proposed by Arditi and Michalski in 1996 [5], a logically consistent model of the two-prey one-predator interaction should take the form 𝑑𝑥(𝑡)𝑑𝑡=𝑎₁𝑥(𝑡)𝑥(𝑡)(𝑡)1−+𝑦𝐾−𝑒₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₁,𝑑𝑦(𝑡)𝑑𝑡=𝑎₂𝑦(𝑡)𝑥(𝑡)(𝑡)1−+𝑦𝐾−𝑒₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₂,𝑑𝑧(𝑡)(𝑡)+𝑚𝑑𝑡=−𝐷𝑧¹𝑒₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₁+𝑚₂𝑒₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₂. (5)

There are a number of factors in the environment to be considered in predator-prey models. One of the important factors is an impulsive perturbation such as fire and flood; these are not suitable to be considered continuously. The impulsive perturbations bring sudden changes to the system. Let us think of prey as a pest and predator as a natural enemy of prey. There are many ways to beat agricultural pests, for example, biological control such as harvesting on prey and releasing natural enemies [6–11] and chemical control such as spreading pesticides. However, integrated pest management (IPM) is a more effective approach to control pests in farm, which has been proved by experiments (see [12–14]). Such tactics are discontinuous and periodical. With the idea of periodic forcing and impulsive perturbations, Baek [15] has investigated the predator-prey model with periodic constant impulsive immigration of the predator and periodic variation in the intrinsic growth rate of the prey. And an ecological model consisting of two preys and one predator with impulsive release of the predator is studied in [16]. In this paper, we will consider an ecological model consisting of two preys and one predator with impulsive control strategy. We give some assumptions as follows.(H1)Two prey species compete the same resource.(H2)Functional response is effected for mutual interference by predators.

Thus, the model can be described by the following impulsive differential equations:𝑑𝑥(𝑡)𝑑𝑡=𝑎₁𝑥(𝑡)𝑥(𝑡)(𝑡)1−+𝑦𝐾−𝑒₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₁,𝑑𝑦(𝑡)𝑑𝑡=𝑎₂𝑦(𝑡𝑥(𝑡)(𝑡))1−+𝑦𝐾−𝑒₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₂,𝑑𝑧(𝑡)(𝑡)+𝑚𝑑𝑡=−𝐷𝑧¹𝑒₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₁+𝑚₂𝑒₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₂,𝑥𝑡𝑡≠𝑛𝑇,₊=1−𝑝₁𝑥(𝑡),𝑦𝑡⁺=1−𝑝₂𝑦(𝑡),𝑧𝑡⁺=1−𝑝₃𝑧(𝑡)𝑥0+𝑞,𝑡=𝑛𝑇,⁺,𝑦0⁺,𝑧0⁺=𝑥₀,𝑦₀,𝑧₀=𝐗₀, (6)

where 𝑥(𝑡), 𝑦(𝑡), and 𝑧(𝑡) are the densities of the two preys and a predator at time 𝑡, respectively. 𝐾 is the total carrying capacity of two prey species; 𝑎_𝑖(𝑖=1,2) are the intrinsic growth rate; 𝑒_𝑖(𝑖=1,2) are the catching rate; 𝑏_𝑖(𝑖=1,2) scale the impact of predators mutual interference; 𝑐_𝑖(𝑖=1,2) are saturation constants; 𝑚_𝑖(𝑖=1,2) denote the efficiency with which resources are converted to new consumers; 𝐷 is the mortality rate of the predator. 𝑇 is the period of the impulsive effect; 𝑛∈𝑁;𝑁 is the set of all nonnegative integers; 𝑝_𝑖(𝑖=1,2,3) represent the fractions of prey and predator which die due to pesticides; 𝑞 is the release amount of the predator at 𝑡=𝑛𝑇.

The paper is arranged as follows. In Section 2, we give some notations and lemmas. In Section 3, we consider the local stability and global asymptotic stability of the two-prey eradication periodic solution by using Floquet theory of the impulsive equation, small amplitude perturbation, and techniques of comparison, and in Section 4 we show that the system is permanent. The paper ends with some interesting numerical simulations and conclusions.

2. Preliminaries

Firstly, we give some notations, definitions, and lemmas which will be useful for our main results.

Let 𝑅₊=[0,+∞), 𝑅³₊={𝑋=(𝑥,𝑦,𝑧)∈𝑅³∣𝑥⩾0,𝑦⩾0,𝑧⩾0}. Denoted by 𝑓=(𝑓₁,𝑓₂,𝑓₃)^𝑇 the map defined by the right hand of the former three equations in system (6) and denoted by 𝑁 the set of all nonnegative integers. Let 𝑉∶𝑅₊×𝑅³₊→𝑅₊; then, 𝑉 is said to belong to class 𝑉₀ if(1)𝑉 is continuous in (𝑛𝑇,(𝑛+1)𝑇]×𝑅³₊ and, for each 𝑋∈𝑅³₊,𝑛∈𝑁, lim_{(𝑡,𝑦)→(𝑛𝑇}⁺_,𝑋)𝑉(𝑡,𝑦)=𝑉(𝑛𝑇⁺,𝑋) exists;(2)𝑉 is locally Lipschitzian in 𝑋;

Definition 1. Let 𝑉∈𝑉₀; then, for (𝑡,𝑋)∈(𝑛𝑇,(𝑛+1)𝑇]×𝑅³₊, the upper right derivative of 𝑉(𝑡,𝑋) with respect to the impulsive differential system (6) is defined as 𝐷⁺𝑉(𝑡,𝑋)=lim_ℎ→0+1supℎ×𝑉𝑡+ℎ,𝑋+ℎ𝑓(𝑡,𝑋)−𝑉(𝑡,𝑋). (7)

The solution of system (6) is a piecewise continuous function 𝑋∶𝑅₊→𝑅³₊, 𝑋(𝑡) is continuous on (𝑛𝑇,(𝑛+1)𝑇], 𝑛∈𝑁, and 𝑋(𝑛𝑇⁺)=lim_{𝑡→𝑛𝑇}⁺𝑋(𝑡) exists. The smoothness properties of 𝑓 guarantee the global existence and uniqueness of solution of system (6) (see [17] for the details).

Definition 2. System (6) is said to be permanent if there exists a compact Ω⊂Int𝑅³₊ such that every solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (6) will eventually enter and remain in the region Ω.
The following lemma is obvious.

Lemma 3. Let 𝑋(𝑡) be a solution of system (6) with 𝑋(0⁺)⩾0; then, 𝑋(𝑡)⩾0 for all 𝑡⩾0. And, furthermore, 𝑋(𝑡)>0,𝑡>0 if 𝑋(0⁺)>0.

We will use an important comparison theorem on impulsive differential equation.

Lemma 4 (comparison theorem [17]). Suppose 𝑉∈𝑉₀. Assume that 𝐷⁺𝑉()((𝑉𝑡𝑡,𝑋≤𝑔𝑡,𝑉𝑡,𝑋)),𝑡≠𝑛𝑇,𝑡,𝑋⁺≤𝜓_𝑛(𝑉(𝑡,𝑋)),𝑡=𝑛𝑇, (8) where 𝑔∶𝑅₊×𝑅₊→𝑅 is continuous in (𝑛𝑇,(𝑛+1)𝑇]×𝑅₊ and, for 𝑢∈𝑅₊, 𝑛∈𝑁, lim_{(𝑡,𝑣)→(𝑛𝑇}⁺_,𝑢)𝑔(𝑡,𝑣)=𝑔(𝑛𝑇⁺,𝑢) exists and 𝜓_𝑛∶𝑅₊→𝑅₊ is nondecreasing. Let 𝑟(𝑡) be the maximal solution of the scalar impulsive differential equation 𝑑𝑢(𝑡)((𝑡𝑢𝑡𝑑𝑡=𝑔𝑡,𝑢)),𝑡≠𝑛𝑇,₊≤𝜓_𝑛(𝑢(𝑡𝑢0)),𝑡=𝑛𝑇,₊=𝑢₀, (9) existing on [0,+∞]. Then, 𝑉(0⁺,𝑋₀)⩽𝑢₀ implies that 𝑉(𝑡,𝑋(𝑡))⩽𝑟(𝑡) for all 𝑡⩾0, where 𝑋(𝑡) is any solution of system (6).

To research the stability of the prey-free periodic solution of (6), we present the Floquet theory for the linear 𝑇-periodic impulsive equations: 𝑑𝑋(𝑡)(𝑡)𝑋(𝑡)𝑑𝑡=𝐴,𝑡≠𝜏_𝑘𝑋𝑡,𝑡∈𝑅,₊=𝐸+𝐵_𝑘𝑋(𝑡),𝑡=𝜏_𝑘,𝑘∈ℤ. (10) Then, we introduce the following conditions. (H2.1)𝐴(⋅)∈𝑃𝐶(𝑅,𝐶^𝑛×𝑛) and 𝐴(𝑡+𝑇)=𝐴(𝑡), (𝑡∈𝑅), where 𝑃𝐶(𝑅,𝐶^𝑛×𝑛) is the set of all piecewise continuous matrix functions which is left continuous at 𝑡=𝜏_𝑘 and 𝐶^𝑛×𝑛 is the set of all 𝑛×𝑛 matrices. (H2.2)𝐵_𝑘∈𝐶^𝑛×𝑛, det(𝐸+𝐵_𝑘)≠0, and 𝜏_𝑘<𝜏_𝑘+1. (H2.3)There exists a 𝑞∈ℕ such that 𝐵_𝑘+𝑞=𝐵_𝑘,𝜏_𝑘+𝑞=𝜏_𝑘+𝑇, and 𝑘∈ℤ.

Let Φ(𝑡) be a fundamental matrix of (10); then, there exists a unique nonsingular matrix 𝑀∈𝐶^𝑛×𝑛 such that Φ(𝑡+𝑇)=Φ(𝑡)𝑀,(𝑡∈𝑅). (11)

For this equality, we call the constant matrix 𝑀 is the monodromy matrix of (10) (corresponding to the fundamental matrix Φ(𝑡))

All monodromy matrices of (10) are similar and have the same eigenvalues. The eigenvalues 𝜇₁,…,𝜇_𝑛 of the monodromy matrices are called the Floquet multipliers of (10).

Lemma 5 (Floquet theory [18]). Let conditions (H2.1)–(H2.3) hold. Then, the linear 𝑇-periodic impulsive equation (10) is(1)stable if and only if all multipliers 𝜇_𝑗(𝑗=1,…,𝑛) of (10) satisfy the inequality |𝜇_𝑗|≤1, and, moreover, to those 𝜇_𝑗, for which |𝜇_𝑗|=1, correspond simple elementary divisors;(2)asymptotically stable if and only if all multipliers 𝜇_𝑗(𝑗=1,…,𝑛) of (10) satisfy the inequality |𝜇_𝑗|<1;(3)unstable if |𝜇_𝑗|>1 for some 𝑗=1,…,𝑛.
Now, one gives some basic properties about the following subsystem of system (6): 𝑑𝑧(𝑡)(𝑡)𝑧𝑡𝑑𝑡=−𝐷𝑧,𝑡≠𝑛𝑇,₊=1−𝑝₃𝑧(𝑡)𝑧0+𝑞,𝑡=𝑛𝑇,₊=𝑧₀. (12)

Then we can easily obtain the following results.

Lemma 6. (1) 𝑧^∗(𝑡)=𝑞exp(−𝐷(𝑡−𝑛𝑇))/(1−(1−𝑝₃)exp(−𝐷𝑇)),𝑡∈(𝑛𝑇,(𝑛+1)𝑇], 𝑛∈𝑁, and 𝑧^∗(0⁺)=𝑞/(1−(1−𝑝₃)exp(−𝐷𝑇)) is a positive periodic solution of (12).
(2) 𝑧(𝑡)=(1−𝑝₃)^𝑛(𝑧₀−(𝑞/(1−(1−𝑝₃)exp(−𝐷𝑇))))exp(−𝐷𝑡)+𝑧^∗(𝑡) is the solution of (12) with 𝑧₀⩾0, 𝑡∈(𝑛𝑇,(𝑛+1)𝑇], and 𝑛∈𝑁.
(3) All solutions 𝑧(𝑡) of (12) with 𝑧₀⩾0 tend to 𝑧^∗(𝑡). That is, |𝑧(𝑡)−𝑧^∗(𝑡)|→0 as 𝑡→∞.

Therefore, we obtain the complete expression for the two-prey eradication periodic solution (0,0,𝑧^∗(𝑡)) of system (6). We will study the stability of the two-prey eradication periodic solution of system (6) in the next section.

3. Prey Eradication Periodic Solution

Firstly, we show that all solutions of (6) are uniformly ultimately bounded.

Theorem 7. There exists a constant 𝑀>0, such that 𝑥(𝑡)⩽𝑀, 𝑦(𝑡)⩽𝑀, 𝑧(𝑡)⩽𝑀 for all 𝑡 large enough, where (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) is a solution of system (6).

Proof. Define 𝑉(𝑡,𝑋(𝑡)) such that 𝑉(𝑡,𝑋(𝑡))=𝑚₁𝑥(𝑡)+𝑚₂𝑦(𝑡)+𝑧(𝑡); (13) then, 𝑉∈𝑉₀. We calculate the upper right derivative of 𝑉(𝑡,𝑋) along a solution of system (6) and get the following impulsive differential inequality: 𝐷⁺𝑉(𝑡)(𝑡)⩽𝑎+𝐿𝑉₁+𝐿𝑚₁𝑥(𝑡)−𝑎₁𝑚₁𝐾𝑥²(𝑡)+𝑎₂+𝐿𝑚₂𝑦(𝑡)−𝑎₂𝑚₂𝐾𝑦²(𝑡)+()𝑧(𝑡)𝑉𝑡𝐿−𝐷,𝑡≠𝑛𝑇,₊⩽𝑉(𝑡)+𝑞,𝑡=𝑛𝑇. (14)
Let 0<𝐿<𝐷; then, 𝐷⁺𝑉(𝑡)+𝐿𝑉(𝑡) is bounded. Select 𝐿₁ and 𝐿₂ such that 𝐷⁺𝑉(𝑡)≤−𝐿₁V(𝑡)+𝐿₂𝑉𝑡,𝑡≠𝑛𝑇,₊≤𝑉(𝑡)+𝑞,𝑡=𝑛𝑇, (15) where 𝐿₁ and 𝐿₂ are two positive constants.
According to Lemma 4, we have 𝑉(𝑡)⩽𝑉0⁺−𝐿₂𝐿₁exp−𝐿₁𝑡+𝑞1−exp−𝑛𝐿₁𝑇1−exp−𝐿₁𝑇exp−𝐿₁()+𝐿𝑡−𝑛𝑇²𝐿₁, (16) where 𝑡∈(𝑛𝑇,(𝑛+1)𝑇]. Therefore, 𝑉(𝑡,𝑋(𝑡)) is ultimately bounded by a constant and there exists a constant 𝑀>0, such that 𝑥(𝑡)⩽𝑀, 𝑦(𝑡)⩽𝑀, 𝑧(𝑡)⩽𝑀 for all 𝑡 large enough. The proof is completed.

Now, we study the stability of the two-prey eradication periodic solution of system (6).

Theorem 8. Let (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be any solution of system (6); then, (0,0,𝑧^∗(𝑡)) is said to be locally asymptotically stable if 𝑎₁𝑒𝑇+¹𝐷𝑏₁𝑏ln1−¹𝑞1−exp(−𝐷𝑇)𝑐₁1−1−𝑝₃exp(−𝐷𝑇)+𝑏₁𝑞1<ln1−𝑝₁,𝑎₂𝑒𝑇+²𝐷𝑏₂𝑏ln1−²𝑞1−exp(−𝐷𝑇)𝑐₂1−1−𝑝₃exp(−𝐷𝑇)+𝑏₂𝑞1<ln1−𝑝₂. (17)

Proof. The local stability of periodic solution (0,0,𝑧^∗(𝑡)) may be determined by considering the behavior of small amplitude perturbation of the solution. Define 𝑥(𝑡)(𝑡)(𝑡)(𝑡),𝑧(𝑡)(𝑡)=𝑢,𝑦=𝑣=𝑤+𝑧^∗(𝑡). (18)
Putting (18) into (6), the linearization of the system becomes 𝑑𝑢(𝑡)=𝑎𝑑t¹−𝑒₁𝑧^∗(𝑡)𝑏₁𝑧^∗(𝑡)+𝑐₁𝑢(𝑡),𝑑𝑣(𝑡)=𝑎𝑑𝑡²−𝑒₂𝑧^∗(𝑡)𝑏₂𝑧^∗(𝑡)+𝑐₂𝑣(𝑡),𝑑𝑤(𝑡)=𝑚𝑑𝑡¹𝑒₁𝑧^∗(𝑡)𝑏₁𝑧^∗(𝑡)+𝑐₁𝑢(𝑡)+𝑚₂𝑒₂𝑧^∗(𝑡)𝑏₂𝑧^∗(𝑡)+𝑐₂𝑣(𝑡)(𝑡),𝑢𝑡−𝐷𝑤𝑡≠𝑛𝑇,₊=1−𝑝₁𝑢(𝑡),𝑣𝑡⁺=1−𝑝₂𝑣(𝑡),𝑤𝑡⁺=1−𝑝₃𝑤(𝑡),𝑡=𝑛𝑇. (19)
Therefore, we have 𝑢(𝑡)𝑣(𝑡)𝑤(𝑡)(𝑡)𝑢(0)𝑣(0)𝑤(0)=Φ,0⩽𝑡<𝑇, (20) where Φ(𝑡) satisfies 𝑑Φ=𝑎𝑑𝑡¹−𝑒₁𝑧^∗(𝑡)𝑏₁𝑧^∗(𝑡)+𝑐₁000𝑎₂−𝑒₂𝑧^∗(𝑡)𝑏₂𝑧^∗(𝑡)+𝑐₂0𝑚₁𝑒₁𝑧^∗(𝑡)𝑏₁𝑧^∗(𝑡)+𝑐₁𝑚₂𝑒₂𝑧^∗(𝑡)𝑏₂𝑧^∗(𝑡)+𝑐₂Φ(𝑡)−𝐷 (21)
Φ(0)=𝐼, the identity matrix, and 𝑢𝑛𝑇⁺𝑣𝑛𝑇⁺𝑤𝑛𝑇⁺=1−𝑝₁0001−𝑝₂0001−𝑝₃𝑢()𝑣()𝑤()𝑛𝑇𝑛𝑇𝑛𝑇. (22)
The stability of the periodic solution (0,0,𝑧^∗(𝑡)) is determined by the eigenvalues of Θ=1−𝑝₁0001−𝑝₂0001−𝑝₃Φ(𝑇). (23)
If absolute values of all eigenvalues are less than one, then the periodic solution (0,0,𝑧^∗(𝑡)) is locally asymptotically stable. All eigenvalues of Θ are as follows: 𝜇₁=1−𝑝₁exp^𝑇₀𝑎₁−𝑒₁𝑧^∗(𝑡)𝑏₁𝑧^∗(𝑡)+𝑐₁,𝜇𝑑𝑡₂=1−𝑝₂exp^𝑇₀𝑎₂−𝑒₂𝑧^∗(𝑡)𝑏₂𝑧^∗(𝑡)+𝑐₂,𝜇𝑑𝑡₃=1−𝑝₃exp(−𝐷𝑇)<1. (24)
According to the Floquet theory of impulsive differential equation, (0,0,𝑧^∗(𝑡)) is locally asymptotically stable if |𝜇₁|<1 and |𝜇₂|<1; that is to say,𝑎₁𝑒𝑇+¹𝐷𝑏₁𝑏ln1−¹𝑞1−exp(−𝐷𝑇)𝑐₁1−1−𝑝₃exp(−𝐷𝑇)+𝑏₁𝑞1<ln1−𝑝₁,𝑎₂𝑒𝑇+²𝐷𝑏₂𝑏ln1−²𝑞1−exp(−𝐷𝑇)𝑐₂1−1−𝑝₃exp(−𝐷𝑇)+𝑏₂𝑞1<ln1−𝑝₂. (25) This completes the proof.

Theorem 9. Let (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be any solution of system (6); then, (0,0,𝑧^∗(𝑡)) is said to be globally asymptotically stable if the conditions of Theorem 8 hold and 𝑎𝑞⩾max¹𝑒₁exp(𝐷𝑇)−1−𝑝₃𝑏₁𝑀+2𝑀+𝑐₁,𝑎₂𝑒₂exp(𝐷𝑇)−1−𝑝₃𝑏₂𝑀+2𝑀+𝑐₂. (26)

Proof. By Theorem 8, we know that (0,0,𝑧^∗(𝑡)) is locally asymptotically stable. In the following, we will prove its global attraction. Let 𝑉(𝑡)=𝑥(𝑡)+𝑦(𝑡); then, we get 𝑉^′∣_(1.6)=𝑎₁𝑥(𝑡)−𝑎₁𝐾𝑥(𝑡)𝑥(𝑡)(𝑡)−𝑒+𝑦₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₁+𝑎₂𝑦(𝑡)−𝑎₂𝐾𝑦(𝑡𝑥(𝑡)(𝑡−𝑒)+𝑦)₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₂. (27) By Theorem 7, there exists a constant 𝑀>0, such that 𝑥(𝑡)⩽𝑀, 𝑦(𝑡)⩽𝑀, and 𝑧(𝑡)⩽𝑀 for each solution 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (6) with all 𝑡 large enough.
Thus, 𝑉^′∣_(1.6)⩽𝑎₁𝑥(𝑡)−𝑒₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑀+2𝑀+𝑐₁+𝑎₂𝑦(𝑡)−𝑒₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑀+2𝑀+𝑐₂. (28)
On the other hand, we have 𝑑𝑧(𝑡)=𝑚𝑑𝑡¹𝑒₁𝑥(𝑡)𝑧(𝑡)𝑏₁𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₁+𝑚₂𝑒₂𝑦(𝑡)𝑧(𝑡)𝑏₂𝑧(𝑡)+𝑥(𝑡)+𝑦(𝑡)+𝑐₂(𝑡)(𝑡)𝑧𝑡−𝐷𝑧⩾−𝐷𝑧,𝑡≠𝑛𝑇,₊=1−𝑝₃𝑧(𝑡)+𝑞,𝑡=𝑛𝑇. (29) By Lemmas 4 and 5, we know that there exists a 𝑡₁>0, and 𝜖>0 small enough, such that 𝑧(𝑡)⩾𝑧^∗(𝑡)−𝜖; for all 𝑡⩾𝑡₁, we have 𝑧(𝑡)⩾𝑞exp(−𝐷𝑇)1−1−𝑝₃exp(−𝐷𝑇)−𝜖≜𝛾. (30) Then, 𝑉^′∣₍₆₎⩽𝑎₁−𝑒₁𝛾𝑏₁𝑀+2𝑀+𝑐₁𝑥(𝑡)+𝑎₂−𝑒₂𝛾𝑏₂𝑀+2𝑀+𝑐₂𝑦(𝑡), (31) if 𝑄₁≜𝑎₁−(𝑒₁𝛾/(𝑏₁𝑀+2𝑀+𝑐₁))<0 and 𝑄₂≜𝑎₂−(𝑒₂𝛾/(𝑏₂𝑀+2𝑀+𝑐₂))<0; that is to say, 𝑎𝑞⩾max¹𝑒₁exp(𝐷𝑇)−1−𝑝₃𝑏₁𝑀+2𝑀+𝑐₁,𝑎₂𝑒₂exp(𝐷𝑇)−1−𝑝₃𝑏₂𝑀+2𝑀+𝑐₂. (32) Then, for 𝑡⩾𝑡₁, we have 𝑉^′∣₍₆₎⩽−𝑄𝑉(𝑡), (33) where 𝑄=min{|𝑄₁|,|𝑄₂|}, so 𝑉(𝑡)→0 and 𝑥(𝑡)→0,𝑦(𝑡)→0 as 𝑡→∞. Notice that the limit system of the system (6) is exactly as system (10); together with Lemma 5, we know that the two-prey eradication periodic solution (0,0,𝑧^∗(𝑡)) is a global attractor. The proof is completed.

4. Permanence

In this section, we investigate the permanence of the system (6).

Theorem 10. The system (6) is permanent if 𝑎₁𝑒𝑇+¹𝐷𝑏₁𝑏ln1−¹𝑞1−exp(−𝐷𝑇)𝑐₁1−1−𝑝₃exp(−𝐷𝑇)+𝑏₁𝑞1>ln1−𝑝₁,𝑎₂𝑒𝑇+²𝐷𝑏₂𝑏ln1−²𝑞1−exp(−𝐷𝑇)𝑐₂1−1−𝑝₃exp(−𝐷𝑇)+𝑏₂𝑞1>ln1−𝑝₂,𝑎𝑞≤max¹𝑒₁exp(𝐷𝑇)−1−𝑝₃𝑏₁𝑀+2𝑀+𝑐₁,𝑎₂𝑒₂exp(𝐷𝑇)−1−𝑝₃𝑏₂𝑀+2𝑀+𝑐₂. (34)

Proof. Let 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be any solution of the system (6) with 𝑋(0)>0. From Theorem 7, we assume that 𝑥(𝑡)⩽𝑀, 𝑦(𝑡)⩽𝑀, 𝑧(𝑡)⩽𝑀, and 𝑀>max{𝑎₁𝑐₁/𝑒₁,𝑎₂𝑐₂/𝑒₂}. Let 𝜁₃=(𝑞exp(−𝐷𝑇))/(1−(1−𝑝₃)exp(−𝐷𝑇))−𝜖,𝜖>0. From Lemma 5, clearly, we have 𝑧(𝑡)⩾𝜁₃ for all 𝑡 large enough. Now, we will find 𝜁₁ and 𝜁₂ such that 𝑥(𝑡)⩾𝜁₁ and 𝑦(𝑡)⩾𝜁₂ for 𝑡 large enough. We will do this in the following two steps.
Step  1. We will prove that there exists a 𝑡₁∈(0,+∞) such that 𝑥(𝑡₁)⩾𝜁₁. Otherwise, 𝑥(𝑡)<𝜁₁ for all 𝑡>0.
We can choose 𝜁₁>0,𝜖₁>0 to be small enough such that 𝑒𝛿=¹𝑎₁𝜁₁𝑏₁+𝑒₂𝑎₂𝑀𝑏₂<𝐷,𝜂=1−𝑝₁exp^(𝑛+1)𝑇_𝑛𝑇𝑎₁1−2𝑀𝐾−𝑒₁𝑣^∗(𝑡)+𝜖₁𝑐₁𝑑𝑡>1. (35) From system (6), we can obtain that 𝑑𝑧(𝑡)⩽()𝑧(𝑡)𝑧𝑡𝑑𝑡−𝐷+𝛿,𝑡≠𝑛𝑇,₊=1−𝑝₃𝑧(𝑡)𝑧0+𝑞,𝑡=𝑛𝑇,₊=𝑧₀. (36) By Lemma 4, we have 𝑧(𝑡)⩽𝑣(𝑡) and 𝑣(𝑡)→𝑣^∗(𝑡) as 𝑡→+∞, where 𝑣(𝑡) is the solution of 𝑑𝑣(𝑡)=()𝑣(𝑡)𝑣𝑡𝑑𝑡−𝐷+𝛿,𝑡≠𝑛𝑇,₊=1−𝑝₃𝑣(𝑡)𝑣0+𝑞,𝑡=𝑛𝑇,₊=𝑧₀, (37) and 𝑣^∗(𝑡)=𝑞exp((−𝐷+𝛿)(𝑡−𝑛𝑇))/(1−(1−𝑝₃)exp((−𝐷+𝛿)𝑇)), 𝑡∈(𝑛𝑇,(𝑛+1)𝑇], and 𝑛∈𝑁. Therefore, there exists a 𝑇₁>0; when 𝑡>𝑇₁, we have 𝑧(𝑡)⩽𝑣(𝑡)<𝑣^∗(𝑡)+𝜖₁ and 𝑑𝑥(𝑡)(𝑡)𝑎𝑑𝑡⩾𝑥₁1−2𝑀𝐾−𝑒₁𝑣^∗(𝑡)+𝜖₁𝑐₁𝑥𝑡,𝑡≠𝑛𝑇,₊=1−𝑝₁𝑥(𝑡),𝑡=𝑛𝑇. (38) Let 𝑁₁∈𝑁 and 𝑁₁𝑇⩾𝑇₁; integrating (38) on (𝑛𝑇,(𝑛+1)𝑇], 𝑛⩾𝑁₁, we can obtain 𝑥((𝑛+1)𝑇)⩾𝑥𝑛𝑇⁺×exp^(𝑛+1)𝑇_𝑛𝑇𝑎₁1−2𝑀𝐾−𝑒₁𝑣^∗(𝑡)+𝜖₁𝑐₁()𝑑𝑡=𝑥𝑛𝑇1−𝑝₁×exp^(𝑛+1)𝑇_𝑛𝑇𝑎₁1−2𝑀𝐾−𝑒₁𝑣^∗(𝑡)+𝜖₁𝑐₁()𝑑𝑡=𝑥𝑛𝑇𝜂. (39)
Then, 𝑥((𝑁₁+𝑘)𝑇)⩾𝑥(𝑁₁𝑇)𝜂^𝑘→∞ as 𝑘→∞, which is a contradiction to the boundedness of 𝑥(𝑡). Hence, there exists a 𝑡₁>0 such that 𝑥(𝑡₁)⩾𝜁₁.
Step  2. If 𝑥(𝑡)⩾𝜁₁, for all 𝑡⩾𝑡₁, then our aim is obtained. Hence, we only need to consider those solutions which leave the region Ω₀={𝑥(𝑡)∣𝑥(𝑡)<𝜁₁} and reenter it again. Letting 𝑡^∗=inf_𝑡⩾𝑡1{𝑥(𝑡)<𝜁₁}, we have 𝑥(𝑡)⩾𝜁₁ for 𝑡∈[𝑡₁,𝑡^∗]; by the continuity of 𝑥(𝑡), we have 𝑥(𝑡^∗)=𝜁₁. In this step, we have only to consider two possible cases.
Case  1.𝑡^∗=𝑛₁𝑇 for some 𝑛₁∈𝑁; then, (1−𝑝₁)𝜁₁⩽𝑥(𝑡^∗+)=(1−𝑝₁)𝑥(𝑡^∗)<𝜁₁. Select 𝑛₂,𝑛₃∈𝑁 such that 𝑛₂𝜖𝑇>ln₁/𝑀+𝑞,−𝐷+𝛿1−𝑝₁^𝑛2exp𝑛₂𝜎𝑇𝜂^𝑛3>1−𝑝₁^𝑛2+1exp𝑛₂+1𝜎𝑇𝜂^𝑛3>1, (40) where 𝜎=𝑎₁(1−(2𝑀/𝐾))−(𝑒₁M/𝑐₁)<0. Let 𝑇^′=𝑛₂𝑇+𝑛₃𝑇. In this case, we will show that there exists 𝑡₂∈(𝑡^∗,𝑡^∗+𝑇^′] such that 𝑥(𝑡₂)⩾𝜁₁; otherwise, 𝑥(𝑡)<𝜁₁ for 𝑡∈(𝑡^∗,𝑡^∗+𝑇^′]. Considering (37) with 𝑣(𝑡^∗+)=𝑧(𝑡^∗+), we have 𝑣(𝑡)=1−𝑝₃^{𝑛−(𝑛1+1)}×𝑣𝑡^∗+−𝑞1−1−𝑝₃)𝑇)()exp((−𝐷+𝛿×exp−𝐷+𝛿𝑡−𝑡^∗+𝑣^∗(𝑡) (41) for (𝑛−1)𝑇<𝑡⩽𝑛𝑇 and 𝑛₁+1⩽𝑛⩽𝑛₁+1+𝑛₂+𝑛₃; then, 𝑣(𝑡)−𝑣^∗(𝑡)<𝑀+𝑞exp(−𝐷+𝛿)𝑡−𝑡^∗<𝑀+𝑞exp(−𝐷+𝛿)𝑛₂𝑇<𝜖₁,𝑧(𝑡)⩽𝑣(𝑡)⩽𝑣^∗(𝑡)+𝜖₁ (42) for 𝑡^∗+𝑛₂𝑇⩽𝑡⩽𝑡^∗+𝑇^′, which implies that (38) holds for 𝑡∈[𝑡^∗+𝑛₂𝑇,𝑡^∗+𝑇^′]. As in Step 1, we have 𝑥𝑡^∗+𝑇^′⩾𝑥𝑡^∗+𝑛₂𝑇𝜂^𝑛3. (43) Since 𝑧(𝑡)⩽𝑀, we have 𝑑𝑥(𝑡)(𝑡)𝑎𝑑𝑡⩾𝑥₁1−2𝑀𝐾−𝑒₁𝑀𝑐₁(𝑡)𝑥𝑡=𝜎𝑥,𝑡≠𝑛𝑇,₊=1−𝑝₁𝑥(𝑡),𝑡=𝑛𝑇, (44) for 𝑡∈[𝑡^∗,𝑡^∗+𝑛₂𝑇]. Integrating (44) on [𝑡^∗,𝑡^∗+𝑛₂𝑇], we have 𝑥𝑡^∗+𝑛₂𝑇⩾𝜁₁1−𝑝₂^𝑛2exp𝜎𝑛₂𝑇. (45)
Thus, 𝑥(𝑡^∗+𝑇^′)⩾𝜁₁(1−𝑝₂)^𝑛2exp(𝜎𝑛₂𝑇)𝜂^𝑛3>𝜁₁, which bring on a contradiction. Now, let 𝑡=inf_{𝑡⩾𝑡∗}{𝑥(𝑡)⩾𝜁₁}. Then, 𝑥(𝑡)⩽𝜁₁ for 𝑡∈[𝑡^∗,𝑡) and 𝑥(𝑡)=𝜁₁. For 𝑡∈[𝑡^∗,𝑡), suppose that 𝑡∈(𝑡^∗+(𝑘−1)𝑇,𝑡^∗+𝑘𝑇], 𝑘∈𝑁, and 𝑘⩽𝑛₂+𝑛₃; from (44), we obtain 𝑥(𝑡)⩾𝑥𝑡^∗+1−𝑝₁^𝑘−1))𝜎𝑡exp((𝑘−1𝜎𝑇×exp𝑡−^∗+(𝑘−1)𝑇⩾𝜁₁1−𝑝₁^𝑘exp(𝑘𝜎𝑇)⩾𝜁₁1−𝑝₁^𝑛2+𝑛3exp𝜎𝑛₂+𝑛₃𝑇≡𝜁^′₁. (46)
Case  2. 𝑡^∗≠𝑛𝑇, 𝑛∈𝑁. Then, 𝑥(𝑡)⩾𝜁₁ for 𝑡∈[𝑡₁,𝑡^∗) and 𝑥(𝑡^∗)=𝜁₁. Suppose that 𝑡^∗∈(𝑛^′₁𝑇,(𝑛^′₁+1)𝑇), 𝑛^′₁∈𝑁. There are two possible cases for 𝑡∈(𝑡^∗,(𝑛^′₁+1)𝑇].
(a) 𝑥(𝑡)<𝜁₁ for all 𝑡∈(𝑡^∗,(𝑛^′₁+1)𝑇]. In this case, we will show that there exists 𝑡₂∈[(𝑛^′₁+1)𝑇,(𝑛^′₁+1)𝑇+𝑇^′] such that 𝑥(𝑡₂)⩾𝜁₁. Otherwise, 𝑥(𝑡)<𝜁₁ for all 𝑡∈[(𝑛^′₁+1)𝑇,(𝑛^′₁+1)𝑇+𝑇^′]. Then, 𝑥(𝑡)<𝜁₁ for all (𝑡^∗,(𝑛^′₁+1)𝑇+𝑇^′]; considering (37) with 𝑣((𝑛^′₁+1)𝑇⁺)=𝑧((𝑛^′₁+1)𝑇⁺), we have 𝑣(𝑡)=1−𝑝₃^{𝑛−(𝑛′1+1)}𝑣𝑛^′₁+1𝑇⁺−𝑞1−1−𝑝₃)𝑇)(𝑛exp((−𝐷+𝛿×exp−𝐷+𝛿)𝑡−^′₁+1𝑇+𝑣^∗(𝑡) (47) for 𝑛𝑇<𝑡⩽(𝑛+1)𝑇 and 𝑛^′₁+1⩽𝑛⩽𝑛^′₁+1+𝑛₂+𝑛₃. So we get 𝑣(𝑡)−𝑣^∗(𝑡<)𝑛)𝑀+𝑞×exp(−𝐷+𝛿𝑡−^′₁𝑇<()𝑛+1𝑀+𝑞exp−𝐷+𝛿₂𝑇<𝜖₁,𝑧(𝑡)⩽𝑣(𝑡)⩽𝑣^∗(𝑡)+𝜖₁ (48) for (𝑛^′₁+1+𝑛₂)𝑇⩽𝑡⩽(𝑛^′₁+1+𝑛₂+𝑛₃)𝑇, which implies that (38) holds for 𝑡∈[(𝑛^′₁+1+𝑛₂)𝑇,(𝑛^′₁+1+𝑛₂+𝑛₃)𝑇]. As in Step 1, we have 𝑥𝑛^′₁+1+𝑛₂+𝑛₃𝑇⩾𝑥𝑛^′₁+1+𝑛₂𝑇𝜂^𝑛3. (49)
Integrate (44) on (𝑡^∗,(𝑛^′₁+1+𝑛₂)𝑇), which yields 𝑥𝑛^′₁+1+𝑛₂𝑇⩾𝜁₁1−𝑝₁^𝑛2+1exp𝜎𝑛₂+1𝑇. (50)
Thus, 𝑥((𝑛^′₁+1+𝑛₂+𝑛₃)𝑇)⩾𝜁₁(1−𝑝₁)^𝑛2+1exp(𝜎(𝑛₂+1)𝑇)𝜂^𝑛3>𝜁₁, which is a contradiction. Let 𝑡=inf_{𝑡⩾𝑡∗}{𝑥(𝑡)⩾𝜁₁}. Then, 𝑥(𝑡)⩽𝜁₁ for 𝑡∈[𝑡^∗,𝑡) and 𝑥(𝑡)=𝜁₁. For 𝑡∈[𝑡^∗,𝑡), suppose that 𝑡∈(𝑛^′₁𝑇+(𝑘^′−1)𝑇,𝑛^′₁𝑇+𝑘^′𝑇], 𝑘^′∈𝑁, and 𝑘^′⩽1+𝑛₂+𝑛₃; we have 𝑥(𝑡)⩾𝜁₁(1−𝑝₁)^{𝑛2+𝑛3+1}exp(𝜎(𝑛₂+𝑛₃+1)𝑇)≡𝜁. Since 𝜁^′>𝜁, so 𝑥(𝑡)⩾𝜁 for all 𝑡∈(𝑡^∗,𝑡).
(b) There exists 𝑡₃∈(𝑡^∗,(𝑛^′₁+1)𝑇] such that 𝑥(𝑡₃)⩾𝜁₁. Let 𝑡̂=inf_{𝑡⩾𝑡∗}{𝑥(𝑡)⩾𝜁₁}; then, 𝑥(𝑡)⩽𝜁₁ for all 𝑡∈[𝑡^∗,𝑡)̂ and 𝑥(𝑡)̂=𝜁₁. Also, (44) holds for 𝑡∈[𝑡^∗,𝑡)̂. Integrating the equation on [𝑡^∗,𝑡)(𝑡^∗⩽𝑡⩽𝑡)̂, we can get that 𝑥(𝑡)⩾𝑥(𝑡^∗)exp(𝜎(𝑡−𝑡^∗))⩾𝜁₁exp(𝜎𝑇)⩾𝜁. Thus, in both cases, we conclude 𝑥(𝑡)⩾𝜁 for all 𝑡⩾𝑡₁.
Similarly, we can prove 𝑦(𝑡)⩾𝜁₂ for all 𝑡⩾𝑡₂.
Set Ω={(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡))∣𝑥(𝑡)⩾𝜁₁,𝑦(𝑡)⩾𝜁₂,𝑧(𝑡)⩾𝜁₃}. Obviously, we know that the set Ω∈Int𝑅³₊ is a global attractor; every solution of system (6) will eventually enter and remain in region Ω. Therefore, system (6) is permanent. The proof is completed.

5. Numerical Simulations and Conclusions

To study the dynamics of an ecological model consisting of two preys and one predator with impulsive control strategy, the solution of the system (6) with initial date (𝑥₀,𝑦₀,𝑧₀)=(0.8,0.6,1) in the positive cone is obtained numerically for biologically feasible range of parametric value. In order to verify our theoretical results, let 𝑎₁=0.8, 𝑎₂=0.6, 𝐾=20, 𝑒₁=0.844, 𝑒₂=0.647, 𝑚₁=0.25, 𝑚₂=0.2, 𝑏₁=1.1, 𝑏₂=1.13, 𝑐₁=0.06, 𝑐₂=0.05, 𝐷=0.027, 𝑝₁=0.3, 𝑝₂=0.25, 𝑝₃=0.01 and 𝑇=10. On the other hand, it follows from Theorem 7 that we can select a positive number 𝑀. Thus, by a straightforward computation, we can obtain the two critical values 𝑞^∗=65.83 and 𝑞_∗=3.98. When 𝑞=66>𝑞^∗, it satisfies the condition of Theorem 9, so the prey eradication periodic solution (0,0,𝑧^∗(𝑡)) is globally asymptotically stable, A type prey eradication periodic solution of the system (6) is shown in Figure 1, where we may observe how the variable 𝑧(𝑡) oscillates in a stable cycle. In contrast, the prey 𝑥(𝑡) and the prey 𝑦(𝑡) rapidly decrease to zero. When 𝑞=3.9<𝑞_∗, it satisfies the condition of Theorem 10, so the system (6) is permanent. That is to say, the preys and the predator can coexist (see Figure 2). Furthermore, we can give the other two critical values 𝑞_∗∗=5.31 and 𝑞^∗∗=65.03. When 𝑞_∗∗<𝑞<𝑞^∗∗, it does not satisfy the conditions of Theorems 9 and 10, but numerical simulations illustrate that the system (6) appears to be the principle of competition exclusion. That is to say, the competition between two prey populations give rise to the result that one prey becomes extinct and the other prey population is persisted. For example, letting 𝑞=6, the result is as in Figure 3; letting 𝑞=60, the result is in as Figure 4.

(a)

(b)

(c)

Figure 1 

Time series of a solution of model (6) with 𝑞=3.9: (a) time series of the prey population 𝑥(𝑡); (b) time series of the prey population 𝑦(𝑡); (c) time series of the predator population 𝑧(𝑡).

(a)

(b)

(c)

Figure 2 

Time series of a solution of model (6) with 𝑞=66: (a) time series of the prey population 𝑥(𝑡); (b) time series of the prey population 𝑦(𝑡); (c) time series of the predator population 𝑧(𝑡).

(a)

(b)

(c)

Figure 3 

Time series of a solution of model (6) with 𝑞=6: (a) time series of the prey population 𝑥(𝑡); (b) time series of the prey population 𝑦(𝑡); (c) time series of the predator population 𝑧(𝑡).

(a)

(b)

(c)

Figure 4 

Time series of a solution of model (6) with 𝑞=60: (a) time series of the prey population 𝑥(𝑡); (b) time series of the prey population 𝑦(𝑡); (c) time series of the predator population 𝑧(𝑡).

In this paper, we have investigated effects of impulsive perturbations on a predator-prey model consisting of two preys and one predator with Beddington-DeAngelis functional response. By using the Floquet theorem and small amplitude perturbation skills, we have proved that the periodic solution (0,0,𝑧^∗(𝑡)) is globally asymptotically stable when the values of impulsive perturbation 𝑞 are greater than the critical value 𝑞^∗. In addition, it has been shown that model (6) is permanent when the values of impulsive perturbation 𝑞 are less than the critical value 𝑞_∗. By numerical simulations, we have found that the system (6) appears to be the principle of competition exclusion, when 𝑞_∗∗<𝑞<𝑞^∗∗. However, our model with the logical consistence is different from the model in [16]; that is, dynamic complexities in [16] does not appear in our model. Thus, our results suggest a new control approach which is more suitable than the classical one in the pest control.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 10771179) and the Natural Science Foundation of Shanxi Province (no. 2013011002-2).