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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.

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 [

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:

If

If model (

According to the logical consistence proposed by Arditi and Michalski in 1996 [

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 [

Two prey species compete the same resource.

Functional response is effected for mutual interference by predators.

Thus, the model can be described by the following impulsive differential equations:

where

The paper is arranged as follows. In Section

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

Let

Let

The solution of system (

System (

The following lemma is obvious.

Let

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

Suppose

To research the stability of the prey-free periodic solution of (

There exists a

Let

For this equality, we call the constant matrix

All monodromy matrices of (

Let conditions (H2.1)–(H2.3) hold. Then, the linear

stable if and only if all multipliers

asymptotically stable if and only if all multipliers

unstable if

Now, one gives some basic properties about the following subsystem of system (

Then we can easily obtain the following results.

(1)

(2)

(3) All solutions

Therefore, we obtain the complete expression for the two-prey eradication periodic solution

Firstly, we show that all solutions of (

There exists a constant

Define

Let

According to Lemma

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

Let

The local stability of periodic solution

Putting (

Therefore, we have

The stability of the periodic solution

If absolute values of all eigenvalues are less than one, then the periodic solution

According to the Floquet theory of impulsive differential equation,

Let

By Theorem

Thus,

On the other hand, we have

In this section, we investigate the permanence of the system (

The system (

Let

We can choose

Then,

Thus,

(a)

Integrate (

Thus,

(b) There exists

Similarly, we can prove

Set

To study the dynamics of an ecological model consisting of two preys and one predator with impulsive control strategy, the solution of the system (

Time series of a solution of model (

Time series of a solution of model (

Time series of a solution of model (

Time series of a solution of model (

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

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).