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We investigate the dynamic behaviors of a two-prey one-predator system with stage structure and birth pulse for predator. By using the Floquet theory of linear periodic impulsive equation and small amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we study the permanence of the investigated model. Our results provide valuable strategy for biological economics management. Numerical analysis is also inserted to illustrate the results.

In the natural world, the predator-prey relationship is one of the important interactions among species, and it has been extensively studied by many authors [

As we know, life history often occurs in natural ecological environments which has significant morphological and behavioral differences between immature and mature species; the dynamics of stage-structured prey-predator system has been widely studied [

The biological meanings of the parameters in (

In view of birth pulse and impulsive control strategy, we formulate the following two prey-predator models with stage-structure and birth pulse for predator:

The organization of this paper is as follows. In the next section, some important lemmas are presented. Sections

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

Let

Consider

The solution of system (

Obviously the smoothness properties of

Suppose

Make a notation as

If

It is easy to obtain the analytic solution of system (

Considering (

For convenience, we choose

(i) If

(ii) If

For convenience, we make a notation as

Obviously, the near dynamics of

(ii) If

Summarizing the above results, we have the following theorems.

The triviality periodic solution

Correspondingly, system (

There exists a constant

Define a function

It follows from the comparison theorem of impulsive differential equations (see Lemma

So

System (

In this section, we investigate the stability of the two-pest prey eradication periodic solution as a solution of system (

Let

Firstly, we prove the local stability of

Define

Hence the fundamental solution matrix is

which are

According to the Floquet theory (see [

In the following, we prove the global attractivity. Choose

Next, we will prove that

From the left hand inequality of (

Set

Therefore, there exists

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

System (

Let us suppose a solution

There exists

There exists

Consider

Similarly, cases (ii) and (iii) can be analyzed as in case (i). Here we omit it. From the above three cases, we conclude that there exist

Let

If

Let

If there exists

From the proof of Theorems

Let

Let

In this section, we carry out numerical simulations of system (

(1) Consider the following choice of parametric values:

Correspondingly,

Globally asymptotically stable prey eradication solution of system (

(2) Consider the following set of parameters:

Since conditions of Theorem

The permanence of system (

In this paper, we study an impulsively controlled three-species prey-predator model with stage structure and birth pulse for predator. Using the comparison theorems, we have shown that there exists a globally asymptotically stable two-pest eradication periodic solution when the impulsive period

Compared to earlier modeling studies on the prey-predator concerning chemical control for pest control at different fixed time, our model considers the predator given birth in regular pules at

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province (PKLHB1524), the innovation projects for undergraduates of Hubei province (201310517014), the soft science research project of Hubei province (2012GDA01309), and key discipline of Hubei province, Forestry.