According to the integrated pest management strategies, a Holling type I functional response predator-prey system concerning state-dependent impulsive control is investigated. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution, and the attractivity of the order one periodic solution by sequence convergence rules and qualitative analysis. Numerical simulations are carried out to illustrate the feasibility of our main results which show that our method used in this paper is more efficient than the existing ones for proving the existence and attractiveness of order one periodic solution.

It is one of the significant problems in the world today to prevent plant pests and pesticide pollution and to protect ecological balance for the sustainable development of agriculture and forestry, which is also an important research topic demanding prompt solution. In recent decades, plant pests worldwide are increasingly serious with the damage of the world's natural ecosystems. In agricultural production, pesticides spaying (chemical control) and release of natural enemies (biological control) are the ways commonly used for pest control. But if we implement chemical control as soon as pests appear, many problems are caused: the first is environmental pollution; the second is increase of costs including human and material resources and time; the third is killing natural enemies, such as parasitic wasp; the last is pests’ resistance to pesticides, which brings great negative effects [

Considering the effectiveness of the chemical control and nonpollution and limitations of the biological one, people have proposed the method of integrated pest management (IPM), which is a pest management system integrating all appropriate ways and technologies to control economic injury level (EIL) caused by pest populations in view of population dynamics and its relevant environment. In the process of practical application, people usually implement the following two schemes for the integrated pest management: one is to implement control at a fixed time to eradicate pests [

In consideration of predator-prey capacity, Holling [

where

Holling type-II and type-III functional responses are as follows, respectively:

Refer to [

As the Lotka-Volterra predator-prey system with Holling functional response is more practical, many authors have studied about it [

This paper is organized as follows. In the next section, we present some basic definitions and important lemmas as preliminaries. In Section

We first consider the model (

We consider the following function:

We can get that

It is easily proved that

Since

Therefore, we observe the straight line:

The system (

Two steady states

The trajectory of system (

A triple

Assume that

function

Then,

For any

We consider state-dependent impulsive differential equations:

Suppose that the impulse set

A trajectory

We get these lemmas from the continuity of composite function and the property of continuous function.

Successor function defined in Definition

In system (

In this section we shall investigate the existence of an order one periodic solution of system (

Phase set

Isoclinic line is denoted, respectively, by lines

For any point

Due to the practical significance, in this paper we assume the set

In the light of the different position of the set

In this case, sets

Take another point

By Lemma

Take that another point

From Lemma

Now we can summarize the above results as the following theorem.

Assuming that

In this case, the set

Since

In light of the different positions of the set

Now we can summarize the above results as the following theorem.

If

If

If

If

Now we can summarize the above results as the following theorem.

Assuming that

From the vector field of system (

For any point below

The trajectory with any initiating point above

Now we can summarize the above results as the following theorem.

Assuming that

In this case, denote the intersection of the line

In this section, under the condition of existence of order one periodic solution to system (

If system (

there exists an odd number of order one periodic solutions of system (

if the periodic solution is unique, then the periodic solution is attractive in region

(I) According to the Subcase 1.3,

(II) By the derivation of Theorem

On the one hand, take a point

Next we will prove

On the other hand, set

Since the trajectory initiating any point of

Any point

The trajectory with any initiating point above

From the above analysis, we know the trajectory initiating any point of

If system (

If system (

Through the derivation of Theorem

We take any two points

Next we prove the attractiveness of the order one periodic solution

Denote the first intersection point of the trajectory with initiating point

Since the trajectory initiating any point of

The trajectory with initiating point between

Assume a point

The trajectory with any initiating point above

From the above analysis, we know there exists a unique order one periodic solution in system (

In this paper, a Holling I predator-prey model with state-dependent impulsive control model concerning different control methods at different thresholds is proposed to find a new method to study existence and attractive of order one periodic solution of such system. We define semicontinuous dynamical system and successor function and demonstrate the sufficient condition that system (

In order to testify the validity of our results, we consider the following example:

Existence and attractiveness of order one periodic solution.

We set

The time series and phase diagram for system (

Existence and attractiveness of positive periodic solutions.

We set

The time series and phase diagram for system (

Existence and attractiveness of positive periodic solutions.

We set

The time series and phase diagram for system (

This project was supported by the National Natural Science Foundation of China (No. 10872118).