According to the integrated pest management strategies, we propose a model for pest control which adopts different control methods at different thresholds. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution of such system, and further, the attractiveness 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. Our results show that our method used in this paper is more efficient and easier than the existing ones for proving the existence of order one periodic solution.

It is of great value to study pest management method applied in agricultural production; entomologists and the whole society have been paying close attention to how to control pests effectively and to save manpower and material resources. In agricultural production, pesticides-spraying (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 instead of working as well as had been expected [

It is worth mentioning that the vast majority of research on population dynamics system with state pulse considers single state pulse, which is to say, only when the amount of population reaches the same economic threshold can measures be taken (e.g., chemical control and biological control); but this single state-pulse control does not confirm to reality. In fact, we often need to use different control methods under different states in real life. For example, in the process of pest management, when the amount of pests is small, biological control is implemented; when the amount is large, combination control is applied. Tang et al. [

First, we give some basic definitions and lemmas.

A triple

Assuming that

function

Then,

For any

One considers state-dependent impulsive differential equations:

Suppose that the impulse set

Successor function defined.

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 (

Next, we consider the model (

two steady states

a unique closed trajectory through any point in the first quadrant contained inside the point

In this paper, we assume that the condition

Illustration of vector graph of system (

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

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

For the convenience, if

Due to the practical significance, in this paper we assume the set always lies in the left side of stable centre

In the light of the different position of the set

In this case, set

On the other hand, the trajectory with the initial point

By Lemma

If

If

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

If

It shows from the proved process of Theorem

In this case, set

(1) (

On the other hand, the trajectory passing through point

If

If

If

(2) (

Take another point

From Lemma

(3) (

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

Assuming that

If

If

If

If

In this case, the set

Since

(1) (

If

If

If

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

Assuming that

If

If

If

(2) (

If

If

If

Assuming that

If

If

If

(3) (

From the vector field of system (

For any point below

Denote the intersection of the trajectory passing through the point

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

Assuming that

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

Assuming that

If

there exists a unique order one periodic solution of system (

if

By the derivation of Theorem

We take any two points

Next, we prove the attractiveness of the order one periodic solution

Take any point

Due to conditions

So the successor function

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

Any trajectory from initial point

There is a unique order one periodic solution (Theorem

Order one periodic solution is attractive (Theorem

Assuming that

Assuming that

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

Attractneness of order one periodic solution (Theorem

(I) According to the Subcase

(II) By the derivation of Theorem

On the one hand, take a point

On the other hand, set

Any point

Denote the intersection of the trajectory passing through the point

Since the trajectory with any initiating point of the

Assuming that

Assuming that

Attractneness of order one periodic solution (Theorem

By the derivation of Theorem

Next, we prove the attractiveness of the order one periodic solution

Denote the first intersection point of the trajectory from initiating point

The trajectory from initiating point between

Denote the intersection of the trajectory passing through point

Assume a point

The trajectory with any initiating point in segment

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

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

These results show that the state-dependent impulsive effects contribute significantly to the richness of the dynamics of the model. The conditions of existence of order one periodic solution in this paper have more extensively applicable scope than the conditions given in [

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 solution.

We set

The time series and phase diagram for system (

Existence and attractive of positive periodic solutions.

We set

The time series and phase diagram for system (

This Project supported by the National Natural Science Foundation of China (no. 10872118).