According to integrated pest management strategies, we construct and investigate the dynamics of a Holling-Tanner predator-prey system with state dependent impulsive effects by releasing natural enemies and spraying pesticide at different thresholds. Applying the Dulacs criterion, the global stability of the positive equilibrium in the system without impulsive effect is discussed. By using impulsive differential equation geometry theory and the method of successor functions, we prove the existence of periodic solution of the system with state dependent impulsive effects. Furthermore, the stability conditions of periodic solutions are obtained. Some simulations are exerted to illustrate the feasibility of our main results.

In 2012, the main maize area in Northern China embraced the large-scale plant diseases and insect pests [

Different control strategies will be applied due to different behavior features of pests and their damages for crops in different stages. For instance, in the spawning period of

Let

Synthesizing systems (

In order to analyze the dynamics of the system (

Consider the state impulsive differential equation:

The dynamic system which is formed by solution mapping of system (

In the system (

Assume that impulsive set

Successor function is continuous.

If there exists a point

Assume that

For the system (

In system (

Assume that

If the multiplier

In the system (

The solutions of the system (

Obviously, the system (

Thus,

If

if

if

Next, we discuss the global stability of

If

From

If

If

Assume that

Illustration of vector graph of system (

In this section, we will discuss the existence and stability of periodic solution of system (

The structure graph of system (

Using successor function and geometric theory of impulsive differential equations and according to different positions of orbit initial points, the existence and stability of periodic solution of system (

Let

The orbit

(a) If

The orbit starting from the point

(b) If

According to Lemma

(c) If

From Lemma

To sum up the above discussed, we get the following.

If the initial point

Next, we will discuss the stability of the above periodic solutions.

Let

Let the orbit

If

Based on existence and uniqueness theorem of differential equations, there exists a unique point

The initial point

The orbit

If

If

If

Based on the discussion above, we get the following.

Assume that the orbit

The isocline

The initial point

The orbit

(a) If

(b) If

If

(c) If

(d) If

(e) If

Based on the discussion above, we get the following.

Assume that the orbit

If

If

The initial point

In this case, the orbit

Next, we discuss the stability of the periodic solution with the initial point on

Assume that

Assume that the periodic orbit

Compared with the system (

If

In this part, we use numerical simulation to confirm the conclusion obtained above. Let

Let

The periodic solution corresponding to Figure

Phase diagram of system (

Time series of system (

Time series of system (

Let

There exists a 1-periodic solution in the area

Phase diagram of system (

Time series of system (

Time series of system (

Let

1-periodic solution in the area

Phase diagram of system (

Time series of system (

Time series of system (

Let

There exists a 1-periodic solution in the area

Phase diagram of system (

Time series of system (

Time series of system (

Let

1-periodic solution in the area

Phase diagram of system (

Time series of system (

Time series of system (

Let

The periodic solution corresponding to Figure

Phase diagram of system (

Time series of system (

Time series of system (

All the simulations above show agreement with the results in Section

This paper establishes a class of integrated pest management model based on state impulse control. In the initial stage of the occurrence of crop pests, that is, the pest density satisfies

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

Wencai Zhao, Tongqian Zhang, and Xinzhu Meng are financially supported by the National Natural Science Foundation of China (no. 11371230), the Shandong Provincial Natural Science Foundation, China (no. ZR2012AM012), and a Project of Shandong Province Higher Educational Science and Technology Program of China (no. J13LI05). Xinzhu Meng is financially supported by the SDUST Research Fund (no. 2011KYTD105).