The theory of impulsive state feedback control is used to establish a mathematical model in the pest management strategy. Then, the qualitative analysis of the mathematical model was provided. Here, a successor function in the geometry theory of differential equations is used to prove the sufficient conditions for uniqueness of the 1-periodic solution. It proved the orbital asymptotic stability of the periodic solution. In addition, numerical analysis is used to discuss the application significance of the mathematical model in the pest management strategy.

Impulse is an interference in the thing at a short time in the course of its development. It is a method of external control. This kind of method is widely used in biological control, prevention of epidemic, cancer cells of chemotherapeutics, and so on. We use impulsive differential equation to reflect the method of external control. We can use impulsive differential equation to describe some biological phenomena in population ecology. There are mainly two kinds of impulsive differential equation. One kind is fixed times impulsive differential equation, and the other kind is differential system with state impulses. In the recent thirty years, many authors have studied the impulsive differential equation [

Pest management is a focus which people are concerned with. Because the technological revolutions have recently hit the industrial world and the experience and lessons are accumulated, the ideology and strategy of pest management have changed a lot. Pest management changes from chemical control to integrated control. It is fully integrated into the development of agriculture and forestry sustainability.

The study of pest management strategy has good application value and significant agriculture production. In the past few decades, many authors have made a lot of research and discussion it [

In recent years, the application of differential system with state impulses in integrated pest management has been greatly developed. Tang used differential system with state impulses in pest management [

System (

For system (

The remainder of this paper is organized as follows. In Section

For the state impulse differential equation

differential equation

If there is a

At the situation of (2), if

For repeated superior surface,

The mapping of the semi-continuous dynamical system:

(1)

If the periodic solution

When there is a point

Successor function: let

The successor function

In fact, successor function

Let continuous dynamical system be as

The

At system (

The intersection point of pulse set

When pulse set is

Let pulse set be

If

If

When pulse set is

Let pulse set is

Let pulse set be

When the pulse set is

if the negative semiorbits of point

The negative semiorbits of point

(1) If the negative semiorbits of point

If

If

(2) The negative semiorbits of point

If

If

If

If

If the condition

From system (

If

The 1-periodic solution of system (

In this part, we use numerical simulation to analyse the dynamical behavior and ecological significance of system (

If there is no impulse, then system has the unique positive equilibrium (5, 4.5) which is globally asymptotically stable.

Next, we consider the existence of 1-periodic solution for system (

From the numerical analyses, we know that it is better to use comprehensive control including chemistry control and biological technique according to different values of economic threshold to pest. When the pulse set is below or equal to the number of the pests of the system at the equilibrium state without pulse, the system has a 1-periodic solution, which is consistent with the proof of the theorem. When the pulse set is more than the number of the pests of the system at the equilibrium state without pulse and the phase set is less than the number of the pests of the system at the equilibrium state without pulse, the system has a 1-periodic solution, which is consistent with the proof of the theorem. When the pulse set is more than the number of the pests of the system at the equilibrium state without pulse and the phase set is more than the number of the pests of the system at the equilibrium state without pulse, the 1-periodic solution of system may not necessarily exist; we must consider different kinds of the number of the natural enemies we released; then the 1-periodic solution exists and has different periods, which is consistent with the proof of the theorem. So we take different release strategies according to different growth periods of the crop. In order to decide how to control the number of the natural enemies we released, the control strategy with impulsive state needs observing and recording the number of the pests and the natural enemies. In theory, we can predict the cycle time without repeated measurements, which can save a lot of manpower and material resources. The model in this paper is closer to the reality than the model that there is no density dependence for the continuous process of pulse points; it is also closer to the reality than the model that did not consider the influence of natural enemies of spraying insecticide.

This work is supported by Natural Science Foundation of Fujian Education Department (JB09078), Minnan Science and Technology Institute and the Young Core Instructor (mkq201006).