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Many existing pest control models, which control pests by releasing natural enemies, neglect the effect that natural enemies may get killed. From this point of view, we formulate a pest control model with stage structure for the pest with constant maturation time delay (through-stage time delay) and periodic releasing natural enemies and natural enemies killed at different fixed time and perform a systematic mathematical and ecological study. By using the comparison theorem and analysis method, we obtain the conditions for the global attractivity of the pest-eradication periodic solution and permanence of the system. We also present a pest management strategy in which the pest population is kept under the economic threshold level (ETL) when the pest population is uniformly permanent. We show that maturation time delay, impulsive releasing, and killing natural enemies can bring great effects on the dynamics of the system. Numerical simulations confirm our theoretical results.

The warfare between man and pests has been on for thousands of years, and pest outbreaks often cause serious ecological and economic problems. With the development of society, human beings have come up with numerous methods to control pests, for instance, biological, cultural, physical, mechanical, and chemical tools in a way that minimize economics. Among those methods, releasing natural enemies is an effective and basic one, and it cannot pollute the environment. Morever, it is relatively simple to implement.

The pest control models by releasing natural enemies have been studied by many researchers [

For these purposes, in Section

A model of single pest population growth incorporating stage structure as a reasonable generalization of the logistic model was derived as follows in [

The basic model that we consider in this paper is the one based on the ideas that the natural enemies may be killed when they are released to control pest population; then we formulate the following two-stage pest control model with stage structure for pest and pulse releasing of natural enemies killed at different fixed time:

The first equation in system (

Consider the following equation:

If

If

For each solution of system (

Let

Consider the following system:

Let

Define

If

From the second equation in system (

Define

If

The first equation of system (

Thus, from the first equation of system (

If

If

If

If

If

Suppose that

The immature pest is hardly any harmful for the crop; so we just consider the effect of the mature pest. From the standpoint of ecological balance and saving resource, we only need to maintain the mature pest population under the economic threshold level (ETL) and not to eradicate the pest totally; then we have the following theorem.

Under the condition of Theorem

From (

In this paper, we discuss a pest control model with stage structure for the pest with constant maturation time delay (through-stage time delay) and periodic releasing natural enemies and natural enemies killed at different fixed time. From Theorems

To verify the theoretical results obtained in this paper, in the following we will give some numerical simulations and take

Time series of system (

Time series of system (

Time series of system (

Time series of system (

Time series of system (

This work is supported by the National Natural Science Foundation of China (10671001 and 10971001) and the Science and Research Project Foundation of Liaoning Province Education Department.