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According to biological strategy for pest control, a mathematical model with periodic releasing virus particles for insect viruses attacking pests is considered. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the amount of virus particles released is larger than some critical value. When the amount of virus particles released is less than some critical value, the system is shown to be permanent, which implies that the trivial pest-eradication solution loses its stability. Further, the mathematical results are also confirmed by means of numerical simulation.

Currently, applications of chemical pesticides to combat pests are still one of the main measures to improve crop yields. Though chemical crop protection plays an important role in modern agricultural practices, it is still viewed as a profit-induced poisoning of the environment. The nondegradable chemical residues, which accumulate to harmful levels, are the root cause of health and environmental hazards and deserve most of the present hostility toward them. Moreover, synthetic pesticides often disrupt the balanced insect communities. This leads to the interest in Biological control methods for insects and plant pests [

Biological control is, generally, man's use of a suitably chosen living organism, referred as the biocontrol agent, to control another. Biocontrol agents can be predators, pathogens, or parasites of the organism to be controlled that either kill the harmful organism or interfere with its biological processes [

Transmission is also key to the persistence of baculoviruses in the environment [

Insight in the epidemiological dynamics, it is necessary to predict optimal timing, frequency, and dosage of virus application and to assess the short and longer term persistence of NPV in insect populations and the environment. Modeling studies can help to obtain preliminary assessments of expected ecological dynamics at the short and longer term. There is a vast amount of literature on the applications of microbial disease to suppress pests [

In this paper, according to the above description, we should construct a more realistic model by introducing additional virus particles (i.e., using viral pesticide) to investigate the dynamical behavior of viruses attacking pests, which is described as follows:

We assume that only susceptible pests

The term

An infective pest

The virus particles

The paper is organized as follows: in Section

We give some definitions, notations, and lemmas which will be useful for stating and proving our main results. Let

Suppose that

Let

Note that if one has some smoothness conditions of

There exists a constant

Define a function

Next, we give some basic property of the following subsystem:

System (

System (

When

Let

The local stability of periodic solution

In fact, for condition (

Let

From (

Note that

In the following, we prove

If

We have proved that, if

System (

Let

Suppose that

Since

If

If

Let us consider the following system:

Dynamical behavior of the system (

Dynamical behavior of the system (

In this paper, we have investigated the dynamical behavior of a pest management model with periodic releasing virus particles at a fixed time. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we establish the sufficient conditions for the global asymptotical stability of the pest-eradication periodic solution as well as the permanence of the system (

From Corollary

This work was supported by National Natural Science Foundation of China (10771179).