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Normally, chemical pesticides kill not only pests but also their natural enemies. In order to better control the pests, two-time delayed stage-structured predator-prey models with birth pulse and pest control tactics are proposed and analyzed by using impulsive differential equations in present work. The stability threshold conditions for the mature prey-eradication periodic solutions of two models are derived, respectively. The effects of key parameters including killing efficiency rate, pulse period, the maximum birth effort per unit of time of natural enemy, and maturation time of prey on the threshold values are discussed in more detail. By comparing the two threshold values of mature prey-extinction, we provide the fact that the second control tactic is more effective than the first control method.

The outbreak of pest often triggers serious ecological and economic problems. In recent years, the management of pest has increasingly become the focus of attention. How to effectively control pest is one primarily concern problem. In practice, lots of factors can affect the efficiency of pest control, for instance, the time of impulsive effects, the number of prey stocked or naturally released, and the proportion of killing or catching pests. Mathematical modelling is one of the main ways for estimating and predicting the range of ecological interactions between pest and predator. Lately, many papers have been devoted to propose and analyze the predator-prey systems [

In one aspect, many species have the life history that goes through two stages, immature and mature, which has significant morphological and behavioral differences between them. Therefore, it is necessary to account for these differences, and the dynamics of mathematical models with stage-structured prey-predator model has been widely studied [

As far as the population dynamics is concerned, most models often considered that the population reproduces throughout all year. However, many species give birth seasonally or in regular pules. In this regard, the continuous reproduction of mature species should be removed from the model and termed this growth form as birth pulse. For instance, Tang and Chen study an age-structured model with density-dependent birth pulse in [

Although, many authors have devoted to study the effects of pesticide on pest and its natural enemies and lots of instructive control strategies also have been given. An optimal time of pesticide applications still seems to be a novel interesting area. Following the practical pest management, we firstly propose the predator-prey model with pulse at the same time. Further, we assumed that the birth pulse and pest control tactics occur at different time. Discussing and comparing the mature prey-extinction of the two models, we get some new effective pest management.

The purpose of this paper is to address how the time of impulsive effects influences the pest control. On the basis of the above discussion and motivated by [

Consider the following equation:

if

if

Consider the following system:

if

if

Integrating the first equation of (

which implies that corresponding periodic solution of (

The mature prey-extinction periodic solution

Let

For simplicity, we assume that (

In view of the comparison differential theorem, for any

Next, we will prove that

Therefore, for any

(i) If

(ii) If

In biological terms, since

In particular, in order to avoid the adverse effects of pesticides on the newly released natural enemies, we consider the following method implemented in practice to avoid such antagonism. That is, we assume that the pulse occurs at

(i) If

(ii) If

Consider system (

Further, we can show that

if

if

By calculation, we get

Now we can deduce that the positive equilibrium

Assume that

Following from the second equation of system (

This is

(i) If

(ii) If

The biological significance of Corollary

In order to investigate the permanence of system (

There exists a constant

Define

It follows from the comparison theorem of impulsive differential equations (see lemma 2.2 [

According to Lemma

The system (

In biological terms, the permanence of (

If

It is seen that the second equation of system (

In the following we show that

By the claim, consider the following two possibilities.

We will show that

Next, from the first, the fourth, and the seventh equation of system (

Define

In the previous sections, we introduced the analytical tools and used them for a qualitative analysis of the system obtaining some results about the dynamics of the system. In this section, we perform a numerical analysis of the model based on the previous results. What we are interested in is how the key factors affect the thresholds

So far, we have considered the global attractive mature prey-eradication solution of systems (

Dynamical behavior of system (

Dynamical behavior of system (

The influence of some key parameters on the threshold level

Complex dynamic properties of system (

Time-series of the system (

Bifurcation diagrams of system (

For system (

Dynamical behavior of system (

Dynamical behavior of system (

We are more interested in how different patterns of insecticide applications affect the two threshold values (

Comparing two kinds of control strategy; the parameter values are as follows:

When using integrated pest management as an approach to control insect pests, one must be committed to a long-term strategy. It is well known that pesticides usually act not only on the pest species but also, with even stronger impact, on their natural enemies; then, despite the fact that pesticides kill the target pest, they are simultaneously reducing the population density of natural enemies; the influence of pesticides on natural enemies is even greater than on insect pests. As a result of this indirect effect, treatments can counterintuitively lead to an effective increase of the pest species, even pest breakout again.

In this paper, our idea is to contribute to pest control programs with pesticide applications and with birth pulse in natural enemies and to provide some strategies; that is, spraying pesticides and natural enemies were born at the same time and at a different time. The threshold conditions which guarantee the existence and stability of the mature prey-extinction periodic solution are provided. If the integrated control methods cannot completely eradicate the mature prey, the pest population can have outbreaks at different scales. By numerical bifurcation investigations, we found that, when choosing different parameter spaces, two attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters. Moreover, the effects of times of spraying pesticides (or the enemy was born) and control tactics on the threshold conditions were carefully investigated. In particular, the effects of the killing rates of pesticides for pest and natural enemy populations and spraying period on the stability of the mature prey-extinction periodic solutions were discussed. The results imply that the modeling methods described can help the pest control specialist to decide appropriate control strategies and assist management decision-making.

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

This work is supported by the Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province (PKLHB1302), the Soft Science Research Project of Hubei Province (2012GDA01309), the Key Discipline of Hubei Province-Forestry, and the National Natural Science Foundation of China (11201433).