We propose an impulsive biological pest control of the sugarcane borer (

One of the challenges for the improvements in the farming and harvesting of cane is the biological pest control. Biological control is defined as the reduction of pest populations by using their natural enemies: predators, parasitoids, and pathogens [

The sugarcane borer

There is an important larvae parasitoid of the sugarcane borer, a wasp named

Mathematical modelling is an important tool used in studying agricultural problems. Thus, a good strategy of biological pest control, based on mathematical modelling, can increase the ethanol production. The applications of host-parasitoid models for biological control were reviewed in [

In [

The proposed model (

Meanwhile, many authors have investigated the different population models concerning the impulsive pest control [

In this paper, we suggest impulsive differential equations [

In this section, we will give some definitions, notations, and some lemmas which will be useful for our main results.

Let

The solution of system (

We will use a basic comparison result from impulsive differential equations.

Let

Next, we consider the following system:

System (

Integrating and solving the first equation of (

After each successive pulse, we can deduce the following map of system (

Consequently,

If

Therefore, system (

To study the stability of the pest-eradication periodic solution of (

There exist a

Let

By equality (

Let conditions (

stable if and only if all multipliers

asymptotically stable if and only if all multipliers

unstable if

In this section, we study the stability of the pest-eradication periodic solution

The pest-eradication periodic solution

The local stability of a periodic solution

Define

Linearizing the system (

Let

The solution of (

There is no need to calculate the exact form of (

Hence, if absolute values of all eigenvalues of

In the following, we prove the global attractivity. Choose sufficiently small

From system, (

Next, we prove that

Then, we have

Assuming that

For numerical simulations of interactions between the sugarcane borer and its parasitoid the following values of model coefficients were used:

Evolution of the egg (a), parasitized egg (b), larvae populations (c), and phase portraits (d) of system (

According to [

From Theorem

Evolution of the egg (a), parasitized egg (b), larvae populations (c), and phase portraits (d) of system (

Choosing the release amount

Evolution of the egg (a), parasitized egg (b), larvae populations (c), and phase portraits (d) of system (

Applying the control strategy

Evolution of the egg (a), parasitized egg (b), larvae populations (c), and phase portraits (d) of system (

In this paper, we suggest a system of impulsive differential equations to model the process of the biological control of the sugarcane borer by periodically releasing its parasitoids. By using the Floquet theory and small amplitude perturbation method, we have proved that for any fixed period

When the stability of the pest-eradication periodic solution is lost, the numerical results show that the system (

If we choose the biological control strategy by periodical releases of the constant amount of parasitoids, the results of Theorem

It is interesting to discuss the result of Theorem

The resetting impulsive condition (

Thus, the results of the present study show that the impulsive release of the parasitoids provides reliable strategies of the biological pest control of the sugarcane borer.

The authors would like to thank the referees for their careful reading of the original paper and their valuable comments and suggestions that improved the presentation of this paper. The first author thanks Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Pesquisas (CNPq) for the financial supports on this research.