We introduce and study a Gompertz model with time delay and impulsive perturbations on the prey. By using the discrete dynamical system determined by the stroboscopic map, we obtain the sufficient conditions for the existence and global attractivity of the “predator-extinction” periodic solution. With the theory on the delay functional and impulsive differential equation, we obtain the appropriate condition for the permanence of the system.

It is well known that Gompertz equation [

Many systems in physics, chemistry, biology, and information science have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes. This dynamical behavior can be modeled by impulsive differential equations. The theory of impulsive differential systems has been developed by numerous mathematicians [

In this paper, we need to consider a function

We introduce the model as follows:

The initial conditions for system (

From the biological point of view, we only consider system (

All outline of this paper is as follows. We give some basic knowledge in Section

Let

Let

Considering the following impulsive differential inequalities:

There exists a constant

Define

We calculate the upper right derivative of

So we have that

Consider a delay equation

If

If

First, we begin analyzing the existence of a “predator-extinction” solution, in which predator is absent from the system, that is,

In this condition, we know the growth of the prey in the time-interval

For system (

Solving the first equation of system (

For the second equation of system (

Since

In the following, we will prove that

System (

Denote that

If

We denote that

We consider the following comparison equation:

Since

In the following, we suppose that

Because the function

Noting that

From Section

There exist a positive integer

From (

Considering the following comparison equation:

Since

Next, we will prove that

Then we have that

By using comparison theorem of impulsive differential equation [

Considering the following comparison system:

Now, we give the following theorem.

Critical values of some parameters of system (

The conditions for global attractivity of | |
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In the above section, we have proved that, when

System (

System (

Denote that

If

Suppose that

Firstly, from the first equation of system (

therefore we have that

Secondly, we will find

The second equation of system (

Define

Since

(1) We claim that the inequality

From the first equation of system (

By comparison theory [

Denote that

We show that

Thus, from the second equation of (

This is a contradiction to

(2) If

We will prove that

When

Since

If

If

If

Since the interval

This proof is complete.

If

Suppose that

Then

The proof is complete.

If

Critical values of some parameters of system (

The conditions for the permanence of system ( | |
---|---|

In this paper, we introduce and discuss a predator-prey system model with Holling III response functional under time delay on the predator and impulsive perturbations on the prey. From Section

In the following, we will analyze the influence of them on the dynamics of system (

Figure

Figure

By Theorems

In Figure