A prey-predator model with Beddington-DeAngelis functional response and impulsive state feedback control is investigated. We obtain the sufficient conditions of the global asymptotical stability of the system without impulsive effects. By using the geometry theory of semicontinuous dynamic system and the method of successor function, we obtain the system with impulsive effects that has an order one periodic solution, and sufficient conditions for existence and stability of order one periodic solution are also obtained. Finally, numerical simulations are performed to illustrate our main results.
The study of the dynamics of prey-predator system is one of the dominant subjects in both ecology and mathematical ecology due to the fact that predator-prey interaction is the fundamental structure in population dynamics. Many scholars have carried out the study of prey-predator system with various functional responses, such as Monod-type and Holling-type. It is well known that Beddington-DeAngelis functional response which was introduced by Beddington and DeAngelis et al. [
Impulsive differential equations have been widely used in various fields of applied sciences, for example, physics, ecology, and pest control. The majority of them just concern the system with impulses at fixed times [
Motivated by the above works, in this paper, we consider the following predator-prey system with Beddington-DeAngelis functional respose:
We nondimensionalize system (
In this paper, we mainly discuss the existence and stability of periodic solution of system (
An outline of this paper is as follows: some definitions and theorems are given for the later use in the next section. The qualitative analysis of the system without impulsive effects is given in Section
Differential equation with impulsive state feedback control
In this paper,
For any
A trajectory
Next we will give the definition of the successor function of semicontinuous dynamical system (
Suppose
If
According to the continuity of compound function, we know the following.
The successor function
In system (
The system (
In this section, we will study the qualitative characteristic of system (
The equilibrium of system (
The system (
Let the straight line
The uniformly bounded region.
In the following, we will analysis the stability of equilibrium
The positive equilibrium point
The Jacobian matrix
Let
In this section, we will investigate the existence of order one periodic solution of system (
In this case, sets
Take another point
Illustration of system (
Take another point
Illustration of system (
Suppose that
In this case, set
If
If
Illustration of system (
If
Illustration of system (
If
Suppose that
Illustration of system (
Suppose that
In the following, we analyze the stability of order one periodic solution in system (
The
The proof of this lemma is referred to Simeonov and Baĭnov [
In the following, we suppose this periodic solution of system (
Then
Set
Suppose
If system (
In order to verify the theoretical results in this paper, we consider the following example
Time series and portrait phase of system (
Time series and portrait phase of system (
Time series and portrait phase of system (
Choosing
The phase portraits of system (
The phase portraits of system (
The phase portraits of system (
The phase portraits of system (
According to the above analysis, it is obvious that the prey can be well suppressed below certain level by using impulsive state feedback strategy for the fact that the system has stable periodic solution under some conditions. The key to the system with impulsive state feedback control is to give the suitable feedback state (the value of
This work is supported by the National Natural Science Foundation of China (11171284), the Fujian Provincial Natural Science Foundation of China (2012J01012), the Fujian Provincial Education Fundation (JA12198), and the Scientific Research Foundation of Jimei University of China (ZC2011003).