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The dynamic complexities of an Ivlev-type prey-predator system with impulsive state feedback control are studied analytically and numerically. Using the analogue of the Poincaré criterion, sufficient conditions for the existence and the stability of semitrivial periodic solutions can be obtained. Furthermore, the bifurcation diagrams and phase diagrams are investigated by means of numerical simulations, which illustrate the feasibility of the main results presented here.

The theoretical investigation of predator-prey systems in mathematical ecology has a long history, beginning with the pioneering work of Lotka and Volterra. During this time, the theory and application of differential equations with impulsive perturbations were significantly advanced by the efforts of Lakshmikantham et al. [

Many factors in the environment must be considered in predator-prey systems [

Generally speaking, there are three possible cases of impulsive perturbation: systems with impulses at fixed times, systems with impulses at variable times, and autonomous impulsive systems. In recent years, most investigations of impulsive differential equations have concentrated on systems with impulses at fixed times [

As is well known, significant developments have recently been achieved in the bifurcation theory of continuous dynamic systems [

Recently, the continuous model with Ivlev-type has been extensively studied [

The rest of this paper is organized as follows. Section

The dynamic behavior of system (

Throughout this paper, it is assumed that

Let

A trajectory

The next step is to construct the Poincaré map. To discuss the dynamics of system (

Poincaré map of system (

Assume that point

Now choose section

In this discussion,

Next, an autonomous system with impulsive effects will be considered:

If the Floquet multiplier

Let

Then

It should be stressed that the semitrivial periodic solution with

When

Now the stability of this semitrivial periodic solution will be discussed.

The semitrivial periodic solution (

In fact,

Therefore,

Set

First, in the case

Second, by examining the bifurcation of map (

A transcritical bifurcation occurs when

The values of

It is easy to see that

Using (

Because (

Finally, (

In this subsection, the existence of a positive periodic solution with

For any

Let point

In conclusion, the following results can be obtained from the previous discussion:

if

if

From (

According to the following discussion, a positive periodic solution exists when

For any

Based on the conclusion of Theorem

From the previously mentioned, it is known that if there exists a

As is well known, system (

To study the dynamic complexity of an Ivlev-type system with state-dependent impulsive perturbation on the predator, a semitrivial periodic solution of system (

Next, two control parameters,

Note that the corresponding focus

Let

Trajectories with initial point (0.02, 0.001) of system (

When

Bifurcation diagram of system (

In Figure

In Figure

From Theorem

Periodic solutions of system (

(a) Phase diagram; (b) time series of

Based on the previous analysis, it can be seen that the impulsive state feedback control can enhance the predator

In this paper, a predator-prey model with Ivlev-type function and impulsive state feedback control has been built and studied analytically and numerically. Mathematical theoretical arguments have investigated the existence and stability of semitrivial periodic solutions of system (

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions on this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 31170338 and no. 30970305) and also by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001).