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The dynamic behavior of a predator-prey model with Holling type IV functional response is investigated with respect to impulsive control strategies. The model is analyzed to obtain the conditions under which the system is locally asymptotically stable and permanent. Existence of a positive periodic solution of the system and the boundedness of the system is also confirmed. Furthermore, numerical analysis is used to discover the influence of impulsive perturbations. The system is found to exhibit rich dynamics such as symmetry-breaking pitchfork bifurcation, chaos, and nonunique dynamics.

In recent years, impulsive control strategies in predator-prey models have become a major field of inquiry. Many authors have studied the dynamics of predator-prey models with impulsive control strategies [

More recently, the author of [

Assume that the top predator also eats the prey, or in other words, that the relationship between the top predator and the mid-level predator is not only that of predator and prey, but also that of competitors. To represent this, a predator-prey model with Holling type IV functional response with respect to an impulsive control strategy can be constructed as follows:

The rest of this paper is organized as follows. Section

First, some useful notations and statements will be provided for use in subsequent proofs. The following definitions will be useful.

Let

Let

System (

Assume that

Let

Now consider a special case of Lemma

Let _{0} are constants, and

For convenience, some basic properties can be defined for the following subsystems of system (

For a positive periodic solution

Next, some main theorems will be proposed.

There exists a constant

Let

Next the stability of a prey and top-predator eradication periodic solution will be examined.

The solution

The local stability of the periodic solution

The solution

By Theorem

System (

Let

By Lemmas

Choose

Therefore,

If

To study the dynamics of system (

First, the influence of the period

Bifurcation diagram of system (

Dynamics of system (

Bifurcation diagram of system (

To see the dynamics of system (

Periodic behavior and chaos corresponding to Figure

Figures

The largest Lyapunov exponent is always calculated to detect whether a system is exhibiting chaotic behavior. The largest Lyapunov exponent takes into account the average exponential rates of divergence or convergence of nearby orbits in phase space [

The largest Lyapunov exponents (LLE) corresponding to Figure

The largest Lyapunov exponents (LLE) corresponding to Figure

To understand the qualitative nature of strange attractors, power spectra can be used [

Strange attractors and power spectra: (a) strange attractor when

In this paper, the dynamic behavior of a predator-prey model with Holling type IV functional response with respect to an impulsive control strategy has been investigated. The conditions for locally asymptotically stable and globally stable periodic solutions and for system permanence have been determined. It has been determined that an impulsive control strategy changes the dynamic behavior of the model. Complex dynamic patterns also have been observed in continuous-time predator-prey or three-species food-chain models [

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. 30970305 and Grant no. 31170338) and also by the Zhejiang Provincial Natural Science Foundation of China (Grant no. Y505365).