^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

The dynamic behaviors of a predator-prey (pest) model with disease in prey and involving an impulsive control strategy to release infected prey at fixed times are investigated for the purpose of integrated pest management. Mathematical theoretical works have been pursuing the investigation of the local asymptotical stability and global attractivity for the semitrivial periodic solution and population persistent, which depicts the threshold expression of some critical parameters for carrying out integrated pest management. Numerical analysis indicates that the impulsive control strategy has a strong effect on the dynamical complexity and population persistent using bifurcation diagrams and power spectra diagrams. These results show that if the release amount of infective prey can satisfy some critical conditions, then all biological populations will coexist. All these results are expected to be of use in the study of the dynamic complexity of ecosystems.

Predator-prey models with disease are a major concern and are now becoming a new field of study known as ecoepidemiology. The disease factor in predator-prey systems has been firstly considered by Anderson and May [

Through the dimensionless transformation (seeing [

In recent decades, technological revolutions have recently hit the industrial world; thus, infected population can now be controlled by many methods such as spraying pesticides and vaccination. It is well known that pest management involves using pesticides and releasing natural enemies, which have been focused by many researchers [

Based on the two aspects discussed, the authors constructed a predator-prey model with disease in prey (a pest) and involving an impulsive control strategy for the purpose of integrated pest management. The impulsive control strategy was used to introduce infected prey (a pest) at a fixed time on the basis of system (

The paper is organized as follows: in the next section, a mathematical analysis of the model is carried out. Section

Some important notations, lemmas, and definitions will be provided, which are frequently used in subsequent proofs.

Let

Let

The solution of system (

system (

System (

Suppose that

There exists a constant

Let

For convenience, some basic properties of certain subsystems of system (

For a positive periodic solution

The solution

The local stability of periodic solution

Clearly,

The solution

Let

Then,

System (

From Lemma

Therefore, it is only necessary to find a

Let

Then

Next, it will be proved that

Otherwise,

So there exists a

Second, if

When

To study the dynamics of system (

First, the influence of the period

Dynamics of system (

Bifurcation diagram of system (

Bifurcation diagram of system (

Bifurcation diagram of system (

To clearly see the dynamics of system (

Periodic and chaotic behavior corresponding to Figure

Periodic and chaotic behavior corresponding to Figure

Figures

To detect whether the system exhibits chaotic behavior, one of the commonest methods is to calculate the largest Lyapunov exponent. 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

The largest Lyapunov exponents (LLE) corresponding to Figure

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

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

In the paper, the dynamic behaviors of a predator-prey (pest) model with disease in prey and involving an impulsive control strategy are presented analytically and numerically. The critical conditions are obtained to ensure the local asymptotical stability and global attractivity of semitrivial periodic solution as well as population permanence. Numerical analysis indicates that the impulsive control strategy has a strong effect on the dynamical complexity and population persistent using bifurcation diagrams and power spectra diagrams. In addition, the largest Lyapunov exponents are computed. This computation further confirms the existence of chaotic behavior and the accuracy of numerical simulation. These results revealed that the introduction of disease and the use of an impulsive control strategy can change the dynamic behaviors of the system. The same results also have been observed in continuous-time models of 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 (NSFC no. 31170338 and no. 30970305) and also by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001).