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The switching discrete prey-predator model concerning integrated pest management has been proposed, and the switches are guided by the economic threshold (ET). To begin with, the regular and virtual equilibria of switching system have been discussed and the key parameter bifurcation diagrams for the existence of equilibria have been proposed, which reveal the three different regions of equilibria. Besides, numerical bifurcation analyses show that the switching discrete system may have complicated dynamics behavior including chaos and the coexistence of multiple attractors. Finally, the effects of key parameters on the switching frequencies and switching times are discussed and the sensitivity analysis of varying parameter values for mean switching times has also been given. The results proved that economic threshold (ET) and the growth rate (

The integrated pest management (IPM) is applied [

For the sake of simplicity, we rewrite system (

In this study, The prey population and the predator population are regarded as the pest and the natural enemy, respectively. we want to discuss discrete-time prey-predator models by utilizing a threshold policy (TP) to control the pest population (prey). When the pest population density is above the economic threshold (ET), the pesticides are applied for pest population and the natural enemies (predator) population is released simultaneously. The IPM subsystem with

By

In the region

We combine system (

In this section, from the point of pest control, we consider the discrete-time switching system (

For the subsystem

If

We have similar analysis for subsystem

Let

Note that the local stability of the fixed point

The characteristic polynomial is denoted as

Then

Let

Then we can calculate the local dynamics of the fixed point

Let

It is a sink if one of the following conditions holds:

It is a source if one of the following conditions holds:

It is a saddle if the following condition holds:

It is nonhyperbolic if the following condition holds:

In the following section, the definition of the real and virtual equilibria concepts is briefly summarized [

Let

For the subsystem

In order to consider the relationship of equilibria for two subsystems, we define five curves as follows:

If

If

If

Real/virtual equilibria. The parameter values are fixed as follows:

In the process of pest control, we should use IPM strategies to prevent pest outbreaks (i.e.,

Parameter bifurcation diagram for the existence of equilibria of system (

In order to show the richer dynamic behavior of discrete switching system (

To discuss the complex dynamics of model (

Bifurcation diagram for system (

From Figure

Several different types of typical solutions of system (

Furthermore, there exists another attractor that tends to infinity (i.e., the prey and predator densities eventually tend to infinity). This attractor is very harmful for pest control. Moreover, in this case, we can not spray insecticides and release natural enemies to keep prey density below the ET. The above results suggest that the control of insect pests may depend on the initial densities of prey and predator populations. Thus, the initial attraction regions of these stable attractors play a pivotal role in pest management. To show this, the basins of attraction with respect to Figure

Basins of attraction of four coexisting attractors of system (

In this section, we will discuss the effects of key parameters on the switching frequencies of system (

For convenience, we provide the definition of switching times and switching frequencies as follows.

For time series of the prey population in system (

The switching times and switching frequencies are critical factors for successful pest control. If switching times (or switching frequencies) of prey population are a constant number with the time passing, then the system is stable, and we can easily design the control strategy. Moreover, the smaller the switching times (or the larger the switching frequencies) are, the more frequently the control tactics should be applied; that is, the pesticide applications and releasing strategies should be implemented frequently, and this is not effective and may result in adverse effects. On the contrary, if the switching times and switching frequencies are dynamic, the system is unstable or has entered a chaotic state. What is more, it is difficult to design suitable control measures for pest control because we do not know when and how control strategies should be adopted.

After that, the switching frequencies are analysed with the spectrogram, which is a visual representation of the spectrum of frequencies in the time series of the prey population as they vary with time. To address the effects of parameters on switching times and switching frequencies, we let the initial values and intrinsic growth rates

Switching times and switching frequencies; (a–c) switching times and (d–f) switching frequencies. The parameter values are fixed as follows:

In order to show more details of the effects of key parameters on switching frequencies, we first introduce the following definition.

Mean switching times are the mean of all switching times between

The effects of key parameters

Mean switching times versus

To examine the sensitivity of mean switching times to parameter variation, we use Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCCs). The LHS method is a type of stratified monte carlo sampling, which was first proposed by McKay et al. [

PRCC results and PRCC scatter plots. The baseline parameters are fixed as follows:

Figure

Figures

PRCCs tell us how the mean switching times are affected if we increase (or decrease) a specific parameter at a time point (see Figure

Mean switching times versus

Specifically, in order to show the relative change in the mean switching times over all generations, we performed a box plot to all PRCCs for comparison (see Figure

It is well documented that the threshold policy (or on-off policy, switching policy) is commonly used in biological system, which is to allow removal of the prey (pest) or increase the predator (natural enemies), such as on-off policy in the herbivore-vegetation dynamics [

The bifurcation diagram of parameter

The key parameters concerning switching frequencies or mean switching times have been analysed, and consequently the relative biological implications with respect to pest control are discussed. If the switching frequencies and switching times are always unstable, it is difficult to design suitable control measures for pest control because we do not know when and how control strategies should be adopted. In order to show more details of the effects of key parameters on unstable states, we first introduce the mean switching times. The effects of key parameters

To examine the sensitivity of mean switching times to parameter variation, we used Latin Hypercube Sampling (LHS) and partial rank correlation coefficients (PRCCs). The results show that the economic threshold (ET) is strongly positively correlated to the mean switching times and the growth rate (

The numerical bifurcation analyses clarify that the switching system could have very complex dynamics including multiple attractor coexistence and chaotic solutions. From the pest control point of view, we have carried out extensively numerical investigations on the switching times and their biological implications have been discussed in more detail. The theoretical/mathematical analysis about subsystem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of Hubei Province of China (2015CFB264), the Key Laboratory of Biologic Resource Protection and Utilization of Hubei Province (PKLHB1506, Changcheng Xiang), and the National Natural Science Foundation of China (no. 11601268, Wenjie Qin).