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A nonlinear discrete switching host-parasitoid model with Holling type II functional response function, in which the switch is guided by an economic threshold (ET), is proposed. Thus, if the weighted density of two generations of the host population increases and exceeds the ET, then integrated pest management (IPM) measures are enacted, i.e., biological and chemical measures are implemented together, assuming that the chemical immediately precedes the biological inputs to avoid pesticide-induced deaths of the natural enemies. First, the existence and local stability of the equilibria of two subsystems were studied, and the existence and coexistence of several types of equilibria of a nonlinear switching system were analysed. Next, the nonlinear switching system was investigated by numerical simulation, showing that the system exhibits quite complex dynamic behaviour. A two-dimensional bifurcation diagram revealed the existence and coexistence regions of different types of equilibria including regular and virtual equilibria. Moreover, period-adding bifurcations in two-dimensional parameter spaces were found. One-dimensional bifurcation diagrams revealed that the system has periodic, quasiperiodic, and chaotic solutions, Neimark–Sacker bifurcation, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the initial densities of hosts and parasitoids associated with host outbreaks and their biological implications are discussed.

Pest control is important in subjects such as agriculture, fisheries, and ecology [

Host-parasitoid models including IPM intervention with fixed and unfixed time pulse effects were first proposed and analysed by Tang et al. [

For a pest population with nonoverlapping generations, the best control threshold depends not only on the density of the current generation but also on the density of the next generation. This means that chemical control and biological control applied together (but with the chemical control immediately preceding the biological inputs to avoid pesticide-induced deaths of the natural enemies) should be guided by the densities of both generation

The structure of this paper is as follows. In Section

A discrete host-parasitoid model with a Holling type II functional response function was numerically investigated by Tang and Chen [

In order to take into account IPM strategies for controlling the pest population, Xiang et al. [

But in reality, the optimal control threshold not only depends on the density of the current generation but also depends on the density of the next generation, which implies that the threshold

Therefore, combining model (

Obviously, model (

In order to reveal the dynamics of the whole switching system (

Then, model (

From now on, we call switching system (

Switching systems have different types of equilibria, and these equilibria play important roles in pest control. In order to show the existence of equilibria of switching system (

A point

If

Obviously, subsystem

The trivial equilibrium

The stability of

The eigenvalues of

In addition, there exists the parasitoid-free equilibrium

Here,

In the following, we address the local stability of the unique positive equilibrium

The unique positive equilibrium

In order to discuss the local stability of the equilibrium, we need to calculate the Jacobian matrix at the equilibrium. Firstly, we can rewrite subsystem

Therefore, we have

Conditions (

If

Similarly,

The host-free equilibrium

The stability of

Accordingly, we find that the eigenvalues are

For the stability of interior equilibrium

The unique positive equilibrium

The detailed mathematical expressions of

The proof of Theorem

In contrast to two subsystems (

According to Definition

By a simple calculation, we conclude that if

Similarly, if

Further, if

Finally, if

The interesting question is how the types of the interior equilibria of model (

We first choose

Bifurcation diagrams for the existence and coexistence of regular and virtual equilibria of system (

The stability and types of equilibria of free system are important for host control. For example, it is finally free of control if the equilibrium of the free system is globally stable. Therefore, in order to design the best control policies to prevent host outbreaks, one of the possible ways is to choose a desirable switching curve such that all equilibria of subsystem

In order to investigate the complex dynamic behaviour of system (

We first choose the instantaneous search rate

Bifurcation diagrams of system (

Phase trajectories of system (

In addition, when we choose the intrinsic growth rate

Bifurcation diagrams of system (

Periods of period point cycles of system (

Bifurcation diagrams of system (

To investigate the effect of different parameter values on the periodicity of solutions for both host and parasitoid populations, according to published methods [

The bifurcation diagram with respect to parameter

It is well known that the initial densities of host and parasitoid populations will have a significant impact on the dynamic behaviour of the system, so they are key to biological control, and in this section, we will investigate how they affect outbreak patterns or final states of the host population.

In order to analyse the relationship between initial densities of populations and host outbreak patterns, Figure

Illustrating the switching effects of initial densities of the host and parasitoid populations of system (

In order to reveal the effects of initial values of both host and parasitoid populations on the outbreak patterns of the host population in more detail, the basin of attraction of the host outbreak frequencies with respect to initial densities is shown in Figure

Dependence of the host outbreak frequencies on the initial values

The initial values not only influence the host outbreak frequencies but can also affect the host final stable states, i.e., multiple attractors could coexist. It follows from the bifurcation analyses that multiple attractors can coexist for a wide range of parameters (see Figure

Three coexisting attractors of system (

In order to illustrate the initial sensitivities more specifically, the basin of attraction with respect to three different host outbreak solutions of coexistence is shown in Figure

Basins of attraction of three coexisting attractors of system (

In this paper, we discussed a nonlinear switching discrete host-parasitoid model with Holling type II functional response function, where the switching strategies and control measures are guided by the weighted density of two generations of the pest population. The important innovation of the model lies in the threshold control strategy. Here, we consider the weighted density of two generations of the pest population as the threshold to judge whether the control strategy is implemented or not, which not only brings new challenges in theory but also makes pest control more operable. The most interesting theoretical analyses are to determine the types of all possible equilibria of the whole switching system and their stabilities because if the real equilibrium of the free subsystem is stable for the whole switching system, then the purpose of pest control can be realized easily. Further, the sensitivity analyses related to the key parameters and initial densities of both host and parasitoid populations can help us to design suitable control measures.

We first investigated the existence and local stability of equilibria of two subsystems and briefly analysed the existence and coexistence of the equilibria of the whole nonlinear switching system. We used the

On the other hand, the one-parameter bifurcation diagrams which were derived from system (

In addition, the relationship between initial densities and pest control was also studied. The results show that the initial densities of the host and parasitoid populations will affect the outcome of an IPM strategy, and the final stable states of the populations depend on their initial densities (Figures

Compared with the basic model and the main published results [

No data were used to support this study. In our study, there are only some numerical simulations to support our main result, and parameter values to support the result of this paper are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This study was supported by the National Natural Science Foundation of China (NSFC) (61772017 and 11631012) and by the Fundamental Research Funds for the Central Universities (GK201901008).