Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 3989625, 14 pages

http://dx.doi.org/10.1155/2016/3989625

## Chemical Control for Host-Parasitoid Model within the Parasitism Season and Its Complex Dynamics

^{1}School of Management, Wuhan University of Technology, Wuhan, Hubei 430070, China^{2}School of Economics and Management, Hubei University for Nationalities, Enshi, Hubei 445000, China

Received 26 November 2015; Accepted 28 February 2016

Academic Editor: Rigoberto Medina

Copyright © 2016 Tao Wang and Youtang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the present paper, we develop a host-parasitoid model with Holling type II functional response function and chemical control, which can be applied at any time of each parasitism season or pest generation, and focus on addressing the importance of the timing of application pesticide during the parasitism season or pest generation in successful pest control. Firstly, the existence and stability of both the host and parasitoid populations extinction equilibrium and parasitoid-free equilibrium have been investigated. Secondly, the effects of key parameters on the threshold conditions have been discussed in more detail, which shows the importance of pesticide application times on the pest control. Thirdly, the complex dynamics including multiple attractors coexistence, chaotic behavior, and initial sensitivity have been studied by using numerical bifurcation analyses. Finally, the uncertainty and sensitivity of all the parameters on the solutions of both the host and parasitoid populations are investigated, which can help us to determine the key parameters in designing the pest control strategy. The present research can help us to further understand the importance of timings of pesticide application in the pest control and to improve the classical chemical control and to make management decisions.

#### 1. Introduction

Integrated pest management (IPM) is the selection and application of pest control actions that can ensure favourable economic and ecological consequences [1–3], which has been successfully applied to agricultural pest management situations. IPM employs a variety of tactics including cultural controls, biological controls, and chemical controls or pesticides, while biological pest control is one of the most common measures applied in IPM through the control and management of natural predators and parasites. For example, mosquitoes are often controlled by putting* Bacillus thuringiensis* (Bt) ssp.* israelensis*, a bacterium that infects and kills mosquito larvae, in local water sources. The aim of biological pest control is to eliminate a pest which can minimize harm to the ecological balance of the environment in its present form.

Pesticide application or chemical control is another important component of IPM measures, which refers to the practical way in which pesticides including herbicides, fungicides, and insecticides are sprayed to the pest population. Public concern about the application of pesticides has highlighted the need to make this process as efficiently as possible which could minimize their release into the environment and human exposure [1–3]. Although pesticide and insecticide applications can cause a number of problems, in most cropping systems they are still the principle means of controlling pests once the economic threshold that defines the lowest population density at which the control actions should be applied has been reached. In practice, pesticides can be relatively cheap, are easy to apply and fast-acting, and in most instances can be relied on to control the pests [1–3].

Undoubtedly, mathematical model is one of key tools to help us to understand those factors in the IPM strategies and the pest control. Recently, the continuous predator-prey models concerning IPM strategy have been developed and investigated [1, 4–10]. In particular, several factors including pest natural enemy ratios, starting densities, timings of natural enemy releases, dosages and timings of insecticide applications, and instantaneous killing rates of pesticides on both pests and natural enemies have been addressed by Tang et al. [10]. Moreover, the importance of timings of pesticide sprays and natural enemy releases has been studied through the stability threshold condition for a pest eradication periodic solution.

However, the discrete or nonoverlapping generation is a common feature among the host and parasitoid populations [11–15]. If so the use of continuous-time models to describe the interaction between the pest population and its natural enemy population becomes questionable. Those show that the discrete host-parasitoid models are much more realistic when the populations have discrete and synchronized generations [16–21]. The complex dynamics of the host-parasitoid model with Holling II functional response function have been investigated by Tang and Chen [19]. The classical Nicholson-Bailey model for a two species’ host-parasitoid system with discrete generations and IPM strategies has been studied by Tang et al. [7]. Note that the pulse IPM control strategies assumed to be applied at the end of each periodic number of generations and some important issues concerning IPM have been addressed.

According to the facts and main results of [7, 10], we know that the timings of pesticide application and releasing natural enemies play a key role in successful pest control. Note that there are several different pesticide spraying methods based on the pest growth generations and the parasitism season: the pesticides are applied at the beginning of each generation; the pesticides are sprayed at the end of each generation; and the pesticides can be applied at any time point within the pest growth generation and the parasitism season. Therefore, the questions are whether there exists an optimal pesticide application time at which the number of the pest populations can be minimized, even eradicated, and how the pesticide application time and efficiency affect the successful pest control and complex outbreak patterns.

To address those questions, it follows from the basic modelling methods proposed in literatures [22–24] that we assume that the chemical control tactic is applied instantaneously within the generation at any time. The main purpose is to extend the Holling II host-parasitoid model with a fraction of survival rate of parasitoid from one generation to next [25, 26] and then investigate the effects of timings of pesticide application and killing rate on this model. In particular, the existence and stability of both the host and parasitoid populations extinction equilibrium and parasitoid-free equilibrium have been investigated. The interesting results indicate that the different pesticide application times could result in significantly different size of the host population and consequently influence the pest control. Moreover, the effects of all important parameters including timings of pesticide application on the threshold conditions have been discussed in more detail. Further, the complex dynamics including multiple attractors coexistence, quasi-periodic windows, chaotic behavior, and initial sensitivity have been studied by using numerical bifurcation analyses. Finally, the uncertainty and sensitivity of all the parameters on the solutions of both the host and parasitoid populations have been investigated, which can help us to determine the key parameters in designing the successful pest control strategy. The present research can help us to further understand the importance of pesticide application times in the pest control and to improve the classical chemical control and to make management decisions.

#### 2. The Host-Parasitoid Model with Chemical Control

##### 2.1. Host Growth Model with Chemical Control

The based discrete map for host or pest population can be described as follows:where is the pest population size at generation and are, respectively, the population production and the per capita production. A common example is the Rocker model with and .

It follows from the basic modelling methods proposed in literatures [22–24] that we assume that the chemical control tactic is applied instantaneously within the generation ; that is, there exists a positive constant with such that the pesticide is sprayed at time point . Further, we assume that a proportional number (denoted by ) of the pest populations has been killed, and thus after , the growth of pest population and its production depends on . Therefore, model (1) with chemical control can be formulated as

According to , we have

##### 2.2. Host-Parasitoid Model with the Effect of Overwintering Parasitoid and Chemical Control

Involving the parasitoid population into (1), the host-parasitoid system with discrete generations can be written [16, 17] as the generalized modelwhere and represent the host and parasitoid population abundance in generation , respectively, denotes the per capita net rate of the increase of the host population in the absence of parasitoid population, and is the proportion of host individuals that escape attack by the parasitoid. Note that is generally interpreted as the abundance of adult parasitoid females and as the abundance of host adults and thus represents the number of parasitoid eggs laid per host, the survival of the parasitoid in the attacked hosts, and the sex ratio of the emerging parasitoid adults, where denotes the density-independent survival of parasitoid propagules at generation . A fraction of pathogen can survive from one generation to the next. This discrete-generation framework characterizes a perfectly synchronized parasitoid interacting with a host that has distinct generations, which is frequent in host-parasitoid systems in temperate regions of the world and even some from more tropical regions when parasitism causes generation cycles within the overlapping generations of some multivoltine hosts [16, 17].

In particular, if then model (4) can be changed as the following impulsive difference equation:which indicates that the chemical control tactics have been applied at the beginning of each generation or parasitism season. For more details of impulsive difference equations, please see [7].

If then model (4) can be changed as the following impulsive difference equation:which shows that the chemical control tactics have been applied at the ending of each generation or parasitism season.

Holling (1959) was the first ecologist to investigate the interaction between host and parasitoid populations and discussed the functional relationship in depth through the experiments of the shrews and deer mice feeding on sawfly cocoons. Holling (1959) presents three different functional response classes, for which the following are nominal forms: type I (linear then constant), type II (decelerating rise to an upper asymptote), and type III (sigmoidal) [16, 17, 25, 26]. In the present work, we assume that the pest population follows the Ricker model in the absence of parasitoid population and the Holling II functional response function for parasitism; that is, we have

Therefore, based on the above special choices we will focus on the existence and stability of the boundary equilibrium which concerns the outbreak of the host population and then address how the timing of spraying pesticides affects the values of this boundary equilibrium. Moreover, the completely numerical bifurcation analyses and sensitivity analyses have been carried out to show the effects of interesting parameters including instant killing rate and pesticide spraying time point on the successful pest control and outbreak.

#### 3. Existence and Stability of Boundary Equilibria

It is easy to see that model (4) has a zero equilibrium and its stability can be determined by the following Jacobian matrix:

It follows from (7) that the stability of can be determined by the eigenvalues of following Jacobian matrix:that is, is stable when the eigenvalues of the above matrix are less than one in magnitude. According to we can see that the stability of is determined by the value of . This indicates that if , then is stable. Moreover, we note that the stability of only depends on the intrinsic growth of the host population and the instant killing rate , which means that both the host and parasitoid populations could die out if the pesticide is effective enough no matter when we spray the pesticides.

Although the timing of pesticide applications (i.e., ) does not affect the stability of , from Figure 1 we note that it can significantly influence the behavior of solutions of model (4). For example, letting the parameter vary and fixing for all others as those in Figure 1 at which holds true and further letting all solutions start from the same initial value, we can see that the host population tends to zero more quickly for small values of . This confirms that we should apply the pesticides at the ending of parasitism season or population generation, which could be best for the host control in practice.