Complexity

Volume 2019, Article ID 8326164, 14 pages

https://doi.org/10.1155/2019/8326164

## Models to Assess the Effects of Nonsmooth Control and Stochastic Perturbation on Pest Control: A Pest-Natural-Enemy Ecosystem

^{1}School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650500, China^{2}College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China^{3}Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China^{4}Laboratory for Industrial and Applied Mathematics, York University, Toronto M3J 1P3, Canada^{5}Department of Mathematics, Hubei Minzu University, Enshi 445000, China^{6}Department of Applied Mathematics, University of Waterloo, Waterloo N2L 3G1, Canada

Correspondence should be addressed to Wenjie Qin; moc.liamtoh@niqeijnew

Received 15 January 2019; Accepted 5 March 2019; Published 14 April 2019

Academic Editor: Xiaopeng Zhao

Copyright © 2019 Xuewen Tan et al. 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

This paper investigates the impact of the threshold control strategy and environmental randomness on pest control. Firstly, a fixed-time impulsive stochastic ecosystem with IPM strategy is proposed, where the local and global existence of positive solution and the boundedness of expectation are discussed in detail. Moreover a sufficient condition for the extinction of the pest population with probability-1 is given. Then, a state-dependent stochastic ecosystem with IPM strategy is proposed. By employing the numerical simulations, the effects of ambient noise intensity on pest-outbreak are discussed. The result shows that there is a close relationship among the frequency of pest-outbreak, ET, the environmental perturbation intensity, and control measures. This study helps us to understand the impact of random factors on pest-outbreak frequency by theoretical derivations and numerical simulations; the results have directive significance in the design of an optimal control strategy for the department of ecological agriculture.

#### 1. Introduction

Pest prevention and control have been attaching great importance by the agricultural departments as well as the management departments due to the outbreak of disaster pests. In the early days, chemical pesticides have played important roles in the pest control since pesticides can quickly kill a significant portion of a pest population, and it is convenient to carry out. However, the abuse of pesticides can also inevitably bring some negative effects such as the pest resistance and environment pollution. Therefore, a lot of control measures including chemical, physical, and biological methods have been proposed through long-term practice; Integrated Pest Management (IPM) is a combination of the above three tactics, which aims at the maximization of economic benefits [1–5] with the combination of the dynamical interaction between pests and their natural enemies and the implement of the comprehensive control strategies, in the hope of keeping the pest population under Economic Injury Levels (EIL).

Mathematical models can assist in designing strategies to control pest-outbreak. The application of IPM strategies has proven to have timeliness and transient effects. Moreover it also has made a crucial difference in the pest control because of the dynamical interaction based on intra-specific cooperation. Recently, many scholars have suggested using impulsive differential equations to investigate the dynamics of pest control models [6–15].

However, the effects of external random noises such as fire, flood, earthquake, and drought on the population cannot be ignored. For example, when the population is small or the noise intensity is high, it is not scientific enough to use the deterministic model. Meanwhile, it is necessary to establish a stochastic dynamic system to study the reality better in ecosystem. There were many scholars such as Kolmogorov [16, 17] and Feller [18, 19] who had done pioneering works in stochastic dynamic system. Meanwhile, in 1951 Itô established a stochastic integration [20–22]. In 1953, Doob published his results [23] and hence built up the mathematical theoretical foundation of stochastic dynamic system. Recently, the theory of the stochastic differential equation has gained renewed interest and also been applied to other fields [24–33].

In fact, the growth rate, environmental capacity, competition factors, and other parameters of the ecosystem are all subject to external random factors such as drought, harvest, fire, earthquakes, floods, deforestation, and hunting. Therefore, stochastic perturbations are introduced in the nonsmooth systems especially for pest-natural-enemy ecosystem being a very practical significance of the topic. The main propose of this paper is to investigate the effects of environmental randomness on IPM strategies. In order to do this, a fixed-time stochastic impulsive ecosystem with IPM strategies and a state-dependent stochastic ecosystem are proposed. Some theoretical derivations, qualitative analysis, numerical simulations are given in order to make the comprehensive and systematic research on those models, which could be in the hope of providing some suggestions for pest control and ecological management. Some theoretical, numerical, and biological analysis are given to investigate how environmental randomness affects the pest-outbreaks.

The organization of the present paper is as follows. In the next section, a fixed-time stochastic pest-natural-enemy ecosystem with IPM strategies is proposed, where the local and global existence of positive solution and the boundedness of expectation are discussed in detail. A sufficient condition for the extinction of pest population with probability is given. In Section 3, with the establishment of a state-dependent stochastic predator-prey ecosystem with IPM strategies and the development of the numerical simulations and methods, the effects of ambient noise intensity on pest-outbreak are discussed. The paper ends with some interesting biological conclusions, which complement the theoretical findings.

#### 2. A Fixed-Time Impulsive Stochastic Ecosystem with IPM Strategy

##### 2.1. Model Formulation

In 2001, in order to investigate the predator-prey interaction when the prey exhibits group defense, Xiao and Ruan [34] proposed the following model:where and represent the densities of prey (pest) and predator (natural enemy) populations, respectively; denotes the intrinsic growth rate and represents the carrying capacity for ; and are the conversion rate and death rate for , respectively; the sigmoidal saturation function, especially (i.e., a simplified Monod-Haldane or Holling IV function [35]), describes the group defence of .

Taking into account the effect of environmental noise and IPM strategies in model (1), then we can obtain the following fixed-time impulsive stochastic modelHere is the pulse control period, is the killing rate for pest population, and is the release constant for natural-enemy; those are the IPM control parameters; , especially, denotes the white noise (i.e., an error term), is a continuous bounded function on representing the intensity of the noise, is a Brownian motion defined on a complete probability space , and .

##### 2.2. Preliminaries

For convenience, we will introduce some notations, definitions, and lemmas for impulsive stochastic differential equations (ISDE) [22, 24], which will be used for establishing our main results later.

*Definition 1 (see [24]). *For the impulsive stochastic differential equations,with initial condition . A stochastic process , , is said to be a solution of model (3) if

(i) is -adapted and each interval ; , where is all valued measurable , a.s. for every ;

(ii) There existswith probability for each ;

(iii) obeys the integral equationfor almost all . And obeys the integral equationfor almost all . Moreover, satisfies the impulsive conditions at each ) with probability .

*Definition 2 (see [22]). *If the pest population satisfies , then is said to be extinction with probability .

Lemma 3 (the strong law of large numbers for martingales [22]). *Let be a local continuous martingale with . Ifholds true, then*

Lemma 4 (Itô’s formula [22]). *Let be defined in , and satisfies which is a -Itô process, and . If , then is Itô’s with*

The comparison theorem of stochastic differential equation is one of the most important technologies to investigate the stochastic mathematical biology models; now it is given as follows.

Lemma 5 (comparison theorem of SDE [22]). *Assume that is the solution for stochastic differential equationwhere if**(i) there exists a function such thatMeanwhile, satisfies and ;**(ii) ;**(iii) .**Then there exists with probability for .*

##### 2.3. Mathematical Analysis for Model (2)

In this section, the local and global existence of positive solution and the boundedness of expectation are discussed in detail, and the sufficient condition for the extinction of pest population with probability is given. Firstly, the local existence of positive solution is given as follows.

Theorem 6. *Model (2) exists a unique local positive solution , for each , being a finite number.*

*Proof. *Consider the following stochastic differential equationwith . Motivated by the main result of [27], (13) provides a unique local positive solution. Meanwhile, assume that with . Since and are continuous on and . For any , it gives that andfor every and . Furthermore, it is easy to derive Analogously, for any , we can derive that and for every and . Meanwhile, it givesAccording to Definition 1, model (2) provides a unique local positive solution. The proof of Theorem 6 is completed.

The existence and uniqueness of the local positive solution of model (2) are derived by Theorem 6, and some numerical simulations are also given, as shown in Figures 1 and 2. In particular, there are two cases representing the different noise density in Figures 1 and 2, which indicate the smaller the environmental disturbance is, the more effective the impulsive strategy is. Meanwhile, if the strong disturbance can cause the dynamical behaviors of system becoming more complex, it will bring some great challenges for pest control.