Complexity

Volume 2019, Article ID 9365293, 12 pages

https://doi.org/10.1155/2019/9365293

## Modelling and Analysis of a Host-Parasitoid Impulsive Ecosystem under Resource Limitation

^{1}Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China^{2}Laboratory for Industrial and Applied Mathematics, York University, Toronto M3J 1P3, Canada^{3}Department of Mathematics, Hubei Minzu University, Enshi 445000, China

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

Received 16 November 2018; Accepted 26 December 2018; Published 14 January 2019

Academic Editor: Toshikazu Kuniya

Copyright © 2019 Peipei Wang 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

With a long history of theoretical development, biological model has focused on the interaction of a parasitoid and its host. In this paper, two Nicholson-Bailey models with a nonlinear pulse control strategy are proposed and analyzed to examine how limited resource affects the pest control. For a fixed-time discrete impulsive model, the existence and stability of the host-free periodic solution are derived. Threshold analysis suggests that it is critical to release parasitoid in an optimal number in case of the happening of the intra-specific competition, which will seriously affect the pest control. Bifurcation analysis reveals that the model exists complex dynamics including period doubling, chaotic solutions, coexistence of multiple attractors, and so on. For a state-dependent discrete impulsive model, the numerical simulations for bifurcation analysis are studied, the results show that how the key parameters and the initial densities of both populations affect the pest outbreaks, and consequently the relative biological implications with respect to pest control are discussed.

#### 1. Introduction

Mathematical models can assist in the design and understanding of basic problems in biology, medicine, and life sciences, which can take many forms including dynamical system, differential equation, difference equation, and so on [1–3]. Different models can describe different biological phenomena, with each having different strengths. Some certain species, including many species of insect, have no overlap between successive generations and so their populations evolve in discrete time-steps. The discrete model governed by a difference equation is more appropriate than the continuous ones taking into account the fact that the population has a short life expectancy, non-overlapping generations in the real world [4–6], so it is reasonable to study biological models governed by difference equations [7–13].

The insect pests’ outbreaks often cause serious ecological and economic problems, requiring complex control measures to reduce the harm brought by insect pests of agriculture and insect vectors of important plant, animal, and human diseases. In order to control pest more effective, at the same time, environmental influences and human interventions are taken into consideration, and the concept of Integrated Pest Management (IPM) strategy has been proposed [14–17], as a consequence it is challenging to evaluate the effectiveness of control measures such as biological (releasing predators), chemical (spraying pesticides), and physical control tactics. Based on the concept of IPM strategy, some mathematical models to describe the interaction between the pest and its natural enemy have been developed [18–21].

One of the IPM strategies is to release natural enemies and spray chemical pesticides at a fixed time, since pesticides can quickly kill a significant portion of a pest population, and it is convenient to carry out. Meanwhile, the IPM strategy can be applied more effectively to achieve maximum effect at the minimum levels of pesticide if the life cycle of a pest is understood; i.e., it is used when the pest is at its most vulnerable. Pesticides can be relatively cheap and easy to apply and fast-acting and in most instances can be relied on to control the pests. Therefore, the periodic control strategy has been applied widely and extended in agriculture and ecology [18, 19].

Another important concept in IPM strategy is the Threshold Policy Control (TPC) [22, 23], which maintains the pests’ density below the Economic Injury Level (EIL) [24–26] by releasing predators and spraying pesticides once the pests’ density reaches the Economic Threshold (ET) [27, 28]; it is also called by state-dependent feedback control. As we know, the main purpose of IPM is to maintain the density of the pests below the EIL rather than seeking to eradicate them, and the suitable tactic can be applied only when the density of pests reaches the given ET; it can minimize the damage of insecticides to non-target pests and preserve the quality of the environment. Therefore the state-dependent impulsive dynamical model could be appropriate to describe such control tactics [20, 21].

However, the above IPM strategy has assumed that the killing rate of pesticide is a constant, which shows that the agricultural resources including chemical pesticides, labor forces, facilities, and costs are very effective and sufficient for pest control. In reality, every community or country has an appropriate or limited capacity for pesticides, costs, etc., especially for developing countries. Although it is critical to understand resource limitation to effective management and conservation of populations, limited resource is difficult to quantify partly because it is a dynamic process [29–32]. The main purpose of this paper is to describe the limited resources based on the classic Nicholson-Bailey model [33]; there are two different models with IPM strategies being derived. In order to characterize the saturation phenomenon of the limited resources, the constant fatality rate is replaced with a nonlinear saturation function which depends on its population density. Some theoretical, numerical, biological analyses are given to investigate how limited resources affect host outbreaks.

The organization of the present paper is as follows: in the next section, a fixed-time impulsive control for a Nichloson-Bailey model with resource limitation is proposed; by employing the qualitative analysis and the discrete dynamical system determined by the stroboscopic map, the threshold conditions which guarantee the existence and stability of the host-free periodic solution are obtained. Meanwhile, 1-dimensional bifurcation analyses are investigated; the key parameters and initial values of both populations affect the pest outbreaks are given by taking advantage of numerical simulation. In Section 3, we describe a Nichloson-Bailey ecosystem with resource limitation by state-dependent impulsive control; by employing numerical simulations, the model is analyzed to understand how resource limitation affects pest population outbreaks; the intersection between the initial densities and pest control will be discussed. The paper ends with some interesting biological conclusions, which complement the theoretical findings.

#### 2. Fixed-Time Impulsive Control for a Nicholson-Bailey Model with Resource Limitation

##### 2.1. Model Formulation

In 1935, Nicholson and Bailey [33] proposed a classic discrete host-parasitoid model, as an intergenerational survival rate for parasitoids was considered. In 2008, Tang [8] extended the basic Nicholson-Bailey model as follows: Here and represent the density of hosts(or pests) and parasitoids(or natural enemies) at -th generation, respectively; is the intrinsic growth rate for the host population; denotes the searching efficiency of the parasitoid population; the terms and represent the probability that the host succeeds and fails in escaping from the parasitism, respectively; is the density-independent survival of the parasitoid at -th generation, .

In order to investigate the dynamics of system (1) with IPM strategies, by introducing periodic spraying pesticide only for hosts, and releasing parasitoids, Tang [8] proposed the following Nicholson-Bailey model: with the initial densities and . Meanwhile, is the pulse period, is the killing rate for hosts, and and are the release rate and constant for parasitoids at -th generation, respectively. and ( and ) denote the densities of both populations at -th generation before (after) the impulsive perturbations, respectively.

Now, taking into account the resource limitation and saturation effects, that is, a linear pulse in model (2) is replaced with a saturation phenomenon of limited resources; especially, a Hill function [34] is used to describe the resource limitation. For convenience and simplification, model (2) can be rewritten as where and represent the maximal fatality rate and the half-saturation constant for hosts, respectively. and represent the maximal release rate and the half-saturation constant for parasitoids, respectively. Assume that and for .

##### 2.2. Mathematical Analysis of the Host-Free Periodic Solution and Threshold Conditions

With the main purpose of IPM strategy, it is necessary to illustrate the existence of the host-free periodic solution of model (3) and focus on determining its global attractivity. In order to do this, we first investigate the host-free set: , which is clearly invariant by model (3); it follows from model (3) and its pest-free set that Obviously, model (4) is a periodic system, so the periodic solution can be defined at the subinterval with the initial value , . Noting that represents the parasitoid’s density after the impulsive perturbation.

From the first equation of model (4), we can use this iteration equation at interval and obtain Especially, it gives that as .

Now, combining the second equation of model (4) and the above one yields Denoting as , then the above equation can be rewritten as the following difference equation: Equation (6) is a stroboscopic map of model (4): it describes the relations of the parasitoids’ density between any two successive pulse points; that is, the existence of a positive steady state of (6) represents the existence of a positive periodic solution of model (4). Therefore, we first explore the positive steady state of the stroboscopic map (6); let us get the derivative of with and yield According to the stability theory of differential equation, it is necessary to ensure that the inequality holds true, and this condition is equivalent to where , , . By a simple calculation, one yields . It is easy to find that if holds true, then , which implies . For simplification, can be rewritten as The above inequality can ensure the stability of a positive solution of the stroboscopic map (6). Meanwhile, it also plays an important role in discussing the existence of the positive fixed point for (6).

Next, we continue to investigate the positive fixed point of the stroboscopic map (6). For convenience, we denote this fixed point by ; then it follows from the stroboscopic map (6) that which means where , , . Obviously, we can obtain that and as holds true. That is, the stroboscopic map (6) exists in only one positive fixed point . Therefore, we have the following result.

Lemma 1. *Model (4) exists as a positive periodic solution provided that holds true. Meanwhile, for every positive solution of model (4), it is given that . Here with .*

Based on Lemma 1, we can obtain the general expression of the host-free periodic solution of model (3) in the interval for all denoted by In order to investigate the global stability of the host-free periodic solution of model (3), the following theorem is derived.

Theorem 2. *Let be any positive solution of model (3); the host-free periodic solution of model (3) is globally asymptotically stable in the first quadrant, provided that *

*Proof. *To prove Theorem 2, the following condition (16) is considered first. It follows from , model (3), and that we have which implies Therefore, from the above inequality, we can assume that there exists a sufficiently small positive such that Next, we continue to prove Theorem 2. Obviously, it is easy to show that . Then the following comparison equation with pulse is considered. By using Lemma 1, it gives that and . Hence, for all , the inequality holds true. Then according to model (3), we have Associated with the condition (16), the following inequality is derived:so it is easy to show that and . It follows from model (3) that , and, for any , we have in other words, .

Thus, we have proved the global stability of the host-free periodic solution of model (3). This completes the proof.

Based on Theorem 2, the global stability of the host-free periodic solution of model (3) is determined by and ; that is, it is crucial that those two threshold values make great effects on the dynamics of model (3). Therefore, in the following, we employ and to investigate the important factors which affect the threshold values most significantly by numerical simulations.

As we know, resource limitation exists extensively in pest control, especially in some developing countries. There are so many factors affecting the limited resource; this study focuses on the thresholds and which ensure the existence and stability of host-free periodic solution, and those thresholds will directly influence the success of pest control. According to the expressions of and , the impulsive periodic , the maximal release rate , and the half saturation constant for parasitoid population are the key parameters that affect , and the limited resource. In Figure 1(a), with the increasing of the pulse periodic , the threshold decreases, while the curve of - is a typical single-peak curve. And only if the pulse periodic , holds true, then the host-free periodic solution can become stable. The higher the frequency of a control measure is, the lower the pulse periodic is. Figure 1(a) indicates that the pest can be out of control by increasing the control frequency (i.e., decreasing the control periodic ), and it is not even realistic to do under the limited resource.