Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 3030723, 10 pages

https://doi.org/10.1155/2018/3030723

## Analysis and Control of the Singular System Model of Aphid Ecosystems

^{1}Department of Mathematics, Northeastern University, Shenyang 110006, China^{2}School of Mathematics and Information Science, Anshan Normal University, Anshan 114007, China^{3}Department of Mathematics, Liaoning Normal University, Dalian 116029, China

Correspondence should be addressed to Tie Zhang

Received 26 May 2017; Revised 14 November 2017; Accepted 28 December 2017; Published 19 February 2018

Academic Editor: Shoudong Huang

Copyright © 2018 Jingna Liu 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

Considering the change of the parameter related to the natural enemy population and the impact on the aphid populations in the fold catastrophe manifold, the singular system model of aphid ecosystems is proposed. Combining singular system theory with catastrophe theory, the corresponding dynamics behaviors and the existence conditions of the impasse points are given by using the qualitative analysis. The biological significance of the analytical results is also discussed. The controllers are designed to make the aphid populations stabilize the refuge level by releasing natural enemy. Some numerical simulations are carried out to prove the results.

#### 1. Introduction

Aphid populations are serious pests of wheat and many crops in the world. In some environment and physiography, aphid populations may lead to yield and economic losses [1–4]. Due to the fact that the fast reproduction of the aphid populations and the outbreak is always happening, insecticides are usually used to manage aphid pests. However, insecticides cause environment pollution and the lower quality of agricultural products. To avoid adverse effects of total reliance on insecticides and to ensure the well balanced ecosystems, there is a need to have ecofriendly management measures like releasing natural enemy [5]. Thus it is important to reveal the outbreak of aphid populations using mathematical models. In [6], the characteristics of catastrophe model are first used to describe and explain the phenomena of the population outbreak. In [7], a fold catastrophe model is proposed to explain the outbreak mechanism of the aphid populations by taking the proportion of winged aphid for the state variable and the effective leaf area for a control variable. In [8], on the basis of the Logistic model, a fold catastrophe model is built to show the complex dynamics behaviors of the aphid populations by taking the density of the aphid population for the state variable and the environmental factor for the main parameter. In [9], based on the fold catastrophe model, a new model is proposed by applying the Allee effect to the logistic equation and the corresponding ecological interpretations are provided. According to the model above, Zhao et al. explain the sudden decrease of the aphid populations after spraying pesticides by means of real data and predict the outbreak of the aphid populations by determining catastrophe regions [10, 11]. The results above show the application of catastrophe theory in aphid ecosystems. In fact, catastrophe theory is the theory related to the bifurcation theory, singularity theory, and structure stability [11].

Catastrophe is a widespread phenomenon in various fields, which frequently appears in engineering systems (such as the voltage load jump phenomenon in power systems). Some singular system models are usually used to describe the jump phenomenon in power systems and some results are obtained. Based on the singularity induced bifurcation of singular system, the complex dynamics behavior of power systems is studied and jump mechanisms are investigated [12–14]. According to the theory related to impasse points, the structure stability of power systems is analyzed and the impasse points in a circuit system are used to show the voltage load jump phenomenon [15, 16].

What is the similarity between aphid populations’ outbreak or sudden decrease and the voltage load jump phenomenon in power systems? Can the singular system models be used to describe the outbreak or sudden decrease phenomenon of aphid populations? It is what we are thinking about the focus of the paper.

The paper is organized as follows: In Section 2, the singular system model of aphid ecosystems is proposed. In Section 3, the dynamics behavior of aphid populations is discussed and the existence conditions of the impasse point are obtained by using qualitative analyses. In Section 4, the controllers are designed to keep aphid populations at refuge level and some numerical simulations are carried out to prove the results.

#### 2. Modelling

Consider the following fold catastrophe manifold of the aphid populations [11]:where is the density of the aphid population; is the parameter related to the natural enemy population; , represent the intrinsic rate and the environmental carrying capacity of the aphid population, respectively; , represent the predation rate of the natural enemy population and the half saturation coefficient, respectively.

In model (1), is the parameter related to the natural enemy population. Without loss of generality, let it be the density of the natural enemy population and change as follows:where is the intrinsic rate of the natural enemy population; is the carrying capacity of the natural enemy population related to the density of the aphid population; and is the carrying capacity of the natural enemy population related to other environmental factor, such as crop and prey (except aphid) populations and so on, and it is the maximum density of the natural enemy population when the aphid population is absented.

Coupling the algebraic equation (1) with the differential equation (2), the singular system model for the aphid ecosystem is got:where , , , , , , and are defined as the above.

For model (3), letapplying transformation (4) to model (3), the following model is got:

Letthe following are given:

In model (5), letIt becomes

Model (9) is discussed on the domain , where is

#### 3. Qualitative Analysis

##### 3.1. Impasse Points

Consider the following model:

According to literature [17], the equilibria and singular points of model (11) are defined as follows:

Lemma 1. *For model (9),*(1)*there may exist five equilibria , , , , and , where and satisfy the following equation set:*(2)*there exist two singular points and , where , , , and .*

*According to literature [16], let be the induced solution curve, then the limit points and impasse points are defined as follows.*

*Definition 2. *For model (11), for any (resp., ), there exists a neighbourhood of such that (resp., ), then the point is called right (resp., left) limit point of the induced solution curve at , where is any of subsets of and

*Definition 3. *For model (11), the point is forward (resp., backward) point, if is a right (resp., left) limit point of the induced solution curve .

*According to the definitions above, in order to find the impasse points, the following conditions are given:*

*(1) The impasse points must be the singular points.*

*(2) The corresponding limit point must be the solution to .*

*(3) The corresponding limit point must be the solution to .*

*(4) The linear coefficient for must not be zero in the Taylor expansion of the at the point .*

*In fact, for singular points of model (9),thus, and are the solution to .*

*Sinceone hasthus, is the solution to , but is not.*

*And the Taylor expansion of the at the point isthus, the following result is got.*

*Theorem 4. For model (9),(1)if , then the point is a left limit point of the induced solution curve , and the point is a backward impasse point;(2)if , then the point is a right limit point of the induced solution curve , and the point is a forward impasse point;(3)if , then there is no limit point or impasse point.*

*3.2. Qualitative Analysis Results*

*3.2. Qualitative Analysis Results*

*Based on the characteristics of the equilibria, singular points, and impasse points, using the geometrical analysis method, the qualitative analysis results of the singular system model (9) are obtained.*

*Theorem 5. The equilibria , are both unstable nodes, and the equilibrium is a stable node.*

*Theorem 6. For model (9), if and , the qualitative analysis results are as follows:(1) If the points and are on either side of point , respectively, then they are both stable nodes and the point is a backward impasse point (see Figure 1).(2) If the points and coincide, then that is a saddle point; the point is a stable node; and there is no impasse point (see Figure 2).(3) If the points and locate below the point , then the point is a stable node; the point is a saddle point; and the point is a forward impasse point (see Figure 3).(4) If the points and coincide, then that is a saddle point and the point is a forward impasse point (see Figure 4).(5) If the points and disappear, then the point is a forward impasse point (see Figure 5).*