Abstract
In order to control pests and eventually maintain the number of pests below the economic threshold, in this paper, based on the nonsmooth dynamical system, a two-stage-structured pest control Filippov model is proposed. We take the total number of juvenile and adult pest population as the control index to determine whether or not to implement chemical control strategies. The sliding-mode domain and conditions for the existence of regular and virtual equilibria, pseudoequilibrium, boundary equilibria, and tangent points are given. Further, the sufficient condition of the locally asymptotic stability of pseudoequilibrium is obtained. By numerical simulations, the local bifurcations of the equilibria are discussed. Our results show that the total number of pest populations can be successfully controlled below the economic threshold by taking suitable threshold policy.
1. Introduction
Insects are the most prosperous animals in the world and they play an important role in the biosphere. And pests are the insects which usually are considered to be harmful to the survival of human beings or their crops and livestock. In fact, it is only when the number of pests reaches a certain level that they can do harm to human beings. Therefore, the control of pests should not be too blind. We take the number of pest populations as the control index and do control measures only when the number of pests results in the economic losses so as to avoid the excessive use of pesticides, environmental pollution, and harm to human beings.
The pest populations have individual differences due to different ages, sizes, and development stages, and such differences have an important impact on the dynamical behavior of the studied population, especially on the success or failure of the pest management [1]. Pest populations usually have two or more life stages; in this paper, we assume that pests have juveniles and adult stages. In recent years, many research works [2–10] have applied the stage-structured models to study population dynamics behavior.
In the growth process of most crops, the use of pesticides is a common method of pest management which can effectively control the number of pests not exceeding the economic injury level (). is defined as the lowest density of pest populations which will cause economic damage [11, 12]. How can the pest be controlled economically? Economic threshold () [13, 14] helps the pest management department to determine whether pest control will have an economic benefit. is defined as the number of pest population at which control action should be taken to prevent an increasing pest population from reaching . In our studies, the total number of the juveniles and adults pest populations has been chosen as a control index for one to decide whether or not to implement chemical control strategies. Once the number of pests exceeds the specified threshold level , pest control strategies are implemented, and otherwise there is no control action, which is called threshold policy [15–18]. This policy creates two systems, in fact, a system of ordinary differential equations with discontinuous right-hand sides (or a piecewise smooth system), and is a special and simple case of variable structure control in the control literature. Such a system has been called Filippov system [19–21] and has many applications in many areas [22–29]. In previous studies [9, 30, 31], scholars modeled the process of pest control by using impulsive differential equations and assumed that the process of killing pests is instantaneous. However, the death of pests due to spraying pesticide is not instantaneous but continuous, and the Filippov system gives a natural description of the process of spraying pesticides with a piecewise continuous system. Now there have been many applications [24–26, 28] on the investigations of the pest control models by using Filippov system, and the threshold index they chose was univariate. Our purposes in this paper are to develop a pest control stage-structured model with threshold policy by choosing two variables as the threshold index and to design simple implementable controls which drive the total number of the pest population below or to a desired globally stable level like .
The organization of this paper is as follows. In the next section, the basic model is proposed and the auxiliary definitions are given. In Section 3, we analyze the global dynamical behaviors of such a system including the sliding-mode domain, the existence interval of real equilibria, virtual equilibria, pseudoequilibrium, boundary equilibria, and tangent points as well as the local stability of pseudoequilibrium. By numerical simulations, the local bifurcations of the equilibria are also discussed. Finally, a brief discussion is given in the last section.
2. Model Formulation and Auxiliary Definitions
In this paper, we assume that the life course of the pest goes through two stages: juveniles and adults; juveniles and adults die with a constant death rate (denoted by ), and the maturation rate of the juveniles is also a constant, denoted by , while adults reproduction rate depends on its density; here we take Beverton-Holt function [4]. So the basic compartmental ODE model employed in this paper is as follows:where and denote the density of juveniles pests and adult pests, respectively; indicates that the effect of internal competition on the pest's birth rate occurs.
In order to control the pest population such that its density does not exceed the , we must carry out the chemical control strategy once the density of pest reaches . Based on the above-mentioned stage-structured pest growth model, a threshold control strategy is chosen, in which the total number of juvenile and adult pest population is taken as the index, and a nonsmooth stage-structured model is established by using piecewise smooth system. The threshold control strategy can be expressed as follows. Once the total number of the pest population exceeds , spraying pesticide is implemented, and a proportion of the juvenile and adult populations will be killed; otherwise, there is no action. Therefore, if , then both the juvenile and adult populations followwhere and denote the proportion of juvenile and adult pests killed by the action of insecticides, respectively. Suppose and the other parameters are the same as those of system (1).
If , the control measures are not necessary, and then both the juvenile and adult populations follow system (1).
Now we introduce some useful properties and definitions on Filippov system according to [28, 29].
Let ( as an index) with , andThen, system (1) and (2) can be rewritten as the following Filippov system:where , . Furthermore, the discontinuity boundary (or manifold) separating the two regions and is defined as is a smooth scale function with nonvanishing gradient on , and . Now we call system (5) defined in region as system (i.e., system (1)) and defined in region as system (i.e., system (2)).
In the regions and , the local trajectories are defined by the vector fields and . In order to extend the definition of a trajectory to the region , we divide the region as follows [29]:(i)crossing region: ,(ii)sliding region: ,(iii)escaping region: ,
where , , represents the standard scalar product.
In Filippov system (5), there exist four types of equilibria: real equilibrium, virtual equilibrium, pseudoequilibrium, and boundary equilibrium, which are defined as follows [29].
Definition 1. If and or and , then is called a regular equilibrium of system (5); if and or and , then is called a virtual equilibrium of system (5).
On , the local trajectories are defined by the convex combination of the Filippov system (5). Letand is called the sliding system (or called the sliding vector field).
Definition 2. If and , then is called a pseudoequilibrium of system (5).
Definition 3. If and or , then is called a tangent point of system (5).
Definition 4. If and or , then is called a boundary equilibrium of system (5).
3. Dynamic Properties of System (5)
In the following subsections, we will focus on dynamic properties of system (5), including the sliding-mode dynamics, the existence of regular and virtual equilibria, pseudoequilibrium, boundary equilibria, and tangent points, the stability of pseudoequilibrium, and local sliding bifurcations of equilibria.
3.1. Sliding-Mode Dynamics and the Existence of Equilibria
From the definitions of and , we have
and, in the following, we assume that holds. Therefore, the sliding region can be obtained asthat is,
By the convex combination method which is introduced in Section 2, the sliding-mode dynamics are determined bywhere , andThere is no need to calculate the exact form of () since it is not required in the analysis that follows.
Equilibria of Filippov system (5): According to the definitions provided in Section 2, under certain conditions, there exist several types of equilibria and one type of special point named as tangent point of system (5).
If andthen system (5) has a unique regular equilibrium in the region ,Conversely, if andthen equilibrium is a virtual equilibrium, denoted by .
If andthen system (5) has a unique regular equilibrium in the region , denoted byOtherwise, if , andthen the equilibrium point is a virtual equilibrium, denoted by .
If is chosen as a bifurcation parameter and the others are fixed, then the main results obtained above with respect to the equilibria of Filippov system (5) are as follows; other cases can be seen in Figure 1.(i)Ifthen Filippov system (5) has a unique regular equilibrium ;(ii)ifthen Filippov system (5) has a unique regular equilibrium ;(iii)ifthen two virtual equilibria and can coexist.
For the existence of the pseudoequilibrium , satisfies the following equationSolving above equation with respect to , we can obtain a possible positive root as follows:where , , . Therefore, ifthen the sliding system has a unique pseudoequilibrium .
The boundary equilibria of Filippov system (5) satisfy and or , that is,
So if and hold, then system (5) has a boundary equilibrium ;
if and hold, then system (5) has a boundary equilibrium
The tangent points of system (5) satisfy the conditions and or , that is,Thus, there exist two tangent points denoted as
In order to determine the effect of the parameters on the existence of the above-mentioned equilibria, we define the following curves for the parameters and ,
The four lines divide the parameter space of and into six regions. In each region, the existence or coexistence of the regular equilibria, virtual equilibria, and pseudoequilibrium is marked, and the boundary equilibrium (or ) only exists on line (or ) (see Figure 1).
3.2. The Stability of the Pseudoequilibrium
In this subsection, we discuss the stability of the pseudoequilibrium.
Theorem 5. If condition (24) holds true, then system (5) has a unique pseudoequilibrium and is locally asymptotically stable in sliding region.
Proof. From the discussion about the existence of pseudoequilibrium in Section 3.1, we know that we only need to prove its locally asymptotic stability.
It follows from (12), we havewhich indicates that the pseudoequilibrium point is locally asymptotically stable. This completes the proof.
Remark 6. In this case, both the unique positive equilibria of system and are virtual, and system (5) has a unique pseudoequilibrium which is locally asymptotically stable (by numerical simulation, we can see it is globally asymptotically stable). Then we can control the total number of pest population to stabilize at a desired level ET by spraying pesticides (see Figure 2(c)).
(a)
(b)
(c)
(d)
3.3. Local Sliding Bifurcation
In this subsection, we discuss the bifurcations of equilibria and the tangent points by using numerical simulations. Take as the bifurcation parameter and let , , , , , , . If ; equilibria and coexist, the pseudoequilibrium does not exist, and is globally asymptotically stable, as shown in Figure 2(a). If , collides with , while and combine to form the boundary equilibrium which is globally asymptotically stable, as shown in Figure 2(b). If , and coexist, the pseudoequilibrium appears which is globally asymptotically stable, and the boundary equilibrium disappears, as shown in Figure 2(c). If , and collide. The pseudoequilibrium disappears and the boundary equilibrium appears, which is globally asymptotically stable, as shown in Figure 2(d). Finally, if , and coexist, the boundary equilibrium disappears, and is globally asymptotically stable.
4. Conclusion
Filippov systems are those in which the trajectories are switched from one system to another at the switching line (or switching surface). In recent years, Filippov systems have been recognized for their wide applications in science and engineering. They provide a natural and convenient mathematical framework for solving some practical problems in real life. In this paper, we use the Filippov system to analyze the global dynamics of a stage-structured pests growth model with threshold control strategy and obtain the conditions of the existence of sliding region, the existence of various types of positive equilibria, and the local stability of pseudoequilibrium. We also carry out numerical simulations to investigate the global stability and local sliding bifurcations of different types of equilibria. Our aims in this paper are to control the total number of juvenile and adult pest population not exceeding a certain critical level and to prevent the outbreak of pests, so we choose parameters in regions , , , , and such that the real equilibrium or the boundary equilibrium ( or ) or pseudoequilibrium is globally stable, which can always be achieved if we strictly restrict the controllable parameter , (i.e., choosing the appropriate dose of pesticide such that ). Note that the economic threshold may also vary with growing conditions. So the control actions will be dynamic and variable, and should not be considered as a constant. Our results obtained in this work can help us to take effective pest management strategies and provide theoretical guidelines for pest control problems.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11371030), the Natural Science Foundation of Liaoning Province (20170540001), and the Doctoral Scientific Research Foundation of Liaoning Province (20170520355).