The Analysis and Control for Singular Ecological-Economic Model with Harvesting and Migration

Abstract

To keep the resources renewable, a singular ecological-economic model is proposed for the populations with harvesting and migration. The local stability and the dynamic behavior of the model are studied. Singular induced bifurcation appears when economic interest is zero, which is different from the ordinary differential models. In order to apply variable structure control to eliminate these complex behaviors, the singular model is transformed into a single-input and single-output model with parameter varying within definite intervals. And then, a variable structure controller is designed to make the model stable. Finally, an inshore-offshore fishery model is given to illustrate the proposed method, and some numerical simulations are shown to demonstrate the control results.

1. Introduction

The management of renewable resources is important for the development of human and society. In exploiting the biological resources, both the economic profit and the environmental effects should be taken into account, which initiates a new research area: biomathematics. Interactions of mathematics and biology promote the development of the biosciences greatly in a certain extent. Since most of biological theories evolve rapidly, it is necessary to develop some useful mathematical models to describe the consequences of these biological models.

Singular model as a branch of modern control theory can describe a class of practical models more accurately. Compared with the ordinary differential models, singular models exhibit more complicated dynamics, such as the impulse phenomenon. They have more applications in power systems, aerospace engineering, chemical processes, social economic systems, biological systems, network analysis, and so forth. With the help of the singular models for the power systems and bifurcation theory, complex dynamical behaviors of the power systems have been extensively studied, which reveal the instability mechanism of power systems [1–3]. Applications of singular models are also found in neural networks [4], fault diagnosis [5, 6], robotics [7, 8] and epidemic [9–11], economics [12, 13], and chemistry [14]. As far as the singular system theory is concerned, there are a few research results in biology. Since a singular biological economics model with stage structure was established to model the biological systems in [15], some singular biological models appeared [16–19]. These ideas are based on the economic theory [20]:NetEconomicRevenue=TotalRevenue−TotalCost.(1.1) This formula presents some solid preliminary on singular biological systems.

In biology, many mathematicians, ecologists, and economists are concerned with the exploitation of renewable resources in recent years, and some results are achieved [21–24]. Though the harvesting can bring economic profit for people, the overexploitation may cause the extinction of some populations. In order to prevent the population from damages, some methods are introduced, such as, to raise taxes or to make the young population forbidden to be harvested. We propose a singular ecological-economic model to model such a problem. Singular model is often strongly nonlinear and unstable. In this case, one of control methods, which are able to perform high-quality automatic control, is demanded.

Variable structure control is considered to be used in this paper. It is a flexible control method to deal with some models with uncertain parameters and external disturbances. The main advantage of this technique is that once the system state variables reach a sliding surface, the structure of the feedback loop is adaptively altered to slide the state variables along the sliding surface. Thereafter, the system response depends on the gradients of the sliding surface and remains insensitive to parameter variations and external disturbances. Variable structure control with sliding mode was first proposed by Emelyanov [25] and was elaborated in the 1970s [26, 27]. In their pioneer works, variable structure controls are used to handle some linear models, and then expanded to nonlinear models, multi-input and multioutput models, discrete time models, infinite-dimensional models and stochastic models [28–33]. In recent years, variable structure control is applied to a wide variety of engineering fields successfully, such as robot control, flight control, motor control, and power control [34–36].

The main contents of the paper are as follows. In Section 1, in order to prevent the extinction of some populations, a singular ecological-economic model is proposed for the populations with harvesting and migration. In Section 2, when the local stability and the dynamic behavior for the model are discussed, singular induced bifurcation appears, and a control method is demanded to eliminate this bifurcation. In Section 3, in order to apply variable structure control, the singular model is transformed into a single-input and single-output model with parameters varying within definite intervals. In Section 4, an inshore-offshore model is given to illustrate the analysis results, and the simulations illustrate the effectiveness of the proposed method.

2. Modeling

In order to model growth of the populations, numerous models have been introduced. The generalized logistic growth model can provide an adequate approximation for the growth of the populations. However, if there is no harvesting, the populations would continue to increase rapidly. Therefore, the harvesting is an effective measure to maintain the diversity of species and protect the renewable resources. The equation of the harvested populations reads 𝑑𝑥𝑑𝑡=𝛾𝑥1−𝐾𝑥−𝑞𝐸𝑥,(2.1) where 𝑥 is the number of population, 𝛾 is a positive constant which is called the intrinsic growth rate, 𝐸 is the harvesting effort, 𝑞 is catch-ability coefficient, and 𝐾 is usually the environment carrying capacity or saturation level. Some papers studied the model (2.1) with a constant effort 𝐸. But it is only suitable for some special case. In practice, the harvesting effort 𝐸 is usually time-varying. For convenience in calculation, the condition 𝑞=1 is usually assumed. If 𝐸>𝛾, a rapid collapse of the populations will occur. The extinction of population is inevitable, and the ecological balance will be destroyed.

In order to keep the resources renewable and prevent the extinction of some populations, the populations can be divided into two regions: Ω₁ and Ω₂. In region Ω₁, the harvesting is permitted, while the harvesting is forbidden in region Ω₂. If there is a difference between region Ω₁ and region Ω₂, the migration can occur between two regions, which is assumed to be proportional to the difference, and the proportional coefficient is positive. To better understand relation of the populations between two regions, a concise schematic is shown in Figure 1.

Figure 1 

The migration between the harvesting permitted region and the harvesting forbidden region.

From Figure 1, it can be seen that the two regions are connected, and the population can migrate freely between two regions. The number of population in region Ω₁ is 𝑥₁, and 𝑥₂ is the number of population in region Ω₂. The ultimate aim in harvesting the biological resources is to get economic profits and practical value. Generally, from the economic aspect, we know that the harvesting behavior changes with many market factors. Therefore, studying the relation between economic profits and the harvesting can help us better protect sustainable resources. If considering the economic profits in the model (2.1), the following mathematical model of the exploited population with protective region, called a singular ecological-economic model, is proposed: 𝑑𝑥₁𝑑𝑡=𝛾𝑥¹1−𝐾𝑥¹₁−𝛼𝑥¹𝑑𝑥₂−𝑥₂−𝐸𝑥₁,𝑑𝑡=𝛾𝑥²1−𝐾𝑥²₂+𝛼𝑥¹0=𝑝𝑥₁−𝑥₂,−𝑐𝐸−𝑚,(2.2) where 𝑝 is the unit price, 𝑐 is the unit cost, 𝑚 is the economic profit, and 𝛼>0 is the migration proportional coefficient between two regions. Considering the practical significance, 𝑝 and 𝑐 are positive constants. 𝐾₁ and 𝐾₂ are the environment carrying capacity of Ω₁ and Ω₂, respectively. The differential equations are the growth rate of the population in region Ω₁ and Ω₂. The algebraic equation is an economic model, which represents the relations of the total income, the total cost, and the economic profits.

Remark 2.1. In the management of sustainable resources, the model (2.2) not only considers the ecological balance but also includes the economic profits when the population is harvested, which combines the biological control problems with the economic problems. The model (2.2) provides an effective way for human being to maintain the ecological resources sustainable when we get economic profit.
Considering the biological significance, the model (2.2) is discussed in the following interval: 𝑅³⁺=𝜒=𝑥₁,𝑥₂,𝐸∣𝑥₁≥0,𝑥₂≥0,𝐸≥0.(2.3)
If 𝑚=0, the economic equilibrium occur, that is, the income is equal to the cost. When the economic profit is zero, the population reaches the maximum harvesting effort, and it is called the overfishing. In the exploitation of population resources, a collapse of the population may occur.
Due to the limitation of the environment, the number of populations cannot exceed the environment maximum carrying capacity. Otherwise, due to the crowded environment, a large number of populations will die gradually. So the state variables and the parameters satisfy the following conditions: 0<𝑥₁<𝐾_1max,0<𝑥₂<𝐾_2max,0<𝐾₁<𝐾_1max,0<𝐾₂<𝐾_2max,0<𝐸<𝛾,(2.4) where 𝐾_1max and 𝐾_2max are the maximum environment carrying capacities of Ω₁ and Ω₂, respectively.

3. Local Stability Analysis

For convenience, the environment carrying capacity in Ω₁ is assumed to be proportional to that in Ω₂, and the ratio is 𝜂, that is, 𝜂𝐾₁=𝐾₂(𝜂>0). For the model (2.2), the equilibrium points are the solutions for the equations: 𝛾𝑥₁1−𝐾𝑥¹₁−𝛼𝑥¹𝛾𝑥₂−𝑥₂−𝐸𝑥₁=0,1−𝜂𝑥𝐾²₁+𝛼𝑥¹𝑝𝑥₁−𝑥₂=0,−𝑐𝐸−𝑚=0.(3.1)

By solving (3.1), we get two equilibrium points for the model (2.2): 𝑝₀=0,0,−𝑚𝑝₁𝑐,=𝑥₁₀,𝑥₂₀,𝐸₀=𝑥₀,𝑚𝛼𝑝𝑥₀−𝑐+1−𝛼𝛾+𝛼𝛾𝐾𝑥⁰₁𝑥⁰,𝑚𝑝𝑥₀−𝑐.(3.2) Here 𝑥₀≠0 is the root of the equation: 𝐶₀𝑥³+𝐶₁𝑥²+𝐶₂𝑥+𝐶₃=0,(3.3) where 𝐶₀=𝛾𝑝(𝐾₁𝛼−𝐾₁𝛾+𝛾)², 𝐶₁=2𝜂𝐾₁𝑝𝑚𝛾(𝐾₁𝛼−𝐾₁𝛾+𝛾)−2𝑝𝑐𝛾(𝐾₁𝛼−𝐾₁𝛾+𝛾)², 𝐶₂ = 𝛾𝑐²(𝐾₁𝛼−𝐾₁𝛾+𝛾)² − 2𝜂𝐾₁𝑐𝑚𝛾(𝐾₁𝛼−𝐾₁𝛾+𝛾)+𝛼𝑝𝜂³𝐾₁₃𝑚(𝛼−𝛾)+𝐾₁₂𝑚²𝛾, 𝐶₃ = −𝛼𝜂³𝐾₁₃𝑚𝑐(𝛼−𝛾).

When the coefficients 𝐶_𝑖(𝑖=0,1,2,3) satisfy certain conditions, there is a positive solution for (3.3). Here, we suppose that the positive equilibrium point 𝑝₁ exists. We are interested in the local stability of the model (2.2) at the equilibrium points 𝑝₀ and the positive equilibrium point 𝑝₁. In order to analyze the local stability of the model (2.2), let 𝐅(𝐗,𝐸)=⎛⎜⎜⎜⎝𝛾𝑥¹1−𝐾𝑥¹₁−𝛼𝑥¹−𝑥₂−𝐸𝑥₁𝛾𝑥₂1−𝜂𝑥𝐾²₁+𝛼𝑥¹𝐺(𝐗,𝐸)=𝑝𝑥₁−𝑥₂⎞⎟⎟⎟⎠−𝑐𝐸−𝑚,(3.4) where 𝐗=[^𝑥1𝑥2]. The local stability of the model (2.2) at the equilibrium points 𝑝₀ is discussed by the following theorem.

Theorem 3.1. If 0<𝛾<𝛼+𝑚/2𝑐 and 𝛾(𝛾−2𝛼)+(𝛾−𝛼+𝑚/𝑐)>0, the model (2.2) is locally stable at 𝑝₀.

Proof. 𝑝₀=(0,0,−𝑚/𝑐) is an equilibrium point of the model (2.2). Since det𝐷_𝐸𝐺|_𝑝0=−c≠0, Jacobian matrix of the model (2.2) at 𝑝₀ is given by 𝐽=𝐷_𝐗𝐅−𝐷_𝐸𝐅𝐷_𝐸𝐺⁻¹𝐷_𝐗𝐺|||_𝑝0=𝛾−𝛼𝛼+𝑚𝑐𝛾−𝛼𝛼,(3.5) where 𝐷_𝐸𝐺 denote the derivative of the function 𝐺 on the variable 𝐸.
The characteristic equation of Jacobian matrix (3.5) can be obtained: 𝜆²−2(𝛾−𝛼)+𝑚𝑐𝜆+𝛾(𝛾−2𝛼)+𝛾−𝛼+𝑚𝑐=0.(3.6) If 0<𝛾<𝛼+𝑚/2𝑐 and 𝛾(𝛾−2𝛼)+(𝛾−𝛼+𝑚/𝑐)>0, the roots of the characteristic equation (3.6) all have negative real part. Therefore, the model (2.2) is locally stable at 𝑝₀.

In order to analyze the local stability at the positive equilibrium point 𝑝₁, a linear transformation 𝝌^𝑇=𝐐𝐒^𝑇 is used, where𝝌=𝑥₁𝑥₂𝐸,𝐒=𝑢𝑣𝐸,𝐐=⎛⎜⎜⎜⎝−𝑝10𝐸₀0100𝑝𝑥₀−𝑐01⎞⎟⎟⎟⎠.(3.7)

Thus, 𝐷_𝝌𝐺(𝝌_𝟎)𝐐=[^00𝑝𝑥₀^−𝑐], 𝑢=𝑥₁, 𝑣=𝑥₂, 𝐸=𝑝𝐸₀/(𝑝𝑥₀−𝑐)+𝐸. The model (2.2) is changed into the following form: 𝑑𝑢𝑑𝑡=𝛾𝑢1−𝐾𝑢₁−𝛼(𝑢−𝑣)−𝐸𝑢+𝑝𝐸⁰𝑝𝑥₀−𝑐𝑢²𝑑𝑣,0𝑑=𝑡(=𝑝𝑢𝛾−𝑣𝑐)1−𝜂𝐾𝑣₁+𝛼(𝑢−𝑣),𝐸−𝑝𝐸⁰𝑝𝑥₀−𝑐𝑢−𝑚.(3.8)

Now the local stability of the model (3.8) at the positive equilibrium point 𝑝₁ will be analyzed. First, the diffeomorphism 𝜓 is defined as follows: 𝑢𝑣𝐸^𝑇=𝜓𝐙=𝐒_0𝑇+𝐔₀𝐙+𝐕₀ℎ𝐙,(3.9) where 𝐔₀=¹⁰⁰¹⁰⁰, 𝐕₀=⁰⁰¹, 𝐙=[^𝑦1𝑦2]^𝑇, 𝐒₀=[_{𝑢0𝑣0𝐸0}], ℎ∶𝑅²→𝑅¹ is a smooth mapping. Jacobi matrix 𝐷𝜓 is a 3×1 real matrix.

Second, by differentiating 𝐺(𝜓(𝐙))=0, the following equation is obtained: 𝐷𝐺(𝝌)𝐷𝜓𝐙=0.(3.10) Differentiating (3.9) and multiplying on the left by 𝐔^𝑇0, it can be obtained that: 𝐔^𝑇0𝐷𝜓𝐙=𝐈₂,(3.11) where 𝐈₂ is a 2×2 unit matrix. From (3.10) and (3.11), the following formula is gotten:𝐷𝜓𝐙=𝐷𝐺(𝐒⁰𝐔_𝑇0)⁻¹𝐈𝟎₂.(3.12)

Furthermore, the following model is further obtained [37]:𝑑𝐙𝑑𝑡=𝐔^𝑇0𝑓𝜓𝐙=𝐔^𝑇0𝐷𝑓𝝌₀𝐷𝐺(𝐒⁰𝐔_𝑇0)⁻¹𝐈0₂+𝑌𝐙,(3.13) where 𝑌(𝐙)=𝑜(𝐙)(𝐙→0⁺).

From the transformation above and (3.13), the coefficient matrix of linear model corresponding to the model (3.8) is gotten as follows: 𝐸𝐒_𝟎=𝐷^𝑆𝐷_𝑆𝑓₁𝐒₀𝑓₂𝐒₀𝐷^𝐬𝐔𝐺(_0𝑇𝐒₀)=⎛⎜⎜⎜⎝𝛾−⁻¹010001𝐸₀−𝛼−2𝛾𝑢⁰𝐾₁+2𝑝𝐸⁰𝑢₀𝛼𝑝𝑥₀−𝑐𝛾−𝛼−𝛼2𝛾𝑣₀𝜂𝐾₁⎞⎟⎟⎟⎠,(3.14) where 𝑢₀=𝑥₀, 𝑣₀=(𝑚/𝛼(𝑝𝑥₀−𝑐)+1−𝛾/𝛼+𝛾𝑥₀/𝛼𝐾₁)𝑥₀, 𝐸₀=𝑝𝐸₀/(𝑝𝑥₀−𝑐)+𝐸₀.

Thus, the characteristic equation of the matrix (3.14) is given by 𝜆²+𝐷₁𝜆+𝐷₂=0,(3.15) where 𝐷₁=2(𝛼−𝛾)+(3𝑝𝐸₀𝑥₀+𝐸₀(𝑝−𝑐))/(𝑝𝑥₀−𝑐)+2𝛾𝑥₀/𝜂𝐾₁((𝑚/𝛼(𝑝𝑥₀−𝑐))+2−(𝛾/𝛼)+(𝛾/𝛼𝐾₁)), 𝐷₂=[𝛾−𝛼+(3𝑝𝐸₀𝑥₀+𝐸₀(𝑝−𝑐))/(𝑝𝑥₀−𝑐)][𝛾−𝛼−(2𝛾𝑥₀/𝜂𝐾₁)((𝑚/𝛼(𝑝𝑥₀−𝑐))+2−(𝛾/𝛼)+(𝛾/𝛼𝐾₁))].

About the local stability of the model (2.2) at the positive equilibrium point 𝑝₁, we have the following theorem.

Theorem 3.2. For the model (2.2):(a)if 𝐷₁>0 and 𝐷₂>0, the model (2.2) is locally stable at the positive equilibrium point 𝑝₁;(b)if 𝐷₁<0 or 𝐷₂<0, the model (2.2) is unstable at the positive equilibrium point 𝑝₁.

Proof. The model (3.8) and the model (2.2) are isomorphic. The local stability of them is discussed by the eigenvalues of the coefficient matrix 𝐸(𝐒₀). When 𝐷₁>0 and 𝐷₂>0, two roots of the characteristic equation (3.15) all have negative real part. The model (3.8) and the model (2.2) are all locally stable at the positive equilibrium point 𝑝₁.
However, when 𝐷₁<0 or 𝐷₂<0, at least one of the eigenvalues of 𝐸(𝐒₀) has nonnegative real part. We can conclude that the model (2.2) is unstable at the positive equilibrium point 𝑝₁. Thus, the proof is completed.

To further study the dynamic behavior of the model (2.2), 𝑥₀ is given a specified value. If 𝑥₀=𝑐/𝑝, the positive equilibrium point of the model (2.2) is 𝑝₁𝑥₁₀,𝑥₂₀,𝐸₀=𝑐𝑝,𝛾−2𝛼𝑝𝛾+𝜃,(𝛾−𝛼)2𝛼𝑐𝛾+1−𝑝𝛾𝐾𝑐₁+𝛼𝜃2𝑐𝛾,(3.16) where 𝜃=(𝛾−𝛼)²𝑝²𝜂²𝐾₁₂+4𝑝𝜂𝐾₁𝛾𝛼𝑐. By analysis, we know that there is a bifurcation at the positive equilibrium point 𝑝₁ for the model (2.2), which is shown in the following theorem.

Theorem 3.3. If 𝛾−(𝛾−𝛼+𝜃)/𝐾₁𝑝−𝛼≠0, there is a singular induced bifurcation for the model (2.2) at the positive equilibrium point 𝑝₁, and 𝑚=0 is a bifurcation value.

Proof. Let 𝑚 be a bifurcation parameter for the model (2.2). 𝑥₁=𝑐/𝑝 makes Δ=det[𝐷_𝐸𝐺]=𝑝𝑥₁−𝑐=0. If 𝛾−(𝛾−𝛼+𝜃)/𝐾₁𝑝−𝛼≠0, the following three conditions are satisfied: (i)trace𝐷_𝐸𝐹adj𝐷_𝐸𝐺𝐷_𝑥1𝐺𝐷_𝑥2𝐺_𝑝1=−𝑝𝐸𝑥¹000_𝑝1=−𝑐(𝛾−𝛼)𝛼2𝑐𝛾+1+𝑝𝛾𝐾𝑐²₁−𝛼𝜃(ii)|||𝐷_𝐗2𝛾≠0;𝐷_𝐗𝐅𝐷_𝐸𝐅𝐺𝐷_𝐸𝐺|||_𝑝0=||||||||𝛾−2𝛾𝐾₁𝑥₁−𝛼−𝐸𝛼−𝑥₁𝛼𝛾−2𝛾𝜂𝐾₁𝑝𝐸0𝑥₂−𝛼𝑝𝑥₁0−𝑐||||||||_𝑝1=𝑐(𝛾−𝛼)𝛼2𝑐𝛾+1−𝑝𝛾𝜂𝑐𝐾²₁+𝛼𝜃(iii)||||𝐷_𝐗2𝛾𝛾−𝛾−𝐾𝛼₁𝑝+𝜃−𝛼≠0;𝐷_𝐗𝐅𝐷_𝐸𝐅𝐷_𝑚𝐅𝐷_𝐗𝐺𝐷_𝐸𝐺𝐷_𝑚𝐺Δ𝐷_𝐸Δ𝐷_𝑚Δ||||_𝑝0=||||||||||𝛾−2𝛾𝐾₁𝑥𝛼₁−𝛼−𝐸𝛾−2𝛾𝛼−𝑥₁0𝜂𝐾₁𝑝𝐸0𝑥₂−𝛼𝑝𝑥₁00𝑝00−𝑐−01||||||||||_𝑝1=−𝑐𝑝𝛾−𝛾−𝐾𝛼₁𝑝+𝜃−𝛼≠0.(3.17) Thus, we can conclude that there exists a smooth curve in 𝐑³ which passes through the positive equilibrium point 𝑝₁, and it is transversal to the singular surface at the positive equilibrium point 𝑝₁. And we can get the following equations: 𝑖=−trace𝐷_𝐸𝐹adj𝐷_𝐸𝐺𝐷_𝑥1𝐺𝐷_𝑥2𝐺_𝑝1=𝑐(𝛾−𝛼)𝛼2𝑐𝛾+1−𝑝𝛾𝐾𝑐²₁+𝛼𝜃𝑗=𝐷_𝑚2𝛾;Δ−𝐷_𝐗Δ𝐷_𝐸Δ𝐷^𝐗𝐷_𝐗𝐹𝐷_𝐸𝐹𝐺𝐷_𝐸𝐺⁻¹𝐷^𝑚𝐷_𝑚𝐹=−𝑝00⎛⎜⎜⎜⎜⎝𝛾−2𝛾𝐺𝐾₁𝑥₁−𝛼−𝐸𝛼−𝑥₁𝛼𝛾−2𝛾𝜂𝐾₁𝑝𝐸0𝑥₂−𝛼𝑝𝑥₁0−𝑐⎞⎟⎟⎟⎟⎠^𝑝−1₁=1−001(𝛾−𝛼)(𝛼/2𝑐𝛾+1)−𝛾𝑐/𝑝𝐾₁+𝛼𝜃/2𝑐𝛾.(3.18) From above we can get that 𝑖/𝑗=𝑐[(𝛾−𝛼)(𝛼/2𝑐𝛾+1)−𝛾𝑐/𝑝𝐾₁+𝛼𝜃/2𝑐𝛾]². Obviously, 𝑖/𝑗>0. According to Theorem  3 in [38], when 𝑚 passes through 0, one eigenvalue of matrix 𝐽=𝐷_𝐗𝐹−𝐷_𝐸𝐹(𝐷_𝐸𝐺)⁻¹𝐷_𝐗𝐺 moves from 𝐶⁻ to 𝐶⁺ along the real axis by diverging through ∞. There is a singular induced bifurcation for the model (2.2), and the model turns to unstable. The proof is completed.

Remark 3.4. When the economic profit is zero, it is called the overfishing in economics. One eigenvalue of the model (2.2) is approaching to endless, and the impulse occurs in the model (2.2). This would lead to the collapse of the population and destroy the ecological balance. It is necessary to find an effective method to make that the population develop sustainably.

4. Controller Design

Variable structure control is often used to deal with some models with internal varying parameters and external disturbances since it provides effective means to design robust state feedback controllers. In this section, variable structure control is introduced to eliminate the bifurcation behavior and ensure the system stable. This approach makes direct use of the nonlinear model and the full biological state information. In order to facilitate the controller design, differentiating the second differential equation in the model (2.2) and substituting the other two equations into it, the model (2.2) is transformed into a second-order differential equation [39]: 𝑑²𝑥₂𝑑𝑡²+𝛼+2𝛾𝑥²𝐾₂−𝛾𝑑𝑥²𝑑𝑡−𝛼²𝑥₂=𝐾¹𝑝𝐸𝛼𝛾𝑚−𝐸𝑐𝛼𝛾−𝛼²𝑚−𝛼²𝑐𝐸−𝛼𝑚𝐸−𝛼𝛾𝑚(𝑚+𝑐𝐸+𝑐𝐸)−𝑐𝛼²𝛾𝐸²𝐾₁𝑝²𝐸²+𝑐𝑝𝐸.(4.1)

Equation (4.1) can be rewritten as a single-input and single-output model with the parameters varying within definite intervals: 𝑑²𝑦𝑑𝑡²+𝑎₁𝑑𝑦𝑑𝑡+𝑎⁰𝑦=𝑏₀𝑢+𝛽,(4.2) where 𝑦=𝑥₂, 𝑢=𝐸, 𝑎₁=𝛼+2𝛾𝑥₂/𝐾₂−𝛾, 𝑎₀=−𝛼², 𝑏₀=𝑐/𝑝, 𝛽 = (𝐾₁𝑝𝐸(𝛼𝛾𝑚−𝐸𝑐𝛼𝛾−𝛼²𝑚−𝛼²𝑐𝐸−𝛼𝑚𝐸)−𝛼𝛾𝑚(𝑚+2𝑐𝐸)−𝑐𝛼²𝛾𝐸²)/𝐾₁𝑝²𝐸².

Obviously, 𝑎₀ and 𝑏₀ are fixed, while 𝑎₁ and 𝛽 change with the parameters and the variables. From the varying intervals (2.4), we can get the varying intervals of the coefficients 𝑎₁ and 𝛽: −𝑐𝛼𝛾−𝛼₂𝛼−𝛾<𝑎₁<𝛼+𝛾,+𝛼𝑚𝑝<𝛽<𝛼𝛾−𝛼²𝑚𝑝𝛾−𝛼𝛾+𝛼²𝑐+𝛼𝑚𝑝−𝛼𝑚𝐾(𝑚₁𝑝+²𝛾2𝑐𝛾)−𝑐𝛼²𝛾𝐾₁𝑝².(4.3)

In order to make the number of the population in protecting region Ω₂ reach the carrying capacity, let 𝑒=𝐾₂−𝑦,(4.4) where 𝑒 is the error of 𝑦 and 𝐾₂. Here 𝑦 is the number of population in Ω₂, while 𝐾₂ is the carrying capacity of region Ω₂.

Differentiating the formula (4.4) twice and considering the model (4.2), the following equation is obtained: 𝑑²𝑒𝑑𝑡²+𝑎₁𝑑𝑒𝑑𝑡+𝑎⁰𝑒=−𝑏₀𝑢+𝑎₀𝐾₂−𝛽.(4.5)

For the differential equation (4.5), 𝑎₀𝐾₂−𝛽 is considered as an external disturbance. According to the transformation, the model (4.1) is considered as a linear uncertain system with the control input. And then the model (4.5) is transformed into 𝑑𝑒₁𝑑𝑡=𝑒²𝑑𝑒₂,𝑑𝑡=−𝑎⁰𝑒₁−𝑎₁𝑒₂−𝑏₀𝑢+𝑎₀𝐾₂−𝛽,(4.6) where 𝑒₁=𝑒.

The model (4.6) can be rewritten as a matrix form: 𝑑𝐰𝑑𝑡=𝐀𝐰+𝐁𝑢+𝐂𝐾²+𝐃,(4.7) where 𝐰=[^𝑒1𝑒2]^𝑇, 𝐀=^01−𝑎_0−𝑎1, 𝐁=[^0−𝑏₀]^𝑇, 𝐂=[^0𝑎₀]^𝑇, and 𝐃=[^0−𝛽]^𝑇.

To stabilize the model (4.7), the variable structure controller is designed as𝑢=−𝜆₁𝑒₁+𝜆₂𝑒₂+𝜆₃𝐾₂Sgn(𝛿(𝐰)),(4.8) where 𝜆_𝑖(𝑖=1,2,3) are switching coefficients and Sgn(𝛿(𝐰)) is a sign function. 𝛿(𝐰) is called sliding surface, which divides the phase plane into two regions. The function 𝛿(𝐰) contains only endpoints of the trajectories of the model (4.7) coming from both sides of the surface and is defined as𝛿(𝐰)=𝑓𝑒₁+𝑒₂,(4.9) where 𝑓>0 is a constant. To suppress the effect of the uncertainty and drive the trajectories of the model (4.7) toward the sliding surface until intersection occurs, the following reachable condition is established:𝛿(𝐰)𝑑𝛿(𝐰)𝑑𝑡<0,for𝛿(𝐰)≠0.(4.10) That is 𝑑𝛿(𝐰)𝑑𝑡=𝑓𝑑𝑑𝑒𝑡¹+𝑑𝑑𝑒𝑡²=𝜆¹−𝜆−₁𝑎₀𝑒₁+𝑓+𝜆₂−𝑎₁𝑒₂+𝑎₀+𝜆₃𝐾₂<0,𝛿(𝐰)>0−𝑎₀𝑒₁+𝑓+𝜆₂−𝑎₁𝑒₂+𝑎₀−𝜆₃𝐾₂>0,𝛿(𝐰)<0.(4.11)

According to the reachable condition (4.10), we get the variable structure controller for the model (4.7): 𝑢=𝑢⁺𝑢=−𝜆₁𝑒₁+𝜆₂𝑒₂+𝜆₃𝐾₂,𝛿(𝐰)>0₋=𝜆₁𝑒₁+𝜆₂𝑒₂+𝜆₃𝐾₂,𝛿(𝐰)<0.(4.12)

Using the controller 𝑢=𝑢⁺ in the model (4.7), the controlled model is ⎡⎢⎢⎢⎣𝑑𝑒¹𝑑𝑑𝑒𝑡₂𝑑𝑡⎤⎥⎥⎥⎦=𝜆01¹−𝑎₀𝜆₂−𝑎₁𝑒𝑒¹₂+𝑎₀+0𝜆₃𝐾²+−0𝛽.(4.13) Let 𝑒^1=𝑒₁+(𝜆₃+𝑎₀)𝐾₂/(𝜆₁−𝑎₀) and 𝑒^2=𝑒₂, then ⎡⎢⎢⎢⎣𝑑𝑒^1𝑑𝑑𝑒𝑡_2𝑑𝑡⎤⎥⎥⎥⎦=𝜆01¹−𝑎₀𝜆₂−𝑎₁𝑒𝑒^12+−0𝛽.(4.14)

Obviously, the model (4.13) and the model (4.14) have the same state matrix 𝑀=^𝜆01_{1−𝑎0𝜆2−𝑎1}. 𝛽 is a bounded constant, and it does not influence the local stability of the controlled model. Thus, we have the following theorem.

Theorem 4.1. If 𝑎₀−𝜆₁>0, 𝑎₁−𝜆₂>0, the model (4.7) can be stabilized by the controller 𝑢⁺.

Proof. When the model (4.7) is controlled by the controller 𝑢⁺, it is transformed into the linear model (4.13). The characteristic equation of the state matrix 𝑀 is ||𝜆𝐸−𝑀||=𝜆²−𝜆₂−𝑎₁𝜆−𝜆₁−𝑎₀=0.(4.15)
According to the Routh-Hurwitz criterion, if 𝑎₀−𝜆₁>0, 𝑎₁−𝜆₂>0, two eigenvalues for the state matrix 𝑀 have negative real part. Therefore, the model (4.13) is locally stable. That is to say, the model (4.7) can be stabilized by the controller 𝑢⁺.

If 𝑢=𝑢⁻ in the model (4.7), the controlled model is⎡⎢⎢⎢⎣𝑑𝑒¹𝑑𝑑𝑒𝑡₂𝑑𝑡⎤⎥⎥⎥⎦=−𝜆¹01−𝑎₀−𝜆₂−𝑎₁𝑒𝑒¹₂+𝑎₀−0𝜆₃𝐾²+−0𝛽.(4.16) Let 𝑒^1=𝑒₁+(𝜆₃−𝑎₀)𝐾₂/(𝜆₁+𝑎₀) and 𝑒^2=𝑒₂, then, ⎡⎢⎢⎢⎣𝑑𝑒^1𝑑𝑑𝑒𝑡_2𝑑𝑡⎤⎥⎥⎥⎦=−𝜆¹01−𝑎₀−𝜆₂−𝑎₁𝑒𝑒^{12}+−0𝛽.(4.17)

The model (4.16) and the model (4.17) also have the same state matrix 𝑁=^01−𝜆_{1−𝑎0−𝜆2−𝑎1}. The transformation does not change the local stability of the model (4.16). Furthermore, we have another theorem.

Theorem 4.2. If 𝜆₁+𝑎₀>0, 𝜆₂+𝑎₁>0, the model (4.7) can be stabilized by the controller 𝑢⁻.

Proof. When the model (4.7) is controlled by the controller 𝑢⁻, it is transformed into the linear model (4.16). The characteristic equation of the state matrix 𝑁 is ||𝜆𝐸−𝑁||=𝜆²−𝜆₂+𝑎₁𝜆−𝜆₁+𝑎₀=0.(4.18)
According to the Routh-Hurwitz criterion, if 𝜆₁+𝑎₀>0, 𝜆₂+𝑎₁>0, two eigenvalues for the state matrix 𝑁 have negative real part. The model (4.16) is locally stable, and the model (4.7) can be stabilized by the controller 𝑢⁻.

From the condition (4.11), Theorems 4.1 and 4.2, we get the varying range of the switching coefficients 𝜆_𝑖(𝑖=1,2,3):𝜆₁≥max||−𝑎₀||,𝜆₂≥max||𝑓−𝑎₁||,𝜆₃≥max||𝑎₀||.(4.19)

According to the condition 𝛿(𝐰)=0 and 𝑑𝛿(𝐰)/𝑑𝑡=0, the equivalent control on the sliding surface 𝛿(𝐰)=0 can be obtained. If 𝛿(𝐰)=0, there is a state variable represented by the remaining state variables. From the condition Sgn(𝛿(𝐰))=0, we have 𝑑𝑒₁𝑑𝑡=−𝑓𝑒¹.(4.20)

Remark 4.3. When applying variable structure control, the singular model is transformed into a linear model with parameters varying within definite intervals. Since the sliding surface can be designed as required and has nothing to do with the parameters and disturbance, it makes the discontinuous control insensitive to internal parameter variations and extraneous disturbance and decreases the chattering phenomenon. Variable structure control can stabilize the nonlinear system effectively.

5. Simulations

Fishery production is an important aspect in human life. In order to guarantee the sustainable development of the fishery, people have taken many necessary measures. Therefore, to study the structure model for the inshore-offshore fishery is necessary. It is a good idea to divide the population into two categories in keeping resources sustainable, a harvesting-permitted category and a harvesting-forbidden category. Some inshore-offshore models in an aquatic environment have ever been studied to keep the fishery sustainable [40–42]. But these papers did not consider the economic profits that the fishery brings for people. In this paper, the sustainable fishery and the economic interest are discussed for the inshore-offshore fishery model.

The sea around Zhoushan is a famous fishing ground in Zhejiang province. The total sea area is about more than 10800 km². The area of the inshore region is about 3700 km², and the offshore region is about 7100 km² [43]. The coiliaspp is a kind of fish, and it is about 1099 million in the whole sea area [44]. To protect the fishery resources, the coiliaspp in the inshore region is permitted to be harvested, while the offshore region is forbidden. In the inshore region, the density of the coiliaspp is greater than that in the offshore region because of the environment effect. So the environment carrying capacity of the inshore region is about 423 million, and the offshore environment carrying capacity is 676 million. The intrinsic growth rate 𝛾 is assumed to be 0.2. When the number of the fish in two regions are different, they migrate between two regions at the proportional 𝛼=0.6. It is supposed that they are sold at the average unit price 𝑝=11, and its unit cost 𝑐 is 6. Considering these conditions, the following singular ecological-economic model can be established: 𝑑𝑥₁𝑑𝑡=0.2𝑥¹1−4𝑥2¹3−0.6𝑥¹𝑑𝑥₂−𝑥₂−𝐸𝑥₁,𝑑𝑡=0.2𝑥²1−6𝑥7²6+0.6𝑥¹0=11𝑥₁−𝑥₂,−6𝐸−𝑚.(5.1)

When the economic profit 𝑚 varies, there are some complex dynamic behaviors for the model (5.1), such as the singular induced bifurcation. When the economic profit 𝑚=0, the model (5.1) has a positive equilibrium point 𝑝^∗(0.545,0.818,0.499). When economic profit 𝑚=0.001, there are two eigenvalues for the matrix 𝐽=𝐷_𝐗𝐹−𝐷_𝐸𝐹(𝐷_𝐸𝐺)⁻¹𝐷_𝐗𝐺, −1.2998 and −0.0002. The eigenvalues became −1.2998 and 0.0017 when the parameter 𝑚=−0.001. It obvious that one eigenvalue remains constant, and the other eigenvalue moves from 𝐶⁻ to 𝐶⁺ along the real axis by diverging through ∞. It is called the overexploitation, and it causes the extinction of the coiliaspp. In order to avoid such phenomena, a variable structure controller is designed to make the coiliaspp in the offshore region reach the environment carrying capacity 676 million. According to the varying range of the switching coefficients (4.19), the variable structure controller is designed as follows:𝑢=−10𝑒₁+76𝑒₂+15800Sgn(𝛿(𝐰)),(5.2) where the sliding surface is chosen as 𝛿(𝐰)=9.6𝑒₁+𝑒₂. By controlling the harvesting effort 𝐸, 𝑥₂ reaches the environment carrying capacity 676 million in Ω₂. Figure 2 shows the control result of 𝑥₁, 𝑥₂, and 𝐸 with variable structure control.

Figure 2 

The state response of 𝑥1, 𝑥2, and 𝐸 when 𝑚=1325 with the controller 𝑢.

In Figure 2, when the harvesting effort 𝐸 is controlled at 0.16 million, the number of fish in region Ω₂ reaches 675.2 million controlled by the controller 𝑢. Due to the migration between Ω₁ and Ω₂, the coiliaspp in inshore region reaches 424.3 million accordingly. The state variables stay in a stable situation, and the singular induced bifurcation is eliminated by the controller 𝑢. In practical, we can regulate the harvesting behavior by the revenue to keep the harvesting and the reproduction in balance. Therefore, the sustainable development of the fishery can be realized by this controller. Further, we know that the corresponding nonlinear singular ecological-economic model can be stabilized by variable structure control.

6. Conclusions

In this paper, the population is divided into the harvesting region and the protecting region, in which the population can migrate between two regions. In harvesting the population resources, when the economic interest and the environmental effects are taken into account, a singular ecological-economic model is established. The local stability and the dynamic behavior for this model are discussed. As the parameters changing, the singular model undergoes the singular induced bifurcation. In order to apply variable structure control to eliminate this complex behavior, the singular model is transformed into a linear single-input and single-output model with parameters varying within definite intervals. Variable structure control with sliding mode is designed to stabilize the model. An inshore-offshore fishery model illustrates the analysis result. Some simulations show the effectiveness of the control method.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (60974004).