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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 973869, 17 pages
http://dx.doi.org/10.1155/2012/973869
Research Article

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

1Institute of Systems Science, Northeastern University, Liaoning, Shenyang 110819, China
2Department of Mathematics, Anshan Normal University, Liaoning, Anshan 114007, China
3School of Science, Dalian Jiaotong University, Liaoning, Dalian 116028, China

Received 12 March 2012; Accepted 11 April 2012

Academic Editor: Zhiwei Gao

Copyright © 2012 Qingling Zhang 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

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 [13]. Applications of singular models are also found in neural networks [4], fault diagnosis [5, 6], robotics [7, 8] and epidemic [911], 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 [1619]. These ideas are based on the economic theory [20]:NetEconomicRevenue=TotalRevenueTotalCost.(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 [2124]. 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 [2833]. 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 [3436].

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 𝑑𝑥𝑥𝑑𝑡=𝛾𝑥1𝐾𝑞𝐸𝑥,(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.

973869.fig.001
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=𝛾𝑥1𝑥11𝐾1𝑥𝛼1𝑥2𝐸𝑥1,𝑑𝑥2𝑑𝑡=𝛾𝑥2𝑥12𝐾2𝑥+𝛼1𝑥2,0=𝑝𝑥1𝑐𝐸𝑚,(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+=𝑥𝜒=1,𝑥2,𝐸𝑥10,𝑥20,𝐸0.(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<𝐾1max,0<𝑥2<𝐾2max,0<𝐾1<𝐾1max,0<𝐾2<𝐾2max,0<𝐸<𝛾,(2.4) where 𝐾1max and 𝐾2max 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: 𝛾𝑥1𝑥11𝐾1𝑥𝛼1𝑥2𝐸𝑥1=0,𝛾𝑥2𝑥12𝜂𝐾1𝑥+𝛼1𝑥2=0,𝑝𝑥1𝑐𝐸𝑚=0.(3.1)

By solving (3.1), we get two equilibrium points for the model (2.2): 𝑝0=𝑚0,0,𝑐,𝑝1=𝑥10,𝑥20,𝐸0=𝑥0,𝑚𝛼𝑝𝑥0𝛾𝑐+1𝛼+𝛾𝑥0𝛼𝐾1𝑥0,𝑚𝑝𝑥0.𝑐(3.2) Here 𝑥00 is the root of the equation: 𝐶0𝑥3+𝐶1𝑥2+𝐶2𝑥+𝐶3=0,(3.3) where 𝐶0=𝛾𝑝(𝐾1𝛼𝐾1𝛾+𝛾)2, 𝐶1=2𝜂𝐾1𝑝𝑚𝛾(𝐾1𝛼𝐾1𝛾+𝛾)2𝑝𝑐𝛾(𝐾1𝛼𝐾1𝛾+𝛾)2, 𝐶2 = 𝛾𝑐2(𝐾1𝛼𝐾1𝛾+𝛾)22𝜂𝐾1𝑐𝑚𝛾(𝐾1𝛼𝐾1𝛾+𝛾)+𝛼𝑝𝜂3𝐾13𝑚(𝛼𝛾)+𝐾12𝑚2𝛾, 𝐶3 = 𝛼𝜂3𝐾13𝑚𝑐(𝛼𝛾).

When the coefficients 𝐶𝑖(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)=𝛾𝑥1𝑥11𝐾1𝑥𝛼1𝑥2𝐸𝑥1𝛾𝑥2𝑥12𝜂𝐾1𝑥+𝛼1𝑥2𝐺(𝐗,𝐸)=𝑝𝑥1𝑐𝐸𝑚,(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(𝛾2𝛼)+(𝛾𝛼+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 det𝐷𝐸𝐺|𝑝0=c0, Jacobian matrix of the model (2.2) at 𝑝0 is given by 𝐷𝐽=𝐗𝐅𝐷𝐸𝐅𝐷𝐸𝐺1𝐷𝐗𝐺|||𝑝0=𝑚𝛾𝛼+𝑐𝛼𝛼𝛾𝛼,(3.5) where 𝐷𝐸𝐺 denote the derivative of the function 𝐺 on the variable 𝐸.
The characteristic equation of Jacobian matrix (3.5) can be obtained: 𝜆2𝑚2(𝛾𝛼)+𝑐𝑚𝜆+𝛾(𝛾2𝛼)+𝛾𝛼+𝑐=0.(3.6) If 0<𝛾<𝛼+𝑚/2𝑐 and 𝛾(𝛾2𝛼)+(𝛾𝛼+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 𝝌𝑇=𝐐𝐒𝑇 is used, where𝑥𝝌=1𝑥2𝐸,𝐒=𝑢𝑣𝐸,𝐐=100010𝑝𝐸0𝑝𝑥0𝑐01.(3.7)

Thus, 𝐷𝝌𝐺(𝝌𝟎[)𝐐=00𝑝𝑥0𝑐], 𝑢=𝑥1, 𝑣=𝑥2, 𝐸=𝑝𝐸0/(𝑝𝑥0𝑐)+𝐸. The model (2.2) is changed into the following form: 𝑑𝑢𝑢𝑑𝑡=𝛾𝑢1𝐾1𝛼(𝑢𝑣)𝐸𝑢+𝑝𝐸0𝑝𝑥0𝑢𝑐2,𝑑𝑣𝑣𝑑𝑡=𝛾𝑣1𝜂𝐾1+𝛼(𝑢𝑣),0=(𝑝𝑢𝑐)𝐸𝑝𝐸0𝑝𝑥0𝑢𝑐𝑚.(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: 𝑢𝑣𝐸𝑇=𝜓𝐙=𝐒0𝑇+𝐔0𝐙+𝐕0𝐙,(3.9) where 𝐔0=100100, 𝐕0=001, [𝐙=𝑦1𝑦2]𝑇, 𝐒0=[𝑢0𝑣0𝐸0], 𝑅2𝑅1 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)𝐷𝜓𝐙=0.(3.10) Differentiating (3.9) and multiplying on the left by 𝐔𝑇0, it can be obtained that: 𝐔𝑇0𝐷𝜓𝐙=𝐈2,(3.11) where 𝐈2 is a 2×2 unit matrix. From (3.10) and (3.11), the following formula is gotten:𝐷𝜓𝐙=𝐷𝐺(𝐒0)𝐔𝑇01𝟎𝐈2.(3.12)

Furthermore, the following model is further obtained [37]:𝑑𝐙𝑑𝑡=𝐔𝑇0𝑓𝜓𝐙=𝐔𝑇0𝝌𝐷𝑓0𝐷𝐺(𝐒0)𝐔𝑇010𝐈2+𝑌𝐙,(3.13) where 𝑌(𝐙)=𝑜(𝐙)(𝐙0+).

From the transformation above and (3.13), the coefficient matrix of linear model corresponding to the model (3.8) is gotten as follows: 𝐸𝐒𝟎=𝐷𝑆𝑓1𝐒0𝐷𝑆𝑓2𝐒0𝐷𝐬𝐺(𝐒0)𝐔0𝑇1=001001𝛾𝐸0𝛼2𝛾𝑢0𝐾1+2𝑝𝐸0𝑢0𝑝𝑥0𝛼𝑐𝛼𝛾𝛼2𝛾𝑣0𝜂𝐾1,(3.14) where 𝑢0=𝑥0, 𝑣0=(𝑚/𝛼(𝑝𝑥0𝑐)+1𝛾/𝛼+𝛾𝑥0/𝛼𝐾1)𝑥0, 𝐸0=𝑝𝐸0/(𝑝𝑥0𝑐)+𝐸0.

Thus, the characteristic equation of the matrix (3.14) is given by 𝜆2+𝐷1𝜆+𝐷2=0,(3.15) where 𝐷1=2(𝛼𝛾)+(3𝑝𝐸0𝑥0+𝐸0(𝑝𝑐))/(𝑝𝑥0𝑐)+2𝛾𝑥0/𝜂𝐾1((𝑚/𝛼(𝑝𝑥0𝑐))+2(𝛾/𝛼)+(𝛾/𝛼𝐾1)), 𝐷2=[𝛾𝛼+(3𝑝𝐸0𝑥0+𝐸0(𝑝𝑐))/(𝑝𝑥0𝑐)][𝛾𝛼(2𝛾𝑥0/𝜂𝐾1)((𝑚/𝛼(𝑝𝑥0𝑐))+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 𝑝1𝑥10,𝑥20,𝐸0=𝑐𝑝,𝛾𝛼+𝜃𝛼2𝑝𝛾,(𝛾𝛼)2𝑐𝛾+1𝛾𝑐𝑝𝐾1+𝛼𝜃2𝑐𝛾,(3.16) where 𝜃=(𝛾𝛼)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(𝛾𝛼+𝜃)/𝐾1𝑝𝛼0, 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[𝐷𝐸𝐺]=𝑝𝑥1𝑐=0. If 𝛾(𝛾𝛼+𝜃)/𝐾1𝑝𝛼0, the following three conditions are satisfied: 𝐷(i)trace𝐸𝐷𝐹adj𝐸𝐺𝐷𝑥1𝐺𝐷𝑥2𝐺𝑝1=𝑝𝐸𝑥1000𝑝1𝛼=𝑐(𝛾𝛼)+2𝑐𝛾+1𝛾𝑐2𝑝𝐾1𝛼𝜃|||𝐷2𝛾0;(ii)𝐗𝐅𝐷𝐸𝐅𝐷𝐗𝐺𝐷𝐸𝐺|||𝑝0=||||||||𝛾2𝛾𝐾1𝑥1𝛼𝐸𝛼𝑥1𝛼𝛾2𝛾𝜂𝐾1𝑥2𝛼0𝑝𝐸0𝑝𝑥1||||||||𝑐𝑝1=𝑐𝛼(𝛾𝛼)2𝑐𝛾+1𝛾𝑐2𝑝𝜂𝐾1+𝛼𝜃2𝛾𝛾𝛾𝛼+𝜃𝐾1𝑝||||𝐷𝛼0;(iii)𝐗𝐅𝐷𝐸𝐅𝐷𝑚𝐅𝐷𝐗𝐺𝐷𝐸𝐺𝐷𝑚𝐺𝐷𝐗Δ𝐷𝐸Δ𝐷𝑚Δ||||𝑝0=||||||||||𝛾2𝛾𝐾1𝑥1𝛼𝐸𝛼𝑥10𝛼𝛾2𝛾𝜂𝐾1𝑥2𝛼00𝑝𝐸0𝑝𝑥1||||||||||𝑐1𝑝000𝑝1𝑐=𝑝𝛾𝛾𝛼+𝜃𝐾1𝑝𝛼0.(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: 𝐷𝑖=trace𝐸𝐷𝐹adj𝐸𝐺𝐷𝑥1𝐺𝐷𝑥2𝐺𝑝1𝛼=𝑐(𝛾𝛼)2𝑐𝛾+1𝛾𝑐2𝑝𝐾1+𝛼𝜃;2𝛾𝑗=𝐷𝑚𝐷Δ𝐗Δ𝐷𝐸Δ𝐷𝐗𝐹𝐷𝐸𝐹𝐷𝐗𝐺𝐷𝐸𝐺1𝐷𝑚𝐹𝐷𝑚𝐺=𝑝00𝛾2𝛾𝐾1𝑥1𝛼𝐸𝛼𝑥1𝛼𝛾2𝛾𝜂𝐾1𝑥2𝛼0𝑝𝐸0𝑝𝑥1𝑐𝑝1100=11(𝛾𝛼)(𝛼/2𝑐𝛾+1)𝛾𝑐/𝑝𝐾1.+𝛼𝜃/2𝑐𝛾(3.18) From above we can get that 𝑖/𝑗=𝑐[(𝛾𝛼)(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=𝐷𝐗𝐹𝐷𝐸𝐹(𝐷𝐸𝐺)1𝐷𝐗𝐺 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+2𝛾𝑥2𝐾2𝛾𝑑𝑥2𝑑𝑡𝛼2𝑥2=𝐾1𝑝𝐸𝛼𝛾𝑚𝐸𝑐𝛼𝛾𝛼2𝑚𝛼2𝑐𝐸𝛼𝑚𝐸𝛼𝛾𝑚(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/𝐾2𝛾, 𝑎0=𝛼2, 𝑏0=𝑐/𝑝, 𝛽 = (𝐾1𝑝𝐸(𝛼𝛾𝑚𝐸𝑐𝛼𝛾𝛼2𝑚𝛼2𝑐𝐸𝛼𝑚𝐸)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: 𝛼𝛾<𝑎1𝛼<𝛼+𝛾,𝑐𝛼𝛾2𝑚+𝛼𝑝<𝛽<𝛼𝛾𝛼2𝑚𝑝𝛾𝛼𝛾+𝛼2𝑐+𝛼𝑚𝑝𝛼𝑚(𝑚+2𝑐𝛾)𝐾1𝑝2𝛾𝑐𝛼2𝛾𝐾1𝑝2.(4.3)

In order to make the number of the population in protecting region Ω2 reach the carrying capacity, let 𝑒=𝐾2𝑦,(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=𝑏0𝑎𝑢+0𝐾2𝛽.(4.5)

For the differential equation (4.5), 𝑎0𝐾2𝛽 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=𝑒2,𝑑𝑒2𝑑𝑡=𝑎0𝑒1𝑎1𝑒2𝑏0𝑢+𝑎0𝐾2𝛽,(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=01𝑎0𝑎1, [𝐁=0𝑏0]𝑇, [𝐂=0𝑎0]𝑇, and [𝐃=0𝛽]𝑇.

To stabilize the model (4.7), the variable structure controller is designed as𝜆𝑢=1𝑒1+𝜆2𝑒2+𝜆3𝐾2Sgn(𝛿(𝐰)),(4.8) where 𝜆𝑖(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=𝑓𝑑𝑒1+𝑑𝑡𝑑𝑒2=𝜆𝑑𝑡1𝑎0𝑒1+𝑓+𝜆2𝑎1𝑒2+𝑎0+𝜆3𝐾2<0,𝛿(𝐰)>0𝜆1𝑎0𝑒1+𝑓+𝜆2𝑎1𝑒2+𝑎0𝜆3𝐾2>0,𝛿(𝐰)<0.(4.11)

According to the reachable condition (4.10), we get the variable structure controller for the model (4.7): 𝑢𝑢=+𝜆=1𝑒1+𝜆2𝑒2+𝜆3𝐾2𝑢,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 𝑑𝑒1𝑑𝑡𝑑𝑒2=𝜆𝑑𝑡011𝑎0𝜆2𝑎1𝑒1𝑒2+0𝑎0+𝜆3𝐾2+0𝛽.(4.13) Let 𝑒1=𝑒1+(𝜆3+𝑎0)𝐾2/(𝜆1𝑎0) and 𝑒2=𝑒2, then 𝑑𝑒1𝑑𝑡𝑑𝑒2=𝜆𝑑𝑡011𝑎0𝜆2𝑎1𝑒1𝑒2+0𝛽.(4.14)

Obviously, the model (4.13) and the model (4.14) have the same state matrix 𝑀=𝜆011𝑎0𝜆2𝑎1. 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 𝑎0𝜆1>0, 𝑎1𝜆2>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 ||||𝜆𝐸𝑀=𝜆2𝜆2𝑎1𝜆𝜆1𝑎0=0.(4.15)
According to the Routh-Hurwitz criterion, if 𝑎0𝜆1>0, 𝑎1𝜆2>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 is𝑑𝑒1𝑑𝑡𝑑𝑒2=𝑑𝑡01𝜆1𝑎0𝜆2𝑎1𝑒1𝑒2+0𝑎0𝜆3𝐾2+0𝛽.(4.16) Let 𝑒1=𝑒1+(𝜆3𝑎0)𝐾2/(𝜆1+𝑎0) and 𝑒2=𝑒2, then, 𝑑𝑒1𝑑𝑡𝑑𝑒2=𝑑𝑡01𝜆1𝑎0𝜆2𝑎1𝑒1𝑒2+0𝛽.(4.17)

The model (4.16) and the model (4.17) also have the same state matrix 𝑁=01𝜆1𝑎0𝜆2𝑎1. 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 ||||𝜆𝐸𝑁=𝜆2𝜆2+𝑎1𝜆𝜆1+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 𝜆𝑖(𝑖=1,2,3):𝜆1||max𝑎0||,𝜆2||max𝑓𝑎1||,𝜆3||𝑎max0||.(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 [4042]. 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=0.2𝑥1𝑥11𝑥4230.61𝑥2𝐸𝑥1,𝑑𝑥2𝑑𝑡=0.2𝑥2𝑥12𝑥676+0.61𝑥2,0=11𝑥16𝐸𝑚.(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=𝐷𝐗𝐹𝐷𝐸𝐹(𝐷𝐸𝐺)1𝐷𝐗𝐺, −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+15800Sgn(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.

973869.fig.002
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).

References

  1. W. G. Marszalek and Z. W. Trzaska, “Singularity-induced bifurcations in electrical power systems,” IEEE Transactions on Power Systems, vol. 20, no. 1, pp. 312–320, 2005. View at Publisher · View at Google Scholar · View at Scopus
  2. M. Yue and R. Schlueter, “Bifurcation subsystem and its application in power system analysis,” IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1885–1893, 2004. View at Publisher · View at Google Scholar · View at Scopus
  3. T. Y. Guo and R. A. Schlueter, “Identification of generic bifurcation and stability problems in power system differential-algebraic model,” IEEE Transactions on Power Systems, vol. 9, no. 2, pp. 1032–1038, 1994. View at Publisher · View at Google Scholar · View at Scopus
  4. R. Riaza and P. J. Zufiria, “Differential-algebraic equations and singular perturbation methods in recurrent neural learning,” Dynamical Systems, vol. 18, no. 1, pp. 89–105, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. Z. W. Gao and S. X. Ding, “State and disturbance estimator for time-delay systems with application to fault estimation and signal compensation,” IEEE Transactions on Signal Processing, vol. 55, no. 12, pp. 5541–5551, 2007. View at Publisher · View at Google Scholar
  6. Z. Gao and D. W. C. Ho, “State/noise estimator for descriptor systems with application to sensor fault diagnosis,” IEEE Transactions on Signal Processing, vol. 54, no. 4, pp. 1316–1326, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. R. J. Spiteri, U. M. Ascher, and D. K. Pai, “Numerical solution of differential systems with algebraic inequalities arising in robot programming,” in Proceedings of the IEEE International Conference on Robotics and Automation, pp. 2373–2380, May 1995. View at Scopus
  8. R. J. Spiteri, D. K. Pai, and U. M. Ascher, “Programming and control of robots by means of differential algebraic inequalities,” IEEE Transactions on Robotics and Automation, vol. 16, no. 2, pp. 135–145, 2000. View at Publisher · View at Google Scholar · View at Scopus
  9. W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,” Journal of Mathematical Biology, vol. 25, no. 4, pp. 359–380, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. N. Yi, Z. Zhao, and Q. Zhang, “Bifurcations of an SEIQS epidemic model,” International Journal of Information & Systems Sciences, vol. 5, no. 3-4, pp. 296–310, 2009. View at Google Scholar
  11. N. Yi, Q. Zhang, K. Mao, D. Yang, and Q. Li, “Analysis and control of an SEIR epidemic system with nonlinear transmission rate,” Mathematical and Computer Modelling, vol. 50, no. 9-10, pp. 1498–1513, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. D. G. Luenberger and A. Arbel, “Singular dynamic Leontief system,” Econometrics, vol. 45, no. 32, pp. 991–995, 1997. View at Google Scholar
  13. J. S. Zhang, Singular System Economic Cybernetics, Tsinghua University Press, Beijing, China, 1990.
  14. A. Kumar and P. Daoutidis, Control of Nonlinear Differential Algebraic Equation Systems, vol. 397 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 1999.
  15. Y. Zhang, Q. L. Zhang, and L. C. Zhao, “Bifurcations and control in singular biological economic model with stage structure,” Journal of Systems Engineering, vol. 22, no. 3, pp. 233–238, 2007. View at Google Scholar
  16. C. Liu, Q. L. Zhang, Y. Zhang, and X. D. Duan, “Bifurcation and control in a differential-algebraic harvested prey-predator model with stage structure for predator,” International Journal of Bifurcation and Chaos, vol. 18, no. 10, pp. 3159–3168, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. Y. Zhang and Q. Zhang, “Optimal control of logistic bioeconomic model with singularity-induced bifurcation,” International Journal of Information & Systems Sciences, vol. 5, no. 3-4, pp. 369–379, 2009. View at Google Scholar
  18. C. Liu, Q. Zhang, and X. Zhang, “Dynamic analysis in a harvested differential-algebraic prey-predator model,” Journal of Mechanics in Medicine and Biology, vol. 9, no. 1, pp. 123–140, 2009. View at Publisher · View at Google Scholar · View at Scopus
  19. X. Zhang, Q.-l. Zhang, and Y. Zhang, “Bifurcations of a class of singular biological economic models,” Chaos, Solitons & Fractals, vol. 40, no. 3, pp. 1309–1318, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. H. S. Gordon, “The economic theory of a common property resource: the Fishery,” Journal of Political Economy, vol. 62, pp. 124–142, 1954. View at Google Scholar
  21. W. J. Reed, “Recruitment variability and age structure in harvested animal populations,” Mathematical Biosciences, vol. 65, no. 2, pp. 239–268, 1983. View at Google Scholar · View at Scopus
  22. T. K. Kar, “Modelling and analysis of a harvested prey-predator system incorporating a prey refuge,” Journal of Computational and Applied Mathematics, vol. 185, no. 1, pp. 19–33, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. L. Liu and J. Sun, “Existence of periodic solution for a harvested system with impulses at variable times,” Physics Letters, Section A, vol. 360, no. 1, pp. 105–108, 2006. View at Publisher · View at Google Scholar · View at Scopus
  24. M. Dieter, “Analysis of trade in illegally harvested timber: accounting for trade via third party countries,” Forest Policy and Economics, vol. 11, no. 8, pp. 600–607, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. S. V. Emelyanov, Variable Structure Control Systems, Nauka, Moscow, Russia, 1967.
  26. Y. Itkis, Control Systems of Variable Structure, John Wiley & Sons, New York, NY, USA, 1976.
  27. V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems, Nauka, Moscow, Russia, 1978.
  28. R. Suarez-Cortez, J. Alvarez-Gallegns, and E. Gonzalez-Mora, “Sliding controller design for a nonlinear fermentation system,” Biotechnology and Bioengineering, vol. 33, no. 4, pp. 377–385, 1989. View at Google Scholar
  29. A. S. Zinober, Deterministic Control of Uncertain Systems, vol. 40 of IEE Control Engineering Series, Peter Peregrinus, London, UK, 1990.
  30. V. I. Utkin, Sliding Modes in Control and Optimization, Communications and Control Engineering Series, Springer, Berlin, Geramny, 1992.
  31. P. Park, D. J. Choi, and S. G. Kong, “Output feedback variable structure control for linear systems with uncertainties and disturbances,” Automatica, vol. 43, no. 1, pp. 72–79, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. C.-C. Wang and J.-P. Su, “A new adaptive variable structure control for chaotic synchronization and secure communication,” Chaos, Solitons & Fractals, vol. 20, no. 5, pp. 967–977, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. F. Juan and L. Gerard, “Variable structure control for power systems stabilization,” International Journal of Electrical Power and Energy Systems, vol. 32, no. 2, pp. 101–107, 2010. View at Publisher · View at Google Scholar · View at Scopus
  34. C.-C. Wang and J.-P. Su, “A new adaptive variable structure control for chaotic synchronization and secure communication,” Chaos, Solitons & Fractals, vol. 20, no. 5, pp. 967–977, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. S. H. Ling, F. H. F. Leung, H. K. Lam, and H. H. C. Iu, “Variable-structure neural network with real-coded genetic algorithm and its application on short-term load forecasting,” International Journal of Information and Systems Sciences, vol. 5, no. 1, pp. 23–40, 2009. View at Google Scholar
  36. J. Fernández-Vargas and G. Ledwich, “Variable structure control for power systems stabilization,” International Journal of Electrical Power and Energy Systems, vol. 32, no. 2, pp. 101–107, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. B. S. Chen, X. X. Liao, and Y. Q. Liu, “Normal forms and bifurcations for differential-algebraic systems,” Acta Mathematicae Applicatae Sinica, vol. 23, no. 3, pp. 429–443, 2000 (Chinese). View at Google Scholar · View at Zentralblatt MATH
  38. V. Venkatasubramanian, H. Schättler, and J. Zaborszky, “Local bifurcations and feasibility regions in differential-algebraic systems,” IEEE Transactions on Automatic Control, vol. 40, no. 12, pp. 1992–2013, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. P. Zlateva, “Variable-structure control of nonlinear systems,” Control Engineering Practice, vol. 4, no. 7, pp. 1023–1028, 1996. View at Publisher · View at Google Scholar · View at Scopus
  40. T. Pradhan and K. S. Chaudhuri, “Bioeconomic modelling of selective harvesting in an inshore-offshore fishery,” Differential Equations and Dynamical Systems, vol. 7, no. 3, pp. 305–320, 1999. View at Google Scholar · View at Zentralblatt MATH
  41. B. Dubey, P. Chandra, and P. Sinha, “A model for fishery resource with reserve area,” Nonlinear Analysis: Real World Applications, vol. 4, no. 4, pp. 625–637, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. M. Jerry, N. Raissi, and A. Rapaport, “A viability analysis for an explicit inshore-offshore model,” Journal of Applied Mathematics, vol. 1, no. 1, pp. 41–60, 2011. View at Google Scholar
  43. http://wiki.cnki.com.cn/HotWord/7044734.htm.
  44. http://info.china.alibaba.com/news/detail/v0-d1023779518.html.