A Model for Influence of Nuclear-Electricity Industry on Area Economy

As a clean energy resource, nuclear electricity is developed more and more quickly in recent years; many nuclearelectricity stations are under construction or planned to construct in China, which will be more beneficial to the development of the economyof the country.Onone hand, the development of economywill not be restricted by the absence of electricity. On the other hand, the output of coal electricity will be reduced; then our resource of coal can be saved largely. With the development of nuclear-electricity speedup in China, many regional governments hope that there are some nuclear-electricity stations which could be established in their provinces or counties. This is because if a nuclearelectricity station is established in their provinces or counties, their regional economy will develop faster and faster. By the time, many authors investigated the nuclearelectricity industry [1–11], and more and more authors are interested in area economy [12–20]. However, few authors have investigated how much influence of nuclear-electricity industry on area economy is. Recently, Lixin [21] investigated the economy of source energy of Jiangsu Province. Using the method of dynamics systems, he brought light to the relation of area economy of Jiangsu Province with energy resource. In this paper, we investigate the influence of nuclearelectricity industry on area economy. First, we establish a mathematics model, show the existence of the solutions of the model, investigate the property of solutions, and obtain how much the influence is. For convenience, we consider the influence of nuclear-electricity industry on a county economy. For example, we consider the influence of nuclearelectricity station of Taojiang on the economy of Taojiang county. The paper is arranged as follows. In Section 2, we will establish a model about nuclear-electricity economy. We predigest the model in Section 3. In Section 4, We will investigate the existence of the solutions of the above model. We will discuss the approximation of solutions in Section 5. Finally, we will give a conclusion.


Introduction
As a clean energy resource, nuclear electricity is developed more and more quickly in recent years; many nuclearelectricity stations are under construction or planned to construct in China, which will be more beneficial to the development of the economy of the country.On one hand, the development of economy will not be restricted by the absence of electricity.On the other hand, the output of coal electricity will be reduced; then our resource of coal can be saved largely.
With the development of nuclear-electricity speedup in China, many regional governments hope that there are some nuclear-electricity stations which could be established in their provinces or counties.This is because if a nuclearelectricity station is established in their provinces or counties, their regional economy will develop faster and faster.
In this paper, we investigate the influence of nuclearelectricity industry on area economy.First, we establish a mathematics model, show the existence of the solutions of the model, investigate the property of solutions, and obtain how much the influence is.For convenience, we consider the influence of nuclear-electricity industry on a county economy.For example, we consider the influence of nuclearelectricity station of Taojiang on the economy of Taojiang county.
The paper is arranged as follows.In Section 2, we will establish a model about nuclear-electricity economy.We predigest the model in Section 3. In Section 4, We will investigate the existence of the solutions of the above model.We will discuss the approximation of solutions in Section 5. Finally, we will give a conclusion.

A Model for the Influence of Nuclear-Electricity Industry on Area Economy
As we know, the construction of nuclear-electricity station is a project which is mastered by the country; at the same time, it is also a big project.The construction of nuclearelectricity station is composed of the following four parts mainly: technology, equipment, component, and material.It is easy to know that technology is usually fetched in, the type of Taojiang nuclear-electricity station is AP-1000, and this technology is fetched from America.Equipment and component are domestic; they will be assembled into a group.Therefore, technology, equipment, and component will not affect the economy of Taojiang county.But material is only associate domain, such as cement and rock.Because there is a big industry of cement in Taojiang, so the export of cement and rock can advance the development of economy of Taojiang county, but it is only for a moment; once the nuclear-electricity station has been built, their influence on the economy of Taojiang county will disappear for a long time.Now, we consider the deriving domains which includes real estate, service, and tour.These domains will influence the area economy of Taojiang county for a long time.
Through this paper, we suppose that  = 0 when we begin to build the nuclear-electricity station.
First, we consider the real estate domain.The real estate only influences the economy of Taojiang until Taojiang nuclear-electricity station has been constructed.Once Taojiang nuclear-electricity station building starts, the influence of real estate on Taojiang will become little.Thus, we do not consider the influence of real estate on Taojiang in this paper and only need to consider service and tour domain.
Second, we consider the service domain.We suppose that the gross economy of the service domain is  1 () at time .The service domain will be developed when we start building the nuclear-electricity station; that is,  1 () > 0, as  > 0. But, during the construction of building nuclear-electricity station, the gross of economy of the service domain is small. 1 years later, the nuclear-electricity station begins running and the gross of economy of the service domain will stabilize.Thus, the gross of economy of the service domain can be expressed as the following: where  11 () is a positive function and  1 is the upper limit of the gross of economy of the service domain  1 (); that is, if  1 () >  1 , then  1 () will be decreasing.Finally, we consider the tour domain.Since the type of pile of Taojiang nuclear-electricity station is AP-1000, which is first used in inner-continent nuclear-electricity station in China.Therefore, many tourists will visit Taojiang nuclearelectricity station.We suppose that the gross of economy of the tour domain is  2 () at time ; then where  22 () is a positive function,  2 > 0, which is the gross of economy of the tour domain  2 () > 0 for  >  2 , and  2 is the upper limit of the gross of economy of the tour domain  2 (); that is, if  2 () >  2 , then  2 () will decreasing.
According to (3)-( 4), the the gross of area economy can be described as where

The Predigestion of Model (5)
It is obvious that the equilibrium points of ( 5) are  = ( 1 ,  2 )  and (0, 0)  , and  = ( 1 ,  2 )  is a positive equilibrium, so we only consider the positive equilibrium  = ( 1 ,  2 )  .Let Then Since Substituting ( 7) into (5), we obtain It is easy to show that (0, 0)  is an equilibrium point of (8).Let Then ( 8) can be rewritten as Let  = max{ 1 ,  2 }.We consider the following initial condition: Therefore, system ( 12) is equivalent to system (5); by the time, it is equivalent to (8) also.Therefore, we only need to investigate system ( 12)-( 13).It is obvious that vector function  satisfies the following: (12) Integrating both sides of (12) from  0 to , it follows that

The Continuous Solutions of Model
where  = max{ 1 ,  2 }.Then () is a continuous solution of ( 12)-( 13) if and only if it is a continuous solution (14).

Approximation of Model (12)
Definition 3. A solution of system ( 12) is oscillatory if and only if there is a nontrivial component of   () ( = 1, 2) being oscillatory; one calls system (12) oscillatory if every solution of ( 12) is oscillatory.
Remark 4. If a vector solution () is nonoscillatory, then every nontrivial component of ( 12) is eventually positive or eventually negative.Definition 5. A solution of system ( 12) is oscillatory strongly if and only if every nontrivial component   () ( = 1, 2) is oscillatory; one calls system (12) oscillatory strongly if every solution of ( 12) is oscillatory strongly.
For convenience, we give some hypothesis as follows: (H):   () (,  = 1, 2) are positive continuous functions, do not vanish to zero, and satisfy lim Usually, we consider the influences of nuclear-electricity industry on area economy after the nuclear-electricity station has been established, and the nuclear-electricity station can be running many years.So, we may ignore the time delays  1 and  2 .Thus, we take  1 =  2 = 0 in this section, and ( 12) can be rewritten as follows: We have the following main theorem.
Proof.We divide the proof into the following two steps.
Thus, the point  2  is a repeated zero point of  1 () and  2 ().We have from ( 29)-(30) that   1 ( 2 ) = 0,   2 ( 2 ) = 0, which means that the point  2 is a maximum point or minimum point of  1 () and  2 ().We consider the following four cases: (a)  2 is a maximum point of  1 () and a maximum point of  2 (); (b)  2 is a maximum point of  1 () and a minimum point of  2 (); (c)  2 is a minimum point of  1 () and a maximum point of  2 (); (d)  2 is a minimum point of  1 () and a minimum point of  2 ().
We only prove case (a) and case (b); since the proof of (b) is similar to (a), thus, we omit it.
Similarly, it is easy to show that cases (c) and (d) are impossible.
By the available, the function  2 () is eventually positive or eventually negative.Suppose that  2 () is eventually negative; by the definition of oscillatory again, there exists a sequence which derive a contradiction.Similarly, we can prove the case of that  2 () is eventually positive.Thus, the function  1 () is nonoscillatory.