Study on tourism development income index calculation of finite element ordinary differential mathematical equation

Mengjie Gong

Détails du journal et du numéro

Applied Mathematics and Nonlinear Sciences

Détails du magazine

Format: Magazine
eISSN: 2444-8656
Première parution: 01 Jan 2016
Périodicité: 2 fois par an
Langues: Anglais

With the acceleration of urbanisation, the population cluster effect begins to appear. Large cities have complete supporting facilities and experience rapid economic development, attracting a large number of people and forming the talent siphon effect [1, 2]. At the same time, the physical pressure and mental pressure of people living in big cities also increase. After living in cities for a long time, people begin to yearn for nature, and tourism develops rapidly. Therefore, rural vacation tourism and scenic spot tourism are highly sought after, and ecological tourism has received widespread attention from government departments, tourism industry and academia around the world; has dominated modern tourism; and has become a “sunrise industry” with vitality and hope [3–6]. At the same time, with the improvement of people’s living standard, consumer demand has started to upgrade, the tourism industry in the strategic position in the development of society began to highlight, and the correlation of the national economy began to improve, and is closely related with other industries and sectors, have high industry relevance, integrated drive ability, high degree of correlation with economic growth. In some regions, the main industry is tourism, and the tourism development income index has become an important pillar of local GDP indicators [7–11].

The phrase clear water and green mountains are mountains of gold and silver not only reflects the relationship between environmental pollution prevention and control costs but also means that a good ecological environment can bring economic benefits and is also closely related to regional economic development [12, 13]. Many scholars have studied the relationship between tourism development and regional economic development using different methods. As shown in Figure 1, the number of articles on tourism economic reports is increasing year by year, especially in the past 15 years. Paramati et al. [14], empirically investigated the dynamic relationship between tourism, economic growth and carbon dioxide emissions by adopting powerful panel econometrics technology, and compared the impact of tourism on economic growth and carbon dioxide emissions between developed and developing economies. Lin et al. [15], used the Bayesian probability model to reveal the factors influencing these different growth patterns. The research results show that regions with a less developed economy, larger economic scale and larger geographical area are more likely to experience tourism-driven growth, while underdeveloped regions are also more likely to experience economy-driven tourism growth. Croes et al. [16], proposed the conceptual relationship between tourism specialisation and human development through the framework of division of labour, and studied the relationship between the economic growth of tourism specialisation and human development in the transition economy. The dynamic comparative advantage, Sen’s capability method and the transition production function represented the conceptual relationship. Nepal et al. [17], evaluated short-term and long-term relationships among tourist numbers, per capita economic output, emissions, energy consumption and capital formation; developed four hypotheses; and tested them using time series econometrics based on the autoregressive distributed lag model and Granger causality test. Lim et al. [18], defined the impact of COVID-19 on the gaming industry by taking Macao, the world’ s largest gambling centre in Yanghua, as an example through a series of quantitative analyses (ARIMA, correlation and regression) of longitudinal data. The survey results also showed that Macao is a gaming-dependent economy because a decline in gambling revenues coincides with declines in economic indicators such as gross domestic product, median wages and employment. Cakmak and Cenesiz [19], used the dynamic general equilibrium model to first estimate the scale of informal tourism economy and then evaluate its relationship with key labour market variables in Thailand. The research shows that the local tourism industry and the regional economic relevance, using different mathematical model or method for data analysis related conclusion finally, illustrates the through mathematical method to evaluate tourism income, forecasts have a certain degree of accuracy and feasibility, can enrich tourism economic theory system, provide a reference for regional tourism economic development.

Based on the background of tourism economy, to a specific mode of regional tourism, the first step is to collect information, for the coupling analysis between the subsystems of the tourism industry and regional economy of synergy, through the tourism development index of income differential mathematical equations of finite element calculation, which can forecast the specific development of regional tourism income index and give countermeasures for the development of the regional tourism economy.

Technical route and method

In order to make the calculation results more accurate, it is necessary to have a deep understanding of the tourism area. Methods include literature analysis, network data mining, and model analysis.

Literature analysis: Wanfang, CNKI, EBSCO, Elsevier and other domestic and foreign databases and related websites were used to sort out, read and analyse the information on the tourism industry income, number of tourists, regional development status including economic conditions, infrastructure and other information of specific tourism regions.

Network data mining method: This method involves searching relevant indexes with the help of network tools, deeply analysing and excavating the attention of the tourism industry in this region and analysing the characteristics of people with tourism intention to this region so as to better understand the current situation of regional tourism development.

Model analysis: This study calculates and predicts the coupling coordination degree between tourism industry development and regional economy through a finite element ordinary differential mathematical equation model, and divides the coupling types according to the coupling coordination degree level in different time periods to analyse the coupling association between the tourism industry and regional economy in order to identify specific reasons for decision-making suggestions.

Figure 2 shows the technical roadmap for this article. Firstly, combining the research background and significance, literature review and data collation are carried out to obtain the theoretical basis. Secondly, the study analyses the current situation of network attention of tourism in this region and the current situation of tourism industry and economic development. Then the coupling coordination analysis of the tourism industry subsystem and regional economic subsystem is carried out, and the development income index is calculated by means of a finite element ordinary differential equation. Finally, according to the calculation results and analysis conclusions, the coordinated countermeasures for the coupling of tourism industry development and economy in this region are given.

Coupling analysis of tourism industry and economic subsystem

In the current context, different conclusions can be drawn on whether tourism and economy will have further impact, and the relationship between them [20–22]. After finishing the data and the preliminary analyses, before the finite element ordinary differential calculation, it is necessary to conduct the coupling analysis of the tourism industry and the economic subsystem, and an empirical study on the coupling coordination relationship between the tourism industry system and the regional economic system. The coupling mechanism between them is shown in Figure 3. The regional economy and tourism industry are mutually catalytic. On the one hand, the tourism industry brings local income and drives local regional economic development; on the other hand, after economic development, investment in the tourism industry is increased, forming a mutual promotion effect. The development of regional economy can significantly increase the local income level, promote the employment level, promote the development of other industries and so on. The development of the tourism industry can stimulate tourism awareness, improve infrastructure and form competitive advantages. Therefore, in general, the tourism industry and regional economy interact with each other. They restrict and influence each other, drive each other inside the system and develop in the direction of coupling and coordination.

Coupling mechanism of tourism industry system and regional economic system

Index system construction and analysis

4.1

Index system

According to the selected research area and the principles of scientificness and typicality, a total of 8 statistical indicators are selected to construct the index evaluation system of the tourism system and regional economic system. Among them, weight is determined using the range standardisation method to deal with the problems of different indicators with different dimensions: 1 $+ u_{i j} = \frac{x_{i j} - min (x_{i j})}{max (x_{i j}) - min (x_{i j})}$ $$ + {u_{ij}} = {{{x_{ij}} - \min ({x_{ij}})} \over {\max ({x_{ij}}) - \min ({x_{ij}})}}$$ 2 $- u_{i j} = \frac{max (x_{i j}) - x_{i j}}{max (x_{i j}) - min (x_{i j})}$ $$ - {u_{ij}} = {{\max ({x_{ij}}) - {x_{ij}}} \over {\max ({x_{ij}}) - \min ({x_{ij}})}}$$ where u_ij is the standardised value of the item j in the subsystem i ( $i = 1, 2$ ; $j = 1, 2, \dots, n$ ); x_ij denotes the original data; and $max (x_{i j})$ $\max ({x_{ij}})$ and $min (x_{i j})$ $\min ({x_{ij}})$ are the maximum and minimum values of the original data of the item j in the four cities in the same year, respectively.

Firstly, the entropy method is used to calculate the weight $γ_{j}$ of the indicator j, and the calculation process is as follows: 3 $M_{i j} = \frac{u_{i j}}{\sum_{i = 1}^{m} u_{i j}}$ $${M_{ij}} = {{{u_{ij}}} \over {\mathop \sum \limits_{i = 1}^m {u_{ij}}}}$$ 4 $N_{j} = - k \sum_{i = 1}^{m} M_{i j} \ln M_{i j}$ 5 $A_{j} = 1 - N_{j}$ 6 $γ_{j} = \frac{A_{j}}{\sum_{j = 1}^{n} A_{j}}$ $${\gamma _j} = {{{A_j}} \over {\mathop \sum \limits_{j = 1}^n {A_j}}}$$ where M_ij represents the proportion of index j data in the overall index in the year i; N_j represent the entropy value of the index j; A_j stands for the difference coefficient; and $γ_{j}$ represents the weight of the index j (Table 1).

Table 1

Index evaluation system for tourism system and regional economic system

Target layer	System layer	Indicator layer (unit)	Weight
Tourism development income index evaluation system	Tourism industry subsystem	Number of domestic tourists (10,000)	0.1231
		Domestic tourism revenue (100 million yuan)	0.1907
		Inbound tourist arrivals (visits)	0.2012
		Foreign exchange income from tourism ($10,000)	0.2035
	Regional economic subsystem	Gross product (100 million yuan)	0.1953
		Output value of tertiary industry (100 million yuan)	0.1177
		Per capita GDP (yuan)	0.1355
		Financial revenue (10,000 yuan)	0.1294

According to the aforementioned formula of entropy value, the number of domestic tourists (10,000 person-times), domestic tourist income (100 million yuan), inbound tourist number (person-times), tourism foreign exchange income ($10,000) in the subsystem of tourism industry, the gross regional economic subsystem (100 million yuan), the third industry production value (100 million yuan), the per capita GDP (yuan), fiscal revenue (10,000 yuan) and other indicators were calculated. It can be seen in the subsystem in the travel industry that the number of inbound tourism and tourist foreign exchange income accounts for a bigger weight of 0.2012 and 0.2035, respectively, and that the regional tourism relies on the contribution of foreign visitor arrivals. In the regional economic subsystem, it can be seen that GDP is larger and weight reached 0.1953, but the tertiary industry output value weight is not large. Although the area is given priority to foreign visitors, the foreign visitors for the first and the second industry obviously promote other assignments. This may be related to the development of primary and secondary industries driven by foreign tourists.

4.2

Analysis of coupling coordination degree

The interaction between two or more systems and the subsequent influence is called the coupling effect [23,24]. The degree of harmony between two or more systems in development and evolution can be regarded as the coupling effect and coordination degree, which show the trend of the factors in the system from disordered discoordination to orderly coordination. In the case studied in this work, the subsystem of the tourism industry and the subsystem of regional economy interact with each other and have a strong correlation. Therefore, it is necessary to conduct research on the interaction between the two systems of tourism industry and regional economy using the coupling coordination model. The specific equation is as follows: 7 $C = 2 {(\frac{u_{1} u_{2}}{(u_{1} + u_{2}) (u_{1} - u_{2})})}^{\frac{1}{2}}$ $$C = 2{\left( {{{{u_1}{u_2}} \over {({u_1} + {u_2})({u_1} - {u_2})}}} \right)^{{1 \over 2}}}$$ where C is the coupling degree of tourism and regional economy, and the value of C ranges from 0 to 1. The larger the C, the higher the correlation degree between the tourism industry and regional economy in the region.

However, the coupling degree cannot reflect the overall efficacy of the research system and the synergistic effect among subsystems in all cases. In different regions, the characteristics of changes and imbalances between tourism and regional economic development are inconsistent, so the comprehensive evaluation value of the subsystems studied is low. Because the coupling degree cannot fully show the degree of coordinated development between subsystems, it can only represent the degree of interaction between systems. Therefore, the coupling coordination degree model of the two is further established, and the equation is as follows: 8 $T = α U_{1} + β U_{2} D$ 9 $D = \sqrt{T * C}$

In the equation, T represents the comprehensive coordination index between the two subsystems of tourism and regional economy; C represents the coupling degree; and D represents the coupling coordination degree of the two subsystems and is the undetermined weight. This indicates that the tourism industry in this region is neither a dominant industry nor a leading force driving regional economic growth. In the interactive development of tourism and urban agglomeration economy, regional economy plays a major role [25, 26]. Interviews with experts also verify this conjecture. Finally, α and β are evaluated as 0.4 and 0.6, respectively, by experts (Table 2).

Table 2

Grade classification of coupling coordination degree

Serial number	Coupling coordination degree interval	Coordination level	Type
1	[0.0.1]	Extreme imbalance	Dissonant recession type
2	[0.0.1]	A serious imbalance between
3	[0.0.1]	Moderate disorders
4	[0.0.1]	Mild disorder	Transitional
5	[0.0.1]	On the verge of disorder
6	[0.0.1]	Barely coordination
7	[0.0.1]	Primary coordination
8	[0.0.1]	Intermediate coordinate	Coordinated ascending type
9	[0.0.1]	Good coordination
10	[0.0.1]	Good coordination

According to the collected data, the tourism region is divided into four regions: A, B, C and D, and the system coupling coordination degree of each region is calculated and analysed. According to the coupling coordination degree model, the calculation results of the coupling coordination coefficients of the two subsystems studied are shown in Figure 4. On the whole, the degree of mutual influence between the tourism industry and regional economic development in this region is constantly increasing, and the coupling coordination degree shows a steady upward trend. The evolution of coupling coordination degree can be divided into four stages according to the time course.

Time evolution diagram of coupling coordination degree of tourism and regional economic subsystem

In the first stage, the overall coordination level was disordered and declined, in which the coordination level of A, B and C was moderate, while the coordination level of D was weaker and serious. It shows that the economic development of the study area has a weak impact on the tourism industry in this stage. Also, the tourism industry development level is relatively low, which cannot promote the steady improvement of the economy. In the second stage, the area A takes the lead in developing to the transitional type, while the other three areas are still in the maladjusted recession type, but the coordination level is increasing, and the development situation is good. The third stage is the type of coupling coordination, which is transitional on the whole. The highest coordination level in place A belongs to the barely coordinated level, which is because it pays more attention to the improvement of comprehensive strength and the development of tourism. The mutual promotion and mutual progress of the two subsystems are obvious. In addition, the span of the coupling coordination level from the second stage to the third stage is the largest in place B, and the value of the coupling coordination degree rises from 0.2874 to 0.4701. This is more likely due to the importance attached to cultural tourism at this stage, which promotes the development of tourism, thus driving the improvement of the regional economic development infrastructure and further promoting the development of tourism. The fourth stage is a coordinated ascending type, indicating that the benign coupling and coordinated development between the two subsystems are formed in this stage, and the degree of mutual influence between the systems is increased. In addition, in 2018, the value of the coupling coordination degree in A was 0.8142, which initially reached the level of good coordination, indicating that the development trend of coupling coordination is good, while the other three regions have not reached the ideal coordination stage at present, and their coordinated development degree still has large room for development.

4.3

Model of tourism development income

The ordinary differential equation has a strong application background, and it is used for solving engineering problems. The stability theory is an important part of the ordinary differential equation, which expresses the motion law of the material system. The stability problem is when practical problems are solved using the ordinary differential equation; this calculation generally only considers the effect of main factors, while the secondary influence factors are generally not considered to simplify the process; the secondary factors are also known as interference factors. The interference factors can, at some time, continue to play a role, from the mathematics point of view. Instantaneous secondary factors cause the change in the initial conditions, and the continuing role of the secondary factors causes a change in the differential equation. Therefore, to study whether small changes in the initial conditions or the differential equation itself in the final results produced small changes is of great significance to the stability of the differential equations, as described in the previous question [27, 28].

The stability analysis of differential equations is generally used in industry, economy and other fields, aiming at formulating better development plans and policies and promoting national economic development. Tourism, as a tertiary industry, has developed by leaps and bounds in the recent ten years, and the income derived from tourism accounts for a large proportion in the national economy. Promoting the development of tourism can help achieve the goal of diversifying and promoting national economic development, and can also strengthen the introduction of regional resources. However, the excessive development of tourism will cause an imbalance in the economic structure of China, thus meeting the goal of balanced development becomes difficult. Therefore, this section will mainly discuss how to develop tourism appropriately.

4.3.1

Model assumptions

Suppose the object of our study is the development of tourism in a fixed region, assume that the tourism income at time t is x(t); Without human intervention, the growth of x(t) follows logistic distribution: 10 $x^{'} (t) = r x (1 - \frac{x}{N})$ $${x^\prime }(t) = rx\left( {1 - {x \over N}} \right)$$ where r is the growth rate of tourism income and N represents the maximum income of tourism under the condition of ensuring the comprehensive and balanced development of regional economy.

Assuming sustainable development of tourism, the tourism income g(x) per unit time is proportional to the tourism income x(t) at time t; and the proportional coefficient is k, which becomes the development force. The expression is shown in Eq. (11): 11 $g (x) = k x$

4.3.2

Model building (revenue model)

According to the hypothesis, the differential equation of tourism income under the condition of sustainable development of tourism can be obtained, and the expression is as follows: 12 $x^{'} (t) = r x (1 - \frac{x}{N}) - k x$ $${x^\prime }(t) = rx\left( {1 - {x \over N}} \right) - kx$$

At present, we pay more attention to how to determine the proportional coefficient k so as to obtain the maximum income of the tertiary industry under the premise of ensuring balanced economic development. We can analyse its stability by solving the equilibrium point of the equation and using the stability theory.

4.3.3

Model solution

If $f (x) = x^{'} (t) = r x (1 - \frac{x}{N}) - k x = 0$ $$f(x) = {x^\prime }(t) = rx\left( {1 - {x \over N}} \right) - kx = 0$$, two equilibrium points can be obtained: 13 $x_{1} = N (1 - \frac{k}{r})$ $${x_1} = N\left( {1 - {k \over r}} \right)$$

Using its derivative, the following expression can be obtained: $\begin{aligned} f_{x}^{'} = r - k - \frac{2 x}{N} \end{aligned}$

Then, $f^{'} (x_{1}) = k - r$ , $f^{'} (x_{2}) = r - k$

The results are discussed as follows:

If $k < r$ , then $f^{'} (x_{1}) < 0$ and x₁ are asymptotically stable points, and $f^{'} (x_{2}) > 0$ and x₂ are asymptotically unstable points.

If $k > r$ , then $f^{'} (x_{1}) > 0$ and x₁ are asymptotically stable points, and $f^{'} (x_{2}) < 0$ $f^{'} (x_{2}) > 0$ and x₂ are asymptotically unstable points.

If $k = r$ , then r is the maximum growth rate.

According to the analysis results of the model, as long as $k < r$ , the development intensity will be less than the maximum investment in tourism; when the income of tourism is stable in x₁ without affecting other industries, a sustainable output $g (x_{1}) = k x_{1}$ can be obtained; when $k > r$ , that is, an excessive development of tourism, the final income of tourism will be zero. Of course, this is an extreme situation, indicating that there is a serious imbalance in regional economic development at this time. Next, we will discuss how to control the development intensity k to maximise the income of tourism when the income of tourism is stable at x₁. A Cartesian coordinate system is established, and the graph of parabolic curve $y = F (x) = r x (1 - \frac{x}{N})$ $y = F(x) = rx\left( {1 - {x \over N}} \right)$ and line $y = g (x) = k x$ of the function expression is drawn, as shown in Figure 5.

Relationship between tourism income and development intensity

The tangent line of $y = F (x)$ at the origin of coordinates is $y = r x$ . It can be seen from the discussion in the previous section that the stable development of tourism and balanced economic development can be guaranteed as long as the development intensity meets $k < r$ . As can be seen from Figure 5, there must be an intersection point between $y = F (x)$ and $y = k x$ , which is set as point M, and the abscissa of point M is x₁. When $y = F (x)$ and $y = k x$ intersect at the parabolic point M between the two lines, maximum benefits can be obtained, and the stable equilibrium point is as follows: 14 $x^{*} = \frac{N}{2}$

The maximum profit per unit time is as follows: 15 $g_{max} = f (\frac{N}{2}) = \frac{r N}{4}$ $${g_{\max }} = f\left( {{N \over 2}} \right) = {{rN} \over 4}$$

At this time, the maximum input force calculated is as follows: 16 $k_{max} = \frac{r}{2}$ $${k_{\max }} = {r \over 2}$$

4.3.1

Model establishment (benefit model)

In the previous section, we analysed the maximum benefits of tourism on the premise of ensuring balanced economic development. But from a macro point of view, in order to ensure stable development, what kind of investment intensity can ensure the best economic returns.

4.3.2

Model assumptions

Suppose the object of our study is the development of tourism in a fixed region, assume that the tourism income at time t is x(t); Without human intervention, the growth of x(t) follows the logistic distribution: 17 $x^{'} (t) = r x (1 - \frac{x}{N})$ $${x^\prime }(t) = rx\left( {1 - {x \over N}} \right)$$ where r is the growth rate of tourism income and N represents the maximum income of tourism under the condition of ensuring the comprehensive and balanced development of regional economy.

Assuming the sustainable development of tourism, the tourism income g(x) per unit time is proportional to the tourism income x(t) at time t; and the proportional coefficient is k, which becomes the development force. The expression is shown in Eq. (11): 18 $g (x) = k x$

it is assumed that economic benefit = income from tourism development – principal of tourism investment.

4.3.1

Model establishment (benefit model)

Assuming that the investment of tourism is P, the cost per unit time is a and the profit per unit time is R, 19 $R = p k x - a k$

4.3.2

Model solution

Under stable conditions $(x - x_{1})$ , substitute $x = x_{1} = N (1 - \frac{k}{r})$ into the equation: 20 $R (k) = p N k (1 - \frac{k}{r}) - a k$ $$R(k) = pNk\left( {1 - {k \over r}} \right) - ak$$

In order to maximise profit, the derivative of this equation is as follows: 21 $R^{'} (x) = p N - \frac{2 p N k}{r} - a$

When $R (k) = 0$ , the maximum investment intensity corresponding to profit R is as follows: 22 $k_{max} = \frac{r}{2} (1 - \frac{a}{p N})$ $${k_{\max }} = {r \over 2}\left( {1 - {a \over {pN}}} \right)$$

By using the equation, the relationship between regional tourism investment intensity and maximum revenue per unit time under maximum profit can be written as follows: 23 $k_{max} = \frac{r}{2} (1 - \frac{a}{p N})$ $${k_{\max }} = {r \over 2}\left( {1 - {a \over {pN}}} \right)$$ 24 $g_{max} = r x_{max} (1 - \frac{x_{max}}{N}) = \frac{r N}{4} (1 - \frac{a^{2}}{p^{2} N^{2}})$ $${g_{\max }} = r{x_{\max }}\left( {1 - {{{x_{\max }}} \over N}} \right) = {{rN} \over 4}\left( {1 - {{{a^2}} \over {{p^2}{N^2}}}} \right)$$

Comparing the output model with the benefit model, it can be seen that the highest tourism income under the premise of ensuring the maximum benefit will increase compared with the highest tourism income under the premise of ensuring balanced economic development. The maximum investment intensity and income growth rate under the principle of maximum benefit are reduced compared with those under the balanced development mode. With the increase in the input cost a per unit time, the proportion of economic income increases, while with the increase in the input cost P, the proportion of economic income decreases. In other words, it is not an ideal choice to blindly pursue the maximisation of tourism income, which may cause negative effects such as high cost and unbalanced economic development in other fields. In order to achieve the optimal effect, the development mode with the best benefit should be selected on the premise of ensuring sustainable and balanced economic development. This section provides an idea and scheme for using the ordinary differential equation to solve the most efficient tourism income.

Next, the stability analysis of the maximum benefit equations of equilibrium development is carried out. Firstly, the differential equations are solved according to the following equations: 25 $f (x) = r x (1 - \frac{x}{N}) - k x = 0$ $$f(x) = rx\left( {1 - {x \over N}} \right) - kx = 0$$ 26 $R (k) = p N k (1 - \frac{k}{r}) - a k$ $$R(k) = pNk\left( {1 - {k \over r}} \right) - ak$$

Based on these, four equilibrium points $P_{1} (N_{1}, 0)$ , $P_{2} (0, N_{2})$ , $P_{3} (\frac{N_{1} (1 - x)}{1 - x k}, \frac{N_{2} (1 - x)}{1 - x k})$ ${P_3}\left( {{{{N_1}(1 - x)} \over {1 - xk}},{{{N_2}(1 - x)} \over {1 - xk}}} \right)$ and $P_{4} (0, 0)$ can be obtained. Where $σ_{1}$ and $σ_{2}$ represent important indicators of the trade-off relationship between the two indicators, which are dimensionless mathematical parameters. $σ_{1} > 1$ represents the maximum return is more obvious than the effective return; when $σ_{2} > 1$ , the effective return is more obvious. The stability conditions for the four equilibrium points are listed in the table. From the data given in the table, only P₁ balances to satisfy $σ_{2} > 1$ , which can be obtained under the balanced development of the best tourism revenues, and P₃ and P₂ are two kinds of benefits which have quite balanced relationship, and with largest gain, there is also the best profit; P₄ balance is not stable state, but it is a kind of limit state. There is only theoretical possibility, but it is not realistic in the actual development process. Therefore, in general, the model has high stability (Table 3).

Table 3

Equilibrium point and stability conditions

Balance	Stability condition
$P_{1} (N_{1}, 0)$	$σ_{1} < 1, σ_{2} > 1$
$P_{2} (0, N_{2})$	$σ_{1} > 1, σ_{2} < 1$
$P_{3} (\frac{N_{1} (1 - x)}{1 - x k}, \frac{N_{2} (1 - x)}{1 - x k})$ ${P_2}(0,{N_2}){P_3}\left( {{{{N_1}(1 - x)} \over {1 - xk}},{{{N_2}(1 - x)} \over {1 - xk}}} \right)$	$σ_{1} < 1, σ_{2} < 1$
$P_{4} (0, 0)$	Unstable

Conclusion

Based on the background of tourism economy, to a specific mode of regional tourism, the first step is to collect information, for the coupling analysis between the subsystems of the tourism industry and regional economy of synergy; through the tourism development income finite element differential mathematical equations, the relationship between the index and development efforts is calculated. This study evaluates and predicts the income index of tourism development in a specific region and provides countermeasures for the development of tourism economy in this region.

Through literature research and analysis, it is found that there is a certain coupling mechanism between the tourism industry system and the regional economic system in this region. Generally speaking, the tourism industry and the regional economy interact with each other, and they restrict and influence each other.

In this study, the coupling coordination degree of tourism and regional economic subsystem in this region can be divided into four stages: maladjusted recession type, transitional type, coupling coordination type and coordinated ascending type. In the second stage, area A takes the lead in developing to the transitional type, while the other three areas are still in the maladjusted recession type, but the coordination level is increased. This is closely related to the improvement of regional comprehensive strength, the emphasis on tourism development and the construction of infrastructure. The mutual promotion and mutual progress of the two subsystems are obvious.

Applying the mathematical differential equation in a fixed area tourism development income situation analysis, we found the tourism income and largest investment do not have a linear relationship. Analysing the stability of the differential equation is known as the investment increase. A tourism revenue decline will be seen after the peak due to excessive development of tourism, which leads to unbalanced economic development, so beyond a certain cost, there is a negative impact on revenue.

In order to take the economic development of other areas in the region into account, seeking the best economic returns on the premise of ensuring balanced economic development is more cost-effective. By solving the differential equation of optimal income, it is found that under the condition of balanced economic development, the highest tourism income is higher than the maximum income, and the investment intensity is reduced. By analysing the stability of the simultaneous cost and benefit model, it is found that the model is unstable only in the case of zero input and zero benefit, which is an extreme case and practically impossible to exist. Therefore, it is considered that the prediction results have high accuracy.

Figures et tableaux

Grade classification of coupling coordination degree

Serial number	Coupling coordination degree interval	Coordination level	Type
1	[0.0.1]	Extreme imbalance	Dissonant recession type
2	[0.0.1]	A serious imbalance between
3	[0.0.1]	Moderate disorders
4	[0.0.1]	Mild disorder	Transitional
5	[0.0.1]	On the verge of disorder
6	[0.0.1]	Barely coordination
7	[0.0.1]	Primary coordination
8	[0.0.1]	Intermediate coordinate	Coordinated ascending type
9	[0.0.1]	Good coordination
10	[0.0.1]	Good coordination

Equilibrium point and stability conditions

Balance	Stability condition
P1(N1,0)	σ1<1,σ2>1
P2(0,N2)	σ1>1,σ2<1
P3(N1(1−x)1−xk,N2(1−x)1−xk) ${P_2}(0,{N_2}){P_3}\left( {{{{N_1}(1 - x)} \over {1 - xk}},{{{N_2}(1 - x)} \over {1 - xk}}} \right)$	σ1<1,σ2<1
P4(0,0)	Unstable

Index evaluation system for tourism system and regional economic system

Target layer	System layer	Indicator layer (unit)	Weight
Tourism development income index evaluation system	Tourism industry subsystem	Number of domestic tourists (10,000)	0.1231
		Domestic tourism revenue (100 million yuan)	0.1907
		Inbound tourist arrivals (visits)	0.2012
		Foreign exchange income from tourism ($10,000)	0.2035
	Regional economic subsystem	Gross product (100 million yuan)	0.1953
		Output value of tertiary industry (100 million yuan)	0.1177
		Per capita GDP (yuan)	0.1355
		Financial revenue (10,000 yuan)	0.1294

Références

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Applied Mathematics and Nonlinear Sciences

Study on tourism development income index calculation of finite element ordinary differential mathematical equation

Mengjie Gong
Mengjie Gong

Publié en ligne: 23 Dec 2022

Volume & Edition: AHEAD OF PRINT

Pages: -

Reçu: 15 Jun 2022

Accepté: 30 Sep 2022

Détails du journal et du numéro

Applied Mathematics and Nonlinear Sciences

Aperçu PDF

Article

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Figures et tableaux

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Grade classification of coupling coordination degree

Equilibrium point and stability conditions

Index evaluation system for tourism system and regional economic system

Références

Articles récents

Planifiez votre conférence à distance avec Sciendo

Applied Mathematics and Nonlinear Sciences

Study on tourism development income index calculation of finite element ordinary differential mathematical equation

Mengjie GongMengjie Gong

Publié en ligne: 23 Dec 2022

Volume & Edition: AHEAD OF PRINT

Pages: -

Reçu: 15 Jun 2022

Accepté: 30 Sep 2022

Détails du journal et du numéro

Applied Mathematics and Nonlinear Sciences

Aperçu PDF

Article

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Figures et tableaux

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Grade classification of coupling coordination degree

Equilibrium point and stability conditions

Index evaluation system for tourism system and regional economic system

Références

Articles récents

Planifiez votre conférence à distance avec Sciendo

Mengjie Gong
Mengjie Gong