Application of Grey Lotka-Volterra Model in Water-Economy-Industry-Technology Innovation System in Beijing-Tianjin-Hebei Region

With the proposal of China’s high-quality development strategy, how to promote regional stability and coordinated development based on a deep understanding of the main contradictions and changes in China’s society has become the focus of research. High-quality development is a brand-new coordinated development concept, which aims to optimize the economic structure, transform the development model, enhance the development momentum, and take innovation as the primary driving force. How to promote the coordinated development of this region has become a hot issue considered by scholars. The Beijing-Tianjin-Hebei region is the capital economic circle of China, and the purpose of this study is to promote the coordinated and stable development of the region. On this premise, this paper firstly adopts the composite Grey Lotka-Volterra (GLV) model and Fractional GM(1,1) (FGM(1,1)) model to research Water Resources system-Economic System-Industrial System-Technology Innovation System in the Beijing-Tianjin-Hebei region. Secondly, by analyzing the research data, it is found that the relationship between the system is very complex, and the stability calculation results are all below 0. Then, the analysis of the research results shows that there is no obvious coordination among the three regions, and they have not yet reached a state of mutual promotion and stable and coordinated development. Finally, four suggestions are put forward for the coordinated development of the Beijing-Tianjin-Hebei region. This can not only provide direction for the future development of the region but also have reference significance for the development of other regions. Further, accelerate the coordination and unity of all factors of production in China and promote China’s development at a deeper and higher level.


Introduction
China's economy has shifted from a stage of rapid growth to a stage of high-quality development. High-quality development needs to focus on and explore the vitality and innovation of development based on a deep understanding of the main contradictions and changes in our society, to achieve the coordination and unity of various institutional elements and promote higher-level development [1]. At present, China has overcapacity in high-pollution and low-tech industries such as chemical, steel, and coal. Low-carbon, clean and high-tech industries are underdeveloped, industrial technology innovation is insufficient and industrial structure is unbalanced [2]. The aggravation of resources and environmental constraints and the lack of innovation power are the key factors restricting the high-quality development of China's industry [3,4]. High-tech industry is the pillar industry of the national economy and social development, which plays a decisive role in system from the two aspects of the balanced development of the three varieties and the evolution of the competition among the three varieties. Anindya S. Chakrabarti [25] applied the Lotka-Volterra equation to simulate the evolution of the technological frontier, proving that the way of technological diffusion has produced repercussions in partner economies. Meng et al. [26] established the government-port-third-party organization evolutionary game LV model for the development of green smart ports, and made suggestions for the development of green smart ports in the future. Zhang et al. [27] investigated the competition and cooperation between Shaanxi Province's TI-RE-EE-IQ by constructing a four-dimensional Grey Lotka-Volterra model, and explored the balance and stability of the four-dimensional model.
According to the findings of the literature in this region, most scholars have begun to study the spatial scope of the Beijing-Tianjin-Hebei region, using different methods such as coupled correlation to study the relationship between systems. Although most scholars use coupling degree to evaluate coordination, the evaluation method of coupling degree is easily affected by extreme data. Lotka-Volterra model is mostly used to describe the relationship between biological populations, but rarely used in regional cooperative development. However, multidimensional Lotka-Volterra model is an in-depth study on the basis of Lotka-Volterra model. In recent years, high-quality development and new development concepts have become the basic orientation and guidelines of China's economic development, and the study of Beijing-Tianjin-Hebei coordinated development from the perspective of highquality development has attracted academic attention. It has become a consensus to study the coordinated development of Jing-Jin-Ji region from the perspective of highquality development, but the index system and regional level of existing research are not detailed enough. Therefore, it is of great theoretical and practical significance to construct a multi-system Lotka-Volterra model to study the coordinated development of Beijing-Tianjin-Hebei region from the perspective of high-quality development.

Study Areas
The coordinated development of Beijing-Tianjin-Hebei is one of the three key areas of major national strategic breakthroughs. Coordinated development between regions has received active policy support from the state and the Beijing-Tianjin-Hebei local government. Therefore, the Beijing-Tianjin-Hebei region is selected as the research object in this paper, and the location of the selected area is shown in Figure 1.
Beijing is located in the north of China and its center located at 116 • 20 east longitude. The terrain northwest high, southeast low. Surrounded by mountains on three sides in the west, north and northeast, the southeast is a plain sloping slowly toward the Bohai Sea and adjacent to Tianjin in the east. By the end of 2021, Beijing had 21.886 million permanent residents.
Tianjin is surrounded by the Bohai Sea in the east, Yanshan Mountains in the north and the capital Beijing in the west. The geographical coordinates are between 38 • 34 -40 • 15 N and 116 • 43 -118 • 4 E. Tianjin is the main node of the China-Mongolia-Russia Economic Corridor, the strategic fulcrum of the Maritime Silk Road, the intersection of the Belt and Road, and the nearest eastern starting point of the Eurasian Land Bridge. It is also a shipping center, logistics center and modern manufacturing base in northern China. By the end of 2021, the city has jurisdiction over 16 districts with a total area of 11,966.45 square kilometers and a permanent population of 13.73 million.
Hebei is located in the north China plain, east of the Bohai Sea, surrounded by Beijing and Tianjin, the Taihang Mountains in the west, Yanshan Mountains in the north, and Zhangbei Plateau in the north. It is the only province in China with plateaus, mountains, hills, plains, lakes and coasts. It is an important grain and cotton producing area in China. By the end of 2021, Hebei had 74.48 million permanent residents.

Data Collection
In this paper, Beijing-Tianjin-Hebei region is taken as the research area, and the coordinated development of the region is studied by constructing a composite Water Resources System-Economic System-Industrial System-Technology Innovation System. Original data of the four systems in the Beijing-Tianjin-Hebei region comes from the statistical yearbooks of the provinces and the statistical bulletins of the provinces. Specific evaluation indexes are shown in Table 1.

Data Collection
In this paper, Beijing-Tianjin-Hebei region is taken as the research area, and the coordinated development of the region is studied by constructing a composite Water Resources System-Economic System-Industrial System-Technology Innovation System. Original data of the four systems in the Beijing-Tianjin-Hebei region comes from the statistical yearbooks of the provinces and the statistical bulletins of the provinces. Specific evaluation indexes are shown in Table 1.

Model Formulation
In this paper, grey Lotka-Volterra model is used to analyze and calculate the relationship between Water Resources System-Economy System-Industrial System-Technology Innovation System in the Beijing-Tianjin-Hebei region. Firstly, the relationship analysis is derived on the current data in the Beijing-Tianjin-Hebei region. Secondly, using the FGM(1,1) model to forecast the existing data. Then, this model analysis is performed on the predicted data. Finally, combining the existing and predicted relationships among the four systems in this region, suggestions for improvement are put forward.

Data Processing
This article uses the entropy method to process the original data, and the processed data is divided according to the system. The specific steps are as follows.
Step 1: Index selection. There are t years, where X i (t) represents the i index in year t.
Step 3: The proportion of indicators is determined.
Step 5: Calculate the redundancy of information entropy.
Step 6: Calculate the weight of each indicator.

Forecasting Model
This paper uses the fractional model (FGM(1,1)) to predict data. Using the fractional model to make predictions can weaken the randomness of the original sequence and make the grey prediction model less disturbed. In addition, FGM(1,1) conforms to the principle of giving priority to new information, which improves the reliability of data prediction results [28,29]. The steps of FGM(1,1) are shown in Figure 2.

Grey Lotka-Volterra Model
Assume that, are positive sequences of original data for the state variables of a system, where,

Forecasting Model
This paper uses the fractional model (FGM(1,1)) to predict data. Using the fractional model to make predictions can weaken the randomness of the original sequence and make the grey prediction model less disturbed. In addition, FGM(1,1) conforms to the principle of giving priority to new information, which improves the reliability of data prediction results [28,29]. The steps of FGM(1,1) are shown in Figure 2.  View x (1) (k) and y (1) (k) as research variables in order to explore the long-term relationship between X (0) and Y (0) . Actually, the data before x (0) (k) often exert an influence on x (0) (k) and y (0) (k), that is to say the accumulation of x (0) (k) will exert an influence on x (0) (k) and y (0) (k). According the grey modeling method, at time k, the grey derivative of the X (1) is, The mean generated sequence of consecutive of , and the mean generated sequence of consecutive neighbors of Y (1) is Z 7 of 24 is called the grey Lotka-Volterra model. Given the error sequence, x (k)z (1) y (k) (13) the least squares estimate of the parameter sequence a 1 , b 1 and c 1 of the grey Lotka-Volterra model is given by,  where, To estimate the unknown parameters a 1 , b 1 and c 1 under the criterion of the minimization of the MAPE, can set the objective function as: Thereby acquiring the estimated valuesâ 1 ,b 1 andĉ 1 . Then, obtain the estimated valuesα 1 ,β 1 , andγ 1 of α 1 , β 1 , and γ 1 . Lastly, can obtain the discrete Grey Lotka-Volterra modelx (1) The 1-AGO isx (0) (k + 1) =x (1) Similarly, can obtain, The 1-AGO isŷ (0) (k + 1) =ŷ (1) Can judge the relationship between X (1) and Y (1) from the signs ofĉ 1 andĉ 2 .

Composite Grey Lotka-Volterra Model
The GLV model of 4 populations can be simplified to, According to the difference between the parameters βij and β ji (i = j), 7 possible relationships between factor i and j can exist [20,30], as shown in Table 2. Let: .
Substituting the original data column when t = 1, 2, . . . , n − 1, into the above equation, the matrix equation can be obtained: where, Similarly, D 2 , D 3 and D 4 can be obtained, and then the least square method can be 3, 4; j = 1, 2, 3, 4; i = j. By calculation, can obtain the parameterÂ i ,B ii ,B ij . Substituting the parameters into equation, the estimated valuex i (t) can be obtained.

Equilibrium Analysis
In order to better understand the evolution results of various factors, the equilibrium and stability of the GLV model are analyzed. GLV model can be expressed as, Therefore, it can be concluded that it has 16 balance points. The balance equation can be written as follows, where, Not all equilibrium points are stable. It can be judged by calculating the eigenvalues of the Jacobian matrix [31]. When the eigenvalues of the Jacobian matrix are all negative real parts, the equilibrium point E reaches a stable state. The Jacobian matrix is as follows, According to the Routh-Hurwitz system stability criterion, the necessary and sufficient condition for the characteristic roots of the system to have a negative real part is that the highest order coefficient of the characteristic equation is greater than 0, and the formulas composed of the coefficients of the characteristic equation are all positive [32]. Assuming that p, q, r and s are the coefficients of the third-order, second-order, first-order, and constant terms of the characteristic equation, then, According to the above analysis, when p > 0, q > 0, qr − ps > 0,qrs − ps 2 > 0, the equilibrium points E reaches a stable state. otherwise, it is unstable.

Prediction and Analysis of Existing Data
Now take the Beijing area as an example, and use the FGM(1,1) model to forecast the original data. It can be seen from Figure 3  Assuming that there is a connection between these four systems, and now calculate and analyze the connection between the four systems.
late and analyze the connection between the four systems.
In Figure 3, each figure is divided into two parts, with known data from 2015 to 2019 and predicted data from 2020 to 2024. Figure 3a shows the change trend of WR in Beijing from 2015 to 2024. Figure 3b shows the change trend of ES in Beijing from 2015 to 2024. Figure 3c shows the change trend of IS in Beijing from 2015 to 2024. Figure 3d shows the change trend of TI in Beijing from 2015 to 2024.  In Figure 3, each figure is divided into two parts, with known data from 2015 to 2019 and predicted data from 2020 to 2024. Figure 3a shows the change trend of WR in Beijing from 2015 to 2024. Figure 3b shows the change trend of ES in Beijing from 2015 to 2024. Figure 3c shows the change trend of IS in Beijing from 2015 to 2024. Figure 3d shows the change trend of TI in Beijing from 2015 to 2024.

Status Quo of Beijing-Tianjin-Hebei
(1) Beijing area Based on the original data of the four systems in the Beijing area, the relationship between the four systems in the Beijing area from 2015 to 2019 is now calculated using composite Grey Lotka-Volterra. The expression formula of the calculation results is as follows.
The competition and cooperation among the four systems in Beijing is detailed in Figure 4.    (2) Tianjin area According to the data of the four systems in Tianjin from 2015 to 2019, Equation (33) and Figure 5 are obtained through relevant calculations. (3) Hebei area By combining the above formulas to calculate the 12 indicators of the four systems in Hebei, Equation (34), Figure 6 are obtained.   (3) Hebei area By combining the above formulas to calculate the 12 indicators of the four systems in Hebei, Equation (34), Figure 6 are obtained.

Equilibrium and Stability
To analyze the equilibrium and stability among Water Resources System, Economy System, Industrial System and Technology Innovation System in these three areas, the parameters (α i , β ii , β ij ) should be estimated first. Second, the 16 equilibrium points (E0-E15) and the equilibrium point stability of the four factors of Water Resources System, Economy System, Industrial System and Technology Innovation System can be calculated.
As mentioned above, the equilibrium point has practical significance only when all four factors exist and are greater than zero [30]. In the equilibrium points E0-E14, no less than 1 factor has a value equal to 0, and only the 4 factors of E15 are all non-zero.

Forecast Data Calculation
Due to the high accuracy of FGM(1,1) prediction, FGM(1,1) is used to forecast data of Beijing-Tianjin-Hebei region in next five years.
After obtaining the forecast data, perform the grey Lotka-Volterra calculation of the composite system on this data, and evaluate the changes in the degree of coordination between the Beijing-Tianjin-Hebei regions in the next five years. The coordinated change trend of the four systems in the future can be obtained by calculation, and a prediction can be made in advance.

Analysis on the Status Quo of Beijing-Tianjin-Hebei
(1) Beijing area Figure 4 shows that there is a predator-predator relationship among Economic System, Industrial System and Technology Innovation System. This phenomenon shows that the current high-tech industry in Beijing still needs to be improved, and it has not attained a state of benign competition with other industries.
In Table 3 that reflects the existence of a predator-predation relationship between the Water Resources System and Economy System. The Water Resources System hinders the development of the Economy System, but the Economy System provides a boost to the Water Resources System. At the same time, the influence coefficient of the Economy System on the Water Resources System is 0.01611328, and the influence coefficient of the Water Resources System on the Economy System is 0.00001788. This shows that the Economy System has a greater impact on the Water Resources System, and the development of the Economic System determines the development of the Water Resources System. In order to achieve healthy competition between the two systems, it is necessary to strictly control the reasonable use of water resources in the Water Resources System under the conditions of rapid economic development.  This shows that there is a mutualistic relationship between Water Resources System and Industrial System. However, it shows that Industrial System controls Water Resources System, and the increase in industrial output value will face an increase in Water Resources System. Under the requirements of high-quality and sustainable development, the total amount of water supply should be controlled within a reasonable range while increasing industrial added value.
From Table 3, the degree of influence of Technology Innovation System on Water Resource System is 0.0439453125, and Water Resource System on Technology Innovation System is 0.0000953674, indicating that there is a competitive relationship between the two. According to the above data, Beijing should increase the investment in technological innovation, and the continuous progress of technological innovation will reduce the total water supply.
It shows that there is a mutualistic between the Economy System and the Industrial System. Based on data, it can be seen that as the total industrial output continues to increase, the economic system also increases.
There is a predator-predator relationship between the Economy System and Technology Innovation System. The impact coefficient of Economy System on Technology Innovation System is greater than the impact of Technology Innovation System on the Economy System, indicating that economic conditions determine the development of technological innovation. Economic foundation determines the ability to innovate. Therefore, to improve the level of local economy, but also pay attention to the development of innovation ability.
Predatory relationship exists between Industrial system and Technology Innovation system. The continuous innovation of science and technology will promote an increase in industrial output value. Thus, paying attention to technological progress, actively adjusting the industrial structure, and eliminating outdated production capacity. By improving industrial efficiency and developing high-tech industries, Beijing's industrial development will be more in line with the needs of the people.
(2) Tianjin area It can be seen from Figure 5 that there is a predator-predator relationship among the Tianjin Water Resources System, Economy System and Industrial System. There is a mutual relationship between the Water Resources System and the Technology Innovation System. There is a competitive relationship between the Economy System and Industrial System systems. Economy System, Industrial System and Technology Innovation System have a predatory relationship. Now analyze it based on the detailed data in Table 4. It indicates the existence of a predation relationship between the system and the system. The Economy System hinders the development of the system, but the Water Resources System provides a boost to the Economy System. The influence coefficient of the Economy System on the Water Resources System is 0.02166748, and the influence coefficient of the Water Resources System on the Economy System is 0.00004959. This shows that the Economy System has a greater impact on the Water Resources System, and the development of the economic system determines the development of water resources.  Predation relationships exist between Water System and Industrial System. The predator-prey relationship means that the development of one system hinders the development of another system. Due to the Industrial System has a greater impact on the Water Resources System. The emergence of this phenomenon shows that the connection between the industrial added value of Tianjin and the water resources system is poor, and the water consumption has not been reasonably controlled under the premise of securing the steady growth of industrial output.
There is a mutual relationship between Water Resources System and Technology Innovation System. The degree of Technology Innovation System's influence on Water Resources System is 0.02306366, and the degree of Water Resources System's influence on Technology Innovation System is 0.00082397, reflecting that technological innovation has a greater influence on the total water supply. Although there is a mutually beneficial relationship between the two, they have not reached a competitive relationship. Only proper competitive relations can foster the high-quality development of various indicators.
Economy System and Industrial System are competing shows in Figure 5. The current economic situation of Tianjin is closely associated with gross industrial output value. To improve the regional economy, it is necessary to grasp the value of industrial output value, but the influence degree between them and the relationship between them still need to be stable.
Predator-predator relationship between Economy System and Technology Innovation System. The effect coefficient of Economy System on Technology Innovation System is greater than the impact of Technology Innovation System on Economy System, indicating that economic conditions determine the development of technological innovation. Economic base determines innovation ability. Therefore, to improve the level of local economy, but also pay attention to the development of innovation ability.
Obviously, there is a predatory relationship between the Industrial System and the Technology Innovation System. However, industrial output has a greater impact on technological innovation, so technological progress must be taken seriously.
(3) Hebei area Through Equation (34), one can find that the value in Industrial System is negative, and the value in Technology Innovation System is less than 1. This value of expresses the parameter of the system itself, and the value is generally greater than 1. However, the data in Hebei showed a negative value. This means that there are serious problems in the industrial system, which need to be improved.
In Table 5, β12 = −0.00002646, β21 = −0.00454712, indicating that there is a competitive relationship between Water Resources System and Economy System. In addition, the influence degree of Economy System on Water Resources System is greater than the influence degree of Water Resources System on Economy System. The Economy System determines the degree of change in the Water Resources System. It can also be said that the level of consumption determines the degree of water consumption, but the growth of the economic system should not be based on sacrificing water resources. The goal wants to accomplish is to decrease the rational use of water resources while the economy grows steadily, so as to achieve a state of win-win cooperation.   Table 5, β 13 > 0, β 31 > 0, a mutualistic relationship is formed between Water Resources System and Industrial System. β 14 > 0, β 41 > 0, a mutualistic relationship is formed between Water Resources System and Technology Innovation System. β 34 > 0,β 43 > 0, there is a symbiotic relationship between Industrial System and Technology Innovation System. The relationship between Water Resources System, Industrial System and Technology Innovation System is symbiosis, which shows that the current general trend in Hebei is conducive to the development of Technology Innovation System.
In accordance with Table 5, β 23 > 0, β 32 < 0, β 24 > 0, β 42 < 0 show that the relationship between Economic System, Industrial System and Technology Innovation System in Hebei is predicated. Due to β 23 = 0.00000089, β 32 = −0.00011921, the influence of the Industrial System on the Economic System is greater than the influence of the Economic System on the Industrial System. Hebei Province is a province with a large proportion of the secondary industry in the Beijing-Tianjin-Hebei region. In recent years, it has been consistently optimizing and upgrading its industrial structure. The upgrading of industrial structure will have a great impact on the economy, but there is a predatory relationship between Economic System and Technology Innovation System. The continuous progress of science and technology requires strong support from the economy. However, in turn, the advancement of science and technology will enhance the manufacturing capacity of industrial enterprises or provide suggestions for the optimization and upgrading of the industrial structure in the province. Technology Innovation System and Economic System should be a state of symbiosis or benign competition, rather than a relationship of predation. Therefore, Hebei Province should pay attention to the relationship between systems, so that the various system levels within the province can form a situation of balanced competition.

Analysis of Equilibrium and Stability
The stability of the system is mainly analyzed according to the calculated values of p, q, qr − ps, and qrs − ps 2 . Tables 6-8, respectively, show the stability calculation of Beijing, Tianjin and Hebei.
From Table 6 that x 1 , x 2 , x 3 , x 4 in E15 is greater than 0, but p, q, qr − ps, qrs − ps 2 in E15 has a value not greater than 0. x 1 , x 2 , x 3 , x 4 are not all greater than 0 from E0 to E14, and the range of values fluctuates greatly. It is believed that the four systems of Beijing Water Resources System, Economic System, Industrial System, and Technology Innovation System have not attained a trend of consistent competition and coordinated growth since 2015.
By calculating the stability of Tianjin data, it is found that the value of p, q in the calculated data at point E15 in Table 7 is less than 0. It shows that in Tianjin, Water Resources System, Economic System, Industrial System, and Technology Innovation System have not attained stable and coordinated development. However, the values of p, q, qr − ps, qrs − ps 2 at point E1, E2 are all greater than zero. It shows that the development of Water Resources System and Economic System in the region is consistently stable. The coordination between systems has not been achieved. It is still necessary to enhance the coordination capabilities between systems to mutually foster high-quality development. Table 6. Estimated equilibrium and eigenvalues in Beijing.

Equilibrium
Stability   Table 8. Estimated equilibrium and eigenvalues in Hebei.

E
Equilibrium Stability In Table 8, Hebei is similar to Beijing and Tianjin, and there are unstable phenomena. In this table, only the E2 data in which p, q, qr − ps, qrs − ps 2 is greater than 0. However, the data of x 1 , x 2 , x 3 , x 4 in E2 only has x 2 greater than 0, and all other data is 0. It shows that the Economic System of Hebei Province has reached a trend of steady growth. However, Hebei lacks the ability to collaborate between systems.

Analysis of Forecast Data
(1) Beijing area Through the above data results, it is found that the relationship between the predicted data and the calculated system will have a lot of changes with the current situation. This phenomenon further proves that the current problems in the Beijing area require the establishment of more effective methods to promote stable competition between systems.
From Tables 9 and 10, find that the relationship between the Water Resources System and the Economic System in Beijing has changed from the previous predation to an incompatible relationship. This phenomenon shows that there is a big problem between the Water Resources System and the Economic System. Thus, a good coupling relationship must be maintained between water conservation and stable economic development. Water Resources System and Technology Innovation System has changed from the previous competition to the current predator relationship, indicating that the development of one system between the two in the future will cause obstacles to the development of the other system. Other systems have developed in the direction of healthy competition, or in symbiosis. However, they must continue to improve and become better at maintaining this good posture.      The constant changes in the relationship between systems confirm the previous stable conclusion. This phenomenon emergence is mostly because each system has not found a more suitable development direction and situation. This needs to fundamentally find a way to improve, to find the direction of coordinated development of each system. From Tables 13 and 14, comparing the previous data, find that the relationship between the five systems has changed. The Water Resources System and Economic System have changed from the previous competitive relationship to a predator relationship. Water Resources System and Industrial System have changed from mutualistic to predation. Water Resources System and Technology Innovation System have changed from symbiosis to competition. Economic System and Industrial System changed from predation to symbiosis. Economic System and Industrial System have changed from predation to a mutualistic relationship.   Half of the relationship between systems in Hebei Province has become symbiosis, which refers to the mutual existence of two systems. This state seems to be in a stable situation, but as resource policies continue to change, the relationship between systems will also change. If the symbiosis relationship has been maintained or tends to a mutualistic state, it will undoubtedly cause problems between systems and affect the development of Hebei Province.

Research Implications
As the coordinated and integrated development of the Beijing-Tianjin-Hebei region is proposed, high-quality development among various regions has also become an issue that needs to study now. High-quality development is inseparable from technological innovation, but the level of technological innovation cannot fully represent whether the region meets high-quality. High-quality development means that all resources between various regions can achieve a reasonable allocation, and can coordinate with each other and promote together.
As the capital economic circle, the Beijing-Tianjin-Hebei region is a key research area. Han et al. [8] used the coupling method to analyze the Economy-Resources-Environment of the region. Hu et al. [10] conducted research on industrial energy and related emissions in the Beijing-Tianjin-Hebei region. Zhang et al. [27] investigated the competition and cooperation between Shaanxi Province's TI-RE-EE-IQ by constructing a four-dimensional Grey Lotka-Volterra model, and explored the balance and stability of the four-dimensional model. On this basis, the Water Resources System-Economic System-Industrial System-Technological Innovation System is constructed, and the Beijing-Tianjin-Hebei region is analyzed and studied by multi-dimensional Grey Lotka-Volterra model.
In the Beijing-Tianjin-Heber regions, most scholars used the grey correlation method or the relationship of calculating the coupling degree of the system to analyze the relationship between the systems. Secondly, few scholars in the region have paid attention to the connection between technological innovation and industry. Moreover, few scholars would think of using the Grey Lotka-Volterra calculation model to analyze the relationship between systems. Although many scholars in Jing-Jin-Ji have put forward their own suggestions and methods for improvement, there are few analyses on the systems of the Beijing-Tianjin-Hebei region. Most scholars study the three aspects of energy, economy, and industry in the Beijing-Tianjin-Hebei, or the relationship between the three, and rarely involve the opposite aspects of technological innovation. Thus, analyzing the relationship among the Beijing-Tianjin-Hebei Water Resources System-Economic System-Industrial System-Technological Innovation System is of great research value. It not only fills the vacancies in this area of the Beijing-Tianjin-Hebei region but can also provide references for other areas in this area.

Research Limitations
This study has certain limitations. In the first place, the phenomenon of agglomeration of manufacturing industries in the Beijing-Tianjin-Hebei region [33]. This article does not consider the impact of the concentration of industrial activities on water resources, the economy, and industry. In future research, we will consider the impact of industrial concentration on water resources and the impact of industrial agglomeration on other systems. Secondly, although have extended the GLV model to four dimensions and selected 12 indicators of the four systems. However, there are still limitations and there is still room for improvement.

Conclusions
This paper takes Beijing-Tianjin-Hebei region as the research area, and first builds a composite Grey Lotka-Volterra model among the Water Resources System-Economic System-Industrial System-Technology Innovation System. Analyze the relationship between the four systems from 2015 to 2019 through the model and calculate the stability of the relationship between the systems. Secondly, the FGM(1,1) model is used to forecast the 12 evaluation indicators under the four evaluation systems. After obtaining the forecast data, the compound GLV model is used to analyze the relationship between the systems on the forecast data.
Through the analysis of the Beijing-Tianjin-Hebei region, the following conclusions are obtained.

1.
Interaction between current systems. The Economic System in Beijing has a positive influence on the development of the Water Resources System, and Technology Innovation System has a positive effect on the development of the Economic System and the Industrial System. In the Tianjin area, Economic System and Industrial System are factors hindering the development of Water Resources System, and the development of Technology Innovation System is influenced by Economic System and Industrial System. There are visible differences between Hebei, Beijing and Tianjin. In Hebei, there is a competitive relationship between the Water Resources System and the Economic System, and there is a symbiotic relationship between the Water Resources System and the Industrial System and Technology Innovation System. 2.
Stability analysis. The relationship between the systems in Beijing-Tianjin-Hebei region is extremely unstable, and the relationship between systems has not formed a steady growth trend.

3.
Predict interaction between systems. In Beijing, there is a state of incompatibility between the Water Resources System and the Economic System, which is seriously inconsistent with the trend of coordinated development between systems. As the Technology Innovation System in Beijing has changed over time, both Water Resources System and Industrial System have become obstacles to it. In Tianjin, the development of the Economic System and Industrial System will inevitably have to sacrifice the Water Resources System, and the Water Resources System has promoted the development of the Industrial System and the Economic System. In Hebei, the Water Resources System hinders the development of the Economic System and Industrial System. Economic System, Industrial System and Technology Innovation System promote each other and make progress together.
Through the analysis of the systems in Beijing-Tianjin-Hebei, it is found that the relationship between the system and the system is poor, and most of the relationship is unstable. At the same time, it is found that the Beijing-Tianjin-Hebei system lacks cohesion and is still far from achieving coordinated development.
Based on the results, this paper puts forward the following suggestions.
(1) Strengthen innovation incentive policies. This is the main way to promote the development of high-quality industries and reduce waste of resources.