## Abstract

In order to assess the water resources carrying capacity (WRCC) of Hubei province, an improved catastrophe progression method based on M-K test and correlation analysis was established. This model includes evaluation, abrupt change test and correlation analysis. It can make a comprehensive assessment of water resource carrying capacity in a certain area. The evaluation results of this model are clear and can effectively avoid the effects of subjective weight and, in addition, it can also streamline the index system. We applied the model to study the WRCC of Hubei province from 2005 to 2016, considering the supply and demand of water resources, ecological environment, economy and society. The results showed that the WRCC of Hubei province is at the **‘**weak’ level, presenting a certain development and utilization potential, but it must be strictly controlled and moderately developed. The WRCC of Hubei province is improving, but must be adjusted by water conservation facilities and long-term management policies to prevent the foreseeable deterioration. Water supply and demand systems and ecological environment systems were found to be the driving factors of WRCC through correlation analysis. This approach gives the decision-makers suggestions about water resource sustainable utilization.

## INTRODUCTION

Water is essential for the development of human society and economy as an irreplaceable natural resource. It also plays an important role in the stability of the ecosystem (Yang *et al.* 2015). In China, the frequencies of water shortages and correlated environmental problems have been increasing and are found throughout the country because of the waste of water resources and rapid growth of the economy (Jia *et al.* 2017). In turn, the shortage and uneven spatial distribution of water resources also seriously restricts the sustainable development of national society and the economy (Ren *et al.* 2016). Water resources have become an important factor restricting sustainable socioeconomic development in China, an effective approach to evaluate and manage water resources is necessary (Elisabeth 2014). This paper introduces the concept of water resources carrying capacity (WRCC) to represent the capacity of water resources to support human society in a region. The concept of ‘resources carrying capacity’ was proposed by United Nations Educational, Scientific, and Cultural Organization (UNESCO) in the early 1980s, while the concept of the WRCC in China was firstly put forward by Xinjiang water resources soft science project team in 1989, after this, scholars started to carry out the continually in-depth research of WRCC (Ren *et al.* 2016). We consider that the WRCC is closely connected with the social, economic and ecological environments. In this study, the WRCC refers to the maximum water supply for the work and life of the population, ecological environmental protection and other water use habits as governed by the local socioeconomic conditions of production and the technology available (Gong & Jin 2009). A water resources system is a complex system with many various uncertainties (Yang *et al.* 2016), and it is very important to correctly evaluate the WRCC of an area, which has become an important aspect of water resources management. Through researching on the WRCC, we can find the relationship between water resources and social economic development, identify the key factors and realize sustainable development of water resources (Yang *et al.* 2015).

At present, research on the evaluation of the WRCC mainly focus on the following three aspects: urban research, regional research, and watershed research. In terms of quantitative methods, there are various methods to quantify WRCC and several common methods are as follows: principal component analysis, fuzzy comprehensive evaluation method, neural network method, and system dynamics model. For example, Wang (2007) identified the key factor which changed WRCC in Jiangxi province through principal component analysis; Gong & Jin (2009) used the method of fuzzy comprehensive evaluation to evaluate the current situation and dynamic trend of WRCC in Lanzhou; Lu *et al.* (2009) applied wavelet neural networks to WRCC prediction and obtained urban water demand; Yang *et al.* (2015) built a system dynamics model on synthesis simulations of coupling effects and feedback mechanisms which can integrally represent system states and assess WRCC. In addition, some other methods are also used in this field: entropy weight method (Xiong *et al.* 2012), set pair analysis model (Mao *et al.* 2012), multi-stage stochastic programming approach (Wang *et al.* 2016). However, there is always a problem in WRCC evaluation if accurate quantification is desired; difficulties will be encountered and the reason is that the factors affecting the WRCC are mostly ambiguous and non-linear. Some of the traditional methods mentioned above involve many assumptions specific to each application, which brings a large impact on the uncertainty of the results. Besides, most methods need to determine weights which are often criticized for its inability to avoid individual opinion (Su *et al.* 2011). Based on such a situation, the improved catastrophe progression method is introduced in this paper, and it is combined with the MK test and correlation analysis to evaluate WRCC.

The main part of the model used in this paper is the catastrophe progression method, a mathematical model proposed by Thom in the 1960s, whose purpose is to deal with incontinuous phenomena (Ahmed *et al.* 2015). Under particular conditions, catastrophe theory can show sudden changes in the steady equilibrium state as a consequence of small changes of input function (Yang *et al.* 2012; Chen *et al.* 2016). In the catastrophe progression method, the dependency of state variables on control variables is determined by catastrophic membership functions rather than weights assigned by the users, which can reduce the impact of human subjectivity on the results (Ahmed *et al.* 2015). For the above reason, the catastrophe progression method has gradually been demonstrated to have a unique advantage in uncertainty calculation and is widely employed in multi-index comprehensive assessment studies. Su *et al.* (2011) developed a catastrophe model to assess the land ecological security of Shanghai and identify the constraints. Wang *et al.* (2018) identified the indices that mostly control the flood intensity by developing a catastrophe progression method. Besides, the catastrophe progression method has also been used in coal and gas outburst prediction (Zhang *et al.* 2009), pollution disaster in near-shore coastal waters (Wang *et al.* 2011), ground water potential zones (Ahmed *et al.* 2015) and so on.

We argue that the change of WRCC can be considered as a catastrophic behavior, a small change in the steady equilibrium state of a subsystem can eventually cause the whole system to reach the crush state. Based on this, our study aims to build a catastrophe progression method based on the M-K test and correlation analysis for WRCC assessment, then apply the model to Hubei province, China. Specifically, our objectives are to: (1) evaluate the WRCC of Hubei province from 2005 to 2016 and determine the main influential factors; (2) find how WRCC changes over time and identify the point of abrupt change; (3) determine the correlation between WRCC and its subsystems.

## STUDY AREA AND DATA COLLECTION

### Study area

Hubei province is located in central China (east longitude 108°21′–116°07′, north latitude 29°01′–33°06′) and the middle reaches of Yangtze River with a land area of 185,900 km^{2}. During 2005–2016, the per capita water resources of Hubei province was 1,605 cubic meters which accounted for 78.7% of the national average. In 2016, the total water resources of Hubei province were 149.8 billion cubic meters and 2,546 m^{3} per capita, which is slightly above the national average level. In general, the water resources of Hubei province are in the middle level. In terms of water consumption, although the water supply is adequate, the shortage of water is becoming increasingly prominent due to climate change and the continuous increase of water demand.

### Data collection

The main sources of data in this paper are as follows: the Hubei Statistical Yearbook (2006–2017), and the Hubei Water Resources Bulletin (2005–2016). In addition, some information was also collected from The Development Report of Hubei Province.

## METHODOLOGY

### Catastrophe progression method

The catastrophe progression method describes the phenomena and general rules for qualitative changes of natural phenomena and the interruption of social activities. In the catastrophe progression method, system function variables are divided into two parts, state variables and control variables (Al-Abadi *et al.* 2016). It takes a potential function ‘’ as a research object and describes the system catastrophe via the state variables and the external control variables . The singular point set equation and the critical point equation of the system can be obtained separately by solving and , then variable *x* is eliminated by solving and , so we can obtain a system bifurcation set. When various control variables in the bifurcation set equation meet the requirements, a catastrophe will occur in the system. Because the numerical units and values of the control variables are not uniform, it is necessary to derive a normalized formula from the bifurcation set. When the state variable is 1, the common catastrophe models are primarily of four types: fold catastrophe, cusp catastrophe, swallowtail catastrophe and butterfly catastrophe, respectively. Some descriptions of these models are shown in Table 1 (Su *et al.* 2011).

Category | Dimension of control variables | Potential function | Bifurcation set | Normalization formula |
---|---|---|---|---|

Fold catastrophe model | 1 | |||

Cusp catastrophe model | 2 | |||

Swallowtail catastrophe model | 3 | |||

Butterfly catastrophe model | 4 |

Category | Dimension of control variables | Potential function | Bifurcation set | Normalization formula |
---|---|---|---|---|

Fold catastrophe model | 1 | |||

Cusp catastrophe model | 2 | |||

Swallowtail catastrophe model | 3 | |||

Butterfly catastrophe model | 4 |

*R*(

_{j}*j*= 1, 2, … ,

*n*), if , then the improved membership value is as follows:

### Sequential Mann–Kendall test

The nonparametric Mann–Kendall (M-K) method is usually used as the base method of detecting trends and abrupt changes and it is widely used in hydrology and meteorology (Rahman *et al.* 2016). In this study, we utilize the sequential M-K test to explore abrupt changes of WRCC and its subsystems, which has been used extensively for detecting the starting point of trends (Rashid *et al.* 2015).

*x*(with

*n*variables), construct a sequence : where the values of are denoted by the number of cases , and are the th (

*i*= 1, 2, … ,

*n*) and th (

*j*= 1, 2, … ,

*i*) data values in time series. Then we can create another sequence , which is the sequence calculated in the order of time series X (X

_{1}, X

_{2}, … , X

_{n}), and it is calculated by Equations (4)–(6): where and are the mean and variance of respectively. Similarly, we can obtain a retrograde sequence by calculating from the last data of the time series

*X*(

*X*,

_{n}*X*

_{n}_{–1}, … ,

*X*

_{1}).

A positive denotes an upward trend while the negative denotes a downward trend. Define a variable which is the critical value of the standard normal distribution with a probability exceeding *α*/2, where *α* is the statistical significance level concerned and is set at 0.05 significant levels (corresponding level is 1.96) in this study. If the absolute value of is greater than , there is a significant trend. The intersection points of the two lines and within the confidence interval are the points of abrupt change (Feng *et al.* 2017).

### Flow diagram of the model

The process of the evaluation model is shown in Figure 1. The catastrophe progression method, M-K test and correlation analysis are both independent and coordinated, which constitute the composite evaluation model together.

## MODEL APPLICATION

### The construction of the index system

The principle of evaluation indexes system is to fully reflect the current situation of water resources and the development between water resources and the social economy system. The standard for index selection is to select the main control index of water resources development based on an evaluation of the objective conditions in a certain area. In this study, the selection of indicators was chosen based on previous studies (Gong & Jin 2009; Ren *et al.* 2016), and the evaluation indexes system is shown in Appendix 1. The index system is divided into three levels: the destination level, the factor level and the index level. After the index system is established, we sort each factor by its importance.

### Data standardization and index sorting

*et al.*2015): where

*i*is the index, is the original value of, and are the maximum and the minimum value of both

*i*and the critical value respectively. Equation (8) is for positive indices (C3, C4, C5, C6, C7, C8, C9, C10, C11) and Equation (9) is for negative indices (C1, C2, C12, C13, C14, C15, C16, C17).

*i*th index;,

*m*is the number of samples; represents the proportion of the standard value of the

*i*th index in year

*j*; represents the weight of each evaluation index. The index ranking results are shown in Appendix 1.

### The construction of the catastrophe model

A catastrophe model for the WRCC evaluation can be established by combining the types of catastrophe models described above and the selected index system, as shown in Figure 2. Both ‘complementary’ and ‘non-complementary’ principles should be considered when judging the indicators, if a significant correlation exists between the control variables, we designate it as the ‘complementary’ type, thus each of them tends to reach the average. In contrast, if no significant correlation between the control variables is present, we designate it as the ‘non-complementary’ type and the state variable value is the smallest of a system (Wang *et al.* 2011; Al-Abadi *et al.* 2016). In this catastrophe model, all the index systems are the ‘complementary’ type.

### Establish scoring intervals

The comprehensive evaluation values are often high, mainly because of the characteristics of normalized formulas used in the catastrophe progression method (Chen *et al.* 2016). We develop the evaluation grading standards of WRCC which is divided into five levels: very weak, weak, middle, strong and very strong (Irankhahi *et al.* 2017). According to the method discussed above under ‘Catastrophe progression method’, the improved scoring intervals are shown in Table 2.

Very weak I | Weak II | Middle III | Strong IV | Very strong V | |
---|---|---|---|---|---|

Ordinary scoring intervals | [0,0.2] | (0.2,0.4] | (0.4,0.6] | (0.6,0.8] | (0.8,1.0] |

Improved scoring intervals | [0,0.8932] | (0.8932,0.9371] | (0.9371,0.9642] | (0.9642,0.9841] | (0.9841,1.0] |

Very weak I | Weak II | Middle III | Strong IV | Very strong V | |
---|---|---|---|---|---|

Ordinary scoring intervals | [0,0.2] | (0.2,0.4] | (0.4,0.6] | (0.6,0.8] | (0.8,1.0] |

Improved scoring intervals | [0,0.8932] | (0.8932,0.9371] | (0.9371,0.9642] | (0.9642,0.9841] | (0.9841,1.0] |

### Normalized calculation

### Correlation analysis

*a*as the mean value and

*b*as the variance of the data, we randomly simulated 1,000 samples by the Monte Carlo method (Wang

*et al.*2009), where

*a*is the annual statistical value and is calculated by the following formula: where is the mean value and variance of the original data respectively. Then we can find the correlation between WRCC and the subsystems.

## RESULTS

### Evaluation results of catastrophe progression method

The results of the WRCC evaluation for Hubei province are shown in Figure 3. First, the WRCC of Hubei province in 2005–2016 as a whole was at the ‘weak II’ level. It was ‘very weak I’ in 2005 and 2006, and reached the ‘middle III’ level in 2008, 2010 and 2016, while the evaluation grade of the remaining years was ‘weak II’. This shows that the water resources of Hubei province have been considerably developed but have not yet reached saturation. There is a certain development potential at present that will satisfy the water demand for some time, however, that development must be appropriate, otherwise the water resources status could very easily deteriorate. Second, according to the changes of the indicators, the WRCC of Hubei province has been improving in recent years; in 2016, 11 indicators performed best during the study period. The ecological environment quality of Hubei province has dramatically improved based on the large-scale environmental investment and feasible ecological environmental governance measures, which also made a large contribution to the WRCC. Finally, the analysis of the membership values indicates that the WRCC of Hubei province changed greatly in 2011 and 2013, which was mainly because of the interannual variability of the rainfall. This is consistent with the drought years and water conservancy construction must be vigorously pursued.

The carrying capacity evaluation results of the subsystem factor are also shown in Figure 3. The economic system carrying capacity shows the best performance with a steady rise, all kinds of water quota have been greatly reduced due to the improvement of science and technology. In the future, we should strengthen the adjustment of industrial structure and give priority to the development of the tertiary industry with low water consumption. The water supply and demand system fluctuated greatly, the trend of this system was similar to the change trend of the WRCC, but the overall system score is much lower than the destination level score mainly due to the water supplies modulus. The increase of water resources consumption requires a set of multi-level water resources utilization systems. Besides, due to the uneven distribution of water resources in Hubei province, obvious interannual and seasonal changes occur, which also have an impact on the water supply and demand system. We should strengthen the construction of water storage facilities, so as to solve the water shortage in the dry period through storing water resources in the flood season. The health score of the ecological environment system was not high, but it is improving all the time. The main task in the future is to increase the urban area green coverage rate, since many green areas have been changed to building land in recent years. The social system shows the worst performance and it is not improving, mainly because of the population growth rate. Therefore, the scale of population growth should be rationally controlled so as to effectively alleviate the social pressure on water resources. In addition to the recommendations for each subsystem, the following three solutions should be implemented: first, protection of the ecological environment should be strengthened to increase the self-production capacity of basic water resources; second, improve the residents' awareness of water conservation on this basis and increase the management of daily water use; third, the utilization of unit water resources and recycling water must be improved.

### M-K test results

M-K test results of WRCC and its subsystems are as shown in Figure 4. The change trend of the WRCC system was divided into two stages, a rising trend during 2006–2011 and a declining trend during 2011–2016. In 2016, the downward trend approached zero, and the situation has improved. There were three points of abrupt change, which were all around 2009 and 2010, this suggested a mutation over the two years. The fluctuation of B1 water supply and demand system was obvious, which experienced a ‘rise–decline’ state, but did not reach a significant level. The point of abrupt change was in 2011. The B2 ecological environment system was always increasing during the study period. Although there were many mutation points, there was no change in the growth trend. B3 economic system was increasing during 2006–2011 and decreasing during 2011–2016. The abrupt change point occurred in 2013; also from this year it began to show a significant upward trend. Similar to the B2 system, the B4 social system also experienced two processes, from a rise (2006–2008) to a decline (2008–2016). The difference is that it reached the significant level from 2013.

### Results of correlation analysis

The correlation analyses between the WRCC and subsystems were carried out using the data generated above under ‘Correlation analysis’, and the cumulative distributions of correlation coefficients are shown in Figure 5. On the basis of analyses, there is no obvious correlation between the four subsystems. The cumulative distributions of correlation coefficient which is greater than 0.5 are all less than 20%, which shows that each of the subsystems has its own relative independence. The correlation between subsystems and WRCC can be also seen from Figure 5. The B1 water supply and demand system and B3 ecological environment system have a certain correlation with WRCC. For system B1, the cumulative distributions of correlation coefficient which is greater than 0.5 is 48%, and for system B3 the ratio is 56%. The B1 system has repeatedly demonstrated its consistency with WRCC in previous analysis and the B3 system can be regarded as another key constraint of WRCC. Therefore, water resources health and water ecological health are the two main aspects of future works as the driving factors of WRCC. It is better to strengthen investment in these two subsystems in the future and transform the advantages of the economic system into these two aspects.

## DISCUSSION

The results of the catastrophe progression method show that the WRCC of Hubei province was at the ‘weak’ level and had been increasing in recent years. The results of the sequential M-K test also illustrate this point. It can be seen from the previous results that the WRCC decreased significantly from 2011 to 2013. According to historical data, the drought in Hubei province was severe at the beginning of 2011, the rainfall was 40–60% less than that in previous years, the lowest in the same period since 1961. The span of two years drought between 2012 and 2013 resulted in a large reduction in agricultural production. It can be seen that the evaluation results of this model not only take into account various factors such as natural and economy, but also change with the mutation of some factors. This model is both comprehensive and sensitive. We found that WRCC is most closely related to the water supply and demand system and ecological environment system, so the urgent task of water resources construction is to improve the unit water use efficiency and maintain the health of the ecological environment at present. The main innovative points of the research are concluded as follows: first, in this study, the WRCC was evaluated by using the improved catastrophe progression method for the first time. Second, a comprehensive evaluation model combined with the catastrophe progression method, M-K test and correlation analysis was established, which provides a feasible method for carrying capacity evaluation.

As the main part of this model, the catastrophe progression method has proved to be reasonable and accurate to evaluate the WRCC. Compared to principal component analysis and the fuzzy comprehensive evaluation method, the advantages of the catastrophe progression method to evaluate the WRCC are obvious because there is no need to calculate the weight and the subjectivity of human rights can be avoided as much as possible. Compared to the neural network method and system dynamics model, the catastrophe progression method can combine qualitative analysis with quantitative analysis. The level is very clear and the evaluation scores of various levels are unambiguous. Of course, it is important to recognize that the network method and system dynamics model have greater advantages in dynamic adjustment, but this advantage is more apparent in the prediction. The prediction ability of the catastrophe progression method can be further developed in the future. In addition, the catastrophe progression method is easier to understand and use. Of course, the catastrophe progression method also has limitations. First, each catastrophe progression method is for a particular evaluation system. When the system index changes, the model structure must be adjusted accordingly. Second, it is impossible to completely avoid subjectivity in the process of index selection and sorting, which is a problem that requires further study.

## CONCLUSIONS

The WRCC in Hubei province from 2005 to 2016 was evaluated using the catastrophe progression method based on the M-K test and correlation analysis. The evaluation results are consistent with the reality and can give the decision-makers' guidance and suggestions regarding the water resources utilization policies and the development of social, economic, ecological and comprehensive systems in the future. This method provides a feasible method for carrying capacity evaluation and we believe that the catastrophe progression method will be applied to a wider range of areas in the future.

## ACKNOWLEDGEMENTS

This work was supported by the National Key Research and Development Program of China (No. 2017YFC0506603, 2016YFC0401305), the Key Program of National Natural Science Foundation of China (No. 41530635), and the General Program of National Natural Science Foundation of China (No. 51379013, 51679007).