(1. Institute of Applied Mathematics, Shandong Univ of Tech, Zibo, Shandong 255049, China; 2. Zibo Water Resources Management Office, Zibo, Shandong, 255032, China; 3. Institute of Environment Sci and Tech, Shandong Univ of Tech, Zibo, Shandong 255049, China; 4. School of Material Sci and Eng, Shandong Univ of Tech, Zibo, Shandong 255049, China; 5. Inspecting Station of Geology and Environment in Zibo, Shandong, 255091, China)

Abstract: Under constraints of limited water resources, and for the goal of sustainable development of society and economy, an optimal model for relationship of water resources with development of population and industry is formulated using domain analysis methods. In practice, we take social development plan of Zibo as a case study. The computation results show that this simple model can reflect a balance relation of the primary indicators in the society development, and can predict some global characteristics of water resources in Zibo with the society economy development in the future years.

Key words: Water problem and water resources, sustainable development, increasing rates of population and industry, optimal control

CLS Classification:N94

        Sustainable development has been widely studied from the end of 20^th century. A lot of system models and assessment indicators have been established and researched in UN and many countries, even for smaller regions (see [1-4]). In actual department fields, such as agricultural sustainable development, industrial sustainable development, financial sustainable development etc., a series of special indicators and evaluating models came forth one after another. Although there have some different explanations to sustainable development, its essentiality is a coordination development between the nature and the society, which compose a moving, opening and complex system. Then, by system analysis method, predictable models may be established on more scientific meanings. These models can be used to supply scientific proofs and suggestions for the social development.

        On the other hand, water resource has become a vital problem of resources and environment all over the world in the 21st century. Since the un-substitutable characteristics of water resource, it plays an important role in the development of society and economy. The sustainable development of a country to some extent relies on continuous utilization and development of water resources. Water problem becomes more serious in China (see [5]). It is a vital problem that how to realize a sustainable development on the base of lower level of water assumption, especially in China in the 21st century.

        So, it has important realistic meanings to research harmonious relations between water resources and social economic sustainable development under the constraint of limited water resources. This paper is concerned with a controllable multi-objective system relating limited water resources with the increase of population and industrial economy with aids of system analysis and optimal method. Taking the social economic system of Zibo city as example, an optimal predicting model is established which reflecting an increasing relationship of population with industrial economy according to statistical data in 1997 and developing plan of the society and economy. By the model and computation with MATLAB software, optimal increasing rates of sustainable development for population and industrial economy of Zibo city in 2005 and 2010 are obtained.

2 Construction of Model

        Without loss of generality, we take the north industry city--Zibo as example to construct the model. By the theory of sustainable development, population, society, economy and environment should be consistent and reach a well-cycled developing mode. Suppose that water consumption for the increasing of population and economy mainly lies on three indicators: population increasing, economy development and agriculture development. We are to determine an optimal increasing model of the population and economy of Zibo city based on the water resources and available water resources in this region. For such purpose, some basic assumptions and known data are given in the follows (see [6]).

        (1) According to the geology and hydrology conditions, we divide Zibo into three sub-districts: middle district (MD in short), south district (SD in short) and north district (ND in short); and the average water resources and available resources in each district are all known.

        (2) Taking 1997 as a datum year, in this year there are 401.03 (unit: ten thousands) people in Zibo, and there are 173.86 ten thousands urban people; and the middle district (MD), south district (SD) and north district (ND) have 157.54, 11.10 and 5.22 ten thousands people respectively. There are 227.17 ten thousands county people in Zibo, and there are 153.19 , 43.89 and 30.09 ten thousands people in the districts MD, SD, and ND respectively. In addition, according to the situation of water demand for each people and the increasing trend of population of Zibo, a ration water supply plan for each people in the three districts is given below. Where D denotes district, Y denotes year, and the unit l/p.d means one litre for each people every day.

        (3) In 1997, the total industrial production value of Zibo is 849.87 billion Yuan, where urban industrial production value is 359.23 billion Yuan, county industrial production value is 490.64 billion Yuan. The production values of the districts MD, SD and ND are listed below respectively.

        According to the actually statistical data, water quantity consumed by Ten thousands Yuan production value (TTYPV in short) is decreasing markedly year after year. In the following table, water quantity consumed by every TTYPV for urban industry and county industry for each sub-district in 2005 and 2010 are predicted. Where the deterministic ration of water quantity consumed by urban industries are predicted by two-order index smooth method according to the developing trend of urban industries, and the water ration for county industries are determined by the actually developing situations of the county industries.

          Tab 2.3 Water ration consumed by TTYPV in 2005 and 2010(unit: m³⁾

        (4) In 1997, there are three millions and thirty thousands farm areas in Zibo. The total irrigating farm is 242.07 (ten thousands areas), where the effective irrigation is 223.44 (ten thousands areas), the water ration for irrigating is 395 m³ per area. According to the economic plan for farm irrigation of Zibo, the irrigating ration is to be decreased gradually. In 2005 and 2010, the farm irrigating consumptions under different precipitation probabilities are given as below, where the unit TT means ten thousands.

          Tab 2.6 Available water supplies in 2005 and 2010(TT m³⁾

        Combining with the available water resources in 2005 and 2010 (see Tab 2.6), and noting to the water assumptions by farm irrigating and stockbreeding listed in Tabs 2.4-2.5, we can work out an optimal increasing rate of IPV and a suitable population increase indicator with aids of MATLAB tool. Take the year of 2010 as example, if precipitation probability is 50%, then the increases of industry and population should satisfy the following inequality:

,

where denote the increasing rates of urban industry and county industry respectively;  represent the increasing rates of urban population and county population respectively; and =10137, =12332, =14476, =9410; =186792,=76020, =746.644.

        On the mathematical meanings, we know by the model (1) that the increasing rate of economy reaches its maximum if population increase becomes zero. However, the increase rate of population should be greater than zero in the present situation. According to statistical data of population of Zibo, the intrinsic rate of population belongs to 4‰-6‰, and with development of the society, this population indicator should be taken on decreasing trend. For convenience to computation, set  and . The predicting results for and  in 2005 and 2010 under different precipitation probabilities are given below.

        Tab 3.2 Prediction of increasing rates

        In order to see more clearly the influence of population indicators to the increase of industrial economy, the changeable trends of increasing rate of industry with the increasing population in 2010 are plotted in cases of CP=50% and CP=75%.

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