中文摘要：

目前已知有2种函数可以用作描述城市化水平与经济发展水平关系的数学模型，即对数函数、幂指数函数。基于Logistic函数发展了第三种模型。这3种函数分别刻画单对数关系、双对数关系和分对数关系。3种数学模型各有不同的建模条件和适用范围，反映的动力学特征也不一样。第一种模型背后的动态演化控制变量是人均产值，第二种模型隐含的控制变量是城市化水平与人均产值的比值，第三种模型暗示的控制变量是城市化水平。第一和第二两种模型反映的变化是先快后慢，第三种模型反映的变化则是两头慢中间快。在实际应用中，可以通过一个区域表现的模型结构揭示系统演化的动力学机制，借助模型参数的变化规律判断系统的演化方向。

英文摘要：

Three functions can be employed to model the relations between the level of urbanization and that of economic development. The first is a logarithmic function, the second is the power function, and the third is the logistic function. The logistic model of the relationships between urbanization and economic development levels is equivalent in mathematics to the exponential model of the relationships between urban-rural ratio （URR） and per capita products such as GDP and GNP. The exponential model is a logit model since URR is defined as the ratio of the urban to the rural population. The logarithmic model suggests that economic variables are control variables of urbanization associated with economic development, the logistic model indicates urban variables are control variables of evolution of regional systems, and the power-law model implies that the ratio of urban variables to economic variables control the system development. The basic dynamical equations of the three models are derived, and the results showed physical properties of the three kinds of systems. The similarities and differences between the different kinds of dynamics are revealed by drawing a comparison between the three models. Among these models, the logistic function presented in this work is applied to the 31 administrative areas of China including provinces, autonomous regions and municipalities directly under the Central Government. The examples based on the data from 2000 to 2008 illustrated how to estimate the parameters of the models for the aims of scientific explanation and prediction. In practice, the models can be used to judge whether or not urbanization keeps in step with economic development in a region.