中文摘要：

城市化水平增长曲线可以分为S形和J形2大类别，其变化特征、动力学背景及研究方法也不尽相同。城市化曲线研究可以分为唯象分析和唯理分析2大途径，唯理研究包括城一乡人口异速生长、城一乡人口替代以及城乡人口迁移和转换3种路径，目前这些方法主要用于S形曲线研究，对J形曲线的探索不够深入。S形曲线是一种奇对称曲线，其模型导数却是一条偶对称曲线，根据这些特征可以将城市化的S曲线分为3或4个阶段。发展中城市化过程通常不能有效拟合S形曲线，却表现出J形曲线的特征。但是，J形曲线及其函数的导数曲线均不对称，其动力学根源和阶段划分方法暂时尚不明朗。中国未来的城市化曲线研究似乎应该将重点放在J形曲线方面，揭示J形曲线更有助于人们从理论上理解中国的城市化动力学。

英文摘要：

The curves on the change of level of urbanization over time are called urbanization curves, which are in fact models of the progress of urbanization based on empirical evidences from developed countries and de- veloping countries. There are two types of urbanization curves. One is S-shaped curve, and the other, J-shaped curve. The S-curve can be modeled with the logistic function, while the J-curve cannot be described by the lo- gistic function. The former is applicable to the developed countries, whereas the latter is applicable to the de- veloping countries. So far, there have been more studies on the S-shaped curve. However, fewer studies have been devoted to the J-shaped curve of urbanization. Consequently, we know little about the general principle on the J-shaped curve and its underlying rationale, which remain to be further explored in the future. In this pa- per, various studies on urbanization curves are systemized to form a framework of models on the level of ur- banization. The logistic curve was once divided into three stages by Northam （1979）： initial stage, acceleration stage, and terminal stage. However, the Northam＇s proposal is just a phenomenological theory with several de- fects： first, the definition of ＇acceleration stage＇ is not correct; second, there are no theoretical basis and critical scale for the divisions. It can be proved that the logistic process falls into four phases. Taking half of urbaniza- tion level capacity as a tipping point, the logistic curve is divided into two stages： acceleration stage （urban ma- jority） and deceleration stage （urban minority）. The tipping point can be determined by two methods of elemen- tary mathematics. Then the two stages are respectively divided into＇two sub-stages by means of the exponen- tial law of urban-rural ratio with a parameter as characteristic length. Now, urbanization process can be divided into four stages： initial stage, acceleration stage, deceleration stage, and terminal stage. Also it can be divided into