基于分形维曲线Logistic模型的城市增长阶段划分
Stage Division of Urban Growth Based on Logistic Model of Fractal Dimension Curves
摘要：
城市形态分维值的时间序列一般呈S型曲线。这些曲线的基本模型是逻辑斯谛函数。从分形维数曲线的逻辑斯蒂模型出发，导出了城市发展的增长率公式和加速度公式。利用分形参数曲线的特征，我们可以识别城市演化的不同阶段。主要结果如下。(1)根据城市形态的分维曲线，城市增长可分为四个阶段：初始缓慢增长、加速快速增长、减速快速增长和末期缓慢增长。三个分界点分别为0.2113Dmax、0.5Dmax和0.7887Dmax，其中Dmax为分形维数的容量。当分形维数达到其容量值的一半，0.5Dmax时，城市增长率达到峰值。(2)根据分形维数比值曲线，城市增长也可以分为四个阶段：初始缓慢填充、加速快速填充、减速快速填充、终止缓慢填充。三个分界点分别为0.2113Zmax、0.5Zmax和0.7887Zmax，其中Zmax=/（2-Dmax）表示分维几率（Dmax<2）的容量。实证分析表明，基于分形维数的方案一适用于年轻城市，基于分形维数比值的方案二适用于成熟城市。结果表明，Logistic函数是基于城市分形参数划分城市增长阶段的重要模型之一。研究结果为认识城市相变的特征和机制提供了新的思路。
英文摘要：
The time series of fractal dimension values of urban form always take on sigmoid curves. The basic model of these curves is logistic function. From the logistic model of fractal dimension curves, we can derive the growth rate formula and acceleration formula of city development. Using the inflexions of the fractal parameter curves, we can identify the different phases of urban evolution. The main results are as follows. (1) Based on the curve of fractal dimension of urban form, urban growth can be divided into four stages: initial slow growth, accelerated fast growth, decelerated fast growth, and terminal slow growth. The three dividing points are 0.2113Dmax, 0.5Dmax, and 0.7887Dmax, where Dmax is the capacity of fractal dimension. When the fractal dimension reaches half of its capacity value, 0.5Dmax, the urban growth rate reaches its peak. (2) Based on the curve of fractal dimension odds, urban growth can also be divided into four stages: initial slow filling, accelerated fast filling, decelerated fast filling, terminal slow filling. The three dividing points are 0.2113Zmax, 0.5Zmax, and 0.7887Zmax, where Zmax=/(2-Dmax) denotes the capacity of fractal dimension odds (Dmax<2). Empirical analyses show that the first scheme based fractal dimension is suitable for the young cities and the second scheme based on fractal dimension odds can be applied to mature cities. A conclusion can be reached that logistic function is one of significant model for the stage division of urban growth based on fractal parameters of cities. The results of this study provide a new way of understanding the features and mechanism of urban phase transition.
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