Stage division of urban growth based on logistic model of fractal dimension curves

Abstract

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 speed formula and acceleration formula of urban growth. Using the inflexions of the fractal parameter curves, we can identify the different phases of city development. 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 speed 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 drawn 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.