LOGISTIC PROCESS OF URBANIZATION FALLS INTO FOUR SUCCESSIVE PHASES: REVISING NORTHAM'S CURVE WITH NEW SPATIAL INTERPRETATION
通过数学变换和指数定律，将城市化的逻辑斯蒂过程从诺瑟姆的三阶段修正为四阶段，并利用分形维数解释各阶段的空间特征，对研究城市化进程的学者有参考价值。
A new pattern of Logistic process of urbanization is illustrated using mathematical transform and exponential laws in this paper.Urbanization course can be described by Logistic curve,which was divided into three stages by Northam(1979): initial stage,acceleration stage,and terminal stage.However,the Northam's is just a phenomenological theory with several defects,e.g.no theoretical basis,no critical scale,etc.The authors demonstrate that the Logistic process falls into four phases.Taking the 50% of urbanization ratio as a tipping point named half,the Logistic curve is divided into two stages: acceleration stage(urban majority) and deceleration stage(urban minority).The tipping point is determined by differential coefficient of Logistic equation through mathematical transform.Then the two stages are respectively divided into two sub-stages by means of the exponential laws with a parameter as characteristic length.The first exponential law coming from Karmeshu's urban-rural population replacement model,which is in fact an equivalent of the Logistic function of urbanization level and can be expressed with an inverse function i.e.X(t)=Cexp(kt),where X(t) represents urban-rural population ratio,C and k are parameters 1/k is just the characteristic length of urbanization time t,which can be turn into an internal scale reflecting the local properties of urbanization course.In order to find an external scale reflecting the whole property of world urbanization in the sense of average,Zhou's model on the relationships between urbanization and economic development level is used as another exponential law,i.e.y=y_0exp(Z/Z_0)=85.6357exp(Z/19.0404),where y denotes the economic development level of a region,and the parameter Z_0=19.0404 is just the characteristic length of world urbanization that can act as an external scale.Taking Z_0/100=19.0404% as one dividing point,the acceleration stage is compartmentalized into two sub-stages: initial phase and acceleration phase;while taking 1-Z_0/100=80.9596% as another dividing point according to symmetry of Logistic curve,the deceleration stages is also curved up into two sub-stages: deceleration phase and terminal stage.Both the acceleration phase and deceleration one can combine to make a celerity phase corresponding to Northam's acceleration stage.The Logistic equation of urbanization is derived from the law of allometric growth of urban-rural population,which is associated with fractal relationships,and fractal dimension is employed to interpret the Logistic course of urbanization falling into four successive phases.Fractal dimension of urban population is a spatial parameter proportional to urbanization level.The higher the urban fractal dimension is,the shorter the internal scale of urbanization.However,the external scale seems to be out of touch with the dimension value of urban population.The idea from fractals provides us with a new insight into the Logistic process of urbanization,but many theoretical problems remain to be solved in the future.