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The fractal dimension growth of urban form can be described with sigmoidfunctions such as logistic function due to squashing effect. The sigmoid curvesof fractal dimension suggest a type of spatial replacement dynamics of urbanevolution. How to understand the underlying rationale of the fractal dimensioncurves is a pending problem. This study is based on two previous findings.First, normalized fractal dimension proved to equal normalized spatial entropy;second, a sigmoid function proceeds from an urban-rural interaction model.Defining urban space-filling measurement by spatial entropy, and defining ruralspace-filling measurement by information gain, we can construct a newurban-rural interaction and coupling model. From this model, we can derive thelogistic equation of fractal dimension growth strictly. This indicates thaturban growth results from the unity of opposites between spatial entropyincrease and information increase. In a city, an increase in spatial entropy isaccompanied by a decrease in urban land availability. An inference is thaturban growth struggles between the force of urban space-filling and the forceof urban space-saving. This work presents a set of urban models of spatialdynamics, which help us understand urban evolution from the angle of view ofentropy and fractals.

Spatial Dynamics of Urban Growth Based on Entropy and Fractal Dimension