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《Journal of Northeast Normal University (Natural Science Edition)》 2004-04
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The spatial complexity of the law of allometric growth and urban population density

LIU Ji-sheng~1, CHEN Yan-guang~2(1.China Northeast Academy, Northeast Normal University, Changchun 130024,China;2.Studay's Center of Geography, Peking University, Beijing 100871,China)  
The relationships of allometric growth between urban area and population should be reduced to urban density models of land-use and population distribution theoretically, and in turn it should be derived from urban density functions. The precondition of the transformation mentioned above is that both urban population and urban land use density follow the same scaling law: either negative exponential or inverse power function. However, the urban population density in real world complies with the Clark's law, namely the negative exponential function, while the urban land use density conforms to the inverse power function. The sticking point lies in that urban population is defined in 3-dimension space with land use in 2-dimension space. Spatial complexity appears while an object in 3-d is projected to 2-d. If population of cities are taken into account in only 2-d space, the space dimension is harmonized and maybe the logic contradiction can be eliminated. In this case, the issue of the dimension consistency must be considered,which can be formulated as a relation b=D/d. Where b is the scaling factor of the model allometric growth, D is the dimension of urban land use, and d the dimension of urban population. When the allometric model is reduced to the power function named Smeed's model, the exponent α must abide by the relation such as α=2-d, where d is dimension of city population. Now that d value comes between 1 and 2, α is supposed to be smaller than 1, namely α1. This is a very important criterion used to judge if urban population of a city conforms to Smeed's model: no matter when the exponent appears greater than 1, it implies that the city fails to comply with the power law. On the other hand, when the observed data follow power-law distribution with the exponent smaller than 1, the urban population can be regarded as self-similar and the fractal geometry can be employed to analyze the city system.
【Fund】: 國家自然科學基金資助項目(40371039)
【CateGory Index】: K901
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