北京大学学报(自然科学版)

城市地理系统结构与功能的分形模型
——关于地理系统异速生长方程与Cobb-Douglas函数的理论探讨与实证分析

陈彦光   

  1. 北京大学环境学院资源与环境地理系,北京,100871
  • 收稿日期:2002-07-01 出版日期:2003-03-20 发布日期:2003-03-20

Derivations and Empirical Analysis of the Allometric Equation and C-D-type Function on Geographical Systems of Cities

CHEN Yanguang   

  1. Department of Geography, Peking University, Beijing, 100871
  • Received:2002-07-01 Online:2003-03-20 Published:2003-03-20

摘要: 从一般城市动力系统出发,推导出城市和城市体系的异速生长方程和Cobb-Douglas函数(即C-D函数),建立了二者之间的数理关系并揭示了其隐含的分形性质;进而证明城市C-D函数中的系数包含有其他各种要素的有关信息,这为借助单要素分析复杂地理动力系统提供了理论依据和实用方法。利用异速生长关系和要素-产出的弹性性质可将C-D函数化为二要素形式,据此发展了城市结构的广义维数方程。基于分形优化思想,运用数学规划原理,从上述模型中导出城市地理系统结构与功能的优化条件表达式。以郑州市为实证对象,对文中有关理论模型进行了实证分析,最后将结论推广到一般地理学领域。

关键词: 城市体系, 异速生长, C-D函数, 分维, 郑州市, 城市结构

Abstract: A Cobb-Douglas-function-type model on cities as systems is derived from general expressions on urban dynamic systems, y=kfi(x1, x2,..., xn), the model can be named C-D function of cities and written as y=μxii^σi,where xi(i=1, 2,..., n) represents some kind of measure on cities, y is output in some sense, k and μ both proportionality coefficients. Throught the allometric relationship between two elements of cities, xixj^αij, the C-D function can be transformed into the two-variable expression, y=η(xi^bi) (xj^bj). A number of equations of fractal dimension are deduced out such as αij=Di/Dj=σj/σi=bj/bi, σiDi=σjDj. It is proved that the equation. σi/(wixi)=σj/(wjxj), should be met in order to optimize urban structure and function. Taking the city of Zhengzhou as example, the models presented and advanced are verified and vindicated.

Key words: urban system, allometric growth, C-D function, fractals, Zhengzhou city, urban structure

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