地理科学 ›› 2001, Vol. 21 ›› Issue (5): 412-416.doi: 10.13249/j.cnki.sgs.2001.05.412

• 论文 • 上一篇    下一篇

城市系统的异速生长关系与位序-规模法则——对Steindl模型的修正与发展

陈彦光1, 刘继生2   

  1. 1. 北京大学城市与环境学系, 北京 100871;
    2. 东北师范大学地理系, 吉林 长春 130024
  • 收稿日期:2000-05-28 修回日期:2001-05-15 出版日期:2001-09-20 发布日期:2001-09-20
  • 基金资助:
    国家自然科学基金资助项目(40071035)的部分内容。

Reconstructing Steindl's Model:from the Law of Allometric Growth to the Rank-Size Rule of Urban Systems

CHEN Yan-guang1, LIU Ji-sheng2   

  1. 1. Department of Urban and Environmental Sciences, Peking University, Peking 100871;
    2. Department of Geography, Northeast Normal University, Changchun, Jilin 130024
  • Received:2000-05-28 Revised:2001-05-15 Online:2001-09-20 Published:2001-09-20

摘要: 基于Steindl模型的建模思想,设置一个时间-年龄变量T,将城市规模(P)-数目(f)异速生长的假设条件重新构造为dP(T)/dT=gP(T),df(T)/dT=-rf(T),据此导出反映城市等级-规模关系的Beckmann-Davis模型:P(m)=P1λm-1,f(m)=f1δ1-m,进而导出三参数Zipf定律:P(r)=C(r-α)-dz,式中g=lnλ,r=lnδ,C=P1〔δ/(δ-1)〕dz,α=1/(1-δ),dz=g/r=lnλ/lnδ。根据几何测度关系建立分维方程:dz=Dp/Df,从而揭示:城市规模分布的分维D=1/dz在本质上乃是城市体系空间结构的分维Df与各城市人口空间分布的平均维数Dp之比。

Abstract: By modifying the assumptions of Steindl's model as allometric relationships: dP(T)/dT=gP(T),df(T)/dT=-rf(T),where P(T) is the average size of cities of age T,f(T) is the number of cities of age T,parameters g and r represent respectively coefficients of growth of P(T) and f(T),we deduce a set of generalized Beckmann-Davis model of city hierarchies and city-size distribution,namely, δn-law advanced by the authors,as follows: Pm=P1λ1-m,fm=f1δm-1,where λ=eg,δ=er,P1 is the size of the largest city(cities), f1 is the number of the largest city(cities),and generally, f1 =1, m is the ordinal of city class (m=1,2,…,N).From the generalized Beckmann-Davis model,a three-parameter Zipf model can be derived as P(r)=C(r-α) -dz ,where r is the rank of a city, P(r) is the size of the rth city,as for parameters, C=P1δ/(δ-1)〕dz, α=1/(1-δ),dz= ln λ/ln δ=g/r.Based on general geometrical measure relationship, Pm1/Dp∝fm-1/Df,an equation of fractal dimension is constructed as dz=g/r=Dp/Df,where Dp is the generalized dimension of Pm,and Df,the dimension of fm.In reality, Dp→D f,g→r,so dz→1,and when dz=1,we have what is called 2n-law presented by K.Davis(1978).

中图分类号: 

  • F290/TU984