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。
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).
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