城市人口密度衰减的分形模型及其异化形式——对Clark模型和Sherratt模型的综合与发展

The Fractal Model of Urban Population Density ——Decay and Its Related Forms:Derivation and Synthesis of Clark Model and Sherratt Model

从城市系统的一般方程出发，导出城市人口密度衰减的分形模型（ρ（ｒ）∝ｒＤ－ｄ），进而提出城市人口空间分布的Ｗｅｉｂｕｌ型公式（Ｐ（ｒ）／Ｐ０＝１－ｅｘｐ［－（ｒ／ｒ０）Ｄ］），基此将传统的城市人口密度衰减模型由指数型（ｅ－ｒ／ｒ０）和Ｇａｕｓｓ型（ｅ－（ｒ／ｒ０）２）推广到一般形式（ｅ－ｒ／ｒ０）δ，并揭示了它与分形模型的内在关系。

The fractal model of urban population density decay,ρ(r)∝r D-d ,was deduced from equations of a general urban dynamic system.On condition that the fractal nature of urban population has degenerated,a Weibull-model-like model of spatial distribution of urban population could be set up as P(r)p 0=1-exprr 0) D],from which a new general model of urban population density,ρ(r)=ρ 0 exp rr 0) D],was derived: when D=1,the new model becomes Clark model,ρ(r)=ρ 0 exp (-rr 0);and when D=2,it becomes Sherratt model,ρ(r)=ρ 0 exp rr 0) 2].The relationships between the fractal model and the other models in the paper were revealed through the Weibull model of urban population.