城市密度分布与异速生长定律的空间复杂性探讨
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摘要
从空间复杂性及空间尺度和空间维数方面研究了异速生长定律与Clark定律存在的模型相容性和模型参数一致性问题.结果表明,城市人口密度分布与城市人口-城区面积异速生长幂次定律的矛盾,本质上是一种空间复杂性问题.三维的空间投影到二维必然导致复杂的结果,不同维数的空间逻辑链接也会表现出复杂行为,解决这个问题的办法是将研究对象的维度保持一致.
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 α<1. 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.
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 α<1. 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.
引文
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[2] ClarkC.Urbanpopulationdensities[J].JournalofRoyalStatisticalSociety,1951,114:490～496.
[3] BattyM.Citiesasfractals:simulatinggrowthandform[A].CrillyAJ,EarnshawRA,JonesH.FractalsandChaos[C],NewYork:Springer-Verlag,1991.43～69.
[4] FrankhauserP.Lafractalit啨desstructuresurbaines[M].Paris:Economica,1994.16～55.
[5] LeeY.AnallometricanalysisoftheUSurbansystem:1960—1980[J].Environment＆PlanningA,1989,21(4):463～476.
[6] LongleyP,MesevV.Beyondanaloguemodels:spacefillinganddensitymeasurementsofanurbansettlement[J].PapersinRegionalSciences,1997,76:409～427.
[7] SmeedRJ.Roaddevelopmentinurbanarea[J].JournaloftheInstitutionofHighwayEngineers,1963,10:5～30.
[8] GoldenfeldN,KadanoffLP.Simplelessonsfromcomplexity[J].Science,1999,284(2):87～89.
[9] BarabasiA-L,BonabeauE.Scale-freenetworks[J].ScientificAmerican,2003(7):50～59.
[10] Allen,PM.Citiesandregionsasself-organizingsystems:modelsofcomplexity[M].Amsterdam:GordonandBreachSciencePub,1997.102～160.
[11] PortugaliJ.Self-organizationandthecity[M].Berlin:Springer-Verlag,2000.96～119.
[12] 罗宏宇,陈彦光.城市土地利用形态的分维刻画方法探讨[J].东北师大学报(自然科学版),2002,34(4):107～113.
[13] 陈彦光,刘继生.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[J].地理科学,2001,21(4):412～416.
[14] 高安秀树 分数维[M].沈步明,常子文译.北京:地震出版社,1988.23～91.
[15] BenguiguiL,CzamanskiD,MarinovM,etal.Whenandwhereisacityfractal?[J] EnvironmentandPlanning:PlanningandDesign,2000,27:507～519.
[16] NordbeckS.Thelawofurbanallometricgrowth[A].AnnArbor.MichiganInter-UniversityCommunityofMathematicalGeographers7[C].Michigan:DepartmentofGeography,UniversityofMichigan,1965.166～171.
[17] NordbeckS.Urbanallometricgrowth[J].GeografiscaAnnaler,1971,53B:54～67.
[18] DuttonG.Criteriaofgrowthinurbansystems[J].Ekistics,1973,36:298～306.
[19] 陈彦光,许秋红.区域城市人口—面积异速生长关系的分形几何模型———对Nordbeck-Dutton城市体系异速生长关系的理论修正与发展[J].信阳师范学院学报(自然科学版),1999,12(2):198～203.
[20] 陈彦光,周一星.基于RS数据的城市系统异速生长分析和城镇化水平预测模型:基本理论与应用方法[J].北京大学学报(自然科学版),2001,37(6):819～826.
[21] LoCP,WelchR.Chineseurbanpopulationestimates[J].AnnalsoftheAssociationofAmericanGeographers,1977,67:246～253.
[22] 冯健.杭州市人口密度空间分布及其演化的模型研究[J].地理研究,2002,21(5):635～646.
[23] 郝柏林.复杂性的刻画与"复杂性科学"[J].科学,1999,51(3):3～8.
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