基于不同的城市定义标准,对山东省城市体系的城市人口和城区面积进行分析,发现只有地级市明确地服从异速生长定律,而县级市的关系则不够明朗。研究表明,山东省城市人口-城区面积异速生长的标度因子均小于1,而且定义的城区越小则标度因子越大。由此可见,山东省城市体系的城区扩展速度整体上比较合理,不过在中小城市郊区地带的土地利用比大城市郊区地带要浪费一些。山东城市人口-城区面积异速生长关系的状态不是很好,这暗示城市体系的发展存在许多不协调的因素,借助数学模型变换揭示了其中结构退化的实质所在。文章指出,只有通过自组织方式在城市之间形成一种自下而上式竞争-合作关系,才能进一步完善城市体系的时空结构。

In this paper, the relationship between urban population size and land area of urbanized area of Shandong Province, PRC, in 2000 by is analyzed means of the fifth Census data.Without an accepted definition of urbanized area in China, we define the urbanized area by means of variable thresholds and two lower limit values are adopted: one is about 500 person/km², and the other is around 600 persons/km², taking into consideration the scale invariance of urban form.The system of cities is divided into three cases: first, larger cities without the towns of counties;second, the towns based only on counties;third, whole cities including both larger and smaller ones.Four simple functions in common use are tested:① linear, A=a+bP;② exponential, A=aexp (bP);③ logarithmic, A= a+blnP, and ④ power, A=aPb.Where A represents the urbanized area, P denotes the population in the corresponding urbanized area, both a and b are parameters regarded as constants under given conditions.Among the four functions, the power, which is most widely used in the literature on fractal and self-organized criticality, fits best for both the data based on different thresholds without considering the counties.However, if the smaller cities based on counties are taken into account, the results make a difference and then the logarithmic one fits better than the power.This implies that only the larger cities in Shandong Province conform to the law of allometric growth.The cause that the smaller ones fail to obey the law can be explained demographically: the data are not based on cities strictly but on districts for administrative purposes.The scaling factor of allometric model,b, is computed such as b =0.711 and b=0.794 for different thresholds, negative allometry results implying that A increases at a slower rate than P.In addition, the data based on non-agricultural population and built-up area are also fitted into the power function, which gives the allometric coefficient as b =0.854 <1, a negative allometry result still.Although the statistical results of the allometric equations in some cases are not satisfactory, we can still draw a clear conclusion that the Shandong urban system is rational state where the urban area-population relationships are concerned.The suggestion is made that self-organizing theories can be employed to optimize the urban system in the studied area.Sometimes the allometric relationships retrogress from the power to the logarithmic one, just as it does in Shandong systems of cities.This phenomenon is interpreted theoretically in the light of mathematical transformation.As a matter of fact, if we suppose that the growth rate of urbanized area is restricted without reason, the allometric equation will inevitably change into the reverse-exponential relationships.