シミュレーションによる都市人口の解析

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We have many models for explanation of existence of the rank-size rule of city. Among them, we have some models in which statistical principles are applied. They are proposed by Berry, Curry, and Fano. Berry pointed out that the distribution of city size (population of city) is truncated lognormal, and when the distribution of city size is such a distribution, the rank-size rule will be applied to the population of cities. Curry applied the information theory to explanation of existence of the rank-size rule. According to the Curry's model, when the entropy of the state of distribution of population in cities become maximum, the relation between city size and the rank will be expressed by rank- size rule. Fano, giving his criticism to the Curry's theory, proposed his theory constructed by transition matrix which expressed the probability of transition of city size from a grade of city size to another. In this paper, the new type of model is proposed based on the law proportionate effect (laloi de l'effet proportional) insisted by R. Gibrat. In this model, the population of a city or a city size at the end of the nth period P_n is expressed by P_n=D_0П^^n__<t=1>(1+d_t)A_0П^^n__<t=1>(1+a_t) (1) where D_t and A_t (t=1,2,……, n) are the population density and the area of a city at the beginning of the t+1st period, respectively, and d_t and a_t (t=1,2,……, n) are the increase ratio of the population density and the area of a city during the tth period. Equation (1) is rewritten as P_n=D_mП^^n__<t=m+1>(1+d_t)A_m П^^n__<t=m+1> (1+a_t) (2) If population density of each city does not change so widely from the m+1st period to the nth period and a_t is affected by D_m (exactly saying, the latter supposition should be expressed by the words that "a_t is affected by D_m, D_<m+1>,……, D_n" but it may be said that a_t is affected by D_m, by the former supposition: "population density of each city does not change so widely,"), the distribution of the product A_m П^^n__<t=m+1>(1+a_t) will be influenced by D_m or D_mП^^n__<t=m+1>(1 d_t) which is equal to D_n. Then, we can write the following expression: P_n=D_<ns>A_<ns> (s=1,2,…… , k) (3) where D_<ns> is D_n which belongs to the sth class of D_n and A_<ns> is a A_n of a city which has a population density D_n which belongs to the sth class of D_n (D_<ns>). The reason why a_t is influenced by D_n is that at should be a function of mobility of persons M of a city and D_n is the index of the mobility. In equation (3), we have two elements, D_<ns> and A_<ns>. The distribution of D_n which consists of D_<ns> (s=1,2,……, k) should be approximately lognormal or unimodal by the central limit theorem, and that of A_<ns> should be also approximately lognormal or unimodal. Using these property, we can try to perform a simulation of generation of city population. A city population is generated by the following steps: 1st step: Determination of population density of a city, D_n^1. We choose one D_n randomly using the distribution of D_n. And the D_n chosen is denoted by D_n^1. 2nd step: Classificat on of the D_n^1. We find the class to which the D_n^1. belongs, and the class is denoted by s_1. And the D_n^1 is denoted by D_<ns1> when we want to express the fact that the D_n^1 belongs to the class s_1. 3rd step: Determination of area of a city, A_<ns1>^1. We choose one A_<ns> randomly using the distribution of A_<ns1> which is the area of cities which have population densities belonging to the class s_1. And the A_<ns> is denoted by A_<ns1>^1. 4th step: Generation of population of a city, P_<ns1>^1. We generate a population of a city P_<ns1>^1 by the equation: P_<ns1>^1=D_n^1A_<ns1>^1. And we regard the P_<ns1>^1 as a population generated by the 1st trial, P_n^1. 5 th step: Generation of populations P_n^t. We generate W city populations P_n^t (t=1,2,……, W) by the process which consists of 4 steps from the 1st step to the 4th step, where P_n^t is a population generated by the tth trial. Using the W city populations, the relation between the city size (city population) and the rank was observed. According to the relation observed, we could conclude that the rank size rule was found in the city population generated by the simulation. On the other hand, the relation between the city population whose distribution is lognormal and the rank was not described by the rank size rule. This fact is very important because this fact shows that in the city populations which generate by the law of proportionate effect, the rank-size rule is not found.
流通経済大学の論文

著者

鈴木啓祐
流通経済大学

シミュレーションによる都市人口の解析

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