人口增长的常用数学模型及其预测方法——兼谈对Keyfitz双曲增长等模型的修正与发展  

Three models for predicting population growth——Theoretical foundation,application methods,and revised expressions

作　　者：陈彦光[1] 余斌[2] 

机构地区：[1]北京大学环境学院城市与区域规划系,北京100871 [2]华中师范大学城市与环境科学学院

出　　处：《华中师范大学学报（自然科学版）》2006年第3期452-456,共5页Journal of Central China Normal University：Natural Sciences

基　　金：国家自然科学基金资助项目(40371039)

摘　　要：预测人口增长的数学模型通常采用3种函数,即指数函数、Logistic函数和双曲函数.3种模型的数学根源都在于二阶Bernoulli式微分方程,它们是不同的生长条件下的Logistic方程的特解.论述了上述3种模型的历史根源、构造思路、数理关系和实用范围,并借助实例对各种模型的应用方法进行了探讨;通过环境参数的适当修正导出了人口预测的反S曲线式人口预测模型和二次双曲线模型,该模型可以作为Keyfitz双曲增长函数的精确表达或者补充形式.Three common mathematic models are always employed to predict the growth of regional or urban population, including exponential function （Malthus model）, inverse hyperbolic function （Keyfitz model）, and logistic function （Verhulst model）. All the three models originate from logistic equation, but under different conditions. The logic relations and distinctions between the three models are expounded and a new model used for population predict is derived from the same mathematic source in the form： 1/P （t）=1/P^1-blnt, which in fact defines an ＂inverse S-shaped curve＇. The Keyfitz model turns out to be the ＂approximator＂ of the inverse S-function advanced by the author. In addition, a new model based on the hyperbola with power is given as 1/P（t） = 1/P^1 - b^a, where α is a parameter. Three sets of data and the corresponding analytical results are taken as examples to illustrate the models and while showing how to use the prediction methods correctly.

关 键 词：人口增长 指数模型 Logistic过程 反双曲关系 反S曲线 

分 类 号：C921.2[社会学—人口学]