在常微分方程下利用二次逼近法探討人口成長模型問題

Abstract

在人口統計領域中，早期習慣將人口變化視為時間的函數，企圖以Deterministic Function來刻劃，例如：1798年Malthus提出的Malthusian Growth Model ；1825年Gompertz提出的Gompertz Model以及1838年Verhulst主張以Logistic Function描述人口成長。而近年來則是傾向於逐項分析各種因素的隨機性模型，例如：1983年Holford加入世代的APC模型；1992年Lee 和Carter提出的Lee-Carter死亡率模型以及2003年Renshaw與Haberman提出改善Lee-Carter死亡率模型的Reduction Factor模型。\r\n\r\n人口變化主要分成自然增加與社會增加，而自然增加是為出生扣掉死亡，社會增加則為移入扣掉移出。首先，本文先不考慮遷移的部分，各別以出生與死亡人口的變化為研究對象，視其變化為一隨時間變動的動態系統，以常微分方程來刻劃。由台灣地區人口統計資料顯示，出生率或死亡率都有逐年下降的趨勢，而且隨著時間而變化加劇的傾向，使得以往使用的模型不易捕捉變化，因此我們提出「二次逼近法」，從出生、死亡人數對時間的變化率與曲度利用數值分析的方式來估計出生與死亡數，進而從中找出在此動態系統背後隱藏的規則。而後再同時考慮其他各種變項，以偏微分方程來刻劃，最後即可建立台灣地區人口變化模型。

In early population statistics, the population changes were regarded as a function of time so that people tended to\r\ndescribe the variations by deterministic functions. For instance, Malthus proposed the Malthusian Growth Model in 1798; Gompertz presented Gompertz Model in 1825; Verhulst advocated using logistic function to describe an increase in population. In recent years, people tend to use the stochastic forecast method to analyse every factor term by term. For instance, the Age-Period-Cohort (APC) Model which was proposed by Holford in 1983; Lee and Carter proposed the Lee-Carter Mortality Model in 2003; and Renshaw and Haberman proposed the Reduction Factor Model in 2003 that improve the Lee-Carter Mortality Model.\r\n\r\nThe population changes equal to nature and social increase, where the nature increase is the difference between birth and death population, and the social increase is the difference between immigrants and emigrants. First, we focus on natural increase rather than social increase. Moreover, we use ordinary differential equation to decribe the variation as a dynamic system over time. From the data obtained from the Ministry of Interior Taiwan, we know that the fertility and mortality has been decreasing, and the change is getting more violent year by year. Under the consideration that previous models are not able to accurately present the changes of birth and death, we proposed \"second-order (or parabola) approximation method.\" From the variation rates and curvatures of birth and death population, we estimated the population size. Furthermore, we want to find the rule in the dynamic system. Later we will consider other factors simultaneously, and describe them by partial differential equation. Finally, the population model is constructed.

Abstract ii\r\n中文摘要 iii\r\n1 Introduction. . . . . . . . . . . . . . . . . . . . . . .1\r\n2 Models. . . . . . . . . . . . . . . . . . . . . . . . .4\r\n2.1 Malthusian Grown Model . . . . . . . . . . . . . . . . 4\r\n2.2 Logistic Model . . . . . . . . . . . . . . . . . . 4\r\n2.3 Gompertz Model . . . . . . . . . . . . . . . . . . . . 5\r\n2.4 Age-Period-Cohort (APC) Model . . . . . . . . . . . . 6\r\n2.5 Lee-Carter Model . . . . . . . . . . . . . . . . . . . 7\r\n3 Parabola Approximation Method in ODE . . . . . . . . . .8\r\n4 Empirical Results. . . . . . . . . . . . . . . . . . .16\r\n4.1 The Empirical Results in Taiwan . . . . . . . . . . . 16\r\n5 Discussion And Suggestion . . . . . . . . . . . . . . 22\r\nReferences. . . . . . . . . . . . . . . . . . . 24\r\nAppendix. . . . . . . . . . . . . . . . . . . 28\r\nPART 1 . . . . . . . . . . . . . . . . . . . 28\r\nPART 2 . . . . . . . . . . . . . . . . . . . . . . 44