人口統計學家在傳統上習慣將人口的種種變化視為時間的函數，皆試圖以決定型（deterministic）的函數來刻劃，例如：1825年Gompertz提出的死力法則、1838年Verhulst以羅吉斯函數描述人口成長。近年則傾向於逐項（item-by-item）分析各種可能因素，例如：1992年Lee-Carter提出的死亡率模型、目前英國實務上使用的Renshaw與Haberman(2003)提出改善Lee-Carter模型的Reduction Factor模型、加入世代（Cohort）因素的Age-Period-Cohort模型等。但台灣地區近年來生育率與死亡率皆不斷下降，且有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。
本文以另一個角度思考生育與死亡變化，將台灣人口視為一隨時間變化的動態系統，使用微分方程來刻劃，找出此動態系統的背後所隱含的規則。人口動態系統的變化，主要來源是出生、死亡與遷移，在建模的過程中，我們先各別針對其中一項，在其他條件不變的情況下，以常微分方程建模，之後再同時考慮各項變動，以偏微分方程建模，找出台灣人口變化的模型。在本文中，我們先介紹使用微分方程模型分別配適與估計出生與死亡。
由台灣地區人口統計資料顯示，不論總生育率或各年齡組的死亡率都有逐漸下降的趨勢，但是每年之間的震盪很大，因此我們提出「二次逼近法」，從出生或死亡對時間的變化率與曲度來估計生育率與死亡率，對於此種震盪幅度較大的資料，可以得到頗精確的估計。唯在連續幾年資料呈現近似線性上升或下降處，非線性的模型容易出現較大的估計誤差，針對此問題我們也提出一些可能的修正方法，以降低整體的模型誤差率。

Conventionally the change of population is considered as a function of time and described by using deterministic functions. The well-known examples are Gompertz law of mortality (1825) and Verhulst’s logistic growth model (1838). Recently demographers favor stochastic models when analyzing factors in an item-by-item fashion. Since 1992, Lee-Carter model is a most commonly used stochastic model in demographic studies. But empirical studies indicate that the rapid declines in both fertility and mortality rates are against the assumptions of Lee-Carter model.

In this study we treat Taiwan population as a dynamic system which changes over time and characterize it by differential equations. Since the changes are from birth, death and migration, we first separately build models using ordinary differential equations. Afterwards the model of Taiwan population can be built by using partial differential equations considering the three main factors simultaneously.

Total fertility and age-specific mortality rates in Taiwan decline over time but with shakes between years. Consequently we propose‘parabola approximation method’and apply it to velocity and acceleration of birth or death to solve the differential equations of Taiwan fertility and mortality. Empirical study shows the method allows us to get accurate estimates of mortality and fertility when the data change a lot in a short period of time. But we found the model may over-fit the data at some time point where the function does not seem to be very continuous.

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