On the Parabola Approximation Method in Ordinary Differential Equation - Modelling Problem on The Population Growth

• Main Authors: Li,Yu Tso, 李育佐
• Other Authors: Li,Meng Rong
• Format: Others
• Language: en_US
• Online Access: http://ndltd.ncl.edu.tw/handle/53534900551128048759

碩士 === 國立政治大學 === 應用數學研究所 === 98 === In early population statistics, the population changes were regarded as a function of time so that people tended to describe 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. The 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.