The Correctional Model of Population Development Equation

The problem of population development has always been the key problem of restricting the development of our country. In order to increase the prediction accuracy, we analyze the exponential model, logistic model and continuous model. Also, the improved discrete population development model is provided to control the quantity and improve the quality of population.


Introduction
Population problem is a common concern to countries around the world.China is a populous country and the population problem is the key problem to restrict the development of our country.The essence of China is to develop.In order to solve the population problem now, we should not only keep the birthrate low, but also plan the quantity, quality, structure and distribution of population.On the base of promoting all-round development of people, we realize that the coordination and positive interaction are beneficial in the population development.At present, the methods of population prediction mainly have simple linear regression [1], multiple linear regression [2], grey forecasting method [3][4][5], time series method [6], neural network method [7] and so on.In many methods, the population development equation model is the most widely used and successful one put forward by Song Jian and Yu Jingyuan [8,9].

Exponential Growth Model-Malthusian Model of Population Growth
Two hundred years ago, an English demographer named Thomas Malthus (1766-1834) surveyed the demographic data of British population more than one hundred years, and found the hypothesis of the population growth rate was costant.On the basis of the hypothesis the famous population exponential growth model was established.Noting the time , the quantity of population is t ( ) x t , the quantity of population is 0 x when the initial time , we suppose the population growth rate is constant 0 t = r , i.e., the increment of ( ) x t per unit time is equal to r multiply by ( ) x t .Therefore we set up the following differential equation [10] ( ) 0 d d 0 By solving the equation we can obtain In other words, the population will be infinitely increase over time exponentially, it is called exponential growth model.
The exponential growth model is good for the population prediction of short-term forecast, but in the long run, the population growth is impossible unlimited at any region, the exponential model can't predict the population evolution process over a long period of time, in fact, the population growth rate is constantly changing, so we have the following population prediction models.

The Model of Retardant Population Growth-Logistic Model
Considering the factors such as natural resources, environment conditions for blocking the population growth, and with the increase of population, the retardant effect is more and more big.Let r be natural growth rate, it denotes that the growth rate of population is low.Let ( ) the population is no longer growth, i.e., the rate .We can obtain the following population development equation 0 ( ) By separating variables, we found The graph of population development over time is a S-shaped curve, at the beginning of the increment x is slow, when

Continuous Mathematical Model of Population Development
Multhus model and Logistic model fail to take into account the age structure of population, they are only adapted not to the population prediction at future but to the past data.In fact, in order to study the future population growth, the population age structure is the important factor that cannot be ignored.Because the fertility and mortality rates of women at different age will cause the very big difference on the population.For example, two countries or regions have the same total population at present, if a country or a region is higher proportion of young people than another country or a region, then the population development status will be very different later.
Considering the age structure of the first order partial differential equation.One is continuous, it is described to use a partial differential equation with boundary control.Let be the total population whose age are less than when the time , is called popula- is the non-negative function of ( ) , N r t two variables and is the increasing function for .When , .Let be the biggest age people can live.Then , ( ) , P r t is called population density function (Also known as the population composition by age, or population state), . So we have ( ) u t is the ratio of the total number of births per unit time and the total population at moments t , is called the relative fertility function.

( ) u t
Defining the relative mortality can be calculated by the statistics according to the population mortality distribution at moments .
( ) , r t ( ) , W r t denotes the perturbation of population caused by migration, natural disasters, war, etc.
By the method of statistics we can obtain , ( ) , the relative fertility function is given, we can get by solving the equation set, it reflects the process of population development is changing along with age and time.In control theory ( )

Discrete Model of Population Development
x t is called state vector, ( )

x
is the largest population.That is to say, when m x x