1 introductionPopulation prediction using Logistic model 代寫

The population problem is one of the most significant issues facing China in the 21st century. Population projection is a highly complex nonlinear prediction problem. To see the population under planning study as a system, the system of population has relations with many other factors like the political, economic and natural conditions. It is a multi-factor complex system of multi-level, with the features of openness, complexity, self-organization and uncertainties [1]. At present, there are a lot of s-change phenomena in nature and society. Logistic model is almost the only mathematical model to describe the s-type growth. This is an s-shaped curve, continuous, monotonically increasing, and parameter k as upper asymptote. The rate of its change at the start is comparatively slow, growth speed of the middle section accelerates, and later the growth rate declines and stabilizes. There are many methods for population projections [2]. Compare the data calculated by using the exponential growth model with the actual data; the early results are better, as for the post period, the gap is big, more suitable for pre-fitting. Population growth rate is related with a number of factors and cannot be a constant. To see the rate of population growth as a constant is obviously not reasonable. How to choose a reasonable population prediction model; scientifically and accurately predict the future population is of great significance on planning for the future the land and water resources utilization and socio-economic development [3].

2 Description and analysis of Logistic model

Before the proposing logistic model, the earliest classic mathematical model given on population ecology is the Malthus model by the British statistician the Malthus (1766 - 1834). Malthus proposed the world-famous Malthus population model in the book

*Principle of Population*in 1798. Suppose at the time t0, the total population is N (t

_{0}) and at the moment t, the total population is N (t), then:

(1)

Population prediction using Logistic model 代寫

However, this model has many limitations: only consider the birth and death rates, without taking into account environmental factors. Actually, resources in the human living environment are not unlimited, and thus the growth of the population can not be unlimited. Practice has shown that Malthus population model is only compatible with the population in the past but can not be used to predict the future of the total population. Scientific and reasonable description of population growth law, predict a certain period of time the number of people has a very important practical significance for us to grasp the initiative to formulate relevant policies.

In population ecology, population growth is a complex issue. Due to that the growth of the population is subject to a number of factors, such as environmental conditions, nutritional status, fertility, mortality, individual base and generation characteristics. Verhulst called it Logistic equation, a transliteration of the Latin Logistic. It has the meaning of some logical reasoning and it is a reasoning model. Ecological significance of each parameter in the model: K is environment carrying capacity, and it means that the maximum growth rate of each individual when not subject to inhibition, N is the number of populations. Verhulst assumed that the relative growth rate of the population law was:

(2)

Using the method of separation of variables of first order differential equation, and to find out the solution

(3)

Logistic equation (2) can also be interpreted as follows: because the resources at most can only maintain K individuals, so the resources needed by each individual on average accounts for 1/K of the total resources. At the time t, N (t) individuals consuming N (t)/K of the total resources, with the remaining 1-N (t)/K. Therefore, logistic equation reflects that, the relative growth rate of the population size is proportional to the amount of the remaining resources. This population density has impact on the inhibition of the growth of population size, it is clear that, without regard to the density constraints, Logistic equation becomes Malthus equation. Analysis shows

According to the knowledge of calculus, the judgment theorem of function monotonicity and concavity and convexity, we can depict the integral curves of the logistic equation. As shown in figure 1:

**Figure 1Logistic integral curve of the equation**

Figure 1, the s-shaped curve below N = K is known as the logistic curve, because it can approximately show the growth process of biological populations, so it is often used as the basis of this theoretical discussion, thus the development of higher plant logistic curve theory. But meanwhile, we also can conduct growth-related factor analysis. The Logistic curve changes divides the biological population into five periods: (l) the starting date, also known as the incubation period, the number of individuals in a small population and slow growth density; (2) accelerated phase, with the increase in the number of individuals, density growth gradually accelerates; (3) transition period, when the number of individuals reached half of saturation density (K/2), fastest growing density appeared; (4) reduction phase, the number of individuals over K/2, the density growth slows slowly; (5) saturation period, the number of population reached the limits of K and saturated.

3 Using Logistic model for population projections

Logistic model is the growth curve model widely used in economic forecasting, used to describe the variation law of economic variables over time. Looking for such a law from the economic activities, and for the future economic forecasts. In recent years, high speed increases production, in the near future will the market be saturated? When saturated? How is the trend of the demand for these products? This is of great concern to businesses. Observed in the economic, social, scientific and technological fields, changes of many things and some of the variables over time are similar to the biological growth process, appearing to be S-shaped. It can be forecasted by Logistic model.

Logistic population model used for the human is the population model. Now use the model for analysis and forecasting of China's population. To get population data from the literature [4] as shown in Table 2, in order to calculate the R, K, in formula (3), and select the population data x 0, x

_{1}, x

_{2}of t

_{0}, t

_{1}, t years wherein t

_{1}- t

_{0}= t

_{2}- t

_{1}=τ, by

and (4)

Draw

and

Draw

(5)

From (4) and (5),

(6)

Because the data was gained from 1985 - 2006, so to choose1985,1995,2005 which have equal spacing years, x0 =10. 585 1 、x1 = 12. 1121 、x2 = 13. 0756 ,τ= 10 , substituting formula (6) was r = 0.06682 and N = 14.2515. Bring the R, N, x 0 into the formula (2)

(7)

(7) is a the population prediction formula in China. Bring each year data into the formula (7), the results are shown in Table 1 by computing.

**Table 1**Actual value, predicted value and prediction error of China population (billion)

Year | Actual value | Predicted value | Prediction error | Percentage (%) |

1985 | 1.05851 | 1.05851 | 0 | 0 |

1986 | 1.07507 | 1.07641 | 0. 013 4 | 0. 124 6 |

1987 | 1.09300 | 1.09371 | 0. 007 1 | 0. 065 0 |

1988 | 1.11026 | 1.11040 | 0. 001 4 | 0. 013 6 |

1989 | 1.12704 | 1.12648 | 0. 005 6 | 0. 049 7 |

1990 | 1.14333 | 1.14195 | 0.013 8 | 0. 120 7 |

1991 | 1.15823 | 1.15681 | 0. 014 2 | 0. 122 6 |

1992 | 1.17171 | 1.1710 6 | 0. 006 5 | 0. 055 5 |

1993 | 1.18517 | 1.1 8471 | 0. 004 6 | 0. 038 8 |

1994 | 1.19850 | 1.19777 | 0. 003 7 | 0. 039 0 |

1995 | 1.21121 | 1.21026 | 0. 009 5 | 0. 078 4 |

1996 | 1.22389 | 1.22217 | 0. 017 2 | 0. 140 5 |

1997 | 1.23626 | 1.23352 | 0. 027 4 | 0. 221 6 |

1998 | 1.24761 | 1.24434 | 0. 032 7 | 0. 262 1 |

1999 | 1.2578 6 | 1.25463 | 0. 032 3 | 0. 256 8 |

2000 | 1.26743 | 1.2 6440 | 0. 030 3 | 0. 239 1 |

2001 | 1.27627 | 1.27369 | 0. 025 8 | 0. 202 2 |

2002 | 1.28453 | 1.28250 | 0. 020 3 | 0. 158 0 |

2003 | 1.29227 | 1.29085 | 0. 014 2 | 0. 109 9 |

2004 | 1.29988 | 1.29876 | 0. 011 2 | 0. 086 2 |

2005 | 1.30756 | 1.30625 | 0. 013 1 | 0. 100 2 |

2006 | 1.31442 | 1.31333 | 0. 010 9 | 0. 082 9 |

From Table 2, the predicted value and the actual population of each year are better in line with the situation.

**Table.2**

**Predicted value of China population**

Year | 2008 | 2020 | 2050 |

predicted value (billion) | 1.32635 | 1.37600 | 1.41833 |

Population prediction using Logistic model 代寫

Conclusion

Logistic model is an S-shaped growth curve, relative to the rest of the population prediction model, to solve the nonlinear simulation and prediction of population development process during the change of the growth rate. It can also obtain by Logistic model the population growth limit. The use of the logistic model establishes the prediction formula of China's population. Using the formula to examine the population of China from 1985 to 2006.Compared with the actual population of China from 1985 - 2006,it can be seen that the predicted results using the formula on the population of China and China's actual population appear better compliance, with the maximum error is 0.26.Using the formula to forecast the population of China in 2008, 2020, 2050, the prediction result is: in 2008, Chinese population is about 1.32635 billion. In 2020, Chinese population is about 1.376 billion. In 2050, Chinese population is approximately 1.4183 300 billion.

Reference

[1] Lutz, W., Vaupel, J. W., & Ahlburg, D. A. (1999).

*Frontiers of population forecasting*. Population Council.

[2] Hosmer, D. W., Hosmer, T., Le Cessie, S., & Lemeshow, S. (1997). A comparison of goodness-of-fit tests for the logistic regression model.

*Statistics in medicine*,

*16*(9), 965-980.

[3] Lee, R., & Tuljapurkar, S. (2001).

*Population forecasting for fiscal planning: Issues and innovations*(Vol. 2). chapter.

[4]