1 Introduction
Most of the people around the world have at least one sister or one brother. Imagine if you were born in China, a country that is leading for many decades now as the most populated country and that has a bigger population than it can actually support. Due to the huge number of people the government created the One-Child Law. This law determines that a couple can only have one child as a measure to downsize the population. The population influences policy, regarding for example how the country can manage to have resources and jobs for everybody, what would be extremely hard due to the amount of people; the economy, regarding the demand and supply to attend at least to the basic necessities of the people; culture, regarding how the population has to change their tradition; if there is education available for so many people and also how it affects the environment, because with more people there is a need for more exploration of resources. The government of China created the One-Child-Law to be able to minimize the many negative effects that an enormous population can cause, as for example conflicts related to resources used by the population that is necessary for them to live, such as land, food, water, and others. Many people want to have baby boys, because they are supposed to take care of their parents when they are old. This leads to an unbalance for the country. Many babies are left in places, where nobody will take care of them and they will starve to death. Other babies are left in relative houses or houses that take care of abandoned children. It is an extremely hard and painful situation for many Chinese people, and described as a ‘’Harsh ‘administrative measure’ for the sake of the nation’’ by Thomas Scharping. (Scharping, 2003). This law is though a necessary evil, otherwise there would not be enough resources as water, food and shelter for everyone. If the government has to invest more in education and health care for example, there would be less income left to invest in the economic sectors of the country, such as industries and the generation of jobs would decrease, there would be more pollution, among other factors.
As the population has been changing over time and it influences greatly the carrying capacity, which refers in this case to how much people the Earth can stand, of the Earth, this essay aims to create a comparative study of population growth models for China since 1965 until 2015, including future predictions. The models that will be used in this investigation are the Malthusian Model, the Logistic Model and the Coalition Model. Therefore we are lead to the question: Which mathematical model best fits the data of China’s population from 1965 to 2015?
2 Models
Models are created and used to analyze mathematically real situations, which are mostly problems that the world is facing. Models take the data of one problem and convert it to mathematical way to find solutions. The solutions obtained are translated to the normal language, so that the problem can be solved.
Some concepts are fundamental to develop this investigation using the population growth models. These concepts will help to understand and gather information that is necessary to the models. The first concept to be defined is the population. Population is defined as a group of individuals and the sum of these groups is called population. The second concept is density. The density of the population of China will be calculated taking the number of individuals per unit area. The size of populations tends to change over time and in the case of China there has been a growth. The growth rate of a population, which is the third concept to be defined, is obtained by measuring the rate of change of a population’s density over time. The death rate is the fourth concept. It will be given by the number of individuals per unit area that are dying in a period of time. The last and fifth fundamental concept is the birth rate. It will be given by the number of individuals per unit area that are born in a period of time.
Based on the population size of China from 1965 to 2015, calculations will be made to determine which model best fits the data.
2.1 Malthusian Model
The Malthusian model is also known as the Thomas R. Malthus’ model. This mathematical model of population growth was proposed by Thomas R. Malthus in the 18th century. His model became a basis for other population models as it is considered to be the simplest one. Malthus recognized many dangers that over-populations could cause, one of them being the scarcity of food in the whole world. So he stated that the number of people cannot surpass the amount of food available. He supposed that population grows geometrically, for example (3, 9, 27, 81, etc.) and that food would grow arithmetically, for example (3, 6, 9, 12, etc.).
Malthus made three claims while creating the growth model. The first one is that population is sufficiently large. The second is that population is homogeneous, in other words, it is equally distributed. The third one is that there are no limitations to grow, for example no limitations of food or space.
Malthus’s model was developed taking into account four principles, which state that food is necessary for human existence; the human population tends to grow faster than resources needed and produced by the Earth; and that the population growth; amount of resources available has to be equal; and that humans are not willing to downsize population voluntarily, so measures had to be taken and seen as positive, once it minimizes poverty, conflicts, famine and diseases.
Malthus created the following model:
dP/dt=rP; r>0 , (2.1.1)
where r is a constant, which shows that the population growth rate in time t is proportional to the current population of the country and is bigger than 0, when calculating growth. Time t has to be equal or bigger than 0. The initial population is represented by the notation P0, population after a certain time P(t). To solve this equation, the formula (2.1.1) will be modified, given
dP* P^(-1)=dt* r, (2.1.2)
then integrating the two sides, to find a pattern, it is obtained:
∫_(t_0)^t▒〖dP* P^(-1) 〗= ∫_(t_0)^t▒〖dt* r〗 (2.1.3)
P(t)=〖(P〗_0 e^(〖-rt〗_0 ))e^rt , t > 0 (2.1.4)
where P_0 e^(〖-rt〗_0 ) is equal to the constant c, giving: ce^rt
Simplifying (2.1.4), it is obtained:
P(t)= P_0 e^rt (2.1.5)
As the growth rate r is positive, the formula (2.1.5) will give an exponential function as time increases. To evaluate how good the Malthusian model fits to the original data given in Table 1 below, the formula (2.1.5) will be used to find the constant r and then calculate the population number for the next 50 years. Taking the year of 1965 as t0, the population given in millions will be P0 = 725.38. The year 1975 will be t10 and P2 = 924.20.
924.20=725.38e^2r (2.1.6)
Solving for r:
r= 1/2 ln〖(763.68/725.38)〗 ≈ 0.0257 (2.1.7)
This means that the growth rate during these 10 years is 0.0257. Putting the value found for r in the formula (2.1.5), it is obtained
P(t)=725.38e^0.0257t (2.1.8)
Using the formula (2.1.8) to calculate predictions, it is get at the following results: