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:
Table 1 – Population of China with predictions based on the Malthusian Model for the years 1965 to 2015
The data from the real population is from the last day of each year. Number of predicted population is rounded to the closest number.
Comparing the actual data to the predicted data, it is observed that after the year 1973 the predicted values start to increase greatly, showing a large difference between the values and in 2015 the difference in the real and predicted value is 1246.87 million of people. This shows that the model may not be accurate for a long period of time, but for calculations for around every 10 years. For calculations for a longer period of time using this model, it would be recommended to calculate a new formula for every 10 years to obtain more accurate predictions.
2.2 Logistic model
Using the Malthusian Model as a base, other growth models were created to improve the accuracy in the results, by adding different variables. The Malthusian model calculates the population through time with an unconstrained growth. Since there are many factors that limit the growth of a population, for example food and water, all populations will have its growth restricted at some point. The mathematician Verhulst then developed the Logistic model, which takes into consideration factors that restrict the growth of a population. Verhulst added to the formula (2.1.1) the variable carrying capacity K – which points out how many people a certain habitat or country can support.
dP/dt=rP (1- P/K); r>0 (2.2.1)
When the growth rate r has a number close to 0, it means that the population number is close to the carrying capacity and when the growth rate r has a number close to 1, it means that the population number is far from the carrying capacity.
dP/(dP(1-P/K))=drt (2.2.2)
then integrating both sides to find a pattern, it is obtained:
∫▒dP/(dP (1-P/K))=∫▒drt (2.2.3)
ln|P|-ln〖|1-P/K|=e^(-rt) 〗 (2.2.4)
which results in the following formula:
P(t)=(KP_0)/(P_0+(K-P_0 ) e^(-rt) ) (2.2.5)
The carrying capacity of China can be predicted based on the resources available for the population, but it can change mainly due to technological advances and if there is a way to increase resources, then the carrying capacity number will increase. As there was not found a number for the carrying capacity of China, the calculations will be proceeded using 10 billion as the carrying capacity K number of people. P0 will be 725380000, giving the following formula:
P(t)=(10000000000(725380000))/(725380000+(10000000000-725380000) e^(-rt) ) (2.2.6)
Simplifying:
P(t)=((7.2538e^18 )(e^rt))/10000000000 (2.2.7)
The formula (2.2.7) will be used to find a value for r according to the carrying capacity, where P(t)=924200000 and t=10:
924200000= ((7.2538e^(18 ))(e^10r))/10000000000 (2.2.8)
Simplifying it is obtained:
ln((9.242e^18)/(7.2538e^18 ))=ln〖(e^10r 〗) (2.2.9)
Solving for r, the following result is obtained:
r≈0.024223
This means that the growth rate during 10 years is 0.024223 approximately. As the value of r is quite a small number, it would not be accurate or recommended to round it more. Substituting this value found for r in the formula (2.2.7), it is obtained
P(t)=((7.2538e^18 )(e^0.024223t))/10000000000 (2.2.10)
Using the formula (2.2.10) and substituting t according to the year that will be calculated, it is obtained the following results:
Comparing the results obtained using the Logistic model to the ones obtained in the Malthusian model, it is clear that the predicted values found for the Logistic model are smaller. Comparing the actual data to the predicted data, it is observed that the numbers did not show a large difference after the first 10 years as in the Malthusian Model, but in the following years the number was increasing gradually. In the year 2015 a difference in the real and predicted value of 1060.21 million of people can be noticed. This shows that this model is more accurate than the Malthusian and could work for a little longer than the period of 10 years to make predictions. As recommended for the Malthusian model, it is also recommended for this model to calculate new formulas for short periods of time to have more accurate predictions.
2.3 The Coalition model
The Malthusian Model was once again used as a base to create other growth models to improve the accuracy in the results, and again by adding different variables. Heinz von Foerster, Larry Amiot and Patricia Mora wrote a paper named “Population Density and Growth”. In this paper they highlight that population growth is due to the improvement in technology, which aims to improve living conditions. Many years ago, people did not have the technology to offer all the health care needed to treat diseases or needed by a pregnant woman. Now with technology, people can live for a longer period of time, the death rate in births has decreased greatly and due to it, the population has increased rapidly and exponentially. The coalition model proposed by the authors has the variable a that represents the productivity rate:
dP/dt=rP^(1+a) (2.3.1)
where r is constant and a has a value bigger than 0, having then a positive value. The derivate of P should have a proportional value in relation to a power of P. The power of P is represented by 1+a. It will give then the formula:
ln dP/dt=ln〖r+(1+a) lnP 〗 (2.3.2)
Solving the formula, it is obtained the following equation:
P(t)= 1/[ar(T-t)]^(1/a) (2.3.3)
where T is a finite time, it means T is what is called the Doomsday – the day the world will no longer exist –. T has to be bigger than t; otherwise the model will not function as it should.
Giving the value X to a and the value Y to T and t being Z, the formula can be rewritten in the following way:
P(t)= 1/[Xr(Y-z)]^(1/X) (2.3.4)
3 Future predictions
Using the three different models analyzed above, predictions for the next 50 years will also be calculated and compared. The value of r in the Malthusian and Logistic model will be calculated again for the predictions.
3.1 Malthusian Model
Based on the formula (2.1.5), which is P(t)= P_0 e^rt, the data used to calculate predictions for the next 50 years are t0 = 2013, P0 = 1360.72, t2 = 2015, P2 = 1375.14, where population P is given in millions. Based on this data it is obtained:
1375.14=1360.72e^2r (3.1.1)
Solving for r:
r=1/2 ln(1375.14/1360.72)≈0.00527 (3.1.2)
The growth rate during these 2 years is 0.00527. Putting the value found for r in the original formula (2.1.5), it is obtained
P(t)=1360.72e^0.00527t (3.1.3)
This data present the future predictions:
As observed in the results from Table 1, the model only works well for a short period of time, so it has to be taken into consideration that predictions for a longer period of time may not be accurate. Therefore, the data for no longer than approximately 10 years for the population of China using the Malthusian model may be near to the future actual numbers, if the population continues to grow at the rate of 0.00527 per year.
3.2 Logistic Model
Based on the formula (2.2.5), which is P(t)=(KP_0)/(P_0+(K-P_0 ) e^(-rt) ) , the data used to calculate the predictions for the next 50 years are t0 = 2013, P0 = 1360720000, t2 = 2015 and P2 = 1375140000. Given that many factors as technology for example, found a way to increase the carrying capacity K to more 3 billion of people in China, calculations will be made using the number 13 billion as K. Based on this data it is obtained:
1375140000= ((13000000000*1360720000)(e^2r))/13000000000 (3.2.1)
Simplifying it is obtained:
ln((1.787682e^19)/(1.768936e^19 ))=ln〖(e^2r 〗) (3.2.2)
Solving for r, the following result is obtained:
r≈0.00527
Putting the value of r in the original formula (2.2.5), it is obtained:
P(t)=((13000000000*1360720000)(e^0.00527t))/13000000000 (3.2.3)
Using the formula (3.2.3) and altering t according to the year calculated, the following data present the future predictions:
When using 13 billion of people as the value of K, it is noticed that the value of r obtained for the future predictions using this model is the same value of r obtained to calculate the future predictions using the Malthusian model. It is a curious coincidence that constraining the population of China to a maximum of 13 billion of people, the growth rate would be the same as not constraining the population, given that both formulas – Malthusian and Logistic model – were calculated using the same values of t0, t2, P0 and P2.