# Best Tool To Predict Future Human Population Growth Accounting Essay

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In this essay, three population models, linear, exponential and logistic were chosen to investigate. The question that was dealt with within this is essay is which function is the best tool to predict future human population growth? Thus each model had to be individually analysed, to see which yields to closest approximation to actual population growth trends.

To do this, 5 Asian countries were arbitrarily chosen: Cambodia, China, India, Japan, and Thailand. The models were found to be in the form of a differential equation, and thus first had to be integrated, in order to use them. Then using data from years past, each of the three models were applied to predict what the current population should be. Through means of 2 goodness of fit test, actual data was compared to the predicted values, in order to see which model produces best results.

It was found that the exponential model yields the results closest to the real data. This was the only model where the null hypothesis was accepted. However, there are several enhancements that could be made to this model to further improve the accuracy of the findings. It could be modified to include growth rate of different age groups, which would account for any infant mortality or maternal mortality or anything else which affects specific age groups. However, there are some things which the model cannot anticipate, exogenous factors that can affect population growth rate.

The perennial enigma of mathematics stems from the Formalism vs. Platonism debate. The question that remains is whether mathematics is a human fabrication, or is it something we have discovered? Whatever the case, the applicability to society and its use in modelling naturally occurring trends is one of mathematics' most intriguing aspects. Its utility derives from the capability math provides to predict future trends. Certainly, this is the case in the study of demographical development. Predicting population patterns is of the utmost necessity to governments. This is usually done through modelling population changes over the past several decades, and extrapolating for future years. Over time, several population models have been developed, attempting to model population growth.

In this essay we will be exploring three such population models, and how accurately they model the data. These models are the linear model, the exponential model, and the logistic model. In order to test the models, five Asian countries have been chosen: Cambodia, China, India, Japan, and Thailand. This essay takes already-existing demographic information for these countries, and utilises it to predict what the current population should be. By simple comparison of predicted vs. actual data, we can conclude which model fits the best.

The above-mentioned models are quite simplistic interpretations of population growth, and don't necessarily take into account all factors that may impact growth. The most comprehensive model is the one with the most factors involved, thus yielding the closest approximation. I therefore believe that the Logistic model will provide the best approximation, as it takes into account growth rate, and what is known as carrying capacity. However, the exponential or linear model could just as easily be the best fit. Thus, this essay will explore which function is the best tool to predict future population growth?

Research Question:

## 'Which function is the best tool to predict future population growth?'

Table 1: Population (thousands)

8,101

9,738

11,368

12,744

14,071

1,070,175

1,155,305

1,219,331

1,273,979

1,315,844

766,053

849,415

935,572

1,021,084

1,103,371

120,837

123,537

125,472

127,034

128,085

## Thailand

50,612

54,639

58,336

61,438

64,233

This table of data was taken from the United Nations Statistic Division. [1] Using the values for 1985 and 1990 and 1995, I will use the data to predict what the population should be in the years 2000, and 2005.

The values in the table above are in thousands. For the sake of the calculations, I will keep the values in the same way.

All the models are in the form of a differential equation. In order to use the model to predict future outcomes, we must first write the equation in the form of a function of time.

## i.) The Linear Model: [2]

, where r is the constant rate, measured in thousands of people per year.

This is the same as stating change in population divided by change in

time is equal to r, i.e.

We can now use this equation to calculate projections of population. Firstly, we must find the constant r for each country. This is why it is necessary to have more than one set of data.

In the equation above, (t-t0) represents change in time. In this case, I will be substituting in values for the years 1985 and 1995 in order to find r, therefore (t-t0) will be equal to 10.

## Cambodia:

11,368 = r(10) + 8,101

11,368-8,101=10r

r= 326.7

## China:

1,219,331=r(10) + 1,070,175

1,219,331-1,070,175= 10r

r= 14915.6

## India:

935,572=r(10) + 766,053

935,572-766,053= 10r

r= 16951.9

## Japan:

125,472=r(10) + 120,837

125,472-120,837= 10r

r= 463.5

## Thailand:

58,336=r(10) + 50,612

58,336-50,612= 10r

r= 772.4

Now that we have the value of the constant, r, we can substitute it into the original equation to predict Pt.

## Cambodia:

P2000 =(326.7)(5) + 11,368

= 13,001.5

## China:

P2000 =(14915.6)(5) + 1,219,331

=1,293,909

## India:

P2000 =(16951.9)(5) + 935,572

=1,020,331.5

## Japan:

P2000 =(463.5)(5) + 125,472

=127,789.5

## Thailand:

P2000 =(772.4)(5) + 58,336

=62,198

## PREDICTION FOR 2005

In this case, we will use P0 generated from prediction calculations.

## Cambodia:

P2005 =(326.7)(5) + 13,001.5

= 14,635

## China:

P2005 =(14915.6)(5) + 1,293,909

=1,368,487

## India:

P2005 =(16951.9)(5) + 1,020,331.5

=1,105,091

## Japan:

P2005 =(463.5)(5) + 127,789.5

=130,107

## Thailand:

P2005 =(772.4)(5) + 62,198

=66,060

## Summary Table

As the value we are predicting is population figures, clearly decimal places do not make sense. Hence the figures in the table below have been rounded up to closest whole number.

Table 2; Values for Linear Model

13,002

12,744

14,635

14,071

1,293,909

1,273,979

1,368,487

1,315,844

1,020,332

1,021,084

1,105,091

1,103,371

127,790

127,034

130,107

128,085

62,198

61,438

66,060

64,233

## Ï‡ 2 goodness of fit test

Now, using the 2 goodness of fit test, we can see how well the linear model has projected population figures.

In this case, the observed value (O) actual data, and the expected value (E) is the predicted data

Null Hypothesis:

O is a good fit with E

We will be using a significance level of 5 %.

## For the year 2000:

=326 (to 3 s.f.)

Degrees of freedom= 5-1=4

The critical region at the 5% level for starts at 9.49. Since 3269.49, we can reject the null hypothesis. [3]

## For the year 2005:

=2130 (to 3 s.f.)

Degrees of freedom= 5-1=4

The critical region at the 5% level for starts at 9.49. Since 21309.49, we can reject the null hypothesis.

## Discussion of Results and Evaluation of Linear Model

As is evident from the 2 value above, the linear model is not a

good fit for the population trends in the five countries. It is in fact a long way off, as the large disparity between the calculated2 value, and the value from the 2 distribution.

This model has predicted lower values for India, and higher values for Cambodia, China, Japan and Thailand. This means that growth in India between 1995 and 2005 occurred at a faster rate than between 1985 and 1995. In the other four countries, growth occurred at a slower rate.

It should be noted that the discrepancy between China's actual and predicted values accounts for a large part of the2 value. For the year 2000, it accounts for 94% of the2 value, and in 2005 it accounts for 95 % of the2 value. Therefore, it can be said that the linear model is much more representative of growth in the other 4 countries than in China. This insinuates at the fact that it is very difficult to have a general function, which charts trends in all countries, as growth in all countries is unique.

It is also important to note that for 2005, the2 value is much greater than for 2000. This is because we used the predicted values from 2000, thus enlarging the difference. Also, the assumption was made that r remains constant, when in fact this is not the case; clearly indicating that population growth does not occur in a linear fashion.

The problem with this model is the fact that it only takes into account previous growth, and no other factors that might affect future growth. The next model, the exponential or Malthusian model is a step up on this, as the rate constant, r, is derived from further demographic information, namely birth rate and death rate.

## ii.) The Exponential Model (Malthusian Growth): [4]

We know that at t= 0, P= P0. From the above equation, we therefore get:

We can therefore substitute for c to get

In this case, we have set 1985 to t=0

In the Malthusian model r is defined as the difference between the birth rate and death rate of countries. Therefore, we need to gather birth rate and death rate information to calculate the value of r.

Table 3; Birth Rate (b) (per 1,000 population)

47.8

39.6

32.3

30.8

22.1

18.3

16

13.6

32.4

30.1

27.2

24.6

10.5

9.7

9.6

9.2

## Thailand

22

19.4

17.3

16.3

Table 4; Death Rate (d) (per 1,000 population)

14.5

12.4

11.1

10.8

6.7

7.3

7

6.8

11.6

10.4

9.4

8.8

6.3

7

7.6

8

## Thailand

6.1

6

6.6

7.3

r= b-dFrom these values, we can calculate r, the rate constant, measured in thousands of people/year. For use in the exponential model, this needs to be divided by one thousand, so that it is measured as growth rate per person per year.

1000r=32.3-11.1

r=21.210-3

1000r=16-7

r=910-3

1000r=27.2-9.4

r=17.810-3

1000r=9.6-7.6

r=210-3

1000r=17.3-6.6

r=10.710-3

1000r=30.8-10.8

r=2010-3

1000r=13.6-6.8

r=6.810-3

1000r=24.6-8.8

r=15.810-3

1000r=9.2-8

r=1.210-3

## Thailand

1000r=16.3-7.3

r=910-3

Now, we can apply the exponential model to predict population figures for 2000 and 2005.

## Cambodia:

P2000= (11,368)e(0.0212)(5)

=12,639.19109

## China:

P2000= (1,219,331)e(0.009)(5)

=1,275,454.196

## India:

P2000= (935,572)e(0.0178)(5)

=1,022,655.656

## Japan:

P2000= (125,472)e(0.002)(5)

=126,733.0146

## Thailand:

P2000= (58,336)e(0.0107)(5)

=61,541.97107

## Cambodia:

P2005= (12,744)e(0.02)(5)

=14,084.29818

## China:

P2005= (1,273,979)e(0.0068)(5)

=1,318,039.063

## India:

P2005= (1,021,084)e(0.0158)(5)

=1,105,021.518

## Japan:

P2005= (127,034)e(0.0012)(5)

=127,798.4952

## Thailand:

P2005= (61,438)e(0.009)(5)

=64,265.85966

## Summary Table

As the value we are predicting is population figures, clearly decimal places do not make sense. Hence the figures in the table below have been rounded up to closest whole number.

12,640

12,744

14,084

14,071

1,275,454

1,273,979

1,318,039

1,315,844

1,022,656

1,021,084

1,105,022

1,103,371

126,733

127,034

127,798

128,085

61,542

61,438

64,265

64,233

## Ï‡2 goodness of fit test

Now, using the 2 goodness of fit test, we can see how well the linear model has projected population figures.

In this case, the observed value (O) actual data, and the expected value (E) is the predicted data

Null Hypothesis:

Ois a good fit with E

We will be using a significance level of 5 %.

## For the year 2000:

=5.87 (to 3 s.f.)

Degrees of freedom= 5-1=4

The critical region at the 5% level for starts at 9.49. Since 5.87<9.49, we can accept the null hypothesis.

## For the year 2005:

=6.79 (to 3 s.f.)

Degrees of freedom= 5-1=4

The critical region at the 5% level for starts at 9.49. Since 6.79<9.49, we can accept the null hypothesis.

## Discussion of Results and Evaluation of Exponential Model

In both the predictions for 2000 and 2005, the null hypothesis has been accepted. This shows us that the exponential model is a good fit for population growth over the last decade.

However a few things were different in this calculation than from the last. Firstly, r was recalculated for the 2005 prediction. This tells us that if we were to use this model for future predictions of population growth, we would firstly have to predict r for that time. Simply using r for the current time will yield an inaccurate prediction. Therefore, another function would have to be developed to predict birth rate and death rate.

Secondly, to predict 2005, P0 values were used from Table 1, not the predicted values for 2000. This also tells us that it is more difficult to predict trends over a longer period of time.

There are several limitations to this model that we must now consider. Because of the nature of the exponential model, it provides us with certainty for a limited period of time. In the short-run, it provides us with values that are plausible, but in the long run, the values become to large to be plausible. The value of r has to be moderated down. This again relates to what has been mentioned about the difficulties of predicting far off into the future.

Furthermore, this model makes several assumptions that are not necessarily true. Firstly, the assumption is made that b and d are continuous over time, and that the values remain constant. This is simply not true as growth rate is always changing. Again this has the effect of distorting values that are predicted for a long time period. Secondly, it also makes the assumption that populations are closed systems, where no immigration or emigration takes place. Thus it is slightly lacking in terms of what it accounts for.

There is also one more factor that this model does not take into account. Population growth occurs at different rates for different age cohorts. For example, if a country had very high infant mortality rate, than r would not be a good representation of the constant of growth. A more comprehensive model would require more variables, to account for growth in different age groups.

## iii.) The Logistic Model [5]

or

In this case, r is the growth constant, measure in thousands of people per year. K is what is known as the 'carrying capacity', or the point at which birth rates and death rates are equal.

In order to simplify the integration, the constant will be replaced with a, i.e.

To integrate this, we first separate the variables

We can write this as partial fractions:

Simplifying,

Upon integration:

, where c is the arbitrary constant

This simplifies to:

(1.)

When t=0, P= P0.

If we substitute this back into equation (1.)

Dividing numerator and denominator by

Substituting in for r and a, we get

Simplifying,

Here is the limiting population size i.e. as,

We must now calculate the value of K for each country, in order to calculate the population growth. We will use the r values calculated for the last model.

-16100

-2540000

-6890000

-140000

## Thailand

-166040

Using the above equation, we can determine the value for K for each country.

1,6873

12,744

1,397,321

1,273,979

1,089,034

1,021,084

125,730

127,034

58,742

61,438

## Ï‡ 2 goodness of fit test

Now, using the 2 goodness of fit test, we can see how well the logistic model has projected population figures.

In this case, the observed value (O) actual data, and the expected value (E) is the predicted data

Null Hypothesis:

O is a good fit with E

We will be using a significance level of 5 %.

## For the year 2000:

=16400 (to 3 s.f.)

## Discussion of Results and Evaluation of Logistic Model

From such a large value for 2 goodness of fit test, it is obvious that the logistic model is a very bad fit for charting population trends in the last decade. This is mostly to do with the relevance of 'K' to human population. K is defined as the carrying capacity, a point where b=d. More significantly, it is a factor that incorporates the relationship between the species and the environment, i.e. the capability of the environment to support the population.

Thus if we consider the approach to finding K, it does not really seem appropriate. In fact, the entire concept of K is not really appropriate to human population, as it cannot fully identify the multi-faceted link between humans and nature.

In practicality, this model was developed for use in ecology, where it is certainly more pertinent. There are simply too many factors to simplify the link between mankind and its environment to one constant. K is more relevant when resources are scarce, and there is a clear relationship between increase in population, and environmental sustainability. At the moment, there are still enough natural resources for humans to survive past the next century.

Additionally, the method of calculating K from past trends is not suitable. K should be calculated independent of population, taking into account factors such as human consumption, impact on environment technology and so on. [6]