Mathematics AI HL

Sample Internal Assessment

Table of content

Introduction

Rationale

Background Information

Methodology:

Actual Model

Evaluation of the derived models -

Bibliography -

18 mins Read

3,558 Words

Logistic functions are used to model the exponential growth of a population through the consideration of factors like carrying capacity, etc (Admin). “The carrying capacity of a population represents the absolute maximum number of individuals in the population, based on the amount of the limiting resource available” (“An Introduction to Population Ecology - the Logistic Growth Equation.”). My interest includes considering to develop a logistic function for the population in the world which takes into account the carrying capacity with a reliable accuracy rate. Considering the constant increase of innovation and technology in the world, the population growth is always increasing. Since we are also including the carrying capacity, it is crucial for the population to be decreasing. Furthermore, having lived and studied in singapore for a few of years shows my personal connection to this particular country which has majorly led me to choose Singapore. In addition, there has been a consistent decrease in population growth rate over a period of time in singapore. Through this, my aim is to understand when Singapore's population will cease to grow and when it will reach the carrying capacity and also finding out the carrying capacity. While doing this research, I need to consider the limitations of the carrying capacity which might have a huge impact on the population growth. For example, looking at the recent covid-19 pandemic which has had a huge impact on the population in Singapore, this model does not include this pandemic and numerous other factors like migrations or epidemics.

The objective of this research is to find the carrying capacity of Singapore as the population increases at a decreasing growth rate. An equation can be used to model this population considering the hypothesis that the carrying capacity exists. In order to do this, I will be using partial fraction decomposition and will solve a differential equation and will also find a solution for this differential equation on the basis of the initial values taken. I will first collect the population growth rate data and population data for Singapore between two intervals. Then, I will solve the differential equations to find a logistic function between these two intervals. After the logistic equation is found, I will be deriving more equations for the population and this can be done by using different data points. Moreover, I intended to find four equations and figure out if samples derived with different data sets have an impact on the final equation that has been derived for the model. Moreover, the three different equations will be compared to an actual model which will be derived using an online graphing tool. By finding multiple models, I will be able to develop a proper conclusion and evaluate the accuracy and reliability of the different models created and how the population data can aid in generating a common model. Furthermore, the research will also include uncovering the limitations of the real-life application of idealistic mathematical models.

Year

Population (Millions)

Growth Rate (%)

2000

4.053602

2.20

2001

4.121337

1.67

2002

4.176794

1.35

2003

4.226413

1.19

2004

4.270401

1.04

2005

4.344637

1.74

2006

4.486583

3.27

2014

5.570502

1.69

2015

5.650018

1.43

2016

5.711933

1.10

2017

5.764487

0.92

2018

5.814537

0.87

2019

5.866405

0.89

2020

5.909869

0.74

2021

5.941060

0.53

2022

5.975689

0.58

My exploration is divided into the following steps-

- Derivation of the logistic model
- Finding a logistic model for the first two consecutive years
- Finding a logistic model for the last two consecutive years
- Finding a logistic model for the spaced-out years in between
- Finding a logistic model for the first and last year
- Evaluating the results

The derivation of different types of models will aid in understanding which logistic model derived best fits the data collected.

The logistic equation can be modelled as

\(\frac{dp}{dt}=hp(\frac{J-p}{J}), or\frac{dp}{dt}=hp(1-\frac{p}{j})\)

( \(\frac{dp}{dt}\)): Growth Rate - Rate at which a population is increasing or decreasing over time dp dt

(*J*): Carrying Capacity - The maximum number of people that can be sustained by the country's limited resources over a period of time, without degrading the environment's ability to withstand the population.

(*D*) - The integration constant of the differential equation -

\(D= \frac{pa}{j-pa}\)

(*h*) - Proportionality constant

(*t*) - Time period

(*p*) - Population of Singapore

The aforementioned differential equation which evaluates the growth rate of population will help me derive the logistic model for each case.

In order to develop the first equation, the two data points that I will be using are the years 2000 and 2001. The population during 2000 is 4.053602 and it has a growth rate of 2.20 while the population during 2001 is 4.121337 and the growth rate is 1.67. I will use these values to find the carrying capacity J and the constant h.

Sample calculation is shown below-

Using the formula-

\(\frac{dp}{dt}= hp(\frac{J-p}{J}) \)

\(⇒2.20=h×4.053602×(\frac{J-4.053602}{J})\)

which gives

\(h=\frac{2.20}{4.053602×(\frac{J-4.053602}{J})}\)

\(⇒1.67=h×4.121337×(\frac{J-4.053602}{J})\)

which gives,

\(h=\frac{1.67}{4.121337×(\frac{J-4.121337}{J})}\)

Using a graphic display calculator, and substituting the above two equations of *h*

*⇒ J = *4. 32

⇒ *h* = 8. 77

Using the h and J values that I found, I substituted them in the logistic equation to model the population of Singapore-

\(⇒ \frac{dp}{dt}=8.77p(\frac{4.32-p}{4.32})= 8.77p(1-\frac{p}{4.32})\)

Integrating this above equation gives us the equation for the population of singapore

\(⇒\frac{dp}{8.77p(1-\frac{p}{4.32})}=dt\)

Partial Fraction Decomposition-

\(⇒\frac{dp}{8.77p(\frac{4.32-p}{4.32})}=dt\)

\(⇒\frac{0.493dp}{p(4.32-p)}=dt\)

Then, we can integrate the equation which gives us two integrated fractions through decomposing the equation \(\frac{0.493dp}{p(4.32-p)}\), this can be rewritten as-

\(⇒\frac{0.493}{p(4.32-p)}=\frac{S}{p}+\frac{R}{4.32-p}\)

\(⇒0.493=\frac{Sp(4.32-p)}{p}+\frac{Rp(4.32-p)}{4.32-p}\)

⇒ 0.493 = S(4.32 - p) + Rp

Substituting 0 (minimum value) in place of *p *-

⇒ 0.493 = S(4.32 - 0) + R(0)

⇒ S = 0.114

Substituting *p* = 8.77 - maximum value -

⇒ 0.493 = S(4.32 - 4.32) + R(4.32)

⇒ R = 0.114

This then gives us the differential equation -

\(⇒\frac{0.114}{p}+\frac{0.114}{4.32-p}dp=dt\)

Further integrating the equation and getting one constant through consolidating the integration constants

⇒ 0.114 In|p| - 0.114 In |4.32 - p| = t + C

\(⇒In|\frac{p}{4.32-p}|=\frac{t}{0.114}+C=8.77t+C\)

\(⇒De^{8.77t }=\frac{p}{4.32-p}\)

To solve for D, the first data point (0, 4.053602), which is the the population of 4.053602 million people in Singapore in the year 2000. The value of t is measured as the number of years elapsed since 2000. This will be substituted in the equation.

\(⇒De^{8.77(0)}=\frac{4.053602}{4.32-4.053602}\)

⇒ D = 15.2

After solving for D, we can solve for p from the linear equation below-

\(⇒\frac{p}{4.32-p}=15.2e^{8.77t}\)

\(⇒ p = 4.32(15.2e^{8.77t}) - 15.2pe^{8.77t}\)

\(⇒ p + 15.2pe^{8.77t} - 65.5e^{8.77t}\)

\(⇒ p (1+15.2pe^{8.77t}) - 65.5e^{8.77t}\)

\(⇒p=\frac{65.5e^{8.77t}}{1+15.2e^{8.77t}}=\frac{4.32}{1+0.066e^{-8.77t}}\)

Using a similar calculation shown above, the rest Logistic models are developed and evaluated.

Thus, this logistic function states that the carrying capacity of the population is approximately 4.32 million, which means the maximum population that can be supported by Singapore with its available resources is 4.32 million. The value of the integration constant is 0.066 and the value of the proportionality constant here is - 8.77.

The above logistic model has been derived using the data points from the years 2001 and 2000. The logistic model graph fits the data points in the early years more accurately than compared to the latter years. This can be because of the close relation the population values the first six years have with the years 2000 and 2001. The change in the logistic model derived can only be observed in the last few years from *x* = 14 to *x* = 22, where the data points have been spaced far away from the graph. This could be because of the time lags between the starting years to the ending, hence this could influence an external factor to result in a change in the total population. For example, the SARS outbreak or the Covid-19 outbreak in singapore.

For the second equation, the two data points that I will be using are the years 2021 and 2022. The population during 2021 is 5.941060 and it has a growth rate of 0.53 while the population during 2022 is 5.975689 and the growth rate is 0.58. Using the aforementioned method and derivation, the logistic function for the population during these two years is as follows -

\(⇒ p=\frac{5.55}{1+0.369e^{-1.26t}}\)

Thus, this logistic function states that the carrying capacity of the population is approximately 5.56 million, meaning that with singapore’s limited resources available, the maximum population it can sustain is 5.56 million. The value of the integration constant is 0.369 and the value of the proportionality constant here is -1.26. The logistic model here has been derived using the data points from the years 2021 and 2022. The derived model almost fits the data points during the last few years as seen on the figure. However, the data points from the first few years are a little farther away from the graph. The reason being that this logistic model has been derived using the data points from the last two years, thus tending to fit the latter data points and spacing away from the starting ones. The unforeseen increase in the growth rate over the years could have been one of the reasons why the model does not fit the data points in the beginning.

For the third equation, the two data points that I will be using are spaced out and chosen randomly. These years are 2006 whose population is 4.486583 with a growth rate of 3.27 and 2016 whose population is 5.711933 with a growth rate of 1.10. Using the method and derivation in logistic model 1, the equation derived for this model is as follows-

\(⇒P=\frac{6.15}{1+0.518e^{-2.69t}}\)

Thus, this logistic function states that the carrying capacity of the population is approximately 6.15 million signifying that highest number of individuals singapore can support is 6.15 million. The value of the integration constant is 0.518 and the value of the proportionality constant here is -2.69. The above logistic model has been modelled using the spaced out data points from the years 2006 and 2016. As seen on the figure, the graph is farther away from the data points. The reason behind taking these data points is to figure out the the logistic function that is fits the best for the data points taken. Since, I have taken data points from the between the data, the logistic model graphed appears to be away from all the data points. The cause for this graph to be away from the data points could be because of the huge difference between the growth rates and the population between the year 2006 and 2016, hence resulting in a function that approximately tries to fit the other data points but practically is a bit farther away from them.

In order to develop the first equation, the two data points that I will be using are the years 2000 and 2022. The population during 2000 is 4.053602 and it has a growth rate of 2.20 while during 2022, the population is 5.975689 and the growth rate is 0.58. Using the similar method of derivation and calculation in logistic model 1, the equation derived for this model is as follows-

\(⇒p=\frac{6.39}{1+0.577e^{-1.48t}}\)

For the actual model, I put the data points on the software logger pro and modelled a logistic function to find the best fit line and a logistic function which fits the data points accurately. The actual model that suffices with singapore's data points is as follows-

The logistic function which best fits the data can be represented as-

\(⇒p=\frac{6.851}{1+0.7899e^{-0.08155t}}\)

As seen on the figure above, the logistic function of the actual model approximately fits most of the data points or is either relatively close to the them as compared to the logistic models derived. The actual model has a carrying capacity of 6.851 million people, this indicates that singapore can only withhold a population of 6.851 million with its current resources present. The integration constant is 0.7899 while the proportionality constant here is -0.08155. The actual model of the logistic function differs from the ones that I have derived is because of the difference in data points used to derive the model.

\(p=\frac{6.851}{1+0.7899e^{-0.08155t}}\)

1.) Taking *t* = 5

\(⇒p=\frac{6.851}{1+0.7899e^{-0.08155(5)}}\) (Using a graphic display calculator to find p)

⇒ *p* ≈ 4.49

On the data table *p* ≈ 4. 34. As observed the population of singapore calculated from the actual model is approximately equal to the population on the data table. The difference in the a few decimal places could be due to the limiting factors like migration, increased competition, etc.

2.) Taking *p* ≈ 5.87

\(⇒5. 87=\frac{6.851}{1+0.7899e^{-0.08155t}}\)

⇒ *t* ≈ 19.05

On the data table *t* = 19, meaning that during the year 2019, singapore had a population of approximately 5.87 million. Through this model, I calculated to find the value of t as 19.05, which rounded off is 19, hence indicating that the time is 19 years. This verifies the accuracy and reliability of the actual model.

When you visually view the graphs of the different models derived and the actual model, all the predicted models have an S-shaped curve/sigmoid curve, which suggests that the models can be hypothetically be considered as logistic models, since the shape is one of the attributes of a logistic model. However for the actual model, it does not depict the S-shaped curve/sigmoid curve which raises questions on the reliability of this model at first glance. The predicted models where I used the spaced out data points (logistic model 3 & 4) consist of a carrying capacity(6.39,6.15) approximately close to the of the actual models(6.81), hence increasing the validity and predictive accuracy of the models derived. The logistic model derived fits into the data points which are closer to the datapoints used to model the function. This mean that graph is successful in fitting the data points, however due to external factors and change in growth rate, the population value may vary inconsistently, thus causing the graph to be farther away from a few points. Since, for all the logistic models, I have used different data points, the logistic models derived only pass through a specific set of data on the graph. On the other hand, the actual model takes all the data points into consideration to develop a common model, hence shows how the accuracy of a model is affected by the variations in implementations of the data points. Moreover, because I did not identify outliers in the data points, this means that there is a chance that my derived model has been affected by these outlier points. This might result in miscalculations and distortions in the estimates of coefficients. Furthermore, another limitation is that I have taken a small sample size of 22 years, and for better predicted probabilities and estimates, it is preferred to take a larger sample size to make my model more accurate. In order to make the model more precise, what I could do in future investigation is to eliminate all the outliers and anomalies which affect the models accuracy. In addition, using a larger sample size with ensembling of different types of models which aid in increasing the predictive power of the logistic model. However, every prediction and model developed will contain some kind of uncertainty which means that not every model can be perfect. For this particular population data another type of model that can be used is the malthusian exponential growth model(Kagan). This model shows a J-shaped curve that suggests that over time population growth occurs at a constant rate. This model also assumes that the population growth is proportional to the size of the population.

The aim of this investigation has been to determine the carrying capacity of singapore and to develop a logistic model and understand how it fits the population data of singapore. To do this, I expanded my learnings of in calculus and derived a common approach for the logistic equation. I used the real-life data of singapore population and growth rate in the derived formula to develop a different types of logistic functions. I was able to find realistic values of the parameters and develop models which had a reasonable explanation behind their shape and relation with the data points. Additionally, I also explored the different features of a logistic function and the role it plays in understanding the population growth of a country. Comparing the model that I developed with the actual logistic model for the data points helped me recognise what I could have done differently to come up with a model which is more accurate and reliable. Furthermore, I learnt to define and represent a logistic model using logger pro. Overall, the mathematics of integration and derivation that I learnt here made my understanding of these concepts more efficient in the classroom.

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