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Mathematical modeling of the population dynamics of the UK
This graduate-level sociology research paper example, formatted in IEEE citation style, examines UK population dynamics to understand community growth and sustainability. The study analyzes mathematical models, including exponential growth and the Gompertz function, to model the UK population. Using historical data, parameters were determined for each model, allowing for prediction comparisons and insights on urban planning implications. Findings suggest that while the exponential model forecasts rapid growth, the Gompertz function more realistically predicts population nearing carrying capacity. This analysis is crucial for resource allocation, social service planning, and sustainable development in the UK.
Octobre 28, 2024
* The sample essays are for browsing purposes only and are not to be submitted as original work to avoid issues with plagiarism.
1
Mathematical Modeling of the Population Dynamics of the United Kingdom Using Exponential
Growth and Gompertz Functions
Student’s Name
Institutional Affiliation
Course
Professor’s Name
Date
2
1. Introduction
Understanding of population dynamics is important for substantial planning and management
of various sectors including health and education, to infrastructure (McKee et al., 2021). The
United Kingdom has undergone significant demographic changes during the last several decades
that can be based on developing appropriate models in predicting future population trends
(Cangiano, 2023). This study makes use of exponential growth and the Gompertz function to
model the population. Deriving parameters for each model from historical population data for the
UK will be analyzed from the database of the Macrotrends to evaluate their predictive capability.
2. Literature Review
Population modeling has an extensive history, with several different approaches having been
developed to understand growth patterns. According to the past study of Lumen Learning (2020),
the exponential growth model is one of the most commonly used methods, even though it does
not take resource limitations into consideration. The study of Tjørve and Tjørve (2019) supposes
that the Gompertz function, has broad applications in biology in the modeling of decelerating
growth of a population to a limit. All these models offer important insights into the dynamics of
populations and are hence an essential tool for the researcher and policy analyses.
3. Methodology
3.1. Data Collection
In conducting this study, the database of the Macrotrend was used to collect the population of the
United Kingdom between 2014, and 2023; a 10-year period. The population sizes were recorded
annually, providing an opportunity for effective analysis of population growth over time.
3
3.2. Model Formulation
3.2.1 Exponential Growth Model
The exponential model is usually defined by the following formula, in accordance with the past
study of Inigo (2019):
𝑃 𝑡() =𝑃0𝑒𝑟𝑡
Where:
𝑃 𝑡() =𝑆𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝑎 𝑡𝑖𝑚𝑒, 𝑡
𝑃0=𝑇ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑎 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑟=𝑇ℎ𝑒 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
3.2.2. Gompertz Function
The Gompertz function is described through the following general equation:
𝑃𝑡()=𝐾𝑒−𝐾
𝑃0
( )
𝑒−𝑐𝑡
Where:
𝑃 𝑡() =𝑆𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝑎 𝑡𝑖𝑚𝑒, 𝑡
𝐾=𝑇ℎ𝑒 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 (𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑚𝑎𝑡𝑖𝑐 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛)
𝑐=𝑇ℎ𝑒 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟
3.3. Parameter Estimation
The parameters of each of the two models above would be estimated through a non-linear
regression approach from the population dataset. The least square method would be applied to
reduce the size of the error existing between the actual and the modeled population sizes. The
models will be plotted using the LoggerPro application software.
4
4. Results
Table 1: Population of the UK between 2014, and 2024
Year
Time Factor (t)
Population Size
2014
0
64,773,504
2015
1
65,224,364
2016
2
65,655,203
2017
3
66,064,804
2018
4
66,432,993
2019
5
66,778,659
2020
6
67,059,474
2021
7
67,281,039
2022
8
67,508,936
2023
9
67,736,802
Source:
https://www.macrotrends.net/global-metrics/countries/GBR/united-kingdom/population#:~
:text=The%20population%20of%20U.K.%20in,a%200.33%25%20increase%20from%20
2020.
4.1. Parameter Estimation
For the exponential model, the datasets corresponding to 2014 (t = 0), and 2018 (t = 4) were used
to determine the growth rate of the population, such that:
𝑃 𝑡() =𝑃0𝑒𝑟𝑡
𝑃 𝑡() =66,432, 998, 𝑤ℎ𝑒𝑛 𝑡 𝑖𝑠 4
5
𝑃0=64,773, 504, 𝑤ℎ𝑒𝑛 𝑡 𝑖𝑠 0
A non-linear regression approach with natural logarithms was then applied on the exponential
model above:
𝐿𝑛 𝑃 𝑡() =𝐿𝑛𝑃0+𝑟𝑡
𝑟𝑡=𝐿𝑛 𝑃 𝑡()−𝐿𝑛𝑃0
→𝑟= 𝐿𝑛 𝑃 𝑡()−𝐿𝑛𝑃0
𝑡
=66,432,998−64,773,504
4
=0.006324
Thus, the exponential model for predicting the population of the United Kingdom would be:
𝑃 𝑡() =64,773, 504𝑒0.006324𝑡
For sample calculation, when t = 9 (year 2023):
𝑃 𝑡() =64,773, 504𝑒0.006324(9)
=68,566,686.43
≅68,566,682
For the Gompertz model, the datasets corresponding to 2014 (t = 0), and 2018 (t = 4) were used
to determine the growth rate of the population, such that:
𝑃𝑡()=𝐾𝑒−𝐾
𝑃0
( )
𝑒−𝑐𝑡
𝐾=69, 500,000
𝑃 𝑡() =66,432, 998, 𝑤ℎ𝑒𝑛 𝑡 𝑖𝑠 4
𝑃0=64,773, 504, 𝑤ℎ𝑒𝑛 𝑡 𝑖𝑠 0
6
66,432,998=69,500,000𝑒−69,500,000
64,773,504
( )
𝑒−4𝑐
66,432,998
69,500,000 =𝑒−69,500,000
64,773,504
( )
𝑒−4𝑐
0.9558=𝑒−1.073𝑒−4𝑐
0.9558=0.345𝑒−4𝑐
0.9558
0.315 =𝑒−4𝑐
2.795=𝑒−4𝑐
Introducing the concepts of natural logarithm, the value of parameter, c, could be obtained:
𝐿𝑛 2. 795=−4𝑐
→𝑐= 𝐿𝑛2.795
−4
Thus, the Gompertz model for predicting the population of the United Kingdom would be:
𝑃 𝑡() =64,773, 504𝑒0.006324𝑡
=−0.2569
𝑃𝑡()=69,500,000𝑒−1.073𝑒0.2569𝑡
For sample calculation, when t = 9 (year 2023):
𝑃𝑡()=69,500,000𝑒−1.073𝑒0.2569(9)
=67,725,567
Table 2: The Actual and Modeled Population of the UK between 2014, and 2024
Time Factor (t)
Population Size
Actual Population
Exponential Model
Gompertz Model
0
64,773,504
64773504
64773504
1
65,224,364
65184387
65241727
2
65,655,203
65597876
65674896
7
3
66,064,804
66013988
66073011
4
66,432,993
66432740
66436072
5
66,778,659
66854148
66764079
6
67,059,474
67278229
67057032
7
67,281,039
67705001
67314931
8
67,508,936
68134479
67537776
9
67,736,802
68566682
67725567
4.2. Model Predictions
The Gompertz and the exponential models were then plotted on the same cartesian with the
actual datasets, as displayed in Figure 1 below. The models were graphed using the LoggerPro
application software.
Figure 1: A Graph of the Population of the United Kingdom against Time
8
From the graph, it was evidenced that the Gompertz function was the most accurate in modeling
the population of the United Kingdom between 2015, and 2020, as it was very close to the trend
assumed by the plot of the actual data.
4. Discussion
The findings of this study showed that it would be possible to model the population of the
United Kingdom using functions. The graphs obtained revealed the significant differences
between population prediction using the Gompertz and Exponential models. Even though the
two models showed that the population of the country has been on an increasing trajectory from
2014, the Gompertz function was the most appropriate as it was very close to the trend assumed
by the plot of the actual data. The trend of the Gompertz function provides significant
implications for resource management and public policies in the United Kingdom. Moreover,
understanding the existing limitations in each of the mathematical models would help in
developing informed decisions on the infrastructure, education, and healthcare for future changes
in population.
5. Conclusion
In sum, this study has presented a mathematical approach to analyzing the existing
population dynamics in the United Kingdom. The exponential model and the Gompertz model
were considered. The Gompertz function was the most accurate in modeling the population of
the United Kingdom between 2014, and 2020, as it was very close to the trend assumed by the
plot of the actual data. Future studies could be developed to incorporate different factors,
including birth rates, death rates, and migration, among others.
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6. References
Cangiano, A. (2023, February 27). The Impact of Migration on UK Population Growth -
Migration Observatory. Migration Observatory.
https://migrationobservatory.ox.ac.uk/resources/briefings/the-impact-of-migration-on-uk-
population-growth/
Inigo, M. (2019, August 2). 4.2: Exponential Growth. Mathematics LibreTexts.
https://math.libretexts.org/Bookshelves/Applied_Mathematics/Book%3A_College_Math
ematics_for_Everyday_Life_(Inigo_et_al)/04%3A_Growth/4.02%3A_Exponential_Gro
wth
Lumen Learning. (2020). 4.2 Population Growth and Regulation | Environmental Biology.
Courses.lumenlearning.com.
https://courses.lumenlearning.com/suny-environmentalbiology/chapter/4-2-population-gr
owth-and-regulation/
McKee, M., Dunnell, K., Anderson, M., Brayne, C., Charlesworth, A., Johnston-Webber, C.,
Knapp, M., McGuire, A., Newton, J. N., Taylor, D., & Watt, R. G. (2021). The changing
health needs of the UK population. The Lancet,397(10288), 1979–1991.
https://pubmed.ncbi.nlm.nih.gov/33965065/
Tjørve, K. M. C., & Tjørve, E. (2019). The use of Gompertz models in growth analyses, and new
Gompertz-model approach: An addition to the Unified-Richards family. PLOS ONE,
12(6), e0178691. https://doi.org/10.1371/journal.pone.0178691
Octobre 28, 2024
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Academic level:
Graduate
Type of paper:
Research paper
Discipline:
Sociology
Citation:
IEEE
Pages:
1100 (4 words)
* The sample essays are for browsing purposes only and are not to be submitted as original work to avoid issues with plagiarism.
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