The research question for this extended essay is “To what extent are differential equations an accurate representation of human population modelling?” Differential equations can effectively be used to predict things in our everyday lives. They are used in many disciplines including biology and physics.
In this extended essay, I will collect data on the Canadian population from the 1900s to the 2000s and compare it to predicted populations given by two models: Malthusian (exponential) and Logistic. I will also analyze the effectiveness of these two models for accuracy and their limitations. In this analysis, I will use statistical tests to evaluate data significance between the actual population data for Canada versus the predicted populations. I will also do analysis on what carrying capacity1 will accurately represent the Canadian population and what this value means. My goal for this exploration is to determine the effectiveness of differential equations in modelling human population growth as well as any ways to improve them. Furthermore, I hope to enhance my skills in mathematics, not only differential equations but its application in our real world.
1 Carrying capacity is the maximum population an organism can sustain denoted by K in the logistic differential equation.
Introduction
In our globalized world, the population is on a rapid increase and it is now important for countries to do the right predictions for the future of their native people as well as their immigrants. Acquiring knowledge about population is necessary for future planning, concerning education, health, job, housing, safety requirements, etc. However, in our current society, a problem which many arise is overpopulation. Overpopulation is where the human population exceeds the carrying capacity of Earth. Overpopulation is caused by a number of factors. Reduced mortality rate, better medical facilities, depletion of precious resources are few of the causes which result in overpopulation. I have also learned about the effect of an increasing population on the environment in many of my classes and the sustainability of the earth. Therefore, I decided to undergo this exploration to see if population growth can be effectively modelled with differential equations to sustain our environment in the future.
Differential equations are simply equations that relate functions with their derivatives. Most continuous models of population dynamics are based on differential equations, which can be solved using a variety of techniques, which will be omitted from this exploration. However, only the simplest of models are algebraically so I will be using the two types of first-order differential equations to predict the population evolution of Canada. Each of the two will be compared for accuracy by coefficient of determination and any ways to improve the model. My dependent variable is the population size of Canada and my independent variable is time. The controlled variable is the country
Table 1 below shows the Canadian Population2 from 1900 to 2000 recorded every decade. The predicted data will be created using the initial population in 1900 and then compared to the actual data in table 1.
Table 1: Actual Canadian Populations from 1900 to 2000
Year Population
1900 5,310,000
1910 6,988,000
1920 8,435,000
1930 10,208,000
1940 11,382,000
1950 13,382,000
1960 17,870,000
1970 21,297,000
1980 24,517,000
1990 27,512,000
2000 30,689,000
2 Censuses of Canada 1665 to 1871: Estimated Population of Canada, 1605 to Present, Statistics Canada, www150.statcan.gc.ca/n1/pub/98-187-x/4151287-eng.htm.
Malthusian Growth Model
A Malthusian growth model or more commonly called a simple exponential model is the population model where growth occurs exponentially, so it increases occurring to the birth rate of the populous. This means the growth rate stays the same regardless of population size, making the population grow faster and faster as it gets larger. This model was introduced by Thomas Malthus in “An Essay on the Principle of Population” in 1798. In this essay, he mentioned that it is only a matter of time before the world population will become too large to the point where it can feed itself.
The Malthusian Model in the form of a first-order differential equation:
dP/dt = kP
Where dP/dt is the population growth rate, which is a measure of the number of individuals added per unit of time, k is the per capita rate of increase or per capita growth rate. If k is positive, the populous is experiencing exponential growth and if it is negative the populous is experiencing exponential decay. The variable P is the population size, or simply the number of individuals in the population at a given t. In this model, we make some assumptions to validate the relationship. Firstly, we assume that the population is living in ideal living conditions including unlimited resources, no factors affecting mortality rate, no land claims etc. We also assume that for at any time t, the growth rate remains constant regardless of the population growth.
We can solve this differential equation for P(t) to give the population size at any time
dP/dt = kP (Multiply both sides by dt)
dP = kP • dt (Divide both sides by P)
1/P dP = k • dt (integrate both sides)
 1/P dP =  k • dt (apply rules of integration)
ln | P | = kt + C
eln | P | = ekt + c (apply exponent rules)
P(t) = Cekt (simplify as ec a constant)
P(0) = Cek(0) (Further simplification)
P0 = C
Therefore,
P(t) = P0ekt
where P0 is the initial population
Constructing The Prediction Model:
Given the base year being 1900 we can use the actual data for population where
P(0) = 5,310,000 and P(10) = 6,988,000 to solve for k.
Substituting respective values gives:
6,988,000 = 5,310,000ek(10)
k = 1/10 ln (6,988,000/5,310,000)
k = 0.0274602557
Therefore the model we will use to predict the Canadian population from 1900 to 2000 is written as:
P(t) = 5,310,000e0.0274602557t
Using the formula we derived above, we can construct the following table:
Table 2: Predicted Population of Canada Using the Malthusian Growth Model
Year Predicted Population
1900 N/A
1920 9,196,260
1930 12,102,348
1940 15,926,781
1950 20,959,764
1960 27,583,207
1970 36,299,709
1980 47,770,690
1990 62,866,589
2000 82,732,904
The actual data and the predicted data can be graphically presented as seen below:
Graph 1
Evaluation:
As seen in the exponential model over predicts the growth rate of the populous. The rate of change gradually increases exponentially however, the actual population increases at a much lower rate. This is due the non-accountability for things that could limit population growth. There are also many assumptions being taken into account for this model such as continuous reproduction (no seasonality), all organisms are the same (no age structure), and the resources are unlimited. This is why the exponential model is unrealistic and not ideal to utilize for long term predictions.
Coefficient of Determination:
The coefficient of determination or R-squared is the percentage of the dependent variable variation that a linear model explains.3 It provides a measure of how correlated a group of actual data is to the predicted data. In this case, we will use it to see how accurate the predicted models are in terms of each other.
R-squared is calculated by applying the following formula:
R^2 = (Variance Explained by the Model)/(Total Variance)
To find the relationship, we will plot the Actual Data as the independent variable, and the predicted data as the dependent variable. This will display the proportion of the variances and we can solve for the according R^2 value. R^2 values are between 0 and 1 and a predicted model which accurately fits the actual data will have a value as close as possible to 1.
The graph for the Actual Data vs the Predicted data by the Malthusian equation can be seen in Graph 2:
3 Frost, Jim, et al. “How To Interpret R-Squared in Regression Analysis.” Statistics By Jim, 30 May 2019, statisticsbyjim.com/regression/interpret-r-squared-regression
Graph 2
Our Actual vs Predicted graph returned a value of about 0.947. This value means it is relatively well predicted but generally our model should return R^2≥0.95. We will not compare this R^2 to the logistic model to see which better predicts the population dynamic of Canada.
Logistic Model
The Malthusian growth model is appropriate for population growth under ideal conditions, but it is important to recognize that a more realistic model must reflect that there are limited resources. Although exponential growth can occur in environments with a very low population and many resources oftentimes, the population will increase or resources run low. The validity of the Malthusian model was investigated in the 1800’s until the Belgian Pierre-Francois Verhulst proposed a revised model that would take into account the occurrence of exponential growth. This model recognizes that when the population increases, there is a tendency for the populous to interfere and alter the number of resources, for example, fighting for food, land, wars. When resources are limited populations are said to be exhibiting logistic growth. In logistic growth, the populous starts by increasing in an exponential manner, but the population levels off as it approaches its carrying capacity, K (the maximum population size that can be supported by an environment). At carrying capacity, it is expected that the
population will neither shrink nor grow, meaning that the population versus time graph will level out. To modify the Malthusian model to take into account for the carrying capacity, Verhulst shows the per capita growth rate, k is proportional to the population size P and its difference from the carrying capacity, K to give the differential equation:
dP/dt = kP ( 1 - P/K )
Where K is carrying capacity, P is the population, k is the per capita growth rate, and dP/dt is the population growth rate.
In the Malthusian model, population growth was mainly dependent on P, as each human added to the populous was contributing to growth equally since the value of k was fixed. However, in the logistic model, the resources available are taken into account so as the population increases, fewer resources are available which causes the birth rate to gradually decrease. We can solve this differential equation for P(t) to create an expression for population size at any given t.
dP/dt = kP ( 1 - P/K ) can be written as:
dP/dt = (kP(K-P))/K (multiply both sides by P(K-P))
dP/(P(K-P)) = k/Kdt (multiply both sides by K and integrate)
 K/(P(K-P)) dP = k dt (Apply partial fraction and integration rules respectively)
ln | P | - ln | K – P | = kt + C
ln | P/(K-P) | = kt + C
P/(K-P) = Cekt
To solve for P:
P = KCekt – PCekt
P + PCekt = KCekt
P(t) = □((KCe^kt)/(1+Ce^kt ))
Now solving for C:
To find the constant C, we represent the initial population as P(0) = P0
P0 = KC/(K+C)
P0 + C P0 = KC
P0 = KC - C P0
C = (P0 )/(K- P0 )
Subbing C back into P(t) gives us the solution for our logistic model:
P(t) = 〖KP〗_0/(P_0+(K- P_(0)) e^(-kt) )
Using over solution of the logistic differential equation we can now create a model to predict the Canadian Population given base year 1900. The carrying capacity of humans is difficult to determine as we have developed new ways of agriculture and other technologies. These revolutions have increased our carrying capacity however, it the earths predicted carrying capacity still is expected to be about 10billion4 so for this prediction model we will start with 300 Million as the carrying capacity of Canada and later alter the value to see what accurately displays the Canadian population growth pattern. Using K as 300,000,000, the derived expression for P(t), and two known points from the given data we can create our model for prediction.
4 “How Many People Can Earth Support?” LiveScience, Purch, www.livescience.com/16493-people-planet-earth-support.html.
Using: P(0) = 5,310,000 and P(10) = 6,988,000, K = 300,000,000 , and P0 = 5,310,000 we form the equation:
6,988,000 = ((300,000,000)(5,310,000))/((5,310,000)+(300,000,000-5,310,000)e^(-10k) )
6,988,000 = (1.593  10^15)/((5,310,000)+(29469000)e^(-10k) )
k = - ln⁡〖(0.7555472564)〗/10
k  0.02803129494
Finally, the equation to solve for population at any given time t with a carrying capacity of 300,000,000 is depicted as:
P(t) = ((300,000,000)(5,310,000))/((5,310,000)+(300,000,000- 5,310,000)e^(-0.02803129494t) )
P(t) = (1.593  10^15)/((5,310,000)+(294,690,000)e^(-0.02803129494t) )
The constructed Logistic model’s predicted populations are displayed in the table 3:
Table 3: Predicted Population of Canada Using the Logistic Growth Model
1920 9,179,743
1930 12,030,687
1940 15,719,189
1950 20,458,212
1960 26,492,812
1970 34,090,373
1980 43,520,058
1990 54,018,393
2000 65,740,462
The actual data and the predicted data from this model can be graphically presented as seen below:
Graph 3
The logistic model, similar to the Malthusian model over predicts the model but by much less. This more improved model takes into account the carrying capacity of the populous and therefore is approaching a levelled off population however, it is too high. We will once again the coefficient of determination to see how accurate the predicted model is.
The logistic model has an R^2 above 0.95 which is defined to be a correlated prediction model.
Evaluation of Both Models:
After evaluating both models individually we can clearly see that the logistic model gave us a more accurate prediction for the population of Canada as it has a higher R^2 value. The three datasets we now have can be illustrated on one plot for further analysis.
As seen in the graph, both models predict the population relatively close until about 1940 where both the lines overpredict the population growth rate. The model cannot effectively predict sudden jumps in the population such as seen from 1950 to 1960The Malthusian model’s plot increases exponentially, and population always grows larger and larger without any finite limit thus, this model is not appropriate to use for long periods of time.
The overall trend of the actual population is increasing gradually with no sudden changes and therefore our model can be improved to better represent this data.
The Malthusian model also only has one parameter, time, and therefore cannot be modified to more accurately represent the trend. However, the Logistic model has K, carrying capacity has another parameter which we can alter to correctly illustrate the population growth. In our original model, we used a carrying capacity of 300 million which may be unrealistic in the 1900 to 2000 time period. Therefore, we will choose a more realistic carrying capacity of Canada by referring to the actual data.
The population of Canada in 2000 was 30,689,000 and therefore we will set our carrying capacity, K to 31 million.
We will once again use our formula for P(t):
6,988,000 = ((3,000,000)(5,310,000))/((5,310,000)+(300,000,000-5,310,000)e^(-10k) )
6,988,000 = (1.6461  10^14)/((5,310,000)+(25690000)e^(-10k) )
k = - ln⁡〖(0.7102411898)〗/10
k  0.03421506627
Which forms our prediction model:
P(t) = (1.6461  10^14)/((5,310,000)+(25690000)e^(-0.03421506627t) )
The constructed Logistic model’s predicted populations are displayed in the table 4:
Table 4: Predicted Population of Canada Using the Improved Logistic Growth Model
1920 9,009,734
1930 11,340,690
1940 13,893,653
1950 16,537,821
1960 19,122,614
1970 21,510,440
1980 23,603,796
1990 25,356,414
2000 26,768,069
From the predicted curve we can tell it is much more accurate compared to the Malthusian and original logistic. The population is underpredicted after 1970 which could be due to the fact that carrying capacity can be increased as a populous develops new technologies or ways to sustain resources. We will once again the coefficient of determination to see how accurate the predicted model is.
The new logistic model has an R^2 above 0.95 which is defined to be a correlated prediction model and it is higher than the previous one with a carrying capacity of 300 million.
Assumptions/ Problems
In this exploration, we were able to create an accurate model with a high R-squared value to show correlation however, there are possible errors in this model. To improve our model, we used the actual data’s final value to approximate the carrying capacity. This would not be an ideal model for population prediction as we do not have future value’s if we were not doing past years predictions. If given an accurate carrying capacity of humans for the future the logistic model could be applied to predict future populations however, we have limited access to data and limited knowledge for the carrying capacity. The logistic model also cannot take into account any advancements or changes to carrying capacity as a populous advance and this would furthermore make it inaccurate.
Conclusion
In this extended essay, differential equations were used and discovered to be accurate in predicting population dynamics of humans. The Malthusian model to predict the population was not very accurate as it began overpredicting more and more. This is due to the fact that it simply predicts the population growth is infinite. This model could be better applied in the population growth of bacteria or any microorganisms as they do not have as much limitations as humans. On the other hand, the Logistic Model effectively predicted the population once we gave it an accurate carrying capacity, K. The carrying capacity takes into account many of the factors that could slow down population growth which the Malthusian model did not for example limitation of resources and/ or pollution. Therefore, if we are able to determine an accurate carrying capacity of a country or populous, we can use the logistic equation to accurately predict the growth.