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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)
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