In this lab, we will investigate population estimates of the number of humans on Earth at various points in human (pre-)history, and consider how closely these estimates match the exponential model for population growth that we learned in Chapter 4.
By the end of this lab, you will be able to:
Work through the following lab. For each of the Exercise blocks, you should:
You are encouraged to work together, but you should state at the top of the sheet with your written answers who you worked with.
If you have any questions, please email me with the Subject line: MA–115 Lab 2 Help.
Our goal for this lab is to see how well an exponential model captures the estimated number of humans alive over the course of human (pre-)history. We discussed in class that the exponential model for population growth works well for populations of bacteria or lily ponds, up to a point. However, we might imagine that this model will break down for more complex organisms, like humans, since:
In this lab, we will see how the exponential model bumps up against these constraints when used for human population data.
We will be considering a collection of estimates for the number of humans alive on Earth at various points in the distant and recent past. These are, of course, estimates, since, for example, no one knows exactly how many people were alive 10000 years before the Common Era. Even today, we do not know how many humans are alive at any given instant, and that number is changing from instant to instant as more humans are born or die.
You can find the original data set from which I created the Excel file for this lab here, along with a description of the data set.
Exercise 1: Read the linked article through the section labeled: Time taken for population to increase by one billion. (The entire article is well-worth reading, but you won’t need anything past that section to finish the rest of this lab.)
Question 1: What are the three periods of population growth the article discusses? Which period are we in now?
Open the file MA115-Lab2.xlsx linked to here. Important: Be sure to save the file to your desktop and regularly save the file as you work through this Lab.
You should start on the Entire Data Set tab, which contains human population estimates in column B at various times relative to the year 1 CE in column A. Note: The population estimates are given in millions of people.
Exercise 2: Look at the first two columns of the spreadsheet.
Question 2: What are the estimates for the number of humans alive 10000 years prior to 0 CE? What is the estimate for the number of humans alive in 2015 CE? Does this match what you would expect?
Recall that our model for the population size \(N(t)\) of a given population at time \(t\) that doubles with a fixed doubling time \(a\) starting with a population size of \(N(0) = N_{0}\) is
\[N(t) = N_{0} 2^{t/a}\]
while the model in terms of the proportional growth rate \(r\) is
\[N(t) = N_{0} e^{rt}.\]
Exercise 3: Create a plot of the human population \(N(t)\) as a function of the time \(t\) in years since 0 CE.
Remember that to create a plot in Excel, you select the columns you want to plot on the horizontal and vertical axes, and then select Insert > Charts, where for this plot you should select the Scatter chart option.
Question 3: Based on the plot of human population size as a function of time, does the human population size look more like an exponential or logarithmic function? Explain.
Notice that if we take the logarithm base–2 of both sides of the equation \(N(t) = N_{0} 2^{t/a}\), we get
\[\begin{align}
\log_{2} N(t) &= \log_{2} N_{0} 2^{t/a} \\
\log_{2} N(t) &= \log_{2} N_{0} + \log_{2} 2^{t/a} \\
\log_{2} N(t) &= \log_{2} N_{0} + t/a \\
\end{align}\]
Thus, for the exponential model, the log-population size \(\log_{2} N(t)\) is a linear function of time \(t\), so when the exponential model is correct, if we plot the logarithm of the population size against time, we should get a straight line, since \(\log_{2} N(t) = m t + b\) where \(m = 1/a\) and \(b = \log_{2} N_{0}\).
Exercise 4: Create a new column in the Excel spreadsheet, labeled log2(Population). To populate that column with \(\log_{2}(\text{Population})\), you should enter
=log(B2)/log(2)
in the first numerical cell of the new column to get the logarithm base–2 of that value. Populate the rest of the cells in the new column with the log-population.
Now, create a plot of the log-population as a function of the time in years since 0 CE.
Question 4: Based on the plot of the log-population versus time, does the exponential model seem like reasonable model for human population growth? Why or why not?
We now have two equivalent representations of the model for \(N(t)\):
\[N(t) = N_{0} 2^{t/a}\]
in terms of the population and
\[\log_{2} N(t) = \log_{2} N_{0} + t/a\]
in terms of the log-population. Both equations have two unknown parameters: \(N_{0}\), the number of humans alive when \(t = 0\), and \(a\), the doubling time for the human population.
Excel can determine a best estimate of both of these parameters based on the available data. To do so, Excel will attempt to minimize the difference between the predictions of the model \(N(t)\) and the estimated population values at each available time point. Excel provides these estimates through its trendline functionality.
Exercise 5: We will first add a trendline to the plot of human population versus time. Right-click (or Control-click, if you are using a Mac) on one of the points in the plot of human population versus time. In the context menu that pops up, select Add Trendline…, then in the context menu that now appears on the right of your screen, select Exponential under the Trendline Options header. This adds a dotted curve-of-best-fit to your plot. Now select the Display Equation on chart in the bottom of the context menu that now appears on the right of your screen. This adds the equation for the exponential curve, in the form \(y = C e^{rx}\). So Excel’s \(x\) is what we’re calling \(t\).
Question 5: What is the function that Excel returns? Is this given in terms of a doubling time model or a proportional growth rate model?
Question 6: Based on the function that Excel returned, what was \(N_{0}\), the number of humans alive when \(t = 0\)?
Question 7: Determine the doubling time \(a\) based on the proportional growth rate \(r\) returned by Excel.
Question 8: Based on the exponential model returned by Excel, how many humans were alive in 1 AD? Does this match the estimates for the number of humans alive in 1 AD?
Exercise 6: We now add the trendline to the plot of the log-population versus time. Right-click (or Control-click, if you are using a Mac) on one of the points in the plot of log-population versus time. In the context menu that pops up, select Add Trendline…. In the context menu that now appears on the right of your screen, select Linear under the Trendline Options header. This adds a dotted curve-of-best-fit to your plot. Now select the Display Equation on chart in the bottom of the context menu that now appears on the right of your screen. This adds the equation for the linear curve to the plot, in the form \(y = mx + b\). Again, what Excel is calling \(x\) is our \(t\), and we need to relate \(m\) and \(b\) to our original function.
Question 9: The coefficient of \(x\) returned by Excel is \(1/a\), the reciprocal of the doubling time \(a\). What is the doubling time \(a\) in years? Does this match the value you determined in the previous exercise? Hint: It should!
Question 10: The constant term returned by Excel is \(\log_{2} N_{0}\). What is \(N_{0}\) then? Does this match the value of \(N_{0}\) you determined in the previous exercise? Hint: It should!
Just by looking at either the exponential model for the population versus time or the linear model for the log-population versus time, we see that the model does a poor job of predicting the estimated population values.
Exercise 7:
Question 11: What is the predicted population based on the exponential model \(N(t)\) for the number of humans alive in 2015 CE? Does this match the estimated number of humans alive in 2015 CE? What is the difference between the predicted value and the estimated value?
Remember from the article from Exercise 1 that human population growth can be divided into three periods: pre-modernity, modernity, and the current period. In the next section of the lab, we will construct models for human population before and after the Industrial Revolution.
The Industrial Revolution was a period of rapid change from pre-industrial manufacturing methods (hand-made products, limited use of chemical processing, man- or animal-powered labor) to industrial manufactoring methods (machine-made products, extensive use of chemical processing, and machine-powered labor) that occurred between 1760 and 1840.
Let’s the divide the data in half, and consider the human population before the industrial revolution and after the industrial revolution.
Exercise 8: Investigate the Excel sheets Pre-Industrial Revolution and Post-Industrial Revolution.
Question 12: What ranges of years are included in each sheet?
We will now repeat the analysis from the previous section separately for each of the periods before / after the Industrial Revolution.
Exercise 9: For each of the Pre-Industrial Revolution and Post-Industrial Revolution sheets (separately!), determine the parameters \(N_{0}\) and \(a\) for the exponential model
\[N(t) = N_{0} 2^{t/a}\]
by fitting a Linear Trendline to the plots of log2(Population) versus time, and extracting \(N_{0}\) and \(a\) from the equation for the trendline.
Question 13: Was the doubling time \(a\) larger or smaller prior to the industrial revolution? What does this mean in terms of the rate of population growth: was the population growing faster before or after the industrial revolution?
Question 14: Does the \(N_{0}\) parameter still correspond to the number of humans at \(t = 0\) for each of the two models you determined? Why or why not?
Question 15: If the doubling time had remained at the pre-Industrial Revolution value, how many humans would be on Earth in 2015? How does this compare to the estimated number of humans on Earth in 2015?
Question 16: Does the exponential growth model do a good job of describing the human population for the post-Industrial Revolution time period? If not, where do their appear to be ‘kinks’ in the log2(Population)? Can you relate this back to the description of different eras of population growth in the original article from Exercise 1?