This lab examines the Malthusian growth models that we are studying in lecture. You will want to be familiar with the material in the lecture notes to work this lab. You may want to download the Excel spreadsheet for discrete dynamical systems to have a blueprint for how to work Malthusian growth models on Excel spreadsheets. The first question examines basic models of Malthusian growth and can largely be completed with minimal use of the computer. The second problem examines the U. S. population over its history and considers two methods of fitting census data to Malthusian growth models. The last question examines the nonautonomous models studied in class and applies those techniques to some European countries.

Question 1: This problem looks at data on populations of two countries. You consider basic Malthusian growth models to find rate of growth, doubling times, and approximate when the populations might be equal in size. The calculations for this problem are similar to the types of calculations you might be expected to perform on an exam, so this problem (except for the graphs) might best be done using nothing more than a basic calculator.

Question 2: This problem expands our work with the Malthusian growth model. We investigate the growth of the U. S. population over two periods of time with our Malthusian growth model. Furthermore, we fit the data with two different techniques. Recall that the Malthusian growth model is given by

P_n+1 = (1 + r)P_n ,

which given an initial population, P₀, has the solution

P_n = (1 + r)ⁿP₀.

We want to find the best fitting parameters P₀ and r for this model for the population data over a given interval of time.

We are given a range of data of the U. S. population that we want to fit with a Malthusian growth model. You create a table in Excel with the appropriate range of dates and their corresponding populations. (Say these are in Columns A and B with the first entries in A2 and B2. I like to label my columns so the labels are in A1 and B1.) Next we insert a column between A and B, so that we have a column for the time n in years after the starting date. Thus, Column B is labeled n and starts with 0 and goes in steps of 10 years, while Column C contains the census data. Create a graph using Chart Wizard from Columns B and C. The next step is to add Trendline, selecting the Exponential fit. This is actually a straight line fit to the logarithm of the y-data or a semi-log fit. As always, you select the option to display the equation on the chart and adjust the formula to have enough significant digits. This equation will have the form:

y = ce^kx,

where the constants c and k are determined by Excel. The solution of the Malthusian growth model has this form if you think P_n = y and n = x. It follows the the constant c = P₀ and e^k = 1 + r. It follows that the exponential fit by Excel easily gives you your values of P₀ and r. To find the sum of square errors, you use the Malthusian growth model to fill down the model values in Column D. Then in Column E, it is easy to compute the square error between the data in Column C and the model in Column D. These are summed at the bottom of Column E to give you the sum of square errors.

The next model fit is the nonlinear least squares best fit to the data. Copy the first 3 columns above (Columns A, B, and C), so that you have the population data and the time n. The next step is to create say in Cells G1 and G2 the names P₀ and r. In Cells H1 and H2 you insert reasonable guesses for these values (perhaps the ones you just found from the Trendline Exponential fit.) Proceed to name these variables so they can be used in your model. In Column D, construct the Malthusian growth model using your named variables. Once again, use Column E to compute the square error between the model and the data, then sum these square errors at the bottom of this column. Click on the cell with the sum of square errors. Go to the Main menu bar and click on Tools. Under Tools, you should find Excel's Solver. (If it is not there, then you may have to go to the option to add this tool.) A window should pop up and your sum of square errors cell should appear in the window for Set Target Cell:. Next you check that you want Equal to: Min. Finally, you click on the box labeled By Changing Cells: and you simply highlight the Cells H1 and H2 to include the parameters that you want Excel to change. After this is completed, you click on the button Solve. You should see the values of P₀ and r change and the sum of square errors get smaller. This is you new model best fit, the nonlinear least square best fit to the data. Answering the remainder of the questions should be fairly easy from the work we have done in class.

Question 3: This problem asks you to repeat what was done in the notes with the U. S. census data to census data from another country. You will be producing similar graphs (see the graph titled "Growth Rate for U. S." and the following graph called "Discrete Growth Models for U. S."), which should give you a guideline. One modification in this Lab Problem is adjusting the starting date (1950) to t = 0.

Use the power of Excel to find your growth rates. Suppose that once again you have entered the dates in Column A (starting in A2) and the population values in Column B (starting in B2). Insert a column between the dates and the census data, then enter the adjusted times t with t measuring the years after 1950 (so t = 0 represents 1950). Pull down these data and times to cover the time period from 1950 to 2050 (with many dates obviously not having data but will be used in you simulation). You will use Columns D-I for simulating the models, computing sum of square errors, and percent errors.

Below the data and space for model simulation, you want Excel to compute the growth rates that will be used for your first graph and the information needed for the model simulations. You could label A14 as t, then fill in the appropriate values for the times, t = 0, 10,...40, in Column A for use with the appropriate growth rates, which you compute in Column B. Label in B14 k, then below this compute the growth rates by entering "= C3/C2 - 1" to let Excel compute the growth rate associated with t = 0. Pull this formula down to obtain the subsequent growth rates associated with their respective times. At the bottom of this Column B you can use Excel's AVERAGE command to find the average growth rate, which will be used in your discrete Malthusian growth model. For graphing purposes, you should simply copy this one value in Column C next to your computed growth rates. Next you graph Columns A-C. Use Excel's Trendline (Linear) to find the best staight line fit to your growth data, which becomes your k(t) for the nonautonomous model. Once again it is very important to adjust this model growth formula to have 5 or 6 significant figures. This graph is adjusted to look similar to the one in the class notes for the U. S. growth rate.

The next stage in this problem is the simulation of the discrete Malthusian growth model using the average growth rate, r, and the nonautonomous growth model using the growth function k(t) that you found with Trendline. In Column D you simulate the Discrete Malthusian growth model by using the 1950 population in D2 (assuming D1 contains a model label). In D3 you insert the formula for the Malthusian growth model "= (1 + r)*D2," where you insert your value of r. This formula is pulled down to give you the model simulation. You can use Columns E and F to compute the sum of square errors and the percent errors needed for the WeBWorK problem.

The last part of the problem (not counting the graphing part) is the simulation of the nonautonomous Malthusian growth model. Start this model similar to the discrete Malthusian growth model with a label in G1 and the 1950 population in G2. The nonautonomous Malthusian growth model uses k(t), which you found earlier and depends on the t values in Column B. In is very important that the formula you put in G3 has all values coming from the second row. That is, if k(t) = a + bt, where a and b are determined by Trendline, then the formula that you put in G3 is given by "= (1 + a + b*B2)*G2." Once again, this discrete dynamical model is readily simulated in Excel by simply pulling down this formula. As above, you use Columns H and I to compute the sum of square errors and the percent errors needed for the WeBWorK problem.