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. Also, to save you some work, you can download an Excel spreadsheet with the U. S. census data.

Question 1: This problem uses an applet similar to the one in the lecture notes. You begin by finding the best fit of the Discrete Malthusian growth model to the data by first adjusting the range of the dates, then adjusting the growth rate. After finding the smallest sum of squares error, you write down this number and the best value for the growth rate. Next 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.) Be sure to include the dates for the decades following the data you analyzed with the applet, as you will be extending your model to these dates. You will put the model in the third column with its initial population agreeing with the actual data, so in C2 write =B2. Say you got a growth rate of 0.263 from the applet, then in C3, you type = 1.263*C2. From there you fill down to complete the simulation of the Malthusian growth model. Create your graph using Chart Wizard, highlighting the first 3 columns. Convert the data points to points and the model to a line, then add the appropriate labels.

Question 2:

Question 3: This problem asks you to repeat what I have 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. Be sure to answer all of the questions carefully.

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). I would suggest that you copy all but the last date again into another column, say Column G. (I like to have different computations away from the main table of data. Note that A2 = G2 are the same dates.) In Column H, you start in H2 and enter "=B3/B2-1" to let Excel compute the growth rate associated with the first date. You simply fill down from here to obtain all the growth rates for your country. At the bottom of the calculations in Column H you can use the Excel AVERAGE command to compute the average growth rate for the country. (I find it easiest to simply put this average value in Column I and fill down, so that I can highlight columns G, H, and I to produce the graph like the one in the notes.) Use trendline on the growth rate data to find the best straight line. In order to get good results, it is very important that after you use trendline to find the best straight line through the growth data, you double click (or right click) on the equation (be careful here), then when the Format menu pops up, you can choose "number" of the 3 folders. Under number, you select "number" again, which should give you a menu to select the way you want your numbers displayed. You should use the scientific notation option. Its this equation that will determine the behavior of your nonautonomous model, so you need 4 significant figures for an accurate model.

Now you return to the Columns A and B in your data set. In Column C, you repeat the procedure you did in Question 1 above using the average value just computed for your growth rate r. In Column D, you will produce the nonautonomous model using the equation you just found from trendline. Say trendline found the equation k(t) = 3.148 - 0.002347 t, then after the first entry in D2 (which is the same as B2 and C2), in D3, you enter "=(1 + 3.148 - 0.002347*A2)*D2." You fill down from here to complete the nonautonomous model. You may want to use my discrete template with the nonautonomous model as a guide, making the appropriate changes. From there you simply need to create tables and graphs of the models and answer the questions posed in this Lab question.