This lab examines the Malthusian growth models that we are studying in lecture along with a breathing model. 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.

Problem 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.

Problem 2: 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. (Note: there is one fewer growth rates than census data, since these are ratios of the census data.) 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.

Problem 3: This question introduces you to the material for linear discrete dynamical systems. The lecture notes (Linear Discrete) have most of the material that you will need to help you through this problem. In Part a, you begin by using the formula to calculate q, then you simply iterate the discrete model much as you have been doing for the last few lab problems. You will want to perform 40-50 iterations (which is very easy in Excel) to get far enough out to answer the question about how many breaths it takes to reduce the level of Ar to 0.01. Part c requires that you use some of your skills in algebra (or skills in Maple to let it do the algebra). You are given c₀ and c₁, which you use in the formula for the model as c_n and c_n+1, respectively. You know g and V_i also, so you are only lacking V_r, which you must solve for. After you have V_r, the problem is very much like Part a.