This lab examines several discrete dynamical models that we are studying in lecture. Once again, you will want to be familiar with the material in the lecture notes to work this lab. If you didn't download the Excel spreadsheet for discrete dynamical systems last week, then you might want to this week.

Question 1: This problem compares a discrete Malthusian growth model to a logistic growth model. The Malthusian growth model is one that you should be able to solve by hand. However, the logistic growth model does not have an exact solution. Thus, you have to simulate this model using an Excel spreadsheet. The only difference in simulating the logistic growth model from the Malthusian growth model (which you simulated in your last lab) is that you enter the equation for logistic growth which includes a quadratic term. This question is a relatively simple extension of the work that you did on your last lab. There is a logistic growth model on the downloadable Excel spreadsheet for discrete dynamical systems, so you can use that as a guideline.

Question 2: This question introduces you to the next section, where we study a more general linear discrete dynamical system. The lecture notes 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 dsicrete 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 b 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.

Question 3: This problem is very similar to the lung problem. Its another linear discrete dynamical model, which can be easily simulated in Excel. In Part b. note that you do NOT have to use Maple to find the answer as I have given you the answer. (I'm willling to show any interested student how to get this answer from Maple.) You simply use the given formula to answer the questions posed. Thus, you are simply substituting into the formula the appropriate numbers to verify that the formula and the simulation in Part a. agree. You can find the doubling time by some good use of algebra or by using Maple's fsolve routine. The techniques of algebra will really show how well you know how to manipulate algebraic expressions, so it is a good exercise. (Note the only variable that is unknown is n, and it appears in the exponent, so you put all constants on one side and all powers of n on the other. By taking logarithms, you can easily derive the answer.) To find when the populations are equal, you will need to use Maple's fsolve routine. You enter your two functions of n, A_n and B_n, then set them equal to each other and solve (using fsolve) for n. You will probably need to tell Maple to search for the solution for n = 0..500 in the fsolve command.