Laboratory Help Page

This page is designed to provide helpful information about the laboratory questions.

Begin this lab and every lab by introducing yourself to your partner. Detemine the times when you can meet together during the week before the lab is due on Friday, Sept. 29. If your schedules are totally incompatible, then notify me immediately.

You will probably want to download your specific lab page (and may want to convert it to a Word document). On the cover page you begin by typing in the name of each team member and your group number.

Question 1: This question is a review of two of the most basic models often used in Biology (which should have been taught in Math 121, but may not have been in sections other than mine). Data are provided from the side of a dog food bag. You enter these data into an Excel worksheet, then use the Trendline feature of Excel twice to create two models (linear and allometric (power law)) and graphs of the models with the data. Since these types of models are so ubiquitous in the Biological Sciences, it is important that you have exposure to working these models. All students should be familiar with the Trendline feature of Excel, so creating and analyzing these models should not be difficult (though you may not have used the power law feature of Trendline before).

Question 2: This problem links this lab to the current studies on optimization. Formulae are given for the physiology and physics of a cough. You begin by combining these formulae algebraically to create the velocity function depending only on the radius of the constricted trachea. (Note that the proportionality constant absorb the mm units measuring the trachea and convert them to the cm/sec units of the air velocity.) This function is simply a cubic polynomial, which is easily differentiated to find the extrema. You find the maximum of this cubic, then graph this function. You may want to work this problem in Maple, like others you have worked recently. Maple is your best tool for diferentiating and setting the derivative equal to zero. You might find that the Conversion tables (under Helpful Tools in the index) help you to convert into miles per hour.

Question 3: The last question combines work from your first lab and the second lab. This question examines the same Paramecium data presented in the first lab, but uses a Ricker's model instead of a logistic growth model. As you did in the first lab, you will want to create one graph of the updating function and another graph showing the simulation of the model. The primary difference is the use of Excel's solver routine to fit the Ricker's function to the data in two distinct ways, one where the Ricker's function is fit as an updating function and another where the model is fit directly to the time dependant data. Below we present more details on how you may want to set up your Excel spreadsheet to solve this problem.

Next (probably on Sheet 2) you should enter the day number in column A, the population data in column B, and the Ricker's model in column C. The Ricker's model starts with the same initial value as the data in column B. If that is in C1, then in C2 you enter then simulation "=$F$1*C1*exp(-$F$2*C1)," then filling down. You will again need to enter new starting values in F1 and F2 (separate from the first sheet). Selecting the values you got in the previous exercise gives a great place to start. Your column D will again sum the square of the error between the data and the model and is once again minimized user the solver routine.

To complete the graph for Part c, you will need to transfer information on the parameters a and b from Sheet 2 to Sheet 1, then graph the updating functions for both. You should create 3 new columns with the first column being P_n ranging from 0 to the roughly highest value of your data and dividing it into 50 even spaces. The next column uses this first column for values in the Ricker's function from Sheet 1, while the second column uses the first column for values in the Ricker's function from Sheet 2. Graph the data first, then use the Add Data feature to add these functions.

For the graph in Part d, you transfer the information on the parameters a and b from Sheet 1 to Sheet 2, then simulate that model and graph it along with the data and the simulation for the model that you already minimized. The last part of the question has you performing the standard equilibrium analysis as we have done in class.