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	Math 121 Calculus for Biology Spring Semester, 2010 Lab Help	28-Apr-10
	San Diego State University

The first problem investigates the continuous logistic growth model, which has its maximum growth potential at a point of inflection during mid-log phase of growth. We find the best fitting model to the data, then use techniques of differential Calculus to find the most rapid growth. The second problem examines three population growth models compared to data for a saw-tooth grain beetle population. You will be fitting functions and graphing them using techniques from the lab and class. Also, you will find equilibria and determine their stability. Finally, you will simulate the discrete dynamical models and fit them to beetle data.

Question 1: This lab examines a yeast culture that begins in exponential growth phase and continues until reaching stationary growth. We use Excel to determine the growth parameters for logistic growth of the yeast culture, identifying several of the key elements of this growth curve with techniques from Calculus and others that have been learned during this course.

You begin the problem by copying the data from the lab to an Excel spreadsheet. Place a label t in Cell A1 and label p in Cell B1. Put the Time data in Column A and Volume data (population) in Column B. These data are graphed (as data points) to see the growth of the culture. In Column C we want to put our logistic growth model given by the formula

We make initial guesses at the growth parameters. Begin with a label P₀ in G1, then guess an initial value (1) in H1, then label G2 as M and put the largest value observed for the volume in the data and place this in H2. Label G3 as r, then as a rough guess, take r = 0.1 and place this in H3. We name these variables, then in Column D, you compute the square error between the model and the data (= (C2-B2)^2). To find the least sum of square errors, we use Excel's Solver. You should see the sum of square errors drop and the values for P₀, M, and r change. This gives you your best model through the data.

The last step for Part a is to graph the model. In Column I, you put time, t, for the model (starting with t = 0), while Column J will have the equation for the model using the parameters in G1:G3 and the time from Column I. You increment the time to make sure that your graph has about 30-60 data points evenly spaced. (For example, if your data goes to 35 hours, then you might increment by one hour, so if I2 has t = 0, then take I3 to have = I2+1.) Fill down in Column J, then add this model to your graph.

Parts b and c follow from the techniques you have learned in class and Lab. You may want to use Maple to help with the differentiation. Make your plots using Excel.

Question 2: This is another problem using real data. The first part of this problem has you finding a discrete logistic growth model for these beetles based on the data. Follow the directions carefully to obtain the right alignment of the data to use Excel's trendline polynomial fit and to get the right quadratic function here. Note that the function you are producing here is the updating function for finding the next population. The second part of Part a has you finding equilibria and determining the stability of the equilibria with derivatives. This part is probably best done using Maple. (You can probably do the logistic model by hand, but the other models will be much easier using Maple.) The next 3 parts of this question have you using Excel's Solver, much like you did in last week's lab to find the best fitting model's, then you will answer the same questions that you did for Part a. In Part e, you exercise your graphing skills that are important on tests. Again this will be made easier if you use Maple to help you accurately find derivatives and second derivatives to find critical points and points of inflection. Part f asks you to step back and compare the different models. Try to get an overview of what you accomplished in this lab on the updating functions.

The last two parts have you simulate the models to compare the models to the time series data (the type most usually found). You take the original data (t in weeks and populations), then simulate each of the 4 models using the best updating functions that you found in the first part of the problem. (Simulation of these models is like the other simulations you did in previous labs using dscrete dynamical models.) You set the initial condition to agree with the actual data at t = 0. Use Excel's Solver to minimize the sum of square errors between the data and the model simulation by changing only the initial value. The rest of the problem is just extracting information and writing a discussion.