This application comes from a text used in Biology 354 (Ecology and Evolution). Carlson (1913) [1] grew yeast in laboratory cultures and collected data every hour for 18 hours. The list below gives the population (p) at representative times (t) and the change in population over the previous hour, G(p), which is the hourly growth of the population.
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This example is a good case of the logistic growth model, where the growth of the population is limited by available space and resources. The growth of this yeast population satisfies a quadratic discrete dynamical system.
a. In the first part of this laboratory exercise you will use the data above to find the rate of growth of the yeast. In particular, you want to use Excel's trendline polynomial fit of order two with the y-intercept set to zero through the data (last 2 columns)
(Note that you ignore the times listed in the table when you find G(p).) Since this is a quadratic equation, you can find the p-intercepts and the vertex. Write these values in your report. Show a graph of the data and the best quadratic G(p) passing through the data. Find the derivative, G '(p), and determine the population that makes G '(p) = 0. This gives the population at the maximum growth rate. What is the maximum growth rate?
b. The growth function that you found in Part a. can be used to simulate the growth of the yeast using a discrete logistic model. The dynamical system for the yeast population is given by the following model:
where G(pn) is the best quadratic function found above. Use your initial population (p1 = 9.6) starting at t = 1 and simulate the growth for 20 hours (20 iterations). (Note you are NOT starting at t = 0.) List the populations at times t = 5, 10, 15, and 20. Plot both the data from your simulation and the data given in the table above. (Note that this time you need to use only the time data and the population data in the table.) Discuss how well your simulation matches the data in the table.
c. Compute the error between the model and the data at times 7 and 16. What does this model say happens to the population of yeast for large t? Find all equilibria for this model. Discuss the stability of these equilibria by observing if the solution approaches the equilibrium (stable) or moves away (unstable).