3. When a monoculture of an organism is grown in a limited (but renewed) medium, then the population of that organism often follows the logistic growth model. Below is a table from a study by G. F. Gause [1] where he grew cultures of Paramecium aurelia and counted the number of individuals/0.5 cc.
|
Day
|
P.
aurelia
|
Day
|
P.
aurelia
|
|
0
|
2
|
7
|
266
|
|
2
|
14
|
8
|
330
|
|
3
|
34
|
9
|
416
|
|
4
|
56
|
10
|
507
|
|
5
|
94
|
11
|
580
|
|
6
|
189
|
12
|
|
a. Find the best Malthusian growth model (P' = rP) that fits the first six days (0-6) of the experiment. (This is most easily done by fitting the data in Excel with the exponential fitting option in Trendline.) Write both the best fit differential equation (including the initial condition) and the solution of this equation. Graph both the data and Malthusian growth model for the first 10 days. Compute the percent error between the model and data at days t = 3, 5, and 8. Where does the model fit the data well and where does it fail? Give a biological reason for this.
b. As noted above, these experiments are best designed to fit the logistic growth model, which is given by
where r is the Malthusian growth rate and M is the carrying capacity. Give the general solution to this equation (with the parameters r, M, and P0.) Write an initial guess to this solution by taking P0 to be the population at day t = 0, r to be the value computed in Part a., and M to be the population at day t = 12.
c. Use a nonlinear least squares best fit to the data (as we have done in previous labs) to fit the general solution of the logistic growth model to the data in the table. List the best fit values for the parameters r, M, and P0. Graph the data, the initial guess model in Part b and the best fit model that was found in this part over the time interval 0 < t < 15. How well do these models fit the data? Compute the percent error between the model and data at days t = 3, 5, 8, and 11. Give a biological explanation why this model improves the fit of the data over the work done in Part a.
[1] G. F. Gause (1934), Struggle for Existence, Hafner, New York.