3. In Lab 1, we analyzed the data of G. F. Gause [1] on Paramecium caudatum with a fairly constant supply of bacteria for food, using the discrete logistic model. The model had similar qualitative features, but the simulation of the model with time as a function of day showed the graph shifted to the left of the actual data.
In this problem we return to these data on Paramecium caudatum using Ricker's model instead and an additional means of fitting the data. Below are data from one of the versions of Lab 1, showing the number of individuals/0.5 cc of P. caudatum from Gause's study (with some minor modifications such as filling in the first day, which had no data).
|
Day
|
P.
caudatum
|
Day
|
P.
caudatum
|
|
0
|
2
|
7
|
94
|
|
1
|
4
|
8
|
142
|
|
2
|
8
|
9
|
175
|
|
3
|
9
|
10
|
189
|
|
4
|
14
|
11
|
217
|
|
5
|
21
|
12
|
199
|
|
6
|
57
|
|
|
Ricker's model for the number of individuals/0.5 cc of P. caudatum Pn can be written
where the constants a and b must be determined from the data.
a. The first part of this question examines finding the best possible updating function from the data. As you did in Lab 1, create columns for Pn+1 vs. Pn, which you can do by entering the number of individuals/0.5 cc of P. caudatum data from times 1-12 for Pn+1 and times 0-11 for Pn. (Be sure that Pn is on the horizontal axis.) This time we want to find the best fit of these data to the function R(P). This requires a nonlinear least squares fit, so you will need to use the Excel solver routine. (Check details on the Help Page for this nonlinear fit.) Find this best updating function and write it in your lab report.
b. In this part of the question, we want to use the Excel solver routine to find the nonlinear least squares best fit to the time data, Pn vs. n. For this part of the problem, you place the days in one column, the data in the next column, a Ricker's simulation in the third column depending on new parameters a and b (different from Part a., but similar), and the square of the differences between columns 2 and 3 in the fourth column. Minimize this simulation with respect to a and b and write this different Ricker's function in your lab report.
c. You should now have two Ricker's functions depending on whether you minimized the fit for the updating function (Part a.) or if you minimized the simulation of the data with Ricker's model (Part b.). First create a graph of the data in Part a. and overlay both updating functions from Parts a and b. How similar are these graphs and how well do they appear to fit the data?
d. Create a graph of the data in Part b. and overlay the simulations of both models. Are there significant differences that you observe between these simulations?
e. Find the equilibria for both models and discuss the behavior of the models near their equilibria. Are there differences in the long time behavior of these models that are experimentally significant when compared to the data on the number of individuals/0.5 cc of P. caudatum (assuming experimental conditions continue)?