3. G. F. Gause [1] studied Paramecium caudatum with a fairly constant supply of bacteria for food. These conditions match the assumptions of the discrete logistic model. The data below show 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
|
|
|
The discrete logistic growth model for the number of individuals/0.5 cc of P. caudatum Pn can be written
where the constants r and m must be determined from the data.
a. Plot 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.) To find the appropriate constants use Excel's trendline with its polynomial fit of order 2 and with the intercept set to 0 (under options). In your Lab, write the equation of the model which fits the data best. (Use the Excel option that finds the equation using scientific notation with 3 significant figures. Graph both f(P) and the data.
b. Find the equilibria for this model. Write the formula for the derivative of the updating function, f '(Pn) and give its value at each of the equilibria. Discuss the behavior of the model near its equilibria. (Recall from lecture notes that if Pe is an equilibrium point, then you can determine the behavior of that equilibrium by evaluating the derivative of the updating function f(Pn) at Pe.) Simulate the model and show this simulation compared to the data from the table above (number of individuals/0.5 cc of P. caudatum vs. time). Discuss how well your simulation matches the data in the table. What do you predict will happen to the number of individuals/0.5 cc of P. caudatum for large times (assuming experimental conditions continue)?