A. C. Crombie [1] studied Oryzaephilus surinamensis , the saw-tooth grain beetle, with an almost constant nutrient supply (maintained 10 g. of cracked wheat weekly). These conditions match the assumptions of the discrete logistic model. The data below show the adult population of Oryzaephilus from Crombie's study (with some minor modifications to fill in uncollected data and an initial shift of one week).
|
Week
|
Adults
|
Week
|
Adults
|
|
0
|
4
|
16
|
405
|
|
2
|
4
|
18
|
471
|
|
4
|
25
|
20
|
420
|
|
6
|
63
|
22
|
430
|
|
8
|
147
|
24
|
420
|
|
10
|
285
|
26
|
475
|
|
12
|
345
|
28
|
435
|
|
14
|
361
|
30
|
480
|
The discrete logistic growth model for the adult population 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 adult population data from times 2-30 for Pn+1 and times 0-28 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. Graph both f(P) and the data.
b. Find the equilibria for this model. Write the derivative of the updating function. 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 (adult population vs. time). Discuss how well your simulation matches the data in the table. What do you predict will happen to the adult saw-tooth grain beetle population for large times (assuming experimental conditions continue)?
c. Another common population model is Ricker's, which is given by
where a and b are constants to be determined. There are no common routines for finding a best fit using a function of this form, so you are just given values of a and b computed by another means. Assume that a = 2.7 and b = 0.0023. Once again plot Pn+1 vs. Pn, using this updating function and show how it compares to the data (much as you did in Part a).
d. Find the equilibria for Ricker's model. Write the derivative of the updating function., then discuss the behavior of these equilibria using this derivative. (Give the value of the derivative at the equilibria.) Simulate the discrete dynamical system using Ricker's model. Show the graphs of the Logistic and Ricker's models with the data. Compare these simulations with the data. Discuss the similarities and differences that you observe between models and how well they work for this experimental situation.