Important:

I will not often be reading email over Thanksgiving break, so please schedule your crises early! You have 3 weeks for this assignment, so, if possible, get it done before the break.

Known typos:

(11-07-06) Third line from bottom on p.1: There is a missing right parenthesis, that should be inserted just before the comma at the end of the expression for L(t+1).

(11-07-06) First line on p.2 after figure should read, "Time t is measured in 14-day units."

(11-10-06) The last column of data in beetledata.m should be labeled Ld, not Lc. The file has now been updated to correct this.

(11-10-06) In determining stability in Problem 2, the wrong matrix was specified. Stability is based on the eigenvalues of that matrix plus I (the identity matrix).

(11-14-06) p1, last bullet, remove the 2nd 'will'.

(11-18-06) In the comments in beetledata.m, delete "and the combined colony".

Question: (11-08-06) On problem 1, do we plot for 100 days or 100 time units?

Answer: Let's make it 50 time units (i.e., 50 * 14 days).

Question: (11-09-06) What does "compare" mean in problem 3b?

Answer: Look at how different your values of the parameters are from those of Desharnais and Liu, and compare the predicted values of the two models with the data.

Question: (11-09-06) How do we do problem 3c?

Answer: There are many ways to do it. For example, you could solve the least squares problem several times with different random perturbations of the data. If you do this, make sure you explain what assumptions you made on the errors in the data in beetledata.m. I expect that the data was gathered by some poor graduate student who began by counting out an initial population of larvae, pupae, and adults into a petri dish of flour. Then every two weeks the graduate student went back and counted the populations again. Your assumptions on errors in the data should be consistent with this.

Question: (11-13-06) In the expression of A_fixed, should we consider natural log or log base 10?

Answer: Natural log.

Question: (11-14-06) I think we can assume that one time unit is 14 days. But, I am a little bit confused about the duration of egg. Assignmnet said that egg duration is 2-4days. To make the time unit constant as 14 days, I think we need to add extra assumption *that the eggs becomes nonfeeding larvas in 14days.* It sounds like a kind of biological question. :)

Answer: The model says that all eggs become larvae within the 14 day period, and that all that matters is how many are "recruited" and survive.

Question: (11-14-06) In introduction, larvae also could be eaten by adults. But, P(t+1) does not seem to consider this fact. Can I just assume that the model is correct, and accept it?

Answer: Adults eat significant numbers of pupae and eggs, but their eating of larvae is insignificant and neglected in the model.

Question: (11-14-06) I'm looking at the homework and I'm confused about the stable/unstable definition. It says: "Each solution can be labeled as stable or unstable depending on whether or not the eigenvalues of the Jacobian matrix of the system all lie within the unit circle." You correct this to be based on the eigenvalues of that matrix plus I. 1) Does the Jacobian still come into play here? How?
2) Which term means that all of the eigenvalues lie in the unit circle?
3) How do we count eigenvalues which lie on the boundary of the unit circle (or sphere)?

Answer: Let J be the Jacobian of the system on p.1. 1) The matrix that you need the eigenvalues for is J+I.
2) "stable".
3) That would mean that the system is "on the stability boundary".

Question: (11-14-06) In the formulation of question 2 (below) [and even adjusted by "(11-10-06) In determining stability in Problem 2, the wrong matrix was specified. Stability is based on the eigenvalues of that matrix plus I"] , you seem to be saying that we should use fsolve() to determine whether the solution (Afixed,Lfixed,Pfixed) is stable. But in principle all we need to do is to compute the eigenvalues of the specified matrix, e.g. using the eig() function. So why do we need to use fsolve() for problem #2?

Answer: You need fsolve to compute A_{fixed}, L_{fixed} and P_{fixed}. I want you to provide fsolve with the Jacobian, which should make fsolve faster and more reliable. Once fsolve finishes, use eig to find the eigenvalues of (J+I), where J is the Jacobian matrix for the system on p.1, evaluated at the solution.

Question: (11-14-06) Should the condition for validity on page 2 actually be: b \geq muA/(1 - muL) since for the first plot in problem 2 muA/(1 - muL) = 1 and we are supposed to try b=1?

Answer: The condition is correct as written. We are not claiming the validity of the solution at b=1 -- we are just using it as a starting guess for fsolve, so we would like it to be close to the solution we seek, but that is all we care about.

Question: (11-14-06) For problem 2, where it says provide fsolve with the Jacobian matrix, does this now mean the Jacobian matrix + I?

Answer: No, it still means the Jacobian.

Question: (11-15-06) For the second question we are suppose to calculate the jacobian of the function and send it to fsolve. Is this using the function jacobian? I am asking because the toolbox that has the jacobian function is not installed on my machine. IS there any other way to calculate the jacobian ? Or I still need to install that toolbox to be able to calculate it?

Answer: You need to write a function that evaluates the function F(x) as well as the Jacobian of F, where the problem you are solving is F(x)=0 and x is a vector containing A_fixed, L_fixed, and P_fixed.

Question: (11-16-06) What do we plot in Problem 2: the values for c_el = 0 or the ones returned by fsolve?

Answer: The ones returned from fsolve.

Question: (11-17-06) About problem 3, when we are doing the recursion to compute A,L,P in the function, what are the initial values for A, L, and P? Do we take the values from beetledata.m at time 0?

Answer: Yes.