# Week 4

### Assignment:

1. Linear maps in the plane: Consider the linear map in two variables depending on a parameter a:
x1 -> 0.75*x1 + a*x2
x2 -> -a*x1 + 0.75*x2
Notice that the origin is a fixed point for all values of a. Determine the parameter values for which the origin is unstable, stable, or asymptotically stable. Draw phase portraits to support your findings. Note: You can use the Linear 2D MAP in the MAP library of Phaser.
Hint: There are two values of the parameter a for which the origin is stable.
Optional: Can you answer the same questions for the linear map:
x1 -> -0.75*x1 + a*x2
x2 -> -a*x1 - 0.75*x2

2. Nicholson-Bailey model: The following pair of difference equations is a famous model that describes the intereaction of host-parasitoid populations (one insect feeds on another):
Hn+1 = k Hn e( -a Pn )
Pn+1 = c Hn [1 - e( -a Pn ) ]

Variables
Hn : Density of host at generation n
Hn+1 : Density of host at generation n+1
Pn : Density of parasitoid at generation n
Pn+1 : Density of parasitoid at generation n+1

Parameters
k : Reproductive rate of host
a : Searching efficiency constant of parasitoid
c : Average number of viable eggs deposited by parasitoid on a single host

In 1941, Debach and Smith in their laboratory experiment started with 36 housefly, Musca domestica, and 18 of its pupal parasite Nasonia vitripennis and followed the populations for seven generations. They arranged the fecundity rate of the host to be 2 (k = 2, c = 1), and determined the searching efficiency constant to be a = 0.045. Please read their original paper at the link below:

DE BACH, P. and SMITH, H.S. [1941]. "Are Population Oscillations Inherent in the Host-Parasite Relation?" Ecology, 22, 363-369. JSTOR URL

In particular examine their data on Table I on page 367 and Figure 1 on page 368 of their article. Their Figure 1 looks like the following Phaser view: Download the following phaser Project file debach_smith.ppf by just clicking on it ( or by right-click and save it to our computer. Now load this file into Phaser).

What are the experimental counts of the host and parasitoid in the second and fifth generations in the paper of DeBach and Smith?
What does the Phaser simulation of the Nicholson-Bailey model predict for these counts?
What are the relative errors in the model predictions?

3. Radioisotopes used in nuclear medicine: Investiagate which radioisotopes are used in nuclear medicine. Select one used in Positron Emission Tomography (PET) imaging, and another one in radiotheraphy for cancer treatment. Look up their half-lives. Use Phaser and the ODE x1' = k*x1, determine the decay constants k of the two radioisotopes you have selected. Use at least two initial conditions for each to make sure that k does not depend on the initial amount. Here is a link to get you started on radioisotopes in nuclear medicine.

4. C-14 dating: The half life of radiactive carbon C-14 is known to be appproximately 5730 years. Using Phaser, determine the decay constant k of C-14.

In 1989, fibres from the Shroud of Turin were found to contain about 92% of the level of C-14 in living matter. Determine the age of the shroud using PHASER. Suppose that there was 0.2% error in the determination of the percentage of C-14 in the sample of the shroud. What is the range of possible dates for the sample?

Some people do not agree with the results of the C-14 dating of the Shroud. Do some research on the Web (see the links above, for example) and identify an objection or raise one of your own. Argue for or against the objection.

5. Libby's Nobel Lecture: Read the Nobel lecture of Willard F. Libby (Chemistry Prize in 1960) linked above. Why was he awarded the Nobel prize? This problem is for your edificiation; you do not have to turn in an answer.