Topics:

• Modeling continuous systems using Ordinary Differential Equations (ODE).
• Application: Radiactive Decay, half-life, C14 dating. Shroud of Turin.
  www.shroud.com
  Radiocarbon Dating of the Shroud of Turin, by P.E Damon et al., Nature, Vol.337 (1989), pp. 611-615.
• Direction Fields
• Logistic diferential equation

Assignment:

1. Read Tutorial 1 in PHASER for computing e using the initial-value problem x' = x, x(0) = 1. Using the method in this tutorial, compute sqrt(e), 2e^3, and -1/e. (hints: stop your computations at the appropriate time; you can solve for negative time; you can change the initial condition)

2. The half-life of C14 is 5730 years. Determine the value of the decay constant k (in the ODE x' = -k x) using PHASER. In 1989, fibres from the Shroud of Turin were found to contain about 92% of the level of C14 in living matter. Determine the age of the shroud using PHASER. Suppose that there was 1% error in the determination of the percentage of C14 in the sample of the shroud. What is the range of possible dates for the sample?

3. Some people do not agree with the results of the C14 dating of the Shroud. Do some research on the Web (see the link above, for example) and identify an objection or raise one of your own. Argue for or against the objection.

4. Consider the continuous anolog of the logistic model using the ODE:
   x' = r x (1 - x/k)
   where x(t) is the population size at time t, r (growth rate) and k (carrying capacity) are parameters. For r = 1 and k = 1.5, several solutions of this equation with various initial conditions are displayed in the picture below. Describe what happens to the population as the initial population size varies.
   Download the following phaser Project file logisticODE.ppf by just clicking on it ( or by right-click and save it to our computer. Now load this file into Phaser). You should see the following XivsTime view:

   

   Fix the parameter r = 1 and vary the parameter k from 0.5 to 3.0. Next fix k = 1.5 and vary r from 0.5 to 4.0. Describe the results of your experiments from the biological viewpoint.

5. Gompertz model of cancer growth: The differential equation
   x' = a*(exp(-b*t))*x
   is used to describe the growth of a tumor, where x(t) is a measure of its size (e.g. weight or number of cells), and a and b are parameters specific to a particular tumor. To get started, let us take a = 3 and b = 2, and x(0) = 5; the solution in the XivsTime view is shown in the image below.

   

   Load the image into your PHASER by clicking on the picture. Explain what this model predicts for the tumor growth. Experiment with several different initial tumor sizes. How does the future size of tumor depend on its initial size? Experiment with several values of the parameters a and b. Explain the roles of these parameters in the tumor growth.

6. Do a literature search on this famous Gompertz cancer growth model and write a brief report of several paragraphs on your findings (no more than a page).