MA 226: LAB 3

The Predator-Prey System Again

This lab is due Thursday, November 1, 2001, by 7PM. Turn it in to your TF or to the TF in the Mac Lab. You may use any technology that you have available: Differential Systems, Mathematica, Matlab, programmable calculator, etc. Interactive Differential Equations in not appropriate. You will be graded on exactly what is asked for in the instructions below. You need not turn in any additional data, graphs, paragraphs, etc. You should submit only what is called for, and in the order the questions are asked. It is perfectly acceptable to hand in hand-drawn figures, in case the queues at the printers become too long. (In fact, there is a bug in the Differential Systems program in the Mac Lab that does not allow you to print certain of the phase planes described below.) So do not print using this software. Remember that you will be graded on your use of English, including spelling, punctuation, logic, as well as the mathematics.

IMPORTANT: The work you submit should be your own and nobody else's. Any exceptions to this will be dealt with harshly.

The goal of this laboratory exercise is to understand the phase plane for a nonlinear system of differential equations, namely the Volterra-Lotka system (predator-prey equations). The particular system you will deal with depends upon several parameters. In this lab, you will need to use two numbers, A and B. These numbers are derived (as usual) from your student ID (not your Social Security number) as follows. The number A is the last nonzero number in your student ID, while the number B is the second last nonzero number in your ID. For example, if your student ID is 123-45-6789, then A = 9 and B = 8. But if your ID is 100-20-3000, then A = 3 and B = 2.

The predator-prey system that you will use is: given by

You should think of x as being the "population" (in some units) of prey (rabbits) and y as the "population" of predators (foxes). The goal is to understand what happens to these populations for various K-values as time increases. Remember that the numbers x and y represent scaled populations --- the units are unspecified, but they may represent hundreds or thousands of predators or prey. As usual, we are only interested in the cases where both x and y are non-negative. Much, though not all, of the action will take place when x and y lie between 0 and 10. You may have to adjust the size of the region that you view on the screen to see all of the behavior discussed below.

Your goal is to investigate different phase portraits for the Volterra-Lotka equations and report what happens.

• 1. First list your student ID and the values of A and B.

• 2. Next find all equilibrium points for this system. Your answer should, of course, depend upon K.

• 3. Now let K = 0. Use a computer to describe the phase plane for this system. In a brief paragraph, discuss what happens in this case to the populations of predators and prey. Do not attempt to print this image if you are using Differential Systems in the Mac Lab.

• 4. Discuss the bifurcation that occurs when K becomes (slightly) positive. Explain what happens now to the populations in a brief essay.

• 5. As K continues to increase (between 0 and 11), a second bifurcation occurs. First find the exact K value at which this bifurcation occurs. Next describe what happens to the populations before and after this bifurcation. Does disaster befall one of the populations? That question was actually a hint!