MA 226: LAB 2

The Predator-Prey System

This lab is due Tuesday, October 22, 2002, in class. You may use any technology that you have available: the HPG system solver on the CD, Mathematica, Matlab,etc. 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 rather than printing out phase planes. Alternatively, you can try this method.

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 predator-prey equations. The particular system you will deal with depends upon several parameters. In this lab, you will need to use the number B, which is the last nonzero number in your BU ID.

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 15. You will definitely have to adjust the size of the region that you view on the screen to see all of the behavior discussed below, as the size of the window definitely depends upon your choice of B.

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

• 1. First list your student ID and the value 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. Provide a sketch of the phase plane and the graphs of x(t) and y(t)

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

• 5. As K continues to increase, 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! Provide appropriate graphs.