MA 226

The Predator-Prey System Lab

This lab is due Thursday, February 24, 2000, by 7PM. Turn it in to your TF or to the Asistant in the Mac Lab. You may use any technology that you have available: Differential Systems, Mathematica, Matlab, programmable calculator, 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, in case the queues at the printers become too long. 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) given by

dx/dt = x(8 -Ax - 3y)
dy/dt = y(-3+x)

where A is a non-negative parameter. This means that you will choose different A values and get correspondingly different differential equations. 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 A-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.

Your goal is to investigate four different phase planes for the Volterra-Lotka equations and report what happens. You will deal with four different A=values:

For each of these A-values, write a brief essay (including phase planes and x(t) and y(t) graphs) describing the fate of the rabbit population (what happens to x(t)) and the fox population (y(t)). Be specific about what your initial conditions are, and what happens to the solution curves through these points. Use different colors for solution curves that behave differently. You should include in your essay a discussion of the location of all equilibrium points, the behavior of solutions near such points, as well as the behavior of solutions when x=0 and when y=0 (i.e., solutions lying on the x and y axis). Each essay should be no more than 2 pages in length (or a total of 8 pages), including figures. As usual, write legibly with proper use of English.

5. In a brief essay, describe the changes (bifurcations) that occur when A becomes positive. (You may wish to look at other A-values besides A = 0 and A = 1.

6. Next describe the bifurcation that occurs between A =2 and A =3 (Hint: Consider the fox population: When does disaster strike?) Can you find the exact A-value of this bifurcation?