MA 226: Spruce Budworm Lab

A Two Parameter Family of Differential Equations

This lab is due Thursday, October 18, 2001, by 7 PM. Turn it in after class or to the TF in the Mac Lab. You will have to use the Interactive Differential Equations package available in the Mac Lab. Please remember the hours of operation of this lab and that there will be 150 students trying to use this software. Please only view the software long enough to understand what is happening and write up your lab elsewhere. Do not print copies of the output; rather just hand draw any picture you wish to use in your report.

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. Again, 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.

In this lab you will investigate the bifurcations that occur in the two parameter family of differential equations given by

x' = rx(1-x/k) - x2/(1 + x2)

This differential equation depends upon two parameters, r and k. Your goal is to investigate the bifurcations that occur as you hold one of the parameters in this equation fixed and vary the other parameter.

Open the tool called Spruce Budworm: kr Plane in IDE. Be sure to open the correct tool, because there are several different spruce budworm tools. You will see three windows displayed.

In the upper left window you see a graph displaying k vs. r. Note that as you move the mouse over this window, things change in the other two windows. Your goal is to understand what this graph is telling you. In the upper right you see a graph involving the graphs of two functions

First answer each of the following questions. In each case your answers should involve several sentences.

1. Drag the mouse over the upper left hand window and observe what happens in the upper right window. Explain in a sentence or two what it means for the differential equation when the two graphs in this picture meet.

2. What does the graph in the lower right window represent? What do the red dots in this picture mean in terms of the differential equation?

3. There are two regions in the upper left window, one called A, the other called B indicated in the above picture. What can you say about the number of equilibrium points for the corresponding differential equation when the point (k,r) lies in the region A? What can you say about the number of equilibrium points for the differential equation when the point (k,r) lies in the region B? What can you say about the number of equilibrium points for the differential equation when the point (k,r) lies on the curve that separates region A from region B?

4. Fix the value of k = 10. Move the cursor in the upper left window so that r increases from 0 to 0.7. Use the information you see to sketch the bifurcation diagram for this family of differential equations as r varies, that is the family

x' = rx(1-x/10) - x2/(1 + x2).

5. Now fix the other parameter r = 0.6. Move the cursor in the upper left window so that k increases from 0 to 16. Use the information you see to sketch the bifurcation diagram for this family of differential equations as k varies, that is the family

x' = 0.6 x(1-x/k) - x2/(1 + x2).

6. In a brief paragraph tell what the upper left hand picture represents in terms of the original differential equation. Include pictures to illustrate your essay.