•The owls-and-squirrels applet

Program notes:
Reverse colors in legend--owls blue, squirrels red.
Start phase-plane curve at initial point.
Discrete points in both graphs, with connecting arcs.

The owls-and-squirrels applet will open in a separate browser window if you click the link.  Your web browser will need to have a Java plug-in installed for this applet to work.  Arrange the browser windows so that you can easily go back and forth between this one and the applet window.  In the applet window you will see a pair of diagrams like the following:

[captured image here]

The graph on the right plots in blue the number of owls and in red the number of squirrels as functions of time .  The graph on the left is a phase-plane plot of the number of (thousands of) squirrels vs. the number of owls.  Thus in the graph on the left you do not explicitly see time.  The small red rectangle in the left graph is the initial point.

What you're seeing above is just a static image.  You are going to begin working with the actual applet, so be sure you have opened the applet in a separate window!

To select an initial point on the phase-plane plot, use the black arrowheads (triangles) to move the default initial point to the left or right, up or down.  The small arrowheads move the initial point in small steps; the big arrowheads move the initial point in big steps.

Try it now:  Using the graph's grid for reference, make the initial point to show roughly 100 owls and 175 (thousands of) squirrels.

To see the coordinates of the initial point, click the Update button.  You will see the actual coordinates of the initial point you selected graphically.  Try it now:  Determine the actual coordinates of the initial point you just selected.

For the questions that begin now, you should write your answers in the computer algebra system document provided--see below.)

Question 1.   Using the applet, find graphically an initial point for which at first the number of squirrels increases; what happens to the number of owls in the immediate future?  In the longer term?
Now find graphically an initial point for which at first the number of squirrels decreases; what happens to the number of owls in the immediate future?  In the longer term?

You can also directly type in the coordinates of the initial point in the input boxes below the graphs that are labelled "Number of owls" and "Number of squirrels".  Then you will see the corresponding initial point on the phase plane.

Try it now:  Type in coordinates of an initial point for which at first the number of squirrels decreases. (Hint:  you might want to look at the graphs you already have to find such a point.)  What is happening at first to the number of owls?

Question 2.  According to the model, the population size in a given month for each species depends on both that species' own size the preceding month as and the size of the other species the preceding month.  Explain briefly why the four parameters ought to have the signs they do.

To form a table of table of times along with corresponding values of and , the applet button Vector sequence.  You'll see the results in a separate, smaller popup window.  If you want a different total time duration from the default offered, type a different duration after "Max. k in vector sequence".

Try it now:  Produce a table of times together with numbers of owls and squirrels for times only.

 Converted by Mathematica  (September 5, 2002)