Computer Assignment 2      Math 375       Uhlenbeck/Lehr


Up to three people are urged to work together. Please put your e-mail
address on all assignments. Please hand in only one assignment per group.


Due Oct 3 (We will not be finished with the theory, but I think you will
    see how much easier the theory is after a few computer pictures!)       


1.  In r2.1, find the equations (3) (4) and (5) on page 16. Make a
    direction field plot for (3) and (4) using the same constants
    you used for problem 5 on computer assignment one.
    Why can you do this without equation 5?

2.  Make a direction field plot for the equation

         dR/dt = R(1 - (R +  .5L))
         dL/dt = L(1 - (L +  .5R)).

Find a couple of solutions.  Describe what happens to most of the solutions
of this equation as t--> infinity.



3.  Now make a direction field plot of  the equations 

         dR/dt = R(1 - (R + aL))
         dL/dt = L(1 - (L + aR))

 with a = 2  (in fact, a = 1/2 is problem 2).  Find a few solutions
 you think are typical    Describe what happens to
 the solutions as t--> infinity.

4. On page 82, there is a discussion of the equation

      dH/dt =  (2 - H - F)H
      dF/dt =  (-1 + H)F.

Make a phase plane portrait of this equation which shows the same
scale as those sketched in the book. What happens to most solutions
as t--> infinity?

5*Computer Experiment: If you have time,  play with different values
of a in the models which you handled in problems 2 and 3.
Watch what happens to solutions as a is 1/2, 3/4,.9, 1, 1.1 etc.
You can   do the assigned problem on "direction fields" but you
will find this more efficient  if you look under  the list of
some "interesting direction fields" on the main page for differential
equations. Here equations depend on parameters are already listed,
and you can put problems 2 and 3 into a one-parameter family. However, watch
your parameter choices...it seemed to me as if the preset ones were
quite weird.