Computer Assignment 2 Math 375 Sadun/Ristov
 

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. (Hopefully, this will clarify the material in chapter 5. It's a lot easier to follow the theory after a few computer pictures!)
 

1. In reading 2.1, 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.