[AMath 383] Homework
Dear class members,
Remember that homework can be turned in late, up to two days, with a
10% penalty per day. The projects may be accepted through the end of
the day tomorrow for full credit.
For the homework, when you are looking at stability of the predator
prey model (prob #1), there are a few tips that help avoid a lot of
mess.
First, there are some special cases for eigenvalues when the
off-diagonal terms are zero.
For upper-triangular
( a b )
( 0 c )
the eigenvalues are listed on the diagonal (a and c). Similarly, for
the lower-triangular matrix
(a 0)
(b c)
the eigenvalues are on the diagonal. This lets you avoid doing some of
the calculation.
This does NOT apply for a matrix that looks like
( 0 a )
( b c )
There is an equilibrium that will give you something like this. Here is
my tip on how to deal with this as painlessly as possible. Try to write
the matrix so that it looks something like this:
( 0 lambda S* )
( -c F* -b F* )
(or something of a similar form, depending on how you ordered your
variables -- I did S and then F)
Now, consider where the equilibrium occurs (1st or 2nd quadrant) to
determine whether the trace or determinant are positive or negative.
I hope this helps out.
And remember that a good project will be much more important than this
homework.
D. Brian Walton
AMath 383 Instructor
Guggenheim 408C
(206) 685-9298