Math 427K-H Homework 8    (Computer homework)   Due  Thursday April 9

For the first four prblems, print out your pplane plot. For the last,
print two plots for b showing the two different behaviors.  Be sure to 
choose the size and placing of the plots to illustrate best what is
happenning.


For the first two problems, find all the fixed points and use pplane
to decide on the classification of the fixed points.


1.  x' = x(1 - x + y) 

    y' = -y( 1 -  1/2 x).


2.  x' = 8y - xy

    y' =  -x + y^2 - 2y.


In the third and fourth problem, convert the second order equation to a system
of first order equations and classify the fixed points.


3.  x" +  x'  +  sin(x) = 0.


4.  x" + (x^2 - 1)x' - x = 0. 

  What is happenning to solutions as time   goes to infinity? 
 Do they go to 0? Out to infinity?  What else can happen? 


5. Consider the (preditor-prey)system:

    x' = x(1- x - y)

    y' = y(-1 + bx).

Determine the fixed points.

There is a fixed point at y = 0, x = 1. Do computer experiments
to determine the stability of this point as a function of the parameter b,
which you can assume is positive.  Determine the value of b for which the
bifurcation between the different behaviors occurs.

Explain what happens to the fixed points of the system at this bifurcation
point.

Does y correspond to the preditor or the prey?  Why?  


Extra Credit

Suppose X = (-R = -N + I + S in the SIRS model)  satisfies the differential
equation 
        X' = -.1X - I(t)

where all we know about I(t) is that it is positive.  Show that if X(0) < 0,
then X(t)< 0 for all t > 0.