Math 427K-H   Assignment 9   Due Wednesday April 14

1.  Find and classify the fixed points for the system


  x' = (2 - x)(y-x),  y' = (4-x)(y+x).


2.  The system
  
    x' = x(1- x - y)

    y' = y(-1 + bx).

Is a model for the preditor-prey interaction.  Does x represent the population
of preditors or prey?  Dtermine the values of b for which the prey does not 
always die out as t --> infinity.

3. The system

  x' = -y, y' = -ay - x(x - .15)(x-2)

results from an approximation of the Hodgkin-Huxley equations for nerve
impulses.

Find the fixed points and classify their stability.

4. Find the fixed points for the system

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

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


5.  Classify the fixed points that you found in problem 4.  (Note that this
is a model for a chemostat.)

Extra Credit:  Due April 15

As we discussed in class, there are models for the spread of disease which
result in the pair of equations with a and b positive constants.

     x' = -xy + a(1 - x  -y)

     y' = xy -  by.

Show that the conditions x > 0, y > 0 and 1 - x - y > 0 are preserved by 
the equations.  Hint:  Draw the triangle given by these conditions, and
compute the direction field on the edges of the triagular region.