Math 374T   Computer Assignment 4   Uhlenbeck  _____________________________


Due March 30


This assignment introduces I hope the last computer program we will
use.  This program will compute solution functions for large systems of 
equations, and some suggestions are in the link on the home page for ode45.
This system can be used for any number of variables, and could  be
helpful in obtaining graphical output for projects.  As usual, any
program which will solve ode's and plot the soutions may be used.


In problems 1-3, give both a direction field curve solution plot  and a
plot of  the components of  the solutions as functions of t for the 
systems with the given initial data.


1.  (Michaelis-Menton)

    N' =  C*N/(10)+C)

    C'  =  -.5*C*N/(10 + C),

    N(0)  = 1,  C(0) =  9.

    Why does the direction field solution look like it does?

2.  (Modification of Michaelis-Menton), see problem 14 on page 154.

    N'  =  C*N/(10+C) - .2*N

    C'  =  (5  - C)   -.5*C*N/(10 + C),

    N(0) = 1; C(0) = 9

3.  (Preditor-Prey)

    N' =  N  -  N*P  - .4*N^2.         

     P' =  -.5*P  + N*P

     N(0) =  1,  P(0) = 1.


4.  For this example, plot the three functions as a function of t (for
obvious reasons, as a direction-field plot would be in three dimensions).
Here you want the graph over a suitable time interval.

    (Preditor-Prey-Grass)

     G'  =  10*G - N*G - G^2 

     N' =   G*N  -  N*P  
     
     P' =  -.5P  + N*P


    G(0) = 1,  N(0) = 1,  P(0) = 1.