Example 6.  Assume that the initial number of foxes and rabbits are   and  , respectively,  and that the coefficients  ,  are used to form the system of D. E.'s
          
and    
        
Solve the system of D. E.'s for x(t)and x(t) over the interval  .  

Solution 6.

Enter the functions    and    and form the vector function  .  

Compute the Runge-Kutta solution.

If we graph the solution over a larger time interval the graph traces over itself, because the solution is periodic.

The graph of the number of prey is formed by using the first coordinate .  

The graph of the number of predators is formed by using the second coordinate .  

The periodic behavior is clearly present in the latter two graphs.  Notice the phase lag for the number of prey.  Did you expect this to happen?

(c) John H. Mathews 2004