Lotka-Volterra Model

The Runge-Kutta method is used to numerically solve O.D.E.'s over an interval  
.

Mathematica Subroutine (Runge-Kutta Method)  To compute a numerical approximation for the solution of the initial value problem  
with
  over the interval
  at a discrete set of points using the formula  

    
,  

where  
,  
,  
, and  
.

Extension to 2D. The Runge-Kutta method is easily extended to solve a system of D.E.'s over the interval  
.

Mathematica Subroutine (Runge-Kutta Method in 2D space)  To compute a numerical approximation for the solution of the initial value problem  

    
  with  
,  
    
  with  
,  

over the interval  
  at a discrete set of points.

Note.  The Runge-Kutta method in 2D is a "vector form" of the one-dimensional method, here the function f is replaced with F.