Module for the Lotka-Volterra Model

Check out the new Numerical Analysis Projects page.

The Lotka-Volterra Equations. The "Lotka-Volterra equations" refer to two coupled differential equations

      
    
    

There is one critical point which occurs when and it is .  

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

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

    ,  

where , , , and .

Example 1.  Solve the I.V.P.     over  .  Use the Runge-Kutta method.
Solution 1.

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

Program (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.

Example 2.  Lotka-Volterra Model.  Solve the I.V.P.  

      with  ,  
and
      with  .    

Use several intervals  .
2 (a).  Use the interval   .
2 (b).  Use the interval   .
2 (c).  Use the interval   .
Can you discover if the solution form an "orbit."
Solution 2.

Example 3.  Lotka-Volterra Model.  Solve the I.V.P.  

      with  ,  
and
      with  .    

Combine the system of D. E.'s to form a separable first-order differential equation and solve the D. E..
Solution 3.

Example 4.  For the  I.V.P.  

       with   ,  
and
       with   .    

Show that the numerical solution in Example 3 and the analytic solution in Example 4 are in agreement.
Solution 4.

Example 5.  For the  I.V.P.  

       with   ,  
and
       with   .    

The implicit solution is  .
Determine if y can be solved as a function of x.
Solution 5.

Research Experience for Undergraduates

The Lotka-Volterra Model  The Lotka-Volterra Model  
Internet hyperlinks to web sites and a bibliography of articles.  
   

Downloads (The Lotka-Volterra Model The Lotka-Volterra Model).  
Download this Mathematica notebook.  

(c) John H. Mathews 2003