Module

for

The Lotka-Volterra Model

     

Background

Math-Model (Lotka-Volterra Equations)  The "Lotka-Volterra equations" refer to two coupled differential equations

        [Graphics:Images/Lotka-VolterraMod_gr_1.gif]  
    
        [Graphics:Images/Lotka-VolterraMod_gr_2.gif]

There is one critical point which occurs when [Graphics:Images/Lotka-VolterraMod_gr_3.gif] and it is [Graphics:Images/Lotka-VolterraMod_gr_4.gif].  

Proof Lotka-Volterra Model  Lotka-Volterra Model  

 

The Runge-Kutta method is used to numerically solve O.D.E.'s over an interval  [Graphics:Images/Lotka-VolterraMod_gr_5.gif].

Computer Programs Lotka-Volterra Model  Lotka-Volterra Model  

Mathematica Subroutine (Runge-Kutta Method)  To compute a numerical approximation for the solution of the initial value problem  [Graphics:Images/Lotka-VolterraMod_gr_6.gif] with [Graphics:Images/Lotka-VolterraMod_gr_7.gif]  over the interval [Graphics:Images/Lotka-VolterraMod_gr_8.gif]  at a discrete set of points using the formula  

    [Graphics:Images/Lotka-VolterraMod_gr_9.gif],  

where  [Graphics:Images/Lotka-VolterraMod_gr_10.gif],  [Graphics:Images/Lotka-VolterraMod_gr_11.gif],  [Graphics:Images/Lotka-VolterraMod_gr_12.gif], and  [Graphics:Images/Lotka-VolterraMod_gr_13.gif].

[Graphics:Images/Lotka-VolterraMod_gr_14.gif]

Example 1.  Solve the I.V.P.   [Graphics:Images/Lotka-VolterraMod_gr_15.gif]  over  [Graphics:Images/Lotka-VolterraMod_gr_16.gif].  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  [Graphics:Images/Lotka-VolterraMod_gr_127.gif].

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

    
[Graphics:Images/Lotka-VolterraMod_gr_128.gif]  with  [Graphics:Images/Lotka-VolterraMod_gr_129.gif],  
    
[Graphics:Images/Lotka-VolterraMod_gr_130.gif]  with  [Graphics:Images/Lotka-VolterraMod_gr_131.gif],  

over the interval  
[Graphics:Images/Lotka-VolterraMod_gr_132.gif]  at a discrete set of points.

[Graphics:Images/Lotka-VolterraMod_gr_133.gif]

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.  
    [Graphics:Images/Lotka-VolterraMod_gr_134.gif]  with  [Graphics:Images/Lotka-VolterraMod_gr_135.gif],  
and
    [Graphics:Images/Lotka-VolterraMod_gr_136.gif]  with  [Graphics:Images/Lotka-VolterraMod_gr_137.gif].    
Use several intervals  
[Graphics:Images/Lotka-VolterraMod_gr_138.gif].
2 (a).  Use the interval   [Graphics:Images/Lotka-VolterraMod_gr_139.gif].
2 (b).  Use the interval   [Graphics:Images/Lotka-VolterraMod_gr_140.gif].
2 (c).  Use the interval   [Graphics:Images/Lotka-VolterraMod_gr_141.gif].
Can you discover if the solution form an "orbit."
Solution 2.

 

Example 3.  Lotka-Volterra Model.  Solve the I.V.P.  
    [Graphics:Images/Lotka-VolterraMod_gr_192.gif]  with  [Graphics:Images/Lotka-VolterraMod_gr_193.gif],  
and
    [Graphics:Images/Lotka-VolterraMod_gr_194.gif]  with  [Graphics:Images/Lotka-VolterraMod_gr_195.gif].    
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.  
    [Graphics:Images/Lotka-VolterraMod_gr_227.gif]   with   [Graphics:Images/Lotka-VolterraMod_gr_228.gif],  
and
    [Graphics:Images/Lotka-VolterraMod_gr_229.gif]   with   [Graphics:Images/Lotka-VolterraMod_gr_230.gif].    
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.  
    [Graphics:Images/Lotka-VolterraMod_gr_249.gif]   with   [Graphics:Images/Lotka-VolterraMod_gr_250.gif],  
and
    [Graphics:Images/Lotka-VolterraMod_gr_251.gif]   with   [Graphics:Images/Lotka-VolterraMod_gr_252.gif].    
The implicit solution is  
[Graphics:Images/Lotka-VolterraMod_gr_253.gif].
Determine if y can be solved as a function of x.
Solution 5.

 

 

Predator-Prey Model

    The study of population dynamics of competing species is attributed two two independently published works by Alfred James Lotka (1880 - 1949) and Vito Volterra (1860-1940).

Consider two two species, the predator is population is y(t) (foxes), and the prey population is  x(t) (rabbits).  It is assumed that the prey, x(t), has adequate food and [Graphics:Images/Lotka-VolterraMod_gr_284.gif], [Graphics:Images/Lotka-VolterraMod_gr_285.gif] are the birth rate and death rates, respectively.  An additional term,  [Graphics:Images/Lotka-VolterraMod_gr_286.gif],  contributing to the decrease of prey is due to successful hunting of the predators.  Combining these 3 quantities, we obtain the rate of change of  

        [Graphics:Images/Lotka-VolterraMod_gr_287.gif].  

The substitution  [Graphics:Images/Lotka-VolterraMod_gr_288.gif], will help simplify this result, and we obtain  

        [Graphics:Images/Lotka-VolterraMod_gr_289.gif].

The birth rate for the predator is proportional to its food supply, x(t),  i.e.the birth rate (predators) is  [Graphics:Images/Lotka-VolterraMod_gr_290.gif],  and the death rate of the predators is [Graphics:Images/Lotka-VolterraMod_gr_291.gif], and we obtain  

        [Graphics:Images/Lotka-VolterraMod_gr_292.gif].

These two equations are an application of the Lotka-Volterra equations.

 

Example 6.  Assume that the initial number of foxes and rabbits are  [Graphics:Images/Lotka-VolterraMod_gr_293.gif] and  [Graphics:Images/Lotka-VolterraMod_gr_294.gif], respectively,  and that the coefficients  [Graphics:Images/Lotka-VolterraMod_gr_295.gif],  are used to form the system of D. E.'s
        [Graphics:Images/Lotka-VolterraMod_gr_296.gif]  
and    
        [Graphics:Images/Lotka-VolterraMod_gr_297.gif]
Solve the system of D. E.'s for x(t)and x(t) over the interval  [Graphics:Images/Lotka-VolterraMod_gr_298.gif].  
Solution 6.

 

Research Experience for Undergraduates

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

 

Download this Mathematica Notebook The Lotka-Volerra Model

 

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(c) John H. Mathews 2004