Module

for

The Lotka-Volterra Model

Background

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

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

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

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   with   over the interval   at a discrete set of points using the formula

,

where  ,  ,  , and  .

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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  .

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.

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.

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 , are the birth rate and death rates, respectively.  An additional term,  ,  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

.

The substitution  , will help simplify this result, and we obtain

.

The birth rate for the predator is proportional to its food supply, x(t),  i.e.the birth rate (predators) is  ,  and the death rate of the predators is , and we obtain

.

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

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.

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

(c) John H. Mathews 2004