The Lotka-Volterra Model
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 .
Proof Lotka-Volterra Model Lotka-Volterra Model
The Runge-Kutta method is used to numerically solve O.D.E.'s over an interval .
Computer Programs Lotka-Volterra Model Lotka-Volterra Model
Method) To compute
a numerical approximation for the solution of the initial value
the interval at
a discrete set of points using the formula
where , , , and .
1. Solve the
I.V.P. over . Use
the Runge-Kutta method.
Extension to 2D. The Runge-Kutta method is easily extended to solve a system of D.E.'s over the interval .
Method in 2D space) To
compute a numerical approximation for the solution of the initial
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.
2. Lotka-Volterra Model. Solve the
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."
3. Lotka-Volterra Model. Solve the
Combine the system of D. E.'s to form a separable first-order differential equation and solve the D. E..
Example 4. For
Show that the numerical solution in Example 3 and the analytic solution in Example 4 are in agreement.
Example 5. For
The implicit solution is .
Determine if y can be solved as a function of x.
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
respectively, and that the
coefficients , are
used to form the system of D. E.'s
Solve the system of D. E.'s for x(t)and x(t) over the interval .
Research Experience for Undergraduates
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(c) John H. Mathews 2004