Lotka-Volterra Equations

The Lotka-Volterra equations describe an ecological predator-prey (or parasite-host) model which assumes that, for a set of fixed positive constants  (the growth rate of prey),  (the rate at which predators destroy prey),  (the death rate of predators), and  (the rate at which predators increase by consuming prey), the following conditions hold.

1. A prey population  increases at a rate  (proportional to the number of prey) but is simultaneously destroyed by predators at a rate  (proportional to the product of the numbers of prey and predators).

2. A predator population  decreases at a rate  (proportional to the number of predators), but increases at a rate  (again proportional to the product of the numbers of prey and predators).

This gives the coupled differential equations

			(1)
			(2)

solutions of which are plotted above, where prey are shown in red, and predators in blue. In this sort of model, the prey curve always lead the predator curve.

Critical points occur when , so

			(3)
			(4)

The sole stationary point is therefore located at .

REFERENCES:

Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 5th ed. New York: Wiley, p. 494, 1992.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 135, 1997.