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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 A (the growth rate of prey), B (the rate at which predators destroy prey), C (the death rate of predators), and D (the rate at which predators increase by consuming prey), the following conditions hold.

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

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

LotkaVolterraEquations

This gives the coupled differential equations

(dx)/(dt)=Ax-Bxy
(1)
(dy)/(dt)=-Cy+Dxy,
(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 dx/dt=dy/dt=0, so

A-By=0
(3)
-C+Dx=0.
(4)

The sole stationary point is therefore located at (x,y)=(C/D,A/B).


See also

Growth, Law of Growth, Logistic Map, Population Growth

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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.

Referenced on Wolfram|Alpha

Lotka-Volterra Equations

Cite this as:

Weisstein, Eric W. "Lotka-Volterra Equations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Lotka-VolterraEquations.html

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