Lotka-Volterra system

The Lotka-Volterra system was derived by Volterra in 1926 to describe the relationship between a predator and a prey, and independently by Lotka in 1920 to describe a chemical reaction.

Suppose that N(t) is the prey population at time t, and P(t) is the predator population. Then the system is

dNdt = N(a-bP)
dPdt = P(cN-d)

where a, b, c and d are positive constants. The term aN is the birth of preys, -bNP represents the diminution of preys due to predation, which is converted into new predators with a rate cNP. Finally, predators die at the natural death rate d.

Local analysis of this system is not very complicated (see, e.g., [1]). It is easily shown that it admits the zero equilibrium (unstable) as well as a positive equilibrium, which is neutrally stable. Hence, in the neighborhoodMathworldPlanetmath of this equilibrium exist periodic solutions (with period T=2π(ad)-1/2).

This system is , and has obvious limitations, one of the most important being that in the absence of predator, the prey population grows unboundedPlanetmathPlanetmath. But many improvements and generalizationsPlanetmathPlanetmath have been proposed, making the Lotka-Volterra system one of the most studied systems in mathematical biology.


  • 1 J.D. Murray (2002). Mathematical Biology. I. An Introduction. Springer.
  • 2 Lotka, A.J. (1925). Elements of physical biology. Baltimore: Williams & Wilkins Co.
  • 3 Volterra, V. (1926). Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI, vol. 2.
Title Lotka-Volterra system
Canonical name LotkaVolterraSystem
Date of creation 2013-03-22 13:22:25
Last modified on 2013-03-22 13:22:25
Owner jarino (552)
Last modified by jarino (552)
Numerical id 9
Author jarino (552)
Entry type Definition
Classification msc 92D40
Classification msc 92D25
Classification msc 92B05