🔗 Lotka–Volterra Equations

🔗 Ecology

The Lotka–Volterra equations, also known as the Lotka–Volterra predator–prey model, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations: $${\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy,\\{\frac {dy}{dt}}&=-\gamma y+\delta xy,\end{aligned}}}$$

where

• the variable x is the population density of prey (for example, the number of rabbits per square kilometre);
• the variable y is the population density of some predator (for example, the number of foxes per square kilometre);
• ${\displaystyle {\tfrac {dy}{dt}}}$ and ${\displaystyle {\tfrac {dx}{dt}}}$ represent the instantaneous growth rates of the two populations;
• t represents time;
• The prey's parameters, α and β, describe, respectively, the maximum prey per capita growth rate, and the effect of the presence of predators on the prey death rate.
• The predator's parameters, γ, δ, respectively describe the predator's per capita death rate, and the effect of the presence of prey on the predator's growth rate.
• All parameters are positive and real.

The solution of the differential equations is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.

The Lotka–Volterra system of equations is an example of a Kolmogorov population model (not to be confused with the better known Kolmogorov equations), which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.