Lotka–Volterra equations
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: d x d t = α x − β x y , d y d t = − γ y + δ x y , {\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); d y d t {\displaystyle {\tfrac {dy}{dt}}} and d x d t {\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.