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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: 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); - and 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 term γy represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey. Numbers of snowshoe hare(yellow, background) and Canada lynx(black line, foreground) furs sold to the Hudson's Bay Company. The maximal value of K is thus attained at the stationary (fixed) point (γδ,αβ){\displaystyle \left({\frac {\gamma }{\delta }},{\frac {\alpha }{\beta }}\right)} and amounts to K∗=(αβe)α(γδe)γ,{\displaystyle K^{*}=\left({\frac {\alpha }{\beta e}}\right)^{\alpha }\left({\frac {\gamma }{\delta e}}\right)^{\gamma },} where e is Euler's number.

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