The Lotka–Volterra equations, also known as the predator-prey equations, are a pair of first-order, nonlinear differential equations that describe the dynamics of biological systems in which two species interact: one as a predator and the other as prey. They were formulated independently by the Italian mathematician Vito Volterra and the American ecologist Alfred James Lotka in the early 20th century.
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The Lotka-Volterra equations model interacting prey and predator populations:with positive parameters. In the absence of predators, prey grows exponentially at rate ; in the absence of prey, predators decline at rate . The terms proportional to represent encounters, suppressing prey and supporting predators.
The positive equilibrium is . On the positive quadrant, the functionis a first integral: differentiating gives . It has a strict minimum at the equilibrium and diverges at the quadrant boundary and at infinity, so its regular levels are closed periodic orbits.
The cycles form a conserved family rather than attracting limit cycles. This neutral behavior is a feature of the idealized interaction model; it should not be inferred for models with additional density dependence or other population effects. The Logarithmic first integral of the Lotka-Volterra equations gives a convenient normalized version.