The Lotka-Volterra predator-prey equations arewith positive parameters. Their positive equilibrium is surrounded by closed level curves of a logarithmic first integral.
For the normalized equations and on the positive quadrant,is a first integral. It is nonnegative, strictly convex, and tends to infinity at the boundary and at infinity. Every positive regular level is therefore a compact simple closed curve traversed by one periodic orbit.
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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.