Population Dynamics

Lotka-Volterra predator-prey model and population oscillations

The Lotka-Volterra equations model the dynamics of biological systems with predator-prey interactions. The prey population x grows exponentially at rate α in the absence of predators, but is reduced by predation at rate βxy (proportional to encounters). The predator population y declines at rate γ without prey, but grows from successful predation at rate δxy. The system produces characteristic oscillations: prey increase → predators increase → prey decline → predators decline → cycle repeats. The amplitude and period depend on all four parameters. The equilibrium point (x̄ = γ/δ, ȳ = α/β) is a center in the phase plane — orbits are closed loops. Real ecosystems add complexity (carrying capacity, refugia, time delays) but the Lotka-Volterra model captures the fundamental mechanism of coupled oscillations.