Predator-Prey Cycles

The Lotka–Volterra Model

The Lotka–Volterra equations describe the dynamics of predator-prey interactions:

• Prey equation: dN/dt = αN - βNP
• Predator equation: dP/dt = δβNP - γP

Parameters

• α (alpha): Prey growth rate (birth rate in absence of predators)
• β (beta): Predation rate (rate at which predators consume prey)
• γ (gamma): Predator death rate (in absence of prey)
• δ (delta): Conversion efficiency (prey biomass to predator offspring)

Key Features

• Cyclic oscillations: Populations oscillate out of phase. Prey peaks lead predator peaks.
• Phase space: Trajectories form closed loops around the equilibrium point (γ/δβ, α/β).
• Neutral stability: The amplitude depends on initial conditions—the system neither converges nor diverges.
• Conservation: The quantity V = δN + P - γ ln(N) - α ln(P) is conserved along trajectories.

Biological Interpretation

When prey are abundant, predators thrive and increase. As predators increase, they overconsume prey, causing prey populations to crash. With less food, predators decline, allowing prey to recover and restart the cycle. This creates the characteristic "predator-prey cycle" observed in nature.