Practice question

Classical Lotka–Volterra predator–prey equations generate coupled cycles because each population changes the growth rate of the other. Prey increase when predators are scarce, creating more food and allowing predator numbers to rise after a lag. Increased predation then drives prey downward; food shortage subsequently reduces predators, permitting prey recovery. In the ideal deterministic model, trajectories form closed orbits around a neutrally stable equilibrium, and cycle amplitude depends on initial conditions. The oscillations are therefore linked rather than independent or random. They are not “always stable” in the sense of returning after perturbation: the equilibrium is neutrally stable, not asymptotically attracting, and realistic stochasticity can alter the cycles. Density dependence, predator saturation, refuges, seasonal forcing, and spatial structure can dampen, amplify, or destabilize oscillations. A wave-like time series is a consequence of the reciprocal feedback, with predator maxima generally lagging behind prey maxima. The model’s value lies in revealing this mechanism even though natural systems rarely satisfy all its simplifying assumptions.