Predator–Prey Dynamics
Lotka–Volterra orbits and the carrying-capacity spiral
Prey grow, predators eat, predators die, prey rebound. Pure Lotka–Volterra closes every initial condition into its own orbit — a marginally stable system with no preferred amplitude. Add a prey carrying capacity and the orbits collapse into a spiral toward a single interior fixed point. Cycles vs. spirals is marginal vs. asymptotic stability.
Phase portrait (prey vs predator)
Population over time
preypredator
Pure Lotka–Volterra orbits are closed — every initial condition is its own conserved cycle. Add a prey carrying capacity and the closed orbits collapse into spirals into the interior fixed point. The qualitative difference between cycles and spirals is the difference between a marginally stable and an asymptotically stable system.