🌿 Population Dynamics — Logistic Growth & Predator-Prey

Logistic growth models a population that cannot grow forever because resources are limited. The equation dN/dt = r·N·(1−N/K) starts out behaving like exponential growth when N is much smaller than K, then slows as N approaches K and stops entirely once N equals K. K is the carrying capacity: the maximum population size the environment can sustain, shown in the simulation as a dashed green line that the N(t) curve approaches but never crosses.

Predator-Prey mode replaces the single logistic equation with the Lotka-Volterra pair: dN/dt = r·N − β·N·P for the prey and dP/dt = δ·N·P − γ·P for the predator. Instead of settling at a carrying capacity, N and P rise and fall in a repeating cycle, prey numbers grow until predators multiply enough to bring them down, then predators decline once prey become scarce, letting prey recover and the cycle repeat, visible as a closed loop on the phase portrait rather than a flat line.

Beta is the predation rate: how quickly predators remove prey from the population as the two interact. Delta is the conversion efficiency: how effectively each prey eaten turns into new predators. Gamma is the predator death rate in the absence of food. Together these three parameters set both the size of the oscillation and the location of the equilibrium point, which the simulation computes as N* = gamma/delta and P* = r/beta and marks with a yellow dot on the phase portrait.

Rabbits only sets beta, delta and gamma to zero so predators have no effect, leaving pure logistic growth toward K = 300. Lynx-Hare and Stable equilibrium both use the full Lotka-Volterra system but with different rate constants, producing cycles of different amplitude and period. Near extinction uses a high predation rate (beta = 0.05) and low conversion (delta = 0.001), which can drive the prey population down to very low numbers before recovering, illustrating how sensitive coupled populations are to their parameters.

The Lotka-Volterra system is a conservative oscillator: it has no built-in damping term, so once started away from the equilibrium point N* = gamma/delta, P* = r/beta, the prey and predator populations orbit around that point indefinitely rather than spiraling into it. Each full loop on the phase portrait corresponds to one complete boom-bust cycle in time; larger loops mean bigger swings between scarcity and abundance for both species.