Logistic Growth:
P(t) = K / (1 + ((K-P₀)/P₀)·e^(-r·t))
Population dynamics is the branch of ecology and mathematics that studies how and why populations change over time. The simplest model — exponential growth — assumes unlimited resources and leads to unbounded increase. The logistic model corrects this by introducing a carrying capacity K: the maximum population the environment can sustainably support. As population N approaches K, the growth rate diminishes, producing the familiar S-shaped sigmoid curve seen in systems from bacterial cultures to invasive species colonizing new territory.
The Verhulst equation dN/dt = r·N·(1 − N/K) has an exact closed-form solution: N(t) = K / (1 + ((K−N₀)/N₀)·e^(−rt)). The inflection point occurs at N = K/2, where population growth is fastest. Below K/2 the population accelerates; above K/2 it decelerates and asymptotically approaches K. This model forms the single-species foundation for more complex extensions including the two-species Lotka-Volterra competition model, epidemiological SIR dynamics, and resource-consumer models.
This interactive visualization solves the logistic ODE numerically in real time and displays the resulting population curve. The control panel lets you modify the growth rate r, carrying capacity K, and initial population N₀. Exploring extreme parameters reveals important behaviors: setting N₀ above K shows the model smoothly declining back to K (the stable equilibrium); a very high r shows faster convergence but potentially oscillatory overshoot near K; and reducing r toward zero slows the sigmoid transition until growth becomes imperceptibly slow.