Compare logistic growth (bounded by a carrying capacity) against unbounded exponential growth. Adjust initial population, growth rate and carrying capacity to see how populations really behave.
dP/dt = r·P·(1 − P/K) describes growth that slows as the population P approaches the carrying capacity K. Its closed-form solution is P(t) = K / (1 + ((K − P0)/P0)·e^(−rt)), an S-shaped (sigmoid) curve that starts near-exponential, then bends over and asymptotically approaches K without ever exceeding it.
P(t) = P0·e^(rt) assumes unlimited resources and grows without bound. It is a reasonable short-term approximation when a population is far below its carrying capacity, but it always eventually diverges from reality — no real population grows forever at a constant rate.
Real human populations pass through stages of high birth and death rates, rapid growth as death rates fall first, then slowing growth as birth rates fall with education and urbanization, ending near replacement level (~2.1 births per woman in developed countries). The logistic curve is a simplified stand-in for this whole multi-stage process.
This simulation compares two textbook population models. Logistic growth, dP/dt = r·P·(1 − P/K), starts out looking exponential when the population is small relative to the carrying capacity K, then bends over and levels off as resources become limiting — the classic S-curve seen in bacterial cultures, island species and, in a rough sense, human demographic transitions. Exponential growth, P(t) = P0·e^(rt), is plotted alongside it as a reminder of what happens when that resource limit is ignored: unbounded growth that becomes physically impossible within a few doublings.
A solid green S-curve is the logistic solution, flattening out as it approaches the dashed yellow carrying-capacity line. A dashed red curve is the exponential comparison starting from the same initial population — watch how quickly it diverges from the logistic curve once the population is no longer small relative to K.
Drag Initial population and Growth rate to set the starting conditions, Carrying capacity to set the environmental ceiling, and Years to extend or shorten the time horizon. The three presets contrast a small, slow-growing island population, a human-like demographic transition, and a fast invasive-species boom.
The United Nations projects world population reaching roughly 9.7 billion by 2050 and 10.4 billion by 2100 — a curve that, after centuries of near-exponential growth, is bending over as fertility rates fall worldwide, much like the logistic curve in this simulation.
Carrying capacity (K) is the maximum population size an environment can sustainably support given its available resources — food, water, space, and other limiting factors. In the logistic model, growth slows as the population approaches K and stops entirely once it is reached.
Exponential growth assumes resources never run out, so the population keeps doubling at a constant rate forever. In reality, every population eventually runs into some limiting factor — food, space, disease, predation, or in human terms, economic and social constraints — which is exactly what the logistic model's carrying-capacity term represents.
The instantaneous growth rate of the logistic model at any moment is r·(1 − P/K). As P approaches K, this rate shrinks toward zero. The simulation scans forward year by year and reports the first year at which this instantaneous rate falls below 1% per year, marked with a pulsing dot on the logistic curve.