This logistic growth calculator models what exponential growth can’t: a population that slows as it approaches the environment’s carrying capacity — the S-shaped curve of real biology, epidemics and market adoption.
The Logistic Equation
N(t) = K ÷ (1 + ((K − N₀)/N₀)·e^(−rt)) — equivalently dN/dt = rN(1 − N/K): growth throttles by the unused fraction of capacity. Early on (N ≪ K) it looks like our Exponential Growth Calculator‘s curve; near K it flattens onto the ceiling. The inflection — fastest absolute growth — sits exactly at K/2, a result with consequences from fisheries quotas to viral marketing. Part of our Biology Calculators collection.
Worked Example (Verified)
- K = 1,000, N₀ = 10, r = 0.5, t = 10: A = 99; N = 1000 ÷ (1 + 99e^(−5)) = 1000 ÷ 1.667 = ≈ 600 — 60% of capacity, already past the inflection
- Inflection time: ln(99) ÷ 0.5 = t ≈ 9.19 — growth peaked just before the readout moment; instantaneous rate at t=10 is 0.5 × 600 × 0.4 = 120 per unit time
- The exponential twin would claim 1,484 by now — a physical impossibility in a K=1,000 world, and a tidy parable about early-curve extrapolation
- Rule of thumb: reaching 95% of K takes about ln(19·A)/r time units — here ≈ 15.1; the last few percent are the slowest residents of the curve
Where the Model Earns Its Keep — and Where It Lies
- Fits: yeast in a vat, reintroduced populations settling, epidemic waves in closed communities, technology adoption S-curves (the Bass model is a cousin).
- Lies: when K itself moves (climate, technology), when time lags cause overshoot-and-crash oscillations, and when age structure matters (matrix models take over).
- Verhulst published it in 1838; Pearl and Reed rediscovered it fitting US census data — and even their beautiful fit eventually broke as K (effective) shifted.
Related Tools
- Exponential Growth Calculator — the no-ceiling model
- Hardy-Weinberg Calculator — the gene-pool side of populations
- Biology Calculators — the full collection
Frequently Asked Questions
What is logistic growth?
Growth that starts exponentially but slows as the population approaches a carrying capacity K, producing an S-curve: N(t) = K/(1 + Ae^(−rt)) with A = (K − N₀)/N₀. The per-capita rate falls linearly with N — crowding is the brake.
What is carrying capacity?
The population size the environment sustains indefinitely given food, space, predation and disease as they are — r-selected booms and logistic ceilings are the two poles of the same ecology. K is a property of the system, not the species alone: the same rabbits have different K on different islands.
Where is growth fastest on the logistic curve?
At N = K/2 (the inflection point), where dN/dt = rK/4 — for the worked example, 0.5 × 1000/4 = 125 per unit time at the peak. Maximum sustainable yield in harvest models sits at exactly this point, which is why fisheries math cares.
Logistic vs exponential — when is each right?
Exponential for short windows far below any limit (a fresh culture, a new niche, compound interest — finance has no K until markets saturate). Logistic the moment density starts biting: depletion, competition, infection of the remaining susceptibles. Fitting the wrong one is the classic forecasting error the comparison card in this calculator exposes.
Can a population exceed K?
Transiently, yes — overshoot happens (reindeer on St. Matthew Island famously ate past K and crashed to near-extinction). The smooth logistic curve has no overshoot built in; time-delayed versions oscillate, and real ecology adds noise on top of both.
Is this logistic growth calculator free?
Yes — free, instant, no sign-up.