Population Growth Calculator
Exponential and logistic population growth, and the difference between them.
Exponential growth assumes unlimited resources: a constant per-capita rate, and a curve that never bends.
How the population growth calculator works
Exponential growth assumes unlimited resources: a constant per-capita rate, and a curve that never bends. It describes a population in a new habitat, or bacteria in fresh medium, and nothing for very long.
Logistic growth adds a carrying capacity. Growth slows as the population approaches it and stops at it, giving the S-shaped curve that real populations follow. The fastest absolute growth happens at half the carrying capacity — a result that matters for harvesting, since maximum sustainable yield sits there rather than at maximum population.
The two models diverge slowly at first and then completely. Below about a tenth of carrying capacity they agree closely, which is why early data can look exponential right up until it does not.
Formula: exponential: N = N₀e^(rt); logistic: N = K / (1 + ((K−N₀)/N₀)e^(−rt))
Worked examples
| Inputs | Logistic population | Note |
|---|---|---|
| 500 growing at 15% toward a cap of 10,000 | 5,139 | logistic well below exponential |
| No carrying capacity | 10,043 (unlimited growth assumed) | pure exponential |
| Early on, the models agree | 663 | barely distinguishable |
FAQFrequently asked questions
What is the difference between exponential and logistic growth?
Exponential assumes unlimited resources and never bends; logistic slows as the population approaches a carrying capacity, giving an S-curve.
What is carrying capacity?
The population an environment can sustain indefinitely, given its resources.
Where does a population grow fastest?
At half the carrying capacity, in absolute terms. That is also where maximum sustainable yield sits.
What is r?
The intrinsic rate of increase — per-capita growth per unit time when resources are unlimited.
Why does early data look exponential?
Because below about a tenth of carrying capacity the two models agree closely. The divergence only becomes obvious later.
Where these figures come from
- NCBI Bookshelf — Molecular Biology of the Cell — growth kinetics and molecular conventions
- Wellcome Sanger Institute — a leading UK genomics institute
Last checked: September 2026. Formulas are the standard textbook forms; assumptions are stated on each page because they are where these models break.