What this calculator does
Exponential growth has no ceiling, which makes it a poor description of anything real for long. The logistic model adds one: growth slows as the population approaches the carrying capacity K, and stops when it reaches it. The result is the S-shaped curve that turns up everywhere from bacterial cultures to the spread of a new technology.
The shape comes from a single term. Growth is proportional to both the population present and the fraction of capacity still unused, so it is slow when the population is small because there are few individuals, and slow again when it is near capacity because there is little room left. It is fastest in the middle, at exactly half of K.
The formula
The model integrates to a closed form, so no step-by-step simulation is needed: N(t) = K divided by 1 plus ((K − N₀)/N₀) times e to the power of minus rt. The growth rate reported alongside is the instantaneous rate at the starting population, rN₀(1 − N₀/K), which is the slope of the curve at t = 0 rather than an average over the period.
| Term | Meaning |
|---|---|
| N₀ | The starting population size. |
| K | Carrying capacity: the population the environment can sustain indefinitely, and the ceiling the curve approaches. |
| r | The intrinsic growth rate, the per-individual rate of increase when the population is small enough that crowding does not bite. |
| Inflection point | K divided by 2, where growth is fastest and the curve changes from accelerating to decelerating. |
The inputs explained
| Field | What to enter |
|---|---|
| Initial population N₀ (individuals) | The population at the start of the period. |
| Carrying capacity K (individuals) | The carrying capacity. It must be above the starting population for the curve to grow. |
| Intrinsic growth rate r (/year) | The intrinsic growth rate per unit time, as a decimal. 0.3 per year means 30% per year under uncrowded conditions. |
| Time elapsed (years) | How far forward to project, in the same time unit the growth rate uses. |
When to use it
Projecting a recovering population
A reintroduced or recovering species starts far below capacity and grows almost exponentially at first, then flattens. The logistic curve is the standard first approximation for how long that takes.
Modelling a culture or a colony
Bacterial and yeast cultures follow the logistic shape closely because the limiting resource is well defined, which is why the model is usually introduced with them.
Sanity-checking an exponential projection
Any projection that grows without limit will overstate the long run. Running the same growth rate with a realistic ceiling shows how quickly the two diverge.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does a population approach its carrying capacity?
The same population projected to a series of time points, from the start out to forty years.
| Years elapsed | Population | % of carrying capacity |
|---|---|---|
| 0 | 50 | 5.00% |
| 5 | 191 | 19.1% |
| 10 | 514 | 51.4% |
| 20 | 955 | 95.5% |
| 30 | 998 | 99.8% |
| 40 | 1,000 | 100.0% |
Questions
What is the difference between logistic and exponential growth?
Exponential growth has a constant per-individual rate and no ceiling, so it rises without limit. Logistic growth reduces that rate as the population fills the available capacity, so it flattens. The two are nearly identical while the population is small relative to K, which is why an early exponential fit can look convincing and still be badly wrong later.
Where does growth happen fastest?
At half the carrying capacity, the inflection point of the curve. Below it each new individual adds more growth than the crowding it causes takes away; above it the balance reverses. This is the basis of maximum sustainable yield in fisheries, where a stock is deliberately held near K/2.
What if the starting population is above the carrying capacity?
The model still works and the population declines towards K rather than growing to it. The same equation covers both directions, since the term that drives growth simply goes negative when N exceeds K.
How realistic is a fixed carrying capacity?
It is a simplification. Real capacity varies with season, weather, disease and habitat change, and populations frequently overshoot and oscillate rather than settling smoothly. The logistic curve is a useful first model and a poor final one.
For an interacting pair of populations rather than one in isolation, see the Lotka-Volterra calculator. For mortality within a managed population, see the animal mortality rate calculator.