Logistic Population Growth Calculator
Real populations cannot grow forever; food, space and competition eventually slow them down. The logistic model captures this by bending the exponential curve into an S-shape that flattens at the environment's carrying capacity. This calculator projects a population using the standard logistic equation N equals K divided by one plus ((K minus N0) over N0) times e to the power of negative r times t. Here K is the carrying capacity, the largest size the habitat can sustain, N0 is the starting population, r is the intrinsic rate of increase when crowding is negligible, and t is the time elapsed. Early on, when numbers are small, growth looks nearly exponential, but a brake strengthens as the population rises, so the curve climbs steeply through the middle and then levels off toward K. A population of one hundred with a carrying capacity of one thousand and a rate of 0.3 reaches about 332 after five time steps, partway up the S. Use it to model wildlife recovery, an invasive species, or any growth bounded by limited resources. Every figure is computed deterministically from the equation, never estimated, with the method and a worked example shown below for verification.
Logistic growth bends toward the carrying capacity: N = K / (1 + ((K - N0)/N0) e^(-rt)). With K = 1,000, N0 = 100, r = 0.3 and t = 5, the population reaches 332.43.
Logistic growth formula
N = K / (1 + A x e^(-r x t))
A = (K - N0) / N0
K = carrying capacity, N0 = initial population
r = intrinsic rate of increase, t = time
The constant A sets the starting gap below K. As time passes the term e^(-rt) shrinks toward zero, so the denominator falls toward one and N rises toward K.
Worked example
A population of 100 has a carrying capacity of 1,000, a rate of 0.3, projected 5 periods ahead.
- A = (1,000 - 100) / 100 = 9
- e^(-0.3 x 5) = e^(-1.5) = 0.223130
- Denominator = 1 + 9 x 0.223130 = 1 + 2.008171 = 3.008171
- N = 1,000 / 3.008171 = 332.43
The projected population is 332.43. These are the calculator's default inputs, so the result above matches the widget exactly.
The S-curve in stages
Growth is slow near the start and end, fastest in the middle.
| Phase | Behaviour |
|---|---|
| Early | Near-exponential |
| Middle | Fastest growth |
| Late | Levels off at K |
Population ecology and modeling: US Geological Survey (USGS).
Logistic population growth calculator: frequently asked questions
What is the logistic growth model?
Logistic growth describes a population that grows quickly at first then levels off as it nears the environment's carrying capacity. The size at time t is K divided by one plus ((K minus N0) over N0) times e to the power of negative r times t, where K is the carrying capacity.
What is the carrying capacity K?
K is the maximum population the environment can support indefinitely given its food, space and other limits. As the population approaches K, growth slows toward zero, producing the familiar S-shaped curve rather than the runaway curve of exponential growth.
How does logistic differ from exponential growth?
Exponential growth assumes unlimited resources and accelerates without bound. Logistic growth starts the same way but adds a brake that strengthens as the population rises, so it flattens out at the carrying capacity instead of climbing forever.
What does r mean here?
r is the intrinsic rate of increase, the per-capita growth rate when the population is very small and resources are abundant. It sets how fast the curve climbs in its early phase before crowding effects slow it down.
Is the result computed automatically?
Yes. The page applies the logistic equation deterministically. No value is estimated or hard-coded, so changing the carrying capacity, starting size, rate or time updates the projected population instantly.
Official sources
- Population ecology and modeling: US Geological Survey (USGS). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.