Logistic Growth Calculator

Logistic growth is exponential growth with a ceiling. Everything grows quickly at first, then the growth rate fades as the population presses against what the environment can support.

Starting populationThe size at time zero. Must be greater than zero.Carrying capacityThe ceiling the growth approaches, written K.Growth rate per unit of time(%)The intrinsic rate r, so 50 means 0.5 per unit of time.Time elapsedHow many units of time have passed, in the same units as the rate.

Population at time t

575.1209

Share of the capacity: 57.51%
Growth rate at that point: 122.1784
Time to reach half capacity: 4.3944
Population at twice the time: 942.8256

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

The formula this calculator uses

P(t) = K / (1 + ((K - P0) / P0) x e^(-rt))

P(t): The population at time t
K: The carrying capacity, the ceiling the growth approaches
P0: The starting population
r: The intrinsic growth rate per unit of time

How to check the result by hand

1
Set the capacity and the starting value
The ceiling is 1000 and the start is 100. Knowing both is what separates the logistic model from plain exponential growth, which needs only a rate.
2
Work out the gap ratio
The room left to grow is 1000 minus 100, which is 900, and 900 over the starting 100 is 9. This single number says the population can multiply nine times over before it is full.
3
Evaluate the decay term
The exponential is e to the minus r times t, which is e to the minus 2.5 for a rate of 50 percent over 5 units of time, giving 0.082085.
4
Divide the capacity by one plus the product
Nine times 0.082085 is 0.738765, and one plus that is 1.738765. A thousand divided by 1.738765 is 575.1209, which is 57.51 percent of the ceiling.

For worked examples, common mistakes and the limits of this formula, read the full How To Calculate Logistic Growth page.