Logistic Growth Calculator (S-Curve with a Carrying Capacity)
Population, or any quantity with a ceiling, after a given time, how close it is to the limit, and how long it takes to reach half and 90 % of it.
Results
The dashed line marks the value you entered.
What this tool does
Unlike plain exponential growth, logistic growth slows as it nears a limit K — food, space, market size — giving the S-shaped curve of bacteria in a dish, a spreading idea or a product being adopted. Growth is fastest exactly at half of K, at a rate r × K ÷ 4, and the curve is symmetric around that point. Early on it looks exactly exponential, so extrapolating the early part overshoots badly. The time to reach half of K is ln((K − P₀) ÷ P₀) ÷ r when you start below half; use the same time unit for r and t.
Formula
P(t) = K ÷ (1 + ((K − P₀) ÷ P₀) × e^(−rt)) · t½ = ln((K − P₀) ÷ P₀) ÷ r · fastest growth = rK ÷ 4
Variables
| Symbol | Meaning | Unit |
|---|---|---|
p0 | Starting population (or amount) | — |
kk | Carrying capacity K (the ceiling) | — |
r | Growth rate r (per unit of time, while still small) | /time |
t | Time | — |
P | Population at that time | — |
PC | Share of the ceiling reached | % |
TH | Time to reach half of K | — |
T9 | Time to reach 90 % of K | — |
GM | Fastest growth (when at half of K), per unit of time | — |
Worked example
- Starting population (or amount)100
- Carrying capacity K (the ceiling)1000
- Growth rate r (per unit of time, while still small)0.5 /time
- Time10
- Population at that time942.83
- Share of the ceiling reached94.3 %
- Time to reach half of K4.39
- Time to reach 90 % of K8.79
- Fastest growth (when at half of K), per unit of time125.00
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.