Population Growth Calculator
Two Very Different Ways a Population Can Grow
Left completely unchecked, a population grows exponentially — each individual adds more individuals at a constant rate, and the curve accelerates without limit. Real environments rarely allow that for long: food, space, and resources cap how large a population can get, which bends the curve into an S-shape that levels off at a carrying capacity. This calculator models both scenarios.
The Formulas
Logistic Growth: N(t) = K ÷ (1 + ((K − N0) ÷ N0) × e(−r × t))
N0 is the starting population, r is the growth rate (as a decimal), t is elapsed time, and K is the carrying capacity used only in the logistic model.
Where This Calculation Matters
- Wildlife and conservation biology — predicting how a species population responds to habitat limits uses the logistic model almost exclusively.
- Microbiology and fermentation — bacterial cultures grow exponentially during log phase before resource limits force a logistic plateau.
- Epidemiology — early outbreak spread is frequently approximated as exponential before susceptible population limits bend the curve.
- Invasive species modeling — forecasting how quickly an introduced population could expand absent natural predators or resource limits.
Exponential vs. Logistic Growth, Side by Side
| Time | Exponential (no limit) | Logistic (K = 1000) |
|---|---|---|
| t=0 | 1,000.0 | 10.0 |
| t=10 | 1,648.7 | 168.7 |
| t=20 | 2,718.3* | 803.0 |
| t=30 | 4,481.7* | 987.9 |
*Exponential figures at t=20 and t=30 use the same 5%/hr rate extrapolated with no cap, shown only to illustrate how sharply the two models diverge once a real population would hit resource limits; the logistic curve levels off near its carrying capacity of 1,000 instead of continuing to climb.
How to Use This Calculator
- Choose the growth model: exponential (unrestricted) or logistic (capped by carrying capacity).
- Enter the Initial Population, Growth Rate (% per time unit), and Time Elapsed.
- For logistic growth, also enter the Carrying Capacity (K).
- Select Calculate to get the projected population.
Related Calculations
Modeling growth by discrete cell divisions rather than a continuous curve? See the Cell Division Time Calculator. The same exponential math also describes the reverse process — see the chemistry section's Half-Life Calculator for decay instead of growth.