The formula
N = N₀ · e^(r · t)
| Symbol | Quantity | Unit |
|---|---|---|
| N | Final population | |
| N₀ | Initial population | |
| r | Growth rate | 1/year |
| t | Time | years |
What it means
The exponential model assumes every individual contributes equally to growth and that resources are unlimited. At 2 % a year, a thousand individuals become 1,221 in ten years and 7,389 in a hundred. It is the same compound interest as in finance under different names, and it shares the property that intuition falls short: nobody estimates an exponential well by eye.
How to work it out by hand
- Convert the growth rate to a decimal
- Multiply it by the time
- Raise e to that product
- Multiply by the starting population
What is worth knowing
No real population grows exponentially for long, because resources run out. The model that describes what happens next is the logistic one, where growth slows as it approaches the carrying capacity of the environment and the curve flattens into an S. The exponential stays useful for the early phase — a species colonising new territory, an epidemic in its first weeks — and for a quick estimate: dividing 70 by the rate as a percentage gives roughly the doubling time. At 2 % a year, about 35 years. That rule of 70 works just as well for populations, savings and inflation.