Exponential Population Growth Calculator
When a population grows with no shortage of food or space, it expands at a constant per-capita rate, and the result is exponential growth. This calculator projects that growth using the standard model N equals N0 times e raised to the power r times t. Here N0 is the starting size, r is the intrinsic rate of increase per unit of time written as a decimal, and t is how far ahead you look. Because each individual contributes the same proportional increase, the population accelerates: a colony of one thousand growing at two percent per year reaches about 1,221 after ten years, and the gap widens the longer it runs. The model is a faithful description only in the early phase, when numbers are far below the environment's carrying capacity, such as bacteria in fresh medium or a species newly arrived in suitable habitat. As resources tighten, real growth slows, which the logistic model captures. Mathematically it is the continuous-compounding cousin of compound interest, applied to organisms instead of money. Enter a starting size, a rate and a time horizon to see the projected population. Every figure is computed deterministically from the formula, never estimated, with the method and a worked example shown below for verification.
Exponential growth multiplies the start by a continuous factor: N = N0 e^(rt). A population of 1,000 growing at 2% per year for 10 years reaches 1,221.40.
Exponential growth formula
N = N0 x e^(r x t)
N0 = initial population
r = per-capita growth rate per period (as a decimal)
t = number of periods, e = 2.71828...
The factor e^(rt) is the continuous growth multiplier. Multiplying it by the starting size gives the projected population at time t.
Worked example
A population starts at 1,000 and grows at 2 percent per year for 10 years.
- r = 2 / 100 = 0.02, rt = 0.02 x 10 = 0.2
- Growth factor = e^0.2 = 1.221403
- N = 1,000 x 1.221403 = 1,221.40
- Increase = 1,221.40 - 1,000 = 221.40
The projected population is 1,221.40. These are the calculator's default inputs, so the result above matches the widget exactly.
Growth factor by rate and time
The multiplier e^(rt) grows quickly as r times t rises.
| r x t | Factor e^(rt) |
|---|---|
| 0.1 | 1.1052 |
| 0.2 | 1.2214 |
| 0.5 | 1.6487 |
| 1.0 | 2.7183 |
Population ecology and modeling: US Geological Survey (USGS).
Exponential population growth calculator: frequently asked questions
What is the exponential growth model?
Exponential growth assumes a population grows at a constant per-capita rate with unlimited resources. The size after time t is the starting size multiplied by e raised to the power of the rate times the time, written N = N0 times e^(rt).
What is r in the formula?
r is the intrinsic rate of increase, the per-capita growth rate per unit of time, expressed as a decimal. A rate of 2 percent per year is r equal to 0.02. It is the birth rate minus the death rate in a closed population.
When does exponential growth apply?
It is a good model only when resources are effectively unlimited and the population is far below the environment's carrying capacity, such as the early phase of colonization or a bacterial culture in fresh medium. Real populations eventually slow as resources tighten, which the logistic model captures.
How does this differ from compound interest?
It is the continuous version of the same idea. Compound interest with continuous compounding uses the same e^(rt) factor. Exponential population growth simply applies that continuous compounding to a count of organisms rather than money.
Is the result computed automatically?
Yes. The page applies the exponential formula deterministically. No value is estimated or hard-coded, so changing the 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.