Exponential Population Growth Calculator
Project a population growing exponentially from its starting size, growth rate and number of periods, with the total increase and doubling time.
How it works
N = N₀ · e^(r·t) · Doubling time = ln 2 ÷ r (r as a decimal per period)
Exponential (or geometric) growth models a population with unlimited resources, where each period multiplies the size by a constant factor. Using the continuous form N = N₀·e^(rt), the growth rate r compounds smoothly over time. The doubling time, ln 2 ÷ r, is independent of the current size — a hallmark of exponential processes, where a fixed percentage rate means a fixed doubling interval.
Worked example
A population of 1,000 growing at 5% per period for 10 periods reaches N = 1,000 × e^(0.05 × 10) = 1,000 × e^0.5 ≈ 1,649, a gain of about 649. At 5% the doubling time is ln 2 ÷ 0.05 ≈ 13.9 periods.
Sources: Exponential growth — Wikipedia
Frequently asked questions
When is exponential growth realistic?
Only when resources are effectively unlimited — such as bacteria in fresh medium or a species colonising new territory. Over time, food, space and predators impose limits, and growth bends toward the logistic S-curve instead.
What is doubling time?
It is how long the population takes to double at the current rate, equal to ln 2 ÷ r ≈ 0.693 ÷ r. Because it does not depend on the current size, a steadily growing population doubles again and again over equal intervals.
Why use e rather than simple multiplication?
The continuous form e^(rt) treats growth as happening at every instant rather than in discrete jumps, which suits biological reproduction that is spread through time. For discrete generations you can instead multiply by (1 + r) each period.
Related calculators
Educational estimate, not medical advice — verify formulas, units and assumptions for your specific case.