Exponential growth means a steady ratio
- The idea
- Predict
- See it
- Why
- Check
The idea
“A population is exponential when its curve is steep, when it doubles, or when it adds a lot each year.”
Textbook exponential curves shoot straight up, and the classic example doubles. A gentle rise that never doubles can still be exponential.
The short version
The mistake: A population is exponential when its curve is steep, when it doubles, or when it adds a lot each year.
What's actually true: It's exponential when each count is the same multiple of the one before, because the per capita growth rate stays constant.
- Exponential growth means the per capita rate, births minus deaths per individual, stays constant. So each count is the same multiple of the one before.
- To test counts taken at equal intervals, divide each by the one before. A steady ratio means exponential, whether it's 1.05 or 3.
- Adding the same amount each time is linear. Start at 100 and add 100 a year, and the ratios go 2, 1.5, then about 1.33. The per capita rate keeps falling.
- A rise alone doesn't prove it. Near carrying capacity a population still rises, but its ratio shrinks, so it's no longer exponential.
On the exam: Divide each count by the one before. The same ratio every time means exponential, however gentle it looks.