Head-to-head
Exponential vs Logistic Growth
The two curves are identical at the start and unrecognisable later. Exponential growth multiplies by a constant factor every period, so the rate of increase itself increases. Logistic growth does the same thing until the population approaches a limit, at which point the growth rate falls away and the curve flattens.
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The difference is a single term: logistic growth multiplies the exponential rate by the fraction of capacity still unused. When the population is small that fraction is close to one and the two curves nearly coincide. When it is near the ceiling the fraction approaches zero and growth stops on its own.
That is why exponential models are useful for short horizons and dangerous for long ones. A 20% rate has a 3.80-period doubling time; run it for forty periods and a population of 100 becomes 146,977, which for anything physical is a signal that a constraint is missing rather than a forecast.
Logistic growth also has a property that surprises people: the steepest absolute increase happens at exactly half the capacity, not at the start. That is where the inflection point sits, and it is why a maturing market can look like it is accelerating just before it stalls.
Exponential
How To Calculate Exponential Growth
Exponential growth happens when a quantity is multiplied by the same factor every period. Interest, population and radioactive decay all follow the same shape once the numbers are set up.
Logistic
How To Calculate Logistic Growth
Logistic growth is exponential growth with a ceiling. Everything grows quickly at first, then the growth rate fades as the population presses against what the environment can support.
How They Differ
| Attribute | Exponential | Logistic |
|---|---|---|
| Shape | Multiplies forever | Multiplies, then flattens |
| Formula | P₀ × (1 + r)^t | K ÷ (1 + ((K − P₀) ÷ P₀) × e^(−rt)) |
| Ceiling | None | K, approached but never crossed |
| Growth rate | Constant | Falls as the population nears K |
| Fastest absolute increase | Always now | At half of K |
| Doubling time | Constant: ln 2 ÷ ln(1 + r) | Not constant — it lengthens |
| Typical use | Early-stage anything, compound interest | Populations, adoption curves, market saturation |
| Failure mode | Absurd numbers at long horizons | A wrong K gives a wrong ceiling |
The Same Numbers, Both Ways
A population starting at 100 with a 20% growth rate, run once without a limit and once towards a carrying capacity of 1,000.
- Starting population
- 100
- Growth rate
- 20% per period
- Carrying capacity
- 1,000
- Periods shown
- 0 to 40
| Line | Exponential | Logistic |
|---|---|---|
| Period 0 | 100.00 | 100.00 |
| Period 5 | 248.83 | 231.97 |
| Period 10 | 619.17 | 450.85 |
| Period 20 | 3,833.76 | 858.49 |
| Period 40 | 146,977.16 | 996.99 |
| Ceiling | None | 1,000 |
| Doubling time | 3.80 periods | Lengthens every period |
At period 10 exponential growth gives 619.17 and logistic growth 450.85; by period 40 the gap is 145,980.17 against a ceiling of 1,000. The curves are identical at the start — the only difference is that one of them has a limit, and the limit decides everything after the inflection at 500.
When To Use Each
Exponential
- The horizon is short enough that no resource constraint binds, such as the first periods of a new market.
- You are modelling compound interest, where the base genuinely has no ceiling.
- You want a worst-case upper bound to show how quickly an unconstrained process diverges.
Logistic
- The thing being modelled consumes a finite resource: a population, a market, a battery, a server pool.
- You are forecasting adoption and expect saturation rather than indefinite growth.
- You need the inflection point, which is where capacity is half used.
Common Questions
How do I know which model fits?
Plot the logarithms. Exponential growth is a straight line on a log scale; logistic growth bends and flattens. If the log plot curves downward at the right-hand end, a constraint is already binding and the exponential fit will over-forecast.
What is the carrying capacity in practice?
Whatever the binding constraint is — food, floor space, addressable customers, grid capacity. It is rarely known in advance, which is the logistic model's weakness: the whole forecast depends on a number that is estimated, not measured.
Why is the inflection point at half the capacity?
Because the growth term is proportional to both the population and the remaining room, P × (K − P). That product is largest when P = K ÷ 2, so the absolute increase peaks there even though the percentage rate has been falling throughout.
Does compound interest ever become logistic?
Not in the arithmetic, but in reality it does: a market cannot exceed the money available to invest in it. The distinction is whether you are modelling the formula or the world, and a model that ignores the constraint is only valid while the constraint is far away.
What is the difference between logistic and exponential decay?
Decay is exponential with a negative rate and approaches zero; logistic growth is positive and approaches a ceiling above zero. Both flatten, which is why a decaying series and a saturating one can look similar over a short window.
More Comparisons
- Geometric Mean vs Weighted Average
Returns of +50%, −50% and +20% average +6.67% arithmetically but compound to −3.45% a year. The geometric mean follows the money; a weighted average follows the weights you chose. On the same three years they disagree by ten points.
- Standard Deviation vs Confidence Interval
With a mean of 100, a standard deviation of 15 and 100 observations, 95% of individuals fall between 70.60 and 129.40 while the 95% interval for the mean is 97.06 to 102.94 — ten times narrower, from the same data.
- Average vs Median
The average uses every value; the median uses the middle one. On five incomes where one is an outlier, the average moves 8.65 and the median moves 1.5 — here is why that matters.
Sources
- Logistic function — derivation and the carrying capacity — Wikipedia
- Exponential growth — doubling time and continuous compounding — Wikipedia
- NIST/SEMATECH e-Handbook — nonlinear growth models — National Institute of Standards and Technology
Both methods sit in Math.