MISSION · Calculus
Logistic Growth
Exponential growth with a brake, and where the brake starts to bite
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The idea in one sentence
Exponential growth assumes unlimited room; logistic growth multiplies it by a factor that fades to zero as the population fills the space available.
The equation, read one piece at a time
\[\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)\]The \(kP\) on its own is plain exponential growth — the rate is proportional to how many there are, which is what you get when every individual reproduces independently.
The bracket is the brake. When \(P\) is small compared with \(L\), the fraction \(\frac{P}{L}\) is near zero, the bracket is near 1, and growth is essentially exponential. As \(P\) climbs toward \(L\), the bracket shrinks toward 0 and growth stalls.
\(L\) is the carrying capacity — the ceiling the system settles at. It is not a limit the population can never cross so much as the level at which births and deaths balance.
The two equilibria
Setting \(\frac{dP}{dt} = 0\) gives \(P = 0\) and \(P = L\), and they behave differently.
\(P = L\) is stable: nudge the population either way and it comes back. Below \(L\) the bracket is positive and it grows; above \(L\) the bracket goes negative and it declines. So a population that overshoots falls back rather than running away.
\(P = 0\) is unstable: any positive start grows away from it.
This is why the solution approaches \(L\) from either direction, and why the ceiling does not need to be enforced by anything outside the equation. It is already in the algebra.
The S-curve, and where it bends
Solving the equation — by separating variables and using partial fractions — gives
\[P(t) = \frac{L}{1 + Ae^{-kt}}\]where \(A\) comes from the starting population. The graph is the sigmoid or S-curve: slow, then steep, then flattening.
The most useful single fact about it is where the steep part peaks. The growth rate \(kP(1 - P/L)\) is a downward parabola in \(P\), with zeros at \(0\) and \(L\), so it is largest exactly halfway between:
\[P = \frac{L}{2}\]That is the inflection point of the S-curve — where the curve stops getting steeper and starts easing off. Before it, growth accelerates; after it, growth continues but slows.
In an epidemic this is the moment new cases per day peak, and it arrives when about half the eventual total has been infected. Total cases keep rising afterwards; the rate does not. Confusing those two is the most common misreading of an epidemic curve.
What it models
- Populations in a bounded habitat — the original use, from Verhulst in 1838.
- Epidemics, where \(L\) is the number who will eventually be infected.
- Technology adoption. Smartphones, cars, electricity all trace S-curves, with \(L\) the size of the market.
- Learning and skill acquisition, giving the “learning curve” its shape.
- Chemical reactions that consume their own reactant.
The recurring theme is a finite pool being used up. Wherever growth feeds on something limited, the curve bends.
Common mix-ups
- Reading the inflection point as the peak. The population is still growing at \(L/2\) — faster than anywhere else. It is the rate that peaks there.
- Expecting \(P\) to reach \(L\). It approaches it asymptotically and never arrives.
- Assuming growth must start below \(L\). Start above and the curve descends to \(L\), which the equation handles without modification.
- Confusing \(k\) with the actual growth rate. \(k\) is the rate the population would have if it were unconstrained; the realised rate is always lower.
- Fitting a logistic curve too early. Before the inflection point, logistic and exponential data look nearly identical, which is why early epidemic forecasts vary so wildly.
Check yourself
What is \(L\) called, at what population is growth fastest, and what does \(P\) approach in the long run?
\(L\) is the carrying capacity. The rate \(kP(1 - P/L)\) is a downward parabola with zeros at 0 and \(L\), so it peaks at \(L/2\). And \(P \to L\), approaching but never reaching it.
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