AP Calculus BC

Exponential vs Logistic Growth

Exponential growth has a rate proportional to the current amount and increases without any ceiling. Logistic growth multiplies that by a braking factor, so growth slows as the quantity nears a carrying capacity and is fastest at exactly half of it.

Exponential

Use when: The rate is proportional to the amount present, with no stated maximum.

Logistic

Use when: The problem names a carrying capacity, a limiting value, or a maximum sustainable population.

Side by side

	Exponential	Logistic
Equation	$\frac{dP}{dt} = kP$	$\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)$
Long-run behaviour	Grows without bound	Approaches $L$
Fastest growth	At the largest $P$	At $P = \frac{L}{2}$
Graph shape	J-shaped	S-shaped

The logistic factor $\left(1 - \frac{P}{L}\right)$ is the brake. When $P$ is small it is close to 1 and the curve looks exponential; as $P$ approaches $L$ it approaches zero and growth stalls.

The inflection point at $P = \frac{L}{2}$ is where growth is fastest, and it is the fact most exam questions turn on. You can read it straight off the differential equation without solving it.

You rarely need the solved form

Carrying capacity, limiting behaviour, and fastest growth all come from the differential equation itself. Solving the logistic equation requires partial fractions and is seldom necessary.

Frequently asked questions

What happens if the population starts above the carrying capacity?

The braking factor turns negative, so the population decreases toward the carrying capacity instead of growing toward it.

In the CED: Unit 7: Differential Equations