AP Calculus BC glossary

Logistic growth

Also called: Carrying capacity

Logistic growth models a quantity that grows nearly exponentially when small but levels off as it approaches a maximum called the carrying capacity. Its growth rate is fastest when the quantity is exactly half the carrying capacity.

\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)

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

The solution curve is S-shaped, with an inflection point at $P = \frac{L}{2}$ . That inflection point is where the population is growing fastest, which is the fact most questions turn on.

You rarely need the solved form

Most exam questions ask for the carrying capacity, the limiting behaviour, or where growth is fastest. All three come straight from the differential equation without solving it.

Appears in: Unit 7: Differential Equations