AP Calculus BC glossary

Carrying capacity

Carrying capacity is the limiting value a logistic model approaches as time increases. Growth is fastest at half the carrying capacity and slows to nothing as the population nears it, giving the logistic curve its S shape.

dPdt=kP(1−PL),lim⁡t→∞P(t)=L\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right), \quad \lim_{t \to \infty} P(t) = L

The inflection point sits at half the carrying capacity. That is where the growth rate peaks, and it is the most commonly asked feature of a logistic curve.

The mistake

Reading the carrying capacity as where growth is fastest. Growth is fastest at half of it; at the carrying capacity growth has stopped.

Appears in: Unit 7: Differential Equations