# What is meant by the carrying capacity of a logistic function?

Apr 10, 2018

The carrying capacity is the limit of $P \left(t\right)$ as $t \to \infty$.

#### Explanation:

The term "carrying capacity" with respect to a logistic function is generally used when describing the population dynamics in biology. Suppose that we are trying to model the growth of a butterfly population.

We'll have some logistic function $P \left(t\right)$ which describes the number of butterflies at time $t$. In this function will be some term which describes the carrying capacity of the system, usually denoted $K = \text{carrying capacity}$.

If the number of butterflies is greater than the carrying capacity, the population will tend to shrink with time. If the number of butterflies is less than the carrying capacity, the population will tend to grow with time. If we let enough time pass, the population should tend toward the carrying capacity.

Thus, the carrying capacity can be thought of as the limit of $P \left(t\right)$ as $t \to \infty$, where $P \left(t\right)$ is a logistic growth function.