# How to manipulate logistic function for carrying capacity K?

So I know that the logistic function is: $P(t)=(P_0 K)/P_0 + (K-P_0)e^-rt$ But I was wondering if there is a way to find carrying capacity which is K if it is not given. How do I go about manipulating the function in order to set it to K= a function? Thank you.

That's not a logistic function, but I will answer the question for the given equation.

The "carrying capacity" $K$ for $P(t)$ is the value it approaches as $t$ approaches infinity. That is,

$$K = \lim_{t\to\infty} P(t)$$

More generally, suppose you were given

$$Q(t) = \frac{A}{B + Ce^{-kt}}$$

as your function, where $A, B$ and $C$ are constants and are not necessarily expressed in terms of $K$ and $P_0$. The exponential term vanishes as $t$ tends to infinity, so the "carrying capacity" is just $\frac{A}{B}$.

• The OP formatted the logistic function badly: it should be $P(t)={P_0 K\over P_0 + (K-P_0)e^{-rt}}$. Fortunately your answer is still correct for the OP's function! – grand_chat Feb 23 '16 at 0:14
• @grand_chat Ahhh, my apologies. I have changed my example to a logistic function instead of an exponential accordingly. – SplitInfinity Feb 23 '16 at 0:29