Motivation/derivation for Logistic Growth formula?

Question

$$P(t) = \frac{c}{1+ae^{-bt}}$$

I see that $ae^{-bt}$ is basically compounding growth formula: $Pe^{rt}$ Not sure what the +1 does. Includes the original 100% quantity? What about the reciprocal $\frac{1}{xxxxxx}$ part?

Just wondering how these transforms turn the expoential growth into logistic/S-curve growth.

Answer 1

From a comment:

The formula comes from solving the differential equation for logisitc growth, which is a standard equation which, being separable, is easily solved. The formula isn't something which directly pops out of the motivation, but instead pops out of a motivated differential equation. I haven't seen a discussion of the differential equation which doesn't discuss its motivation. See the Wikipedia article for a discussion. –

Answer 2

For a semi-mechanistic derivation (motivation) of the logistic equation in the case of epidemics, which I believe is what you're asking, you can find it in:

https://www.sciencedirect.com/science/article/abs/pii/S0025556418300853

For a general derivation of the logistic equation from resource model point of view, please find:

https://www.researchgate.net/profile/Yang_Kuang2/publication/254324683_Some_mechanistically_derived_population_models/links/53db0ad80cf2e38c63397ea3.pdf

If you search the article in Google Scholar, you should find a free pdf version somewhere.