I pointed out that there is a way to derive precisely the logistic function via a stochastic spread in characteristics here:
https://forum.azimuthproject.org/discussion/comment/21938/#Comment_21938

This essentially says that an aggegration of sub-populations will generate a logistics sigmoid if the individual populations follow a Max Entropy spread in characteristics.

I have never found anything like this elsewhere in the research literature -- trying "stochastic logistic function" on Google Scholar:
https://scholar.google.com/scholar?q=%22stochastic+logistic+function%22