> "Note that In the Petri net model for SI, the logistic equation is a result of the analysis of the rate equation not a premise."
The classic logistic equation is not strictly a stochastic derivation, and at best assumes a mean value for the measure of interest, with no uncertainty in the outcome. In any realistic situation there would be a spread in rates and constraints and so that's what my derivation calculates.
In the comment before yours at #32, I described an experiment and model for a process that is purely stochastic, the popping of a popcorn kernel. As an exercise, see if you can describe the behavior of the amount popped as a function of time just by assuming a mean value of one kernel popping. I originally tried to fit a mean-value model and it didn't come close to Figure C.8 above. That's because the variability in popcorn kernel characteristics was large enough to skew the expected temporal behavior away from that assuming a single mean value.
I just looked up the stochastic logistic equation and a recent paper on that is here: https://www.sciencedirect.com/science/article/pii/S0893965913000050
This uses an Ito calculus formulation which is a noise perturbation on the mean value approach.
I can elaborate more about the "quirky coincidence" and "heuristic" aspects with another example that I will place in another comment.