The Sigmoid has no meaning other than as a cheap heuristic to subjectively describe an S-curve, see my rant way below. I know that you aren't applying it to describe something like a URR, yet I don't want to propagate the myth that it has some brilliant deeper meaning.

I mean that the prices will have an S curve with a sharp rise in the middle. The low part of the S is when there are more producers than consumers. They bid against each other holding the price low. Prices rise slowly with production costs.

Then peak oil happens.

Now you have more consumers than producers. They quickly bid up the price for this non-replaceable resource. Prices slew rapidly toward the top of the S. I think that is what we are seeing now and why price elasticity is decreasing - which is opposite of expectations.

Prices stop rising when most of the value of a barrel of oil is going to the producer. Prices stop rising because consumers have no extra margin to trade for more oil. The consumer economy has little surplus to grow. Demand destruction is causing contraction (holding the price lower).

I don't think we are at the top of the S yet. I think at $300.00 we will be there.

I agree and this happens for reasons completely different than what happens in a Birth-Death model which produces the textbook definition of a Sigmoid.