These are natural questions for someone in the early stages of addressing the issue.

Individual region profiles tend to have non symmetrical profiles more close to the first derivative of a Gompertz curve. But the Logistic is very simple to use and can in fact identify fairly the peak of a Gompertz like curve.

If so, one would expect to see increasing production for at least another few decades given current demand trends.

This is the crux of the matter. Every year, beyond the extra demand coming t the market, production has also to address the declining production for mature fields. CERA estimates this decline in 4.5% per year. That's almost 4 Mb/d every year (or a new Nigeria) to stay in the same place. If production goes up much higher it won't be soon before we a need a new Saudi Arabia every year to cope with depletion and extra demand. Where is that Oil?

Is there something in economics that studies the behavior of prices of goods for which there is a relatively fixed demand, a finite supply, and no obvious alternative?

When I studied Economics these were called Giffen goods. Geffen found that some goods would go up in price during times of Economic hardship, like bread. He thought these were indispensable goods that would rise in price because demand for them would increase from poorer families.

Today I understand this a completely different way. Geffen goods are those that get expensive first during times of resource driven resource constrains. Just like rice these days.

Vencit omnia veritas.