What spurs the growth of bacteria in a fish tank (see details!)

I’m looking for a very specific answer here. It might be a stretch.

Those familiar with the intricacies of keeping a healthy aquarium know about the nitrate cycle. My understanding is that a new tank will have problems at first because of ammonia, but that bacteria will soon grow to convert it to nitrites, and then different bacteria will grow that convert it to nitrates.

What I want to know is, what prompts the growth of these bacteria? Is it the ammonia/nitrites themselves that feeds their growth? Or do they grow as a side effect of something else in the tank chemistry?