Here's one that looks fairly persuasive to me...
This was sand that was cycled with biospira, ramped the capacity way up, then left it "curing" (circulating in the dark without input) for a couple of months, the capacity for nitrification dropped considerably but did not disappear.
Then I pulled some of the sand, added it to tank water spiked with ~1ppm total ammonia-N.
The green is the ammonia, but look at the blue - NO2-N. It's remarkable how cleanly you can fit it to an exponential for ~3 days.
The exponential function for that fit is y = 0.0486*e^(.824*t) - 0.048
That rate constant implies a nitrite doubling time of 0.841 days = 20.2hrs.
It stops fitting at 3+ days because then NO2->NO3 becomes significant, and by day 3.5 there's significantly lower ammonia concentration left to process.
Note that the ammonia decrease is much less clean - no evidence of exponential population growth, because there are a bunch of ways to consume ammonia. The ammonia oxidizers weren't the only path. But they were the only path to NO2 production and that capacity expanded in exactly the relationship you would expect for population growth.
If you just want to see pretty exponential curves, start with a tiny population and track nitrite until the NO3 producers catch up.