A lot of people don't get further than Malthus, and don't realize that he was just the first pioneer. They think "Malthus was wrong", and don't realize the rabbit hole that opens up once you start treating population dynamics mathematically.
on the one hand, yes ecology and math bio is cool. On the other, the demographic transition does not fall out of these models whatsoever. Humans decided to do something very weird for whatever reason.
The actual population size during demographic transition looks very logistic-y. You'd be forgiven for thinking Verhulst applies. (though K is very much an empirical constant in that case, since you can't easily predict it from anything I don't think.)