I note you failed to actually illustrate species with stable numbers. What the logistic function has to do with anything, I don't know. When reading through the reason they choose this function to model population growth it is exactly to construct a stable population growth, not because this was observed in practice.

And most certainly the human population curve looks nothing like the logistic function. It was going up much faster than the logistic function predicts. And the current slowdown of human population growth is so much faster than the ramp-up it's ridiculous to compare them. In the logistic function the slowdown mirrors the ramp-up. That's obviously not happening to humans.

I can give plenty of popular examples of wildly fluctuating population numbers. Cats versus kangaroos/wallabies versus rabbits versus birds in Australia. Buffalo's versus beavers versus cats versus ... in Yellowstone park. Those are well studied examples of major flips in population density. I also grew up in an area with lots of farms, and I know that for example the number of wolves living in nature follows this pattern. Some years it is so bad they actually attempt to eat adult humans (and sometimes succeed), sometimes there are none. We had a rabbit problem while I was at university, and the biology department investigated. There was an exchange between the number of cats (ex-housecats) and the number of rabbits. First the cats would eat nearly all the rabbits, then they'd starve to the point that few cats remained (and human interference like milk bowls does indeed make things worse), after that the rabbit population multiplied in two years by a factor of more than 1000, after which the cat numbers started growing again.

In all these cases you will observe massive swings in the population numbers. The swings were so large that just the swings themselves were at risk of making the species die off entirely (if a swing hits the number 1 or 0, the species never recovers, and has to be reintroduced. In practice that number is 9-10 according to the biology department. For rabbits or cats it never takes long for the species to be reintroduced of course). The biology department paper also claimed that while the timeframes vary wildly from species to species, these swings are the status-quo (within one habitat, the global population is more stable), not the exception.

I do know this more from an algorithmic point of view, and I learn biology only as I need it, or if I'm bored or stuck. Which is why I asked you for a few examples of these stable population numbers, as I wouldn't really know. In genetic algorithms stable population numbers don't occur.

Normally, because from time t -> t+1 you either let every current try duplicate n -> kn (higher k is better, but requires more resources), which is of course exponential growth. Then you kill all the ones that underperform (which is always basically all of them), or you select the k best performers. Or you can try to be smarter and model sexual reproduction, at which point you grow like n -> k n * (n - 1), which is even faster.

There are plenty of variations, but almost all the ones I've ever seen work have exponential growth + 90%+ die offs. Even then genetic algorithms are disappointingly slow in most cases. On the plus side, genetic algorithms will find answers no other algorithm is able to find (much better coverage of the search space). However this comes at a truly massive cost in resources (as you would expect, but it'd be nice if it didn't).

So you have an exponential growth, followed by a 99% dieoff (the number that die off has to increase the greater the population size becomes due to memory/cpu constraints).