Consider that surviving severe overpopulation is trauma. Survivors are physically and/or psychologically sick. They're not good breeders and at this point.
Happens frequently in aquariums. Guppies breed like guppies. Overpopulate, then die back. The proximate causes can be varied. Maybe there's an ammonia spike, oxygen deficiency or other consequence of overabundance.
When the "crash" happens, survivors tend to be a precocious bunch. They're past breeding prime, and probably injured by the events of the crash and its precedents. Starting a colony with "poor stock" is always more difficult, and that's what you now have.
It doesn't have to be overpopulation that causes a crash. Say you accidentally pollute the water with soapy hands. Some fish will die today. Some tomorrow. Dead fish rotting is also polluting. Fish surviving the ordeal will probably have reduced lifespans and breeding potential.
My unqualified guess is that most extirpation events leave survivors, usually in less than ideal shape. The technical extirpation happens later, as survivors gradually drop off without successful reproduction.
Good question about minimum population size for boom/bust cycles to occur. I suspect this is more about environmental richness than pure numbers. Some predation probably stabilizes things, and some boom/bust might be possible. If you had multiple colonies connected by 500m of tubing, this "distance" between territories might allow for "local" booms and busts to occur.
Recently there was an article posted on HN about how trauma experienced by parents are passed down to children through (a term that means it's related to activation of genes, I believe, but not DNA itself... I thought it was metagenomics but upon looking at the definition I don't think that's it). Perhaps the trauma of overpopulation activated or deactivated some genes which in turn reduced reproduction rate in offspring.
I was thinking the same thing. From the article I got the impression that the final collapse happened because all the survivors were the same age, and too old to spawn a new generation. Semi-separation would at least break the synchronization.
Between the two moments of equal population something must have changed in the population, either genetically or, IMO more likely, culturally.