Why populations boom and crash
Populations almost never settle at a comfortable number and stay there. They overshoot, collapse, recover, and overshoot again. The reason is not that nature is cruel or badly designed. It is that growth responds to conditions that have already changed, and a delay of any length turns a stable system into an oscillating one.
Nothing grows exponentially for long
Give a population unlimited resources and its growth compounds: more individuals produce more offspring, which produce more still. The curve is the same one that describes compound interest, and it is spectacular and brief, because unlimited resources do not exist.
What actually happens is that as numbers rise, the per-individual share of whatever is scarce falls. Births slow, deaths rise, and growth flattens toward a ceiling set by the environment rather than by the organism. That ceiling is the carrying capacity, and it is not a fixed property of a place. It moves with the weather, the season, and with what the population itself has already consumed.
The delay is the whole story
Here is the part worth understanding properly, because it generalises far beyond biology.
A population responds to conditions with a lag. Offspring conceived while food was plentiful are still being born after it has run out. An organism that built a large body during good times still has that body, and its costs, during bad ones. So numbers keep climbing for a while after the resource has already turned, and the population sails past the level the environment can support.
Then the correction arrives, and it is not gentle. The population is not merely above capacity, it is above a capacity that its own overshoot has damaged, because the resource was consumed faster than it regenerated. The crash therefore goes well below where it would have needed to stop, and the recovery has to wait for the resource to rebuild.
Any system where the feedback arrives later than the action it should have moderated will do this. The size of the swings is set by how long the delay is relative to how fast things grow. Short delays damp out into the smooth curve. Long delays produce sustained boom and bust, and long enough ones produce collapse with no recovery at all.
When the recovery does not come
The oscillating case assumes the resource can rebuild and enough individuals remain to use it. Neither is guaranteed.
If overshoot degrades the environment rather than merely depleting it, the carrying capacity itself drops, and the population oscillates around a lower ceiling than before, or keeps stepping down. And below a certain number, small populations face problems that have nothing to do with resources: chance alone can finish them, and variety is lost in ways that cannot be recovered, which is the subject of genetic drift and small populations.
This is why extinction is so often a two-stage event. Something reduces a population, and then something entirely unremarkable finishes it, and the second thing gets recorded as the cause.
What this looks like in a world you can watch
Every ingredient is present here, which is why population is the single most informative number on the site.
Resources deplete where organisms feed and regrow over time, so a good patch becomes a poor one through being good, and then recovers while nothing is eating from it. Reproduction responds to an organism's own stored energy, which reflects conditions it experienced in the recent past rather than conditions right now. The lag is built in, because it is built into the situation rather than into the code.
So the population graph oscillates, and the oscillation is not a bug or a tuning failure. It is the expected behaviour of a delayed feedback system, and a world whose population sat perfectly flat would be the more suspicious result.
One earlier candidate world went to zero at tick 2042 and stayed there, with the service running an empty map for thousands of ticks afterwards. Because every run here is reproducible from its seed, that death could be re-run offline with full instrumentation rather than guessed at. The account is in everything that has gone wrong.
A useful habit if you watch: check the population number, then come back in a few hours and check it again. A single reading tells you almost nothing, because you cannot tell a peak from a plateau without a second point. Two readings tell you which part of the curve you are standing on, and that is the entire skill. More on that in how to watch a world.