First off, slavery didn’t abruptly end when the western empire collapsed—it was already in decline beforehand, and it persisted to some extent for many centuries afterward. And it wasn’t unique to the Romans—slavery was common in nearly all European cultures at the time.

Secondly, it’s important to distinguish between Roman slavery and the sort of slavery we’re more familiar with from the American South, where slaves and freeborn were permanent, separate communities. In the Roman Empire, many slaves weren’t slaves from birth—there were also war captives, convicts, poor people who voluntarily sold themselves into slavery, and abandoned infants who were found and raised as slaves. Likewise, many slaves didn’t remain slaves for life—they could purchase their own freedom, or be freed by their owners when they were too old to work, or as an act of religious piety (this was increasingly common in the late empire).

Given these additional sources of influx and outflux of the Roman slave population, and the relative lack of historical documentation, it’s impossible to say anything definitive about its demographics; but most estimates put the reproductive rate of the slave population at well below replacement level. So the slave population would be expected to naturally disappear over time in the absence of new sources—or in the absence of continuing demand.

Lastly, the later empire saw the rise of other categories of semi-free workers that gradually replaced agricultural slavery. Faced with recurrent labor shortages, imperial laws made occupations like farming a hereditary obligation, which gave rise to the medieval institution of serfdom. And tribes migrating into the empire were given agricultural land to work, on the condition that they provide continuing recruits to the Roman army.