Scholars are hard pressed to get accurate data for the fertility of slaves both within slavery and after manumission.
The following is generally accepted among historians of the period:
"Owing to heavy disease loads, life was short even in the top echelons of Roman society. For that reason alone, slaves need not have lived significantly shorter lives simply because of the hazards inherent their legal status. However, the use of slaves in particularly unhealthy rural locales and especially their disproportionate concentration in large and therefore infection-rich cities may well have lowered their overall mean life expectancy even further, thereby impeding natural reproduction at or near replacement level."
Also, " Several factors militated against slave reproduction at or near replacement level: imbalanced sex ratios if and when they persisted; higher mortality in cities and mines and on malarial estates; family break-ups through sale or inheritance; and the manumission of slave women of childbearing age.”
Indeed, while the average life expectancy of the poor in the empire was, according to certain scholars from 20-30 years, that of slaves was under 20. Miners, galley slaves, and men being worked to death under insalubrious conditions in latifundia weren't procreating and keeping slave numbers stable. Only house slaves and commercial slaves would have fared better, but most of them were living in crowded urban environments and would have been subject to the periodic outbreaks of disease that entailed.
For the viewpoint of another respected historian, we have John Madden:
http://www.ucd.ie/cai/classics-irela.../Madden96.html
"However, on closer analysis, this reasoning is flawed. True, some of the more fortunate city slaves and certain rural ones as well enjoyed a secure home life. And undoubtedly these together with the many female slaves who had children by their masters (or other free men) will have contributed considerably to the number of new slaves entering the system each year. Nevertheless, the belief that the total slave-body was more or less self-propagating is unsound. There are a number of reasons for this.
"1. First of all it is clear that males were in the majority where work was difficult and weighty - in building, in mining, in numerous types of industry, in a wide variety of services such as loading and unloading at docks, portage, transportation, etc. In agriculture also male slaves would have been more in demand. Small landowners would have to be content with whatever slaves were available irrespective of their sex, while large landowners would undoubtedly have needed some female slaves e.g. for weaving, cloth making, cooking. However, it is clear from passages in Varro and Columella, where the question of which of the more reliable agricultural slaves should be allowed a female companion is treated, that permission for such a partner was a special concession. Varro recommended that praefecti ['overseers'], as an incentive to their faithfulness, should be granted female slaves with whom they could have children, while lesser slaves should have to do with less (Rust. 1.17.5,7). In Columella, on the other hand, it is the vilicus ['steward'] who should be given a female partner (Rust. 1.8.4). In the ergastula - the private prisons belonging to many Roman farms where slaves were forced to work in fetters - the inmates would have been very largely male. It is evident from this that among agricultural slaves males surely outnumbered females.
When we turn to domestic staff the evidence suggests that there too male slaves were more numerous. S. Treggiari in her analysis of the 79 members of the city household staff of Livia has noted that 77% were male (the percentage was similar among freedpersons and slaves). This is a very revealing figure since we would expect a domina to have a higher number of female staff than a dominus. And in her study of the city familiae of the Statilii and the Volusii Treggiari has shown that about 66% of the freedpersons and slaves were male, while of the thirty child slaves whose names were inscribed on the tombs of these two families 80% at a minimum were male. "
So, contrary to an assertion made upthread, the statements of Razib Khan in his comments about this paper are in total accord with the pronouncements of recognized authorities and it is those who assert the contrary who are obviously posting unsupported opinions motivated by repugnant world views.
According to a recognized specialist in this field, " I allow for a reproductive shortfall of up to 50% in late Republican Italy, at a time when the slave population was greatly expanding and dynamically unstable."
Note that this is at a time when military conquests were increasing the number of slaves.
So, how was the deficit at least partly made up?
Walter Scheidel makes short shrift of the role of pirates in supplying slaves. You may wish to read his explanations on page ten of his work:
https://www.princeton.edu/~pswpc/pdf...del/050704.pdf
Other sources included those enslaved as the result of conviction of crimes, among which were avoiding the census and the draft, freedwomen cohabiting with slaves, and ungrateful freedmen. Foundlings were sometimes enslaved, and poor parents sometimes sold either themselves or their children or both into slavery, and increasingly as time went on, but it's impossible to get completely accurate numbers for this local source of slaves.
"It is unlikely that Roman fathers ever had a formal right to sell their children; in classical law, family members could not be sold into slavery or pawned. As in the case of enslaved foundlings, the state favored a pragmatic compromise position: the sale of minors did not affect their status and was technically void; therefore, redemption remained possible, with or sometimes without compensation. This focus on redemption accounts for prohibitions of the sale of such slaves overseas.
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As a result, there were no clear boundaries between sale, pawning, and lease: given the formal inviolability of free status, ‘sale’ might merely amount to an extended lease of minors in times of hardship. "
Bondage for debt, which could be for life given the life expectancy of the average slave, also increased as time went on and the Empire came under more stress.
Some of the short fall was made up because slave trading was conducted with peripheral areas once the maximum extent of the empire had been reached and no new conquests were filling the slave marts.
“The Black Sea region and the Caucasus had been well established as a major source of slaves since the archaic Greek period (see Chapter 6), and this tradition continued into late antiquity. Together with free Germany, that northeastern periphery must have accounted for most imports once the Roman empire had reached its maximum extension. Black slaves from as far away as Somalia and the occasional import from India made for comparatively rare but consequently high-prestige retainers.”
It should no longer need to be repeated that these slaves would have gone to wealthy estates in the provinces as well as Italia, whether the estates were owned by Italic Romans or Gallic Romans etc. The enslavement of foundlings, debtors etc. would have been the enslavement of locals.