The population cycles of which you speak were of origin in many very different situations. Most of what we think we know about them -- their extent, duration and effects -- amounts to educated guesses. Among the causes were regular climate-cycles in temperature and rainfall distribution (ice ages and droughts, etc), one-time natural events like polar shifts or meteor strikes or extreme volcanic activity, and man-made conditions like deforestation, desertification (stripping of grasslands by overgrazing), protracted warfare or the collapse of top-heavy and interdependent political/economic systems. At least some are thought to have been "multifactoral", which means that two or more of these situations may have coincided accidentally, bringing down a civilization which was too secure to be felled by one factor alone. Such is thought to have been the case with the "Dark Ages" of the eastern Mediterranean, when virtually EVERY population in the region nosedived at the same time (within 30 years of each other, which is really in a blink of the eye because of how slowly things changed back then) during the late 13th/early 12th century BC. Egypt alone seems to have been spared. After each of these population "troughs", our adaptable species usually regrouped, found new venues or even fixed the problems that caused there to be more dying than reproducing, whereupon repopulation got moving. This was the case in the Dark Ages mentioned above; The Greeks, who had lost a greater proportion of their population and more of their knowledge/skills than any other bunch around the Med (even lost their original alphabet, Linear B), were up and at it again within less than 250 years -- again, a REALLY short time on the scale of how big a butt-kicking they'd taken! And finally, it should be noted that a recovery from depopulations often brought along with it advances that might never have happened otherwise (or at least not that soon).