Here’s the introduction to a paper which seeks to determine when and for what reasons modern human populations began to undergo rapid growth spurts at various times during the Late Pleistocene and on into the Neolithic:
Reconstructing the timing and magnitude of changes in human population size is important for understanding the impact of climatic fluctuation, technological innovation, natural selection, and random processes in the evolution of our species. With census population sizes estimated to be only in the millions during most of the Pleistocene , , it is obvious that human population size has increased dramatically towards the present.
A major unanswered question is whether expansion began with hunter-gatherer groups, perhaps as a result of the invention of particular technologies or behavioral innovations, or much more recently with the advent of agriculture . Early mtDNA studies suggested that humans experienced a burst of population growth between 30 and 130 thousand years ago (kya)—well before the start of agriculture . More recent results have extended the timeframe for sub-Saharan African growth to 213–12 kya, depending in part on mtDNA haplogroup , .
However, it is populations—not haplogroups—that are subject to growth, and many present-day hunter-gatherer groups, including those in Africa, do not exhibit any mtDNA signal of demographic expansion at all . On the other hand, Y chromosome sequence data are compatible with a model of constant size for both hunter-gatherer and farming populations in Africa . Autosomal microsatellites tend to indicate an early (pre-Neolithic) start to population growth, but there is disagreement among studies on the time of expansion and whether or not the expansions involved African populations , . Zhivotovsky et al.  examined a large autosomal microsatellite dataset in 52 worldwide populations and concluded that African farmers, but not hunter-gatherers, exhibit the signal of population growth.
Unfortunately, inferences of demographic parameters based on the above mentioned loci may be unreliable due to the possible confounding effects of natural selection or evolutionary stochasticity (for the haploid loci), or uncertainty in our understanding of mutation rates or the underlying mutation process (for mtDNA and microsatellites) , .
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Reference: Autosomal Resequence Data Reveal Late Stone Age Signals of Population Expansion in Sub-Saharan African Foraging and Farming Populations – PLoS ONE
Cox MP, Morales DA, Woerner AE, Sozanski J, Wall JD, et al. 2009 Autosomal Resequence Data Reveal Late Stone Age Signals of Population Expansion in Sub-Saharan African Foraging and Farming Populations. PLoS ONE 4(7): e6366. doi:10.1371/journal.pone.0006366