Relevant to an article on which I'm slowly working, an important new study looking at human population growth and the Quaternary Megafauna Extinction shows our population has grown at the direct expense of other species. Here's marine scientist Emmett Duffy's summary and discussion of the study. And Here's the study.
We must be cautious though in interpreting these data. They do not mean pre-agricultural people were causing extinctions at rates resembling those we're seeing today. In fact, the evidence is clear that for nearly all of human history prior to agriculture, hunting-gathering ways of living came much, much closer than today's society to true sustainability -- in part because population sizes were so much smaller. (Evidence suggests, in fact, that many hunting-gathering societies made efforts to hold their populations to within ecological limits. [1]) As mentioned in the article, species extinction rates were modest until not long before the advent of agriculture when the growth of the human population began to accelerate.
But once the size of the human population began to take off there began a pronounced impact on large vertebrates, exacerbated, in some instances, by a changing climate. The subsequent shift to agriculture followed by industrialization and consequent massive increases in the human population have made matters far worse with a much wider array of species now affected. We are in the midst of the Sixth Mass Extinction in Earth's history, marked at its start by the Quaternary extinction event but having accelerated enormously in recent history.
This work adds well to that of anthropologist Jeffrey McKee and colleagues who found that after "species richness" human population was the single best predictor of the number of threatened species in a country. Here's Jeffrey's book elaborating on their work:
Sparing Nature: The Conflict Between Human Population Growth and Earth's Biodiversity
Here's the original study (PDF):
http://www.johnfeeney.net/storage/documents/McKee_2003_BiolConserv.pdf
I believe the Sixth Extinction is the most unjustifiably underreported issue in the ecological crisis. It's potential consequences are as severe as those of climate change, perhaps more so. (Of course the two overlap, but the Sixth Extinction would still be occurring in the absence of climate change.) And importantly, its link to the size and growth of the human population is direct and fundamental.