Yep, we overrun and consume resources, and eventually we'll kill what we're living on, good analogy.
The problem is our technology. Yes, it's put us at the top, but nothing is reducing our numbers naturally. That is an UN-natural state for anything that lives on this planet. The state of affairs for any species is natural control, either through environment (droughts, floods, tornadoes, etc.) or another species eating them. Being on the upper level of the food chain, we have less things that kill us to begin with, and thanks to our technology, we're MUCH better at killing them faster. Technology is enabling us to be better and better at surviving (or avoiding) natural disasters, and technology improves the number of people who survive birth. Nothing slows our growth down.