Well, it does seem to have some valid arguments in regards to us using up resources faster than they replenish. What I find far more questionable is its claims of an actual collapse of civilization and mass death as a result.
Let's return to one of the most basic bits of economic knowledge: As supply shrinks, prices go up. As prices go up, people buy less of the product. So it seems to me that when we start hitting a point where we're actually really overdoing it with things (such as, say, fishing, which it argues is becoming unsustainable, as well as oil given that there's only a finite amount of the stuff), prices go up, people buy less, and that fixes the problem. It does mean people would have to accept a somewhat lower standard of living, but I'm not seeing societal collapse and mass death as a result. When we start running out of oil, oil prices go up, and people do things to use less oil.
The big analogy it brings up is a small island where, absent predators, the deer got overpopulated, overate the food, but rather than the population becoming stable or dropping, it caused them to die in droves. But deer don't have the market to constrain them; when there's food, they just eat it. There being an actual market and money constrains that in human society. So if something starts to become scarce, the price goes up, people buy less, and they try for other things that are cheaper and more plentiful. Food that is more sustainable will become more popular because its prices won't go up.
This is an important point, and the article never addresses it. It seems to assume we'll just wake up one day and find we're almost out of food and resources and then chaos reigns, when the market itself would prepare us for that simply via the laws of supply and demand.
Now, don't get me wrong, shortages would cause problems. People would have to cut back on various things they might like due to lower supply. But that's an inconvenience (even if a big one), not a collapse of civilization.