I'm not sure how much this helps, but the
Doomsday Argument might be worth considering.
Basically, it uses statistical sampling to take a subset of a group and use it to find out how large the entire group is. By applying that to humans (e.g. "all humans who've ever lived up until now" being the subset, and "all humans who are living, have ever lived, or will ever live" being the total), you can then - presuming a baseline regarding the average lifespan and total number of people existing resources can support (e.g. capping the total number that can be alive at a given time) - chart out a rough figure for how long humans have before we go extinct.
Obviously this will shift depending on how you calculate things like the total number of humans who've lived so far, what the average life expectancy will be, and how many humans the planet can support at once, but the numbers generated don't seem to vary that widely. The baseline prediction given in the article (which, using the math given there, is listed as only 95% likely to be accurate) is that we'll last just over
9,100 years.
This might be useful if you want to figure out how long a civilization lasts - that is, instead of trying to figure if something brings a species to an end, and then trying to determine what, this will let you figure out a date for when an extinction event occurs, and then you can work backwards to determine how their end came about.