Isn't this what the Fibonacci sequence is for, except with rabbits and not humans? Anyway.
I wrote a script in Python to simulate this. It makes a few assumptions:
| Attribute | Assumption made |
|---|---|
| Lifespan | 65 years |
| Minimum age of fertility | 15 years |
| Maximum age of fertility | 50 years |
| Pregnancy period | 9 months |
| Recovery period afterwards | 0 months |
| Male to female ratio | perfect 50:50 split |
It also assumes that they breed the maximum possible amount. All these attributes together should give us a fairly good estimate of the lower boundary for a problem like this.
I ran the program to a population of 1 000 000, because Python isn't exactly the fastest language in the world, and I'm not the best a competent programmer. Python told me it took 1270 months to get to a population of 1 million. The death toll for this was only 469, but that's presumably because they only started dying 65 years in. Plotting the values on a graph gave me this. Here's the same graph, but with only the trend line so that /r/dataisugly don't burn me at the stake for data point overlap. And then here's the same graph, but with a natural log y scale. It probably won't stop the angry mobs from /r/dataisugly because it doesn't intercept the y axis where it should, and I've probably labelled the y axis incorrectly for that type of graph.
But anyway, using these graphs we can solve the "why am I such an inefficient programmer" problem through extrapolation. Excel helpfully tells us that the equation of the trend line for the first/second graph is y=3.8529e0.0098x. If we solve this for x, we get a nasty looking equation, but one that we can substitute in values for. If we substitute in y=106 then we get 1.272. Python told us it should take 1270 months, so it's not perfect but surprisingly good.
So, to finally solve the original question, we substitute in y=109 to get our answer: 1977 months or 165 years.
This DOES seem very low and I'd wager that it's an error with the maths on my part but it assumes a relatively high life expectancy, and also assumes that everybody is always breeding all the time.
Here is my raw data. I suggest just copying and pasting it into excel or the headings are all messed up.
Here is my python program.
