It's kind of the idea that the most successful parasites can't kill their hosts too efficiently or quickly, because then they run out of places to live. If you screw up your environment like that, it'll come back to haunt you.
You may not be surprised to hear that that was my original, much desired goal, but after much thought and experiment I realized that the abrupt discontinuity caused by the exhaustion of the nonrenewables is real, for the reason that, in practice, the rate of nonrenewable use increase as their exhaustion approaches, creating an abrupt knee that resists conversion to a closed form.
> At least, this should remove the sudden discontinuity in the slope of the nonrenewable resources function.
But that abrupt end to nonrenewables is real -- it reflects the fact that the growing colony becomes increasingly reliant on them as their exhaustion approaches. So the knee can't be removed without changing the meaning of the equation.
Oh, well -- as it turns out, there are many very common differential equations that resist conversion to closed form. Orbital systems with more than two bodies are a classic example. Another one is the integral to the common exponential function that gives us the normal distribution -- very commonly used in statistics and elsewhere, but no closed form. All applications of the normal distribution use a numerical algorithm called the "error function" to produce results -- all of them approximate, and having the drawback that they cannot be symbolically differentiated or integrated.
Maybe a much more skilled mathematician that I am could find a way around this obstacle, but I doubt it.
Nature will always be fine, whether humanity as we know it survives is the problem.