I'll give you a hint: Each one of those equations has an assumption embedded in it. The first and last one seem pretty solid. The middle two, though- The second assumption is that maintenance on capital is some constant proportion of total capital. That might be true, but also might not be, especially if you consider knowledge to be a form of capital. Maybe maintenance should scale with the log of capital, or something like that. But even then, I'm pretty sure that unless maintenance goes to zero, this doesn't affect the general shape of the curve.

The third assumption, that capital increases as a constant proportion of resources extracted, is even more questionable. I think it's fair to say that we are able to do more with less now than we could in the past. You might say that that is a function of the existing capital too. So what happens if you try something like C += r*q*(1 + .01*C)? Do you still get a peak?

And of course, there's the assumption that the next bit of resources you extract is a little harder than the last bit. Statistically this has been true for some time now, but you could imagine this rule reversing, at least temporarily, if we gained access to some new resource, say, asteroid mining.

So there's plenty in the model to attack, if you take the time to really consider it.

Well, I majored in physics, so to me the cow is always a sphere.. but you know, all models of reality are imperfect, no matter how complicated. And it’s true that I have no idea where on the curve we might be. I would certainly not use this model to make specific predictions- that would be silly. It’s more the general behavior of the system I’m interested in. But you can definitely make refinements. For example, if you add renewables then you still get a peak, but it doesn’t go all the way to zero afterwards.

I do realize too that my model makes assumptions- which have not been demonstrated to be true. So, that’s where I would attack it. Which assumptions are false?