The relationship I’ve identified between consumption and population is analogous to the relationship between power and current in a direct current electrical circuit. In the analogy, per-capita consumption corresponds to electrical potential (voltage) and the multiplier of population to get per-capita consumption corresponds to resistance (as in Ohm’s Law).
My mathematical modeling suggests that the overall “resistance” is unchanging in the world “circuit” and that “voltage” is the primary variable that affects “current.” The voltage varies exponentially with changes in the amount of available “energy” (resources), a variation that is primarily offset by control of the “voltage source” (resource extraction and distribution technology).
Because in a closed system “energy” (non-renewable resources) cannot increase, “voltage” must inevitably decrease – and with it, “current.” The world, in a sense, is like a light bulb and a fixed number of batteries; first one battery is attached to the light bulb, then another (in “series” with the first), and then another, until all the batteries are connected. Just as a battery has internal resistance that increases over time, causing the voltage across the battery to drop, the resources consumed by humanity become waste which inhibits further consumption, causing per-capita consumption (and population) to decrease.