The current land speed record of 763 miles per hour was reached in 1997, which is about as fast as passengers or cargo can be expected to fly. If my hypothesis is correct that transaction speed is proportional to the square of the population, then starting at 0.7 mph in 0 A.D. (the best fit to historical data, with a starting population of 300 million), the world will need to exceed the current record speed by 2040 for the population to continue growing. Interestingly, the timing of the projected population peak corresponds well with the peak that my consumption model projects from the depletion of ecological resources.
There are several differences between the projections of this new “speed model” and my consumption model. For one, the peak population values are different: 9.9 billion for the speed model versus 7.8 billion for the consumption model. The speed corresponding to the consumption model’s peak would be about 480 mph, or around two-thirds of the record speed and close to the Root Mean Square value that can be expected from observations of physical systems. The speed model does not inherently project the time that a population value is reached; the limit at 2040 assumes that the population would grow at a constant exponential rate starting in 2000 (a bit faster than reality, since the rate is decreasing). Also, the speed model does not require a decrease in population after the maximum speed is reached; if anything, the population should stay constant.
To properly compare the two models, note that speed is proportional to consumption. For every transaction in the speed model two resources are required to be exchanged, and the total number of transactions is one-half the square of the population (everyone in the group trades with everyone in the group directly or indirectly). For each resource, an average distance must be traversed to move the resource from its original location to a member of the group. Since we're dealing with a fixed amount of time, the amount of resources consumed (traded) is proportional to the number of transactions, which is proportional to the speed (distance traveled in the fixed unit of time).
When the maximum speed is attained, any increase in the number of transactions will result in fewer resources being consumed per transaction (and per person). Conversely, increasing the resources per transaction will force a reduction in the number of transactions and therefore the number of people (population).