Q: What’s the highest population growth rate that the Earth can support?

Physicist: Zero.

Populations tend to grow exponentially, which is why the growth rate is defined as R in P = Ae^Rt, where A is the population at time t=0, and P is the population at any other time t. If the average growth rate is greater than zero, then the population will grow exponentially forever, which is sadly impossible. Here’s why:

If the population could grow forever, then eventually the total mass of Humans would be greater than the mass of the Earth, which makes no damn sense. What were we eating?

The population could grow forever if we found a way to colonize other star systems. However, even with speed-of-light ships we could only colonize something like $\frac{4}{3} \pi (Ct)^3$ planets in time t, since we can’t travel faster than light (this is the volume of a sphere that expands at the speed of light, C). The population density in the colonized area of the universe would then look like $\frac{\textrm{Population}}{\textrm{Volume}} = \frac{3A e^{Rt}}{4 \pi (Ct)^3} \propto \frac{e^{Rt}}{t^3}$ , where “ $\propto$ ” means “proportional to”. You’ll notice that if R is greater than 0, then as time increases the population density goes up forever, which makes no damn sense. If you don’t notice, then just graph it.

What this ultimately means is that the average, over all time, of the number of children that a person has is 2 or less. No way around it.

We’ve had a lot of visits from reddit.com with a little confusion over the line “populations tend to grow exponentially” (my bad). What I should have written is “populations tend to grow exponentially under the assumption that they have not yet begun to saturate the available resources”, but I figured that might be pushing the discussion. The less that resources are available, the more the population will tend to level off. Both the exponential growth and leveling off can be modeled using various scalings of the logistic function. Try graphing: $\frac{1}{1+e^{-t}}$ , and $e^t$ . You’ll find that they line up almost exactly until near $t=0$ , where resources begin to dwindle.