# 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 = AeRt, 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.

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### 5 Responses to Q: What’s the highest population growth rate that the Earth can support?

1. Bob says:

The wording of your statement that people must have two or fewer children could be misconstrued as a mathematical validation that it is okay to have two children. The important points are: 1) people should not have children faster than the rate at which other people die, and 2) those children must consume less than the person they are replacing on average long term if the population does not drop or renewable resources are not protected or developed.

2. Physicist says:

At any one time the growth rate can be higher than zero, of course. This post was is an exploration of the very long term effects of a positive population growth rate. If the average person has less than two children, then eventually we’ll run out of people. If the last several generations have been sticking to the “everyone has two kids” rule, then you’ll find that the death and birth rates match. However, you are correct in that if everyone were to suddenly commit to having only two children starting today, the growth rate would stay positive, but slowly drop to zero over about 80 years. If the growth rate has been positive (as has been the case), then even having less than two children on average will cause the population to increase. This has been the case in China. Even with the implementation of the “one child” rule the population has continued to grow, since there are more than twice as many people of baby-bearing-age as there are people of getting-old-and-dying-age.
Resources, on the other hand are more of a “total number of people” problem, than a “rate at which there are more people” problem. Happily, resources regenerate. Not stupid ones like “coal”, but real ones like wind, and farmland.

3. pdf23ds says:

If you’re talking about “over all time ever” then you should bring up the possibility that eventually death from aging will be cured. In that situation, you couldn’t have any children at all.

4. Flavian Popa says:

IMHO, better have less population on the Globe but that lives in better conditions, rather than billions of billions of people fighting crazy for a tiny piece of bread…Birth control would be an additional thing differentiating human kind from other mammals that simply reproduce, reach saturation, then fight for resources and eventually decrease to cope with decreasing resources…

5. Since the universe appears to be finite, even if we were not constrained by the speed of light, populations could not grow forever. Eventually, all mass in the universe would be contained in human bodies.