Calculating population growth in immortal society

[Bobakka]

Hi!

I have a question regarding this paper:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3192186/

Can you please explain this part to me in a way that I can explain it to my grandma and my 7 year old daughter?

"Let us start with the worst-case scenario (for overpopulation)—physical immortality (no deaths at all). What would happen with population numbers then? Common sense and intuition suggest that there should be a demographic catastrophe if immortal people continue to reproduce. However, a deeper mathematical analysis leads to paradoxical results. Consider a situation, when parents produce less than 2 children on average, so that each next generation is smaller than the previous one:

Then even if everybody is immortal, the final size of the population will not be infinite, but just 1/1 − r times larger than the initial population. For example, one-child practice (r = 0.5) will only double the total immortal population, because 1/(1 − 0.5) = 2. In other words, a population of immortal reproducing organisms can grow indefinitely in time, but not necessarily indefinitely in size, because asymptotic growth is possible. This conclusion does not require any complex calculations and questionable assumptions, but follows directly from the calculus, and the property of infinite geometric series to converge when the absolute value of the common ratio, r, is less than one:

So, the fears of overpopulation based on lay common sense and uneducated intuition are in fact grossly exaggerated. In fact, immortality, the joy of parenting, and sustainable population size, are not mutually exclusive."


I guess my uneducated intuition fails me here gravely, lol.
Thanks!

 

[HallsofIvy]

A couple having less than 2 children does not "replace itself". Each generation is smaller than the previous and eventually you are down to one person who cannot reproduce.

Suppose we start with 1024 couples and that each couple has only 1 child. Then the next generation has 512 children, the next 256, then 128, 64, 32, 16, 8, 4, 2, 1, to 1/2 couple, and, since a single person cannot reproduce, that ends with a finite number of people.

That is, of course, assuming that given generation cannot reproduce with a person from a previous generation even though they are still alive.

 

[Bobakka]

Thank you I think I get it. So does that mean anything less than average of 2 kids per couple, (or less than 1 per person) would result in a finite growth either?

 

[Bobakka]

A couple having less than 2 children does not "replace itself". Each generation is smaller than the previous and eventually you are down to one person who cannot reproduce.

Suppose we start with 1024 couples and that each couple has only 1 child. Then the next generation has 512 children, the next 256, then 128, 64, 32, 16, 8, 4, 2, 1, to 1/2 couple, and, since a single person cannot reproduce, that ends with a finite number of people.

That is, of course, assuming that given generation cannot reproduce with a person from a previous generation even though they are still alive.

Thank you that was illuminating. About my follow up comment, reading the question again I realized it said it as much already, that is anything less than 2 kids per couple would cap out.
Thanks again!