# Mechanism to limit the exponential growth of the number of ancestors that are n-generations away from an organism?

The context is organisms with sexual reproduction, with 2 parents per organism. Construct the family tree of one such organism, back into geologic time.

Take a(n) as being the number of ancestors that are n generations distant. The function a(n) has an exponential character (at least in the beginning, with n sufficiently small).

For 2^n, simple math gives:

• After 20 generations, you have about 10^6 ancestors.
• After 30 generations, you have about 10^9 ancestors.
• After 50 generations, you have about 10^15 ancestors.
• After 1000 generations, you have about 10^300 ancestors.

I've read one estimate that there are about 10^80 atoms in the Universe. What does this mean?

It's important to note the the members of generation n usually aren't all alive at the same time. As an example, for humans, the birth-dates of generation-20 are roughly between the years 1720-1520, for example.

It doesn't take very many generations for a(n) to exceed p(t), the population as a function of time. This is accounted for (at least for sufficiently low n) by the fact that a(n) represents organisms that aren't alive simultaneously, while p(t) does. I understand that. But, this effect, important as it is, seems to be linear with time. As such, it doesn't seem capable of limiting the exponential growth of a(n).

In this context, inbreeding is often raised. But, in my opinion, inbreeding is not germane. Here, all that matters is that each organism has 2 parents. Period. Whether or not inbreeding was involved doesn't change that basic fact. All inbreeding means is that one ancestor is a member of more than 1 generation. That's not relevant to my question.

Edit later: No, inbreeding is germane. It means that 2 members of a(n) can have the same two parents in a(n+1). That reduces the exponential growth of a(n).

It seems that a(n) must be limited by something, for sufficiently large n. What is that something?

• In addition to @mgkrebbs answer, note that "inbreeding" refers to breeding among closely related individuals. Individuals several generations removed are not closely related. Oct 7 '19 at 19:38

Of course a(n) is limited, and much more tightly than by the number of atoms that exist. The analysis in the question goes off track with "inbreeding is not germane. Here, all that matters is that each organism has 2 parents. Period."

Consider a person whose parents are first cousins. Then:

• a(1) = 2
• a(2) = 4
• a(3) = 6, not 8

While each of the two parents has four grandparents, together the two parents have only six grandparents because they share one pair of grandparents. This is due to pedigree collapse.

This table shows the actual collapse for some people with ancestry known to nine generations and with significant inbreeding (from de.Wikipedia):

• I just realized that it's germane. I have edited the question a second ago. So is inbreeding the only mechanism for limiting the exponential growth of a(n)?
– John
Oct 7 '19 at 19:36
• I see. It's this pedigree collapse idea which is the key. Thanks!
– John
Oct 7 '19 at 19:39
• @JohnO at a point all organisms are a product of inbreeding since we all evolved from the same LUCA.
– John
Oct 7 '19 at 19:50