Initial population when I count backwards?

Question

My question deals with the number of ancestors it took to produce me:

• --> To produce me, it took 2 people (my parents) = ($2^1$) people = 2
• --> One generation further included = ($2^1 + 2^2$) people = 6
• --> In the past 500 years, my pedigree chart contains 15 generations, so $2^{16}$ people = 65.532 people to produce me (-2 to be completely correct)
• --> When I calculate back to 35.000 year ago I get a larger number *)
• --> When I calculate to all humans 2.400.000 ago I get a ridiculous large number *)
• --> When I include up to all mammal grandfathers and grandmothers 60.000.000 years ago the total number takes pages to scroll *)

*) see: http://ed.je/2L6 for the actual numbers did not want to clutter here.

*) I took 3 generations per 100 years to let the numbers be small as possible.

So, I think I'm making a mistake somewhere. Only I don't know where: to create a child you need 2 parents. And to create a parent you need 2 further parents. But somehow calculating back to "all mammals" just related directly to me would mean $2^{1.800.001}$ mammals walking around 60 millions years ago being my direct predecessor back then.

So, where am I going wrong? Somehow, I can't figure out where my error lies (I've not studied Biology though).

I visualize a pedigree tree (reverse pyramid) starting with me and then logically each parent needs to be born out of 2 other parents just calculating my way up.

Answer 1

Your calculations are the following. Assuming non-overlapping generations, the number of ancestors you have in the last $t$ generation is given by:

$$\sum_{i=1}^t 2^t$$

This sounds correct. But there are some very strong assumptions:

• Generations are non-overlapping. A more realistic model would need to consider $t$ as a continuous variable a give a probability of reproducing which depend on the age of the individual.

• Never two related individuals mate together. Otherwise, you paternal grandfather might be your maternal grandfather. In reality we are all somehow related, we have a common ancestor (Hey Bro). For example if you consider that your parents are siblings and that their parents are siblings and so forth… then you don't need more than 2 great-great-great-great-grandparents. In order to keep track of such events (inbreeding), you need a stochastic model and you would need to know the population size $N(t)$ as a function of $t$.

Tracking the change of population size through time is part of a very big in biology which is called population biology. For example you will find in this post a model of population growth of two interacting species (prey-predator)

Answer 2

An easy way to visualize the mistake in your thought experiment is to consider a bottleneck event, when the ancestral population was very small, maybe just a few individuals. This would mean that the entire current population is descending from just a few individuals. Your thought experiment is assuming that the "pyramid" of your ancentors is expanding all the time, while the bottleneck event will essentially result in the opposite situation with an inversion of the pyramid, such as:

Your assumption:            Bottleneck:  

 xx xx xx xx                     x
   xx xx                        x x
    x x                        xx xx     
     x                      xx xx xx xx
           (present time)

The reason why the number of ancestors is smaller in the bottleneck event is naturally that the population is inbred (all individuals are related). In contrast, your thought experiment is assuming that all individuals forming the pyramid of your ancestors have completely unrelated ancestry. This is equivalent to assuming that the subpyramids for all ancestors never overlap.

Your assumption is unreasonable since human generations are overlapping, the fecundity distribution of individuals is skewed, inbreeding is common and reproduction between more distantly related individuals will happen by pure chance. We also know that humans have historically gone through several population bottlenecks, see e.g. Hawks et al. (2000) and Lui et al. (2006) for an overview. An realistic model of your ancestry is more likely to be (highly idealized and compacted):

       x
      x x
     xx xx
  xx xx xx xx
  xx xx xx xx
     xx xx
      x x
       x
     (you)

The issue we are discussing here is actually known as pedigree collapse, where you will eventually find overlap between your ancestors, which will mean that the pedigree collapses instead of growing exponentially.

Beside historical bottlenecks, there are also modern examples of extreme human inbreeding. Even if this is not directly related to your thought experiment, you might for instance want to look the pedigree of the Habsburg royal family. Here is the genealogy of Charles II - remember, narrow loops in pedigrees are seldom a good idea:

Here you can for instance see that Charles' father is also the uncle of his own wife. You can also see that only 5 persons enter in the family tree from the outside through marriages, which means that Charles only had 7 completely independent ancestors (ignoring relationships further back, outside of this tree). Going back 3 generations (sort of - hard to count with overlap), it is also apparent that Charles had a total of 8 ancestors instead of the 14 expected under your model. Another modern example of extreme human inbreeding is the Fundamentalist Church of Jesus Christ of Latter-Day Saints, where marriages between close relatives have been common along with high levels of rare birth defects. Both these examples definately show, in an extreme way, that you cannot assume that all your ancestors are unrelated.

Answer 3

Another way to demonstrate that the underlying reasoning in the calculation is not correct is to project into the future.

Given the assumptions, the approximately 7 billion people on Earth today could only produce 3.5 billion people (two parents per offspring). Projecting forward, after approximately 32 generations, there would be just one human in the world.

However, we know this is not accurate because a woman can have more than one offspring.