human genetics - Initial population when i count backwards?

My question is dealing with how many 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

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)