Before attempting to calculate the kinship distribution of such population analytically, I have written a small C++ program to estimate it via simulation. You will need Boost to compile the code. The simulation proceeds as follows:
- A pool of individuals is maintained, with new offspring being inserted after their parent generation.
- At each iteration, females of the last generation are traversed and mated with available males at random. The couple is assigned a number of children randomly generated according to a Poisson distribution with median 2.
- After a number of iterations the kinship distribution of the population is assumed to stabilize. The kinship histogram is calculated from a random sample of pairs of members of the last generation.
The figure shows the results for generation sizes N = 1,000, 10,000 and 100,000 with simulations of 100 generations each. We only depict values of K(n) for n even, since, because of the stratification property, K(n) = 0 if n is odd.
The kinship distribution K(n), n even, initially grows exponentially (in fact, as a·2n, as we will see when we derive analytical expressions for the function) until a maximum value between 0.4 and 0.5, from where it decays abruptly. The most probable kinship relationships between arbitrary individuals for N = 1,000, 10,000 and 100,000 are fifth, sixth and eighth cousins, respectively.
In a later entry we will try to determine K(n) analytically from basic probability theory.