I ran calculations for a model with a single "neutral" DNA base, that can be A or B; so two possibilities instead of the usual four. Neutral means that individuals with base A have an equal chance of survival and reproduction as individuals with base B.

In this model, the population size N is fixed. Each step consists of picking one random individual and copying it, with μ probability of a mutation happening for the copy. At the same time, another random individual dies to keep the population size fixed.

N steps count as 1 generation.

The first two graphs show equilibrium distribution for the DNA for different population sizes N, for a "mutation rate" of μ = 10

^{-7}.

Equilibrium for different popularion sizes.

Same graph in

logarithmic scale.

I found that when N = μ

^{-1} = 10

^{7} (ten million), the equilibrium distribution is a perfect uniform, flat line.

For larger populations, the equilibrium distribution seems to approximate the normal distribution, as the central limit theory predicts.

For the graphs below, each individual in the initial population has base A. I defined mutation penetration to be the fraction of the population that has base B, which starts at zero for these graphs.

The gray lines are for the distribution after 1, 3, 7, 15, 31, 63, 127, etc generations until equilibrium has been reached.

N = 1024

N = 2048

N = 8192

N = 16384

This

graph shows the average similarity between two random individuals, when equilibrium has been reached.

Here

this graph shows how the average of mutated base changes over time (blue curve).

For larger populations, N ≫ mu

^{-1}, the similarity of DNA between two random individuals will drop towards 50% (red curve).

*Edited by sensei, : Added images in seperate tags*

*Edited by sensei, : Added explanation of what is meant by neutral.*

*Edited by sensei, .*

*Edited by sensei, .*