# Variance of frequency in expanding populations

I'm trying to understand the variance in frequency of some strain after some time as passed in a growing population. The idea is to then plug this in kolmorov foward equations and get the classic results (probability of fixation, time to fixations, mean fitness). However, I got a weird result.

I started with the model definde here

$N(t) = N(0)W^{t}$

and from the answers I got that the variance of $N(t)$ would be

$\text{Var}[N(t)] = N(0) W^t(W^t-1)$

Now, the frequency of strain $N_{m}$ is

$f_{m}(t) = \dfrac{N_{m}(t)}{N_{m}(t) + N_{b}(t)}$

assuming only two sub populations. From there I need only the variance of $f_{m}(t)$ and this and this helped me build a formula. However, the variance of the frequency is not $0$ when $f_{m}(0) = 1$ and it has units much larger than 1...

My question is, have I done something wrong?

• You suddenly talk about two strains while it was no present in the beginning of your model. Can you please link the existence of these two strains to the beginning of your model. Where is your calculation of the variance of this frequency which you state is wrong? Sep 15, 2016 at 18:29
• Indeed I had made a mistaken when calculating the variance. As the formula is very large, I will just point to the fact that using the references I gave it is possible to get the variance of the frequency. Oct 11, 2016 at 11:23