So we have the genetic diploid model for the spread of an advantageous gene:
| genotype | $AA$ | $Aa$ | $aa$ |
|---|---|---|---|
| zygote frequency | $P$ | $Q$ | $R$ |
| relative fitness | $1+s$ | $1$ | $1$ |
Now that this model applies to the species that selfs, can we express $P'$ and $p'$, the next gen frequency of $AA$ genotype and $A$ allele respectively (where $p = P + Q/2$) in terms of $P$ and/or $p$? I tried using $P = p^2$ (as if random mating) to deduce $$p' = (p+sp^2)/(1+sp^2)$$ so this equation has fixed points at least at $0$ and $1$ as well, and then $P' = P + Q/4$ because of selfing.
I feel like it is not that appropriate to keep the assumption that $P = p^2$ that comes from random mating, but I'm not so sure putting fitness or selfing first when I deduce $P'$ and $p'$. Thanks in advance.
Let me know if I do it wrong.
$P' = P + Q/4 = P/2 + (P/2 + Q/4) = P/2 + p/2$,
$p' = [(1 + s)P + Q/2]/(1 + sP) = [P + Q/2 + sP]/(1 + sP) = (p + sP)/(1 + sP)$.