General solution to Wright-Fisher model - Haploid selection Wright-Fisher models are classical theoretical results in evolutionary biology. There are two models, one for haploid selection and one for diploid selection (the meaning of these models does not matter for the purpose of my question). My question is: **What is the general solution of the below haploid selection model?** Haploid selection: $$p(t+1) = \frac{W_Ap(t)}{W_Ap(t) + W_aq(t)}$$ Note that in the above equation $q(t) = 1-p(t)$ by definition Same question for diploid selection model can be found here * * * By general solution, I mean an equation expressing $p(t)$ in function of $p(0)$, $t$, $W_{A}$ and $W_{a}$ $W_{A}$ and $W_{a}$ are different variables. I could have called them $X$ and $Y$. One should not try to infer one from the other or anything like this.

Note that $p(t+1)=u(p(t))$ where $u(x)=\dfrac{W_Ax}{W_Ax+W_a(1-x)}$ hence $$ \frac{u(x)}{1-u(x)}=\frac{W_A}{W_a}\cdot\frac{x}{1-x}, $$ which trivializes the iteration of $u$. To wit, for every integer $t\geqslant0$, $$ \frac{p(t)}{1-p(t)}=\frac{u^{\circ t}(p(0))}{1-u^{\circ t}(p(0))}=\left(\frac{W_A}{W_a}\right)^t\cdot\frac{p(0)}{1-p(0)}, $$ that is, $$ p(t)=\frac{(W_A)^tp(0)}{(W_A)^tp(0)+(W_a)^t(1-p(0))}. $$