The Moran model is a model for genetic drift. Basically, it is a finite Markov chain (more precisely, a birth-death chain) with state space $S:=\{0,...,N\}$ and the following transition probabilites:
$$p_{00} = p_{NN} = 1 \ \ \text{ (absorbing states)},$$
$$ p_{i,i+1} = p_{i,i-1} = \frac{i(N-i)}{N^2}, $$
$$ p_{i,i} = 1-2p_{i,i+1} .$$
Now we define a probability vector $q(n):=(q_1(n),...,q_{N-1}(n))$, where $q_i(n)$ is the probability of being in the state $i \in \{1,...,N-1\}$ under the condition that we have not been absorbed yet, that is,
$$ q_i(n) := \text{P}(X_n=i \ \vert \ X_n \in \{1,...,N-1\}) .$$
If we have the situation that $q(n+1)=q(n):=q$ for all times $n$, then we call $q$ a quasistationary distribution for this Markov chain.
It is precisely this $q$ that I want to find for the Moran model. I saw a proof about its existence and uniqueness, but so far I shipwrecked when I tried to explicitly compute it. Maybe one can somehow construct a nonlinear recursion and solve it. So far my ideas are very vague. Thanks for any advice!