Survival function of birth-death process

Question

There is a linear birth-death process with $N$ states + an absorbing state $0$, with $$\Pr[X_{t+1}=0|X_{t}=0]=1, \\ \Pr[X_{t+1}=i+1|X_{t}=i]=\Pr[X_{t+1}=i-1|X_{t}=i]=q_i, i\in [1..N-1],$$ and $X_0=k$. I want to calculate or lower-bound the survival function of this Markov chain $S_t=\Pr[X_t=0]$ with explicit dependence on time $t$. Specifically I am interested in Moran process with $q_i=(N-i)i/N^2$. The generic expression $S_t=\underline{e_k} P^t \overline{1}$, where $P$ is the transition matrix for $N$ states (as here), does not help, because there is no analytic way to estimate how big $P^t$ is.