The life-periods of critical branching processes with random migration

Let $\{\xi_{ij}(t)\}_{j,t=1}^\infty$ ($i=1,2$), $\{\eta_t\}_{t=1}^\infty$ be independent integer-valued random variables with $F_i(s)=\mathbf Es^{\xi_{ij}(t)}$,
$$ \mathbf P\{\eta_t=-1\}=p,\ \mathbf P\{\eta_t=0\}=q,\ \mathbf P\{\eta_t=1\}=r,\ p+q+r=1. $$
The branching process with random migration $\mu_t$ is defined by the following formula: if $\mu_t=n$ then
$$ \mu_{t+1}=\sum_{j=1}^{\varphi_1(t;n)}\xi_{1j}(t+1)+\sum_{j=1}^{\varphi_2(t;n)}\xi_{2j}(t+1),\quad t=0,1,\dots, $$
where $\varphi_1(t;n)=\max\{\min\{n,n+\eta_t\},0\}$, $\varphi_2(t;n)=\max\{0,\eta_t\}$. In the critical case ($F'_1(1)=1$) we investigate the asymptotical behaviour of the probability $\mathbf P\{\tau>t\}$, $t\to\infty$, where $\tau$ is the life-period defined by the conditions $\mu_{T-1}=0$, $\mu_t>0\,(T\le t<T+\tau)$, $\mu_{T+\tau}=0$.