Two-type linear fractional branching processes in varying environments with asymptotically constant mean matrices

Consider two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices. Let $\nu$ be the extinction time. Under certain conditions, we show that both $P(\nu=n)$ and $P(\nu>n)$ are asymptotically the same as some functions of the products of spectral radii of the mean matrices. We also give an example for which $P(\nu=n)$ decays with various speeds such as $\frac{c}{n(\log n)^2},$ $\frac{c}{n^\beta},\beta >1$ et al. which are very different from the ones of homogeneous multitype Galton-Watson processes.
Comment: 29 pages; In this version, we prove the results for general mean matrices and some technical conditions are removed