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Consider two-type linear-fractional branching processes in varyingenvironments with asymptotically constant mean matrices. Let $\nu$ be theextinction 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 ofspectral 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 ofhomogeneous multitype Galton-Watson processes.

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