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

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AbstractConsider 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 $\mathbb{P}(\nu=n)$ and $\mathbb{P}(\nu>n)$ are asymptotically the same as some functions of the products of spectral radii of the mean matrices. We also give an example for which $\mathbb{P}(\nu=n)$ decays with various speeds such as ${c}/({n^{1/2}\log n)^2}$ , ${c}/{n^\beta}$ , $\beta >1$ , which are very different from those of homogeneous multitype Galton–Watson processes.