In this paper,we study a 2-type linear-fractional branching process in varying environments with asymptotically constant mean matrices.Let ν be the extinction time and for k ≥ 1,let Mk be the mean matrix of offspring distribution of individuals of the(k-1)-th generation.Under certain conditions,we show that P(ν=n)and P(ν＞n)are asymptotically equivalent to some functions of products of spectral radii of the mean matrices.We complement a former result of Wang and Yao(2022)which requires in addition a condition that∀k ≥ 1,det(Mk)＜-ε for some ε＞0.Such a condition requires that individuals in the system are more likely to produce children of another type so that it excludes a large class of mean matrices.However,our results do not need such an additional assumption.As byproducts,we also get some results on the asymptotics of products of nonhomogeneous matrices and limit periodic continued fractions that have their interests.
关键词
变环境下分枝过程/灭绝时/非齐次矩阵乘积/谱半径/连分数
Key words
branching process in varying environment/extinction time/product of nonhomogeneous matri-ces/spectral radius/continued fraction
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