The Asymptotic Behaviour of the Extinction Probabibity in Bisexual Galton-Watson Branching Processes with Independent and Identically Distributed Random Environments

In bisexual Galton-Watson branching process with independent and identically distributed random environments, it is shown that under certain conditions there exists $0<\alpha<+\infty$ and $0<c<+\infty$ such that the extinction probability starting with $k$ individuals is bounded above by $ck^{-\alpha}$ for sufficiently large $k$.