A decomposable branching process in a Markovian environment

A population has two types of individuals, with each occupying an island. One of those, where individuals of type 1 live, offers a variable environment. Type 2 individuals dwell on the other island, in a constant environment. Only one-way migration (1$\to$2) is possible. We study then asymptotics of the survival probability in critical and subcritical cases.