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 ( ) is possible. We study then asymptotics of the survival probability in critical and subcritical cases. 1. Introduction Multitype branching process in random environment is a challenging topic with many motivations from population dynamics (see, e.g., [1–3]). Very little is known in the general case and in this paper we consider a particular two-type branching process with two key restrictions: the process is decomposable and the final type individuals live in a constant environment. The subject can be viewed as a stochastic model for the sizes of a geographically structured population occupying two islands. Time is assumed discrete, so that one unit of time represents a generation of individuals, some living on island 1 and others on island 2. Those on island 1 give birth under influence of a randomly changing environment. They may migrate to island 2 immediately after birth, with a probability again depending upon the current environmental state. Individuals on island 2 do not migrate and their reproduction law is not influenced by any changing environment. Our main concern is the survival probability of the whole population. An alternative interpretation of the model under study might be a population (type 1) subject to a changing environment, say in the form of a predator population of stationary but variable size. Its individuals may mutate into a second type, no longer exposed to the environmental variation (the predators do not regard the mutants as prey). Our framework may be also suitable for modeling early carcinogenesis, a process in which mutant clones repeatedly arise and disappear before one of them becomes established [4, 5]. See [6] for yet another possible application. The model framework is that furnished by Bienaymé-Galton-Watson (BGW) processes with individuals living one unit of time and replaced by random numbers of offspring which are conditionally independent given the current state of the environment. We refer to such individuals as particles in order to emphasize the simplicity of their lives. Particles of type 1 and 2 are distinguished according to the island number they are occupying at the moment of observation. Our main assumptions are(i)particles of type 1 form a critical or subcritical branching process in a random environment,(ii)particles of type 2 form a critical branching
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