Extinction of population-size-dependent branching processes in random environments

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Zn}(n greater than or equal to 0) is associated with the stationary environment <(xi)over bar> = {xi(n)}(n greater than or equal to 0), let B = {omega : Z(n)(omega) = 0 for some n}, and q(<(xi)over bar>) = P(B \ <(xi)over bar>, Z(0) = 1). The result is that P(q(<(xi)over bar>) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(<(xi)over bar>) = 1) = 1) and for non-certain extinction (i.e. P(q(<(xi)over bar>) < 1) = 1) are obtained for the model.