On a decomposable branching process with two types of particles

A decomposable Galton–Watson branching process with two types of particles is considered. It is assumed that particles of the first type produce particles of both the first and the second types, and produce them in equal amounts, while particles of the second type only produce particles of the same type. An asymptotic formula is obtained for the probability that the total number of particles of the second type up to time $N$ is greater than $\theta N$, where $\theta$ is a positive constant and $N\to\infty$. A limit theorem is established for the total number of particles of the first type considered under the condition that the total number of particles of the second type up to time $N$ is greater than $\theta N$.