 Multitype Poisson branching process

A particle of type $i$ has independently a Poisson distribution of type-$j$ offspring with mean ${\lambda }_{ij}$. Different particles reproduce independently. If $p_i^{(t)}$ is the probability generation $t$ is nonempty from one type-$i$ ancestor, then $p^{(0)}=1$ and $p^{(t+1)}=T(p^{(t)})$, where $T_i(x)=1-\exp [-(\Lambda x{)}_i]$. This iteration decreases to the survival probability of a branching process, the greatest nonnegative fixed point of $T$. The expected type-count row vector in generation $t$ is $e_i^T{\Lambda }^t$.