Multitype Poisson branching process ($B_i(\Lambda )$)

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 $e_i^T{\Lambda }^t$.

This criterion holds even for reducible nonnegative matrices. If a fixed point $p\ne 0$ exists, $1-e^{-z}<z$ for $z>0$ gives $(\Lambda p{)}_i>p_i$ on its support. Some $\alpha >1$ therefore satisfies $\Lambda p\ge \alpha p$, forcing the spectral radius above one by the Perron–Frobenius theorem. Conversely take $v\ge 0$, $v\ne 0$, with $\Lambda v=\rho v$ and $\rho >1$. For sufficiently small $\eta >0$, $1-e^{-\eta \rho v_i}\ge \eta v_i$. Thus $\eta v$ is a subsolution; monotone iteration under $T$ gives a positive fixed point below $1$. Strictness of $1-e^{-z}<z$ excludes survival at the critical value for this Poisson offspring law.