Critical parameters of germ-monotone families of branching random walks

We introduce a broad class of families of branching random walks on a set $X$. The processes in each family are parametrized by a positive parameter $\lambda$ and they are monotonically increasing in $\lambda$ with respect to the germ order, a notion that extends classical stochastic domination. We define a general notion of critical parameter $\lambda(A)$ associated with a subset $A \subseteq X$ and investigate how modifications of reproduction laws affect these critical parameters.