Conditions of extinction of a $\varphi$-branching process with random $\varphi$

We introduce a new class of $\varphi$-branching processes, in which the controlling functions $\varphi_t(n)$ $(t,n=0,1,2,\dots)$ are random variables. We give conditions on $\varphi_t(n)$ sufficient for extinction or non-extinction of the $\varphi$-process.