Non-linear equations for special classes of Markovian processes

For a special class of Markovian processes with a discrete phase space, namely branching random processes, the first-order non-linear ordinary differential equation for the generating function of the transition probability, is known. The analogous non-linear partial differential first-order equations for two processes from another special class of Markovian processes for a discrete phase space, that is for the branching random processes with interaction of particles, are derived.