Stochastic operation induced by group allocation of particles and generalized branching processes

We introduce a stochastic operation induced by group allocation of particles in cells, which belongs to the class of Archimedean operations. We analyze its properties and prove limit theorems. We also introduce and study branching processes with this operation, which can describe various phenomena in biological populations and computer networks, including virus propagation. In the limit, a nonlinear dynamical system occurs.