Front propagation of branching random walk with periodic branching sources

We consider the model of branching random walk on an integer lattice $\mathbb{Z}^d$ with periodic sources of branching. It is supposed that the regime of branching is supercritical and, for a jump of the random walk, the Cramér condition is satisfied. The theorem established describes the rate of front propagation for particles population over the lattice as the time increases unboundedly. The proofs are based on fundamental results related to the spatial spread of general branching random walk.