Branching random walks on $\mathbf{Z}^d$ with periodic branching sources

We consider a continuous-time branching random walk on $\mathbf{Z}^d$ with birth and death of particles at a periodic set of points (the sources of branching). Spectral properties of the evolution operator of the mean number of particles at an arbitrary point of the lattice are studied. The leading term of the asymptotics as $t\to\infty$ of the mean number of particles at a given point is obtained. Under an additional moment condition, an asymptotic series expansion of the mean number of particles is derived.