Asymptotic behavior of the mean number of particles of branching random walk on $\mathbf Z^d$ with periodic sources of branching

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 an evolution operator of the mean number of particles are studied. We derive a representation of the mean value of particle number in a form of asymptotic series.