Periodic Multitype Symmetric Branching Random Walks on $\mathbf{Z}^{{d}}$

We consider the model of symmetric branching continuous-time random walks on the lattice $\mathbf Z^d$ with $n$ types of particles and periodically located branching sources. It is assumed that initially there is only one particle of type $\mathcal T_s$ at some point. For this process, we construct a periodic operator describing the evolution of the mean number of particles of type $\mathcal T_j$ and study its spectral properties. We also obtain the asymptotics of the mean number of particles of type $\mathcal T_j$ at a fixed point of the lattice as $t \to\infty$.