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We consider a new model of a branching random walk on a multidimensionallattice with continuous time and one source of particle reproduction and death,as well as an infinite number of sources in which, in addition to the walk,only absorption of particles can occur. The asymptotic behavior of the integermoments of both the total number of particles and the number of particles at alattice point is studied depending on the relationship between the modelparameters. In the case of the existence of an isolated positive eigenvalue ofthe evolution operator of the average number of particles, a limit theorem isobtained on the exponential growth of both the total number of particles andthe number of particles at a lattice point.

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point