Summary: The model of age-dependent population dynamics was for the first time described by McKendrick (1926). This model was based on the first-order partial differential equation with the standard initial condition and the non-local boundary condition in integral form. M. E. Gurtin
and R. C. MacCamy
in their paper [Arch. Ration. Mech. Anal. 54, 281–300 (1974; Zbl 0286.92005
)] analyzed a more general model, where the progress of the population depends on its number. They proved the existence of the unique solution to their model for all time. In our paper the results of Gurtin and MacCamy will be generalized on the case, when the dependence on a number of a population is delayed.