A model system of two differential rate equations is proposed to describe the temporal characteristics of the radiation of a solid-state laser with high-energy pumping. Using a laser head with a YAG:Nd active element and a side-pumped laser diode array unit as an example, a method is discussed of determining the excitation levels of the active elements at which the amplified luminescence and the nonaxial (stray) lasing modes begin to manifest themselves in the output radiation dynamics all the way to the limitation of the maximum pulse energy generated by a solid-state laser in the Q-switched regime. The proposed approach to analyzing the role of the processes associated with the formation and evolution of amplified luminescence fluxes and nonaxial (stray) lasing modes in the active media can also be applied to other types of laser radiators.