Magnetohydrodynamic dynamos operating in stellar interiors produce the diverse range of magnetic activity observed in solar-like stars. Sophisticated dynamo models including realistic physics of convection zone flows and flux tube dynamics have been built for the Sun, for which appropriate observations exist to constrain such models. Nevertheless, significant differences exist in the physics that the models invoke, the most important being the nature and location of the dynamo α-effect and whether it is spatially segregated from the location of the Ω-effect. Spatial segregation of these source layers necessitates a physical mechanism for communication between them, involving unavoidable time delays. We construct a physically motivated reduced dynamo model in which, through the use of time delays, we mimic the generation of field components in spatially segregated layers and the communication between them. The model can be adapted to examine the underlying structures of more complicated and spatially extended numerical dynamo models with diverse α-effect mechanisms. A variety of dynamic behaviors arise as a direct consequence of the introduction of time delays in the system. Various parameter regimes give rise to periodic and aperiodic oscillations. Amplitude modulation leads to episodes of reduced activity, such as that observed during the Maunder minima, the length and duration of which depend on the dynamo number. Regular activity is more easily excited in the flux transport-dominated regime (when the time delay is smaller than the dissipative timescale), whereas irregular activity characterizes solutions in the diffusion-dominated regime (when the time delay is larger than the dissipative timescale).