Abstract: Population cycles in small mammals have attracted the attention of several generations of theoretical and experimental biologists and continue to generate controversy. Top-down and bottom-up trophic regulations are two recent competing hypotheses. The principal purpose of this paper is to explore the relative contributions of a variety of ecological factors to predator–prey population cycles. Here we suggest that for some species – collared lemmings, snowshoe hares and moose in particular – maturation delay of predators and the functional response of predation appear to be the primary determinants. Our study suggests that maturation delay alone almost completely determines the cycle period, whereas the functional response greatly affects its amplitude and even its existence. These results are obtained from sensitivity analysis of all parameters in a mathematical model of the lemming–stoat delayed system, which is an extension of Gilg’s model. Our result may also explain why lemmings have a 4-year cycle whereas snowshoe hares have a 10-year cycle. Our parameterized model supports and extends May’s assertion that time delay impacts cycle period and amplitude. Furthermore, if maturation periods of predators are too short or too long, or the functional response resembles Holling Type I, then population cycles do not appear; however, suitable intermediate predator maturation periods and suitable functional responses can generate population cycles for both prey and predators. These results seem to explain why some populations are cyclic whereas others are not. Finally, we find parameterizations of our model that generate a 38-year population cycle consistent with the putative cycles of the moose–wolf interactions on Isle Royale, Michigan.