Predator Maturation Delay and Functional Response Controls Population Cycles in Lemmings and Snowshoe Hares

Population cycles in small mammal has attracted the attention of several generations of theoretical and experimental biologists and continues to generate controversy. Top-down and bottom-up trophic regulations are two recent competing hypotheses. The principal purpose of this paper is to answer the following simple question: What ecological factors control population cycles? For some species — collared lemmings and snowshoe hares in particular — maturation delay of predators and the functional response of predation appear to be the primary determinants. Maturation delay alone almost completely determines the cycle period, whereas the functional response greatly affects its amplitude and even its existence. This result is obtained from sensitivity analysis of all the parameters in the lemming-stoat delayed system, that also explains why lemming has a 4-year cycle whereas snowshoe hare has a 10-year cycle. This modeling approach is mainly based on, but generates more insights beyond, Gilg’s work. Our parameterized model is a sort of confirmation and extension of May’s assertion on the impact of time delay on the 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; suitable intermediate predator maturation periods and suitable functional responses can generate population cycles for both prey and predators. These results clearly explain why some populations are cyclic whereas others are not. Subject headings: Population Cycle, Snowshoe Hare, Lemming, Stoat, Lynx, Holling Type, Period, Amplitude, Predator-prey, Delay Differential Equation