Regulation of population cycles by genetic feedback: Existence of periodic solutions of a mathematical model

Populations of voles, and lemmings of the Northern hemisphere exhibit cyclic fluctuations with a cycle of three to four years. Krebs et al. presented evidence that the cycles are driven by changes in the genotypic structure of the population 9]. Incorporating some of their hypotheses we present a mathematical model of a one locus two allele population with density dependent selection and assuming a slow selection hypothesis, the existence of periodic solutions is proved. These solutions arise by Hopf bifurcation in ₁/¦₁¦, the ratio of the residual death and birth rates of the density sensitive homozygote.Partially supported by NSF Grant # MCS-8005777