Stable long-period cycling and complex dynamics in a single-locus fertility model with genomic imprinting

Jeremy Van Cleve, Marcus W. Feldman

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


Although long-period population size cycles and chaotic fluctuations in abundance are common in ecological models, such dynamics are uncommon in simple population-genetic models where convergence to a fixed equilibrium is most typical. When genotype-frequency cycling does occur, it is most often due to frequency-dependent selection that results from individual or species interactions. In this paper, we demonstrate that fertility selection and genomic imprinting are sufficient to generate a Hopf bifurcation and complex genotype-frequency cycling in a single-locus population-genetic model. Previous studies have shown that on its own, fertility selection can yield stable two-cycles but not long-period cycling characteristic of a Hopf bifurcation. Genomic imprinting, a molecular mechanism by which the expression of an allele depends on the sex of the donating parent, allows fitness matrices to be nonsymmetric, and this additional flexibility is crucial to the complex dynamics we observe in this fertility selection model. Additionally, we find under certain conditions that stable oscillations and a stable equilibrium point can coexist. These dynamics are characteristic of a Chenciner (generalized Hopf) bifurcation. We believe this model to be the simplest population-genetic model with such dynamics.

Original languageEnglish
Pages (from-to)243-264
Number of pages22
JournalJournal of Mathematical Biology
Issue number2
StatePublished - Aug 2008


  • Fertility selection
  • Frequency dependence
  • Gene frequency cycling
  • Generalized Hopf bifurcation
  • Population genetics

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


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