Analytical study of a triple Hopf bifurcation in a tritrophic food chain model

Jean Pierre Françoise, Jaume Llibre

Research output: Contribution to journalArticleResearchpeer-review

16 Citations (Scopus)


We provide an analytical proof of the existence of a stable periodic orbit contained in the region of coexistence of the three species of a tritrophic chain. The method used consists in analyzing a triple Hopf bifurcation. For some values of the parameters three limit cycles born via this bifurcation. One is contained in the plane where the top-predator is absent. Another one is not contained in the domain of interest where all variables are positive. The third one is contained where the three species coexist. The techniques for proving these results have been introduced in previous articles by the second author and are based on the averaging theory of second-order. Existence of this triple Hopf bifurcation has been previously discovered numerically by Kooij et al. © 2011 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)7146-7154
JournalApplied Mathematics and Computation
Publication statusPublished - 1 May 2011


  • Averaging theory
  • Hopf bifurcation
  • Limit cycle
  • Population dynamics


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