Bifurcation structures in a 2D exponential diffeomorphism with Allee effect

  • J. Leonel RochaEmail author
  • Abdel-Kaddous Taha
Original Paper


An embedding of one-dimensional generic growth functions into a two-dimensional diffeomorphism is considered. This family of unimodal maps naturally incorporates a key item of ecological and biological research: the Allee effect. Consequently, the presence of this species extinction phenomenon leads us to a new definition of bifurcation for this two-dimensional exponential diffeomorphism: Allee’s effect bifurcation. The stability and the nature of the fixed points of the two-dimensional diffeomorphism are analyzed, by studying the corresponding contour lines. Fold and flip bifurcation structures of this exponential diffeomorphism are investigated, in which there are flip codimension-2 bifurcation points and cusp points, when some parameters evolve. Numerical studies are included.


Allee’s effect bifurcation Fold and flip bifurcations Diffeomorphisms Contour line 



Research partially funded by FCT - Fundação para a Ciência e a Tecnologia, Portugal, through the project UID/MAT/00006/2013 (CEAUL), the research Grant SFRH/BSAB/128144/2016 and ISEL. The authors thank the editor and the anonymous referees for their careful reading of the manuscript.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.CEAUL. ADM, ISEL-Engineering Sup. Institute of LisbonPolytechnic Institute of LisbonLisboaPortugal
  2. 2.INSAFederal University of Toulouse Midi-PyrénéesToulouseFrance

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