Bifurcation structures in a 2D exponential diffeomorphism with Allee effect
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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.
KeywordsAllee’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.
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Conflict of interest
The authors declare that they have no conflict of interest.
- 15.Rocha, J.L., Taha, A-K.: Allee’s effect bifurcation in generalized logistic maps. Int. J. Bifurc. Chaos (to appear) (2019)Google Scholar