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- Generalized r-Lambert function in the analysis of fixed points and bifurcations of homographic 2-Ricker mapsPublication . Rocha, J. Leonel; TAHA, Abdel-KaddousThis paper aims to study the nonlinear dynamics and bifurcation structures of a new mathematical model of the γ-Ricker population model with a Holling type II per-capita birth function, where the Allee effect parameter is γ = 2. A generalized r-Lambert function is defined on the 3D parameters space to determine the existence and variation of the number of nonzero fixed points of the homographic 2-Ricker maps considered. The singularity points of the generalized r-Lambert function are identified with the cusp points on a fold bifurcation of the homographic 2-Ricker maps. In this approach, the application of the transcendental generalized r-Lambert function is demonstrated based on the analysis of local and global bifurcation structures of this three-parameter family of homographic maps. Some numerical studies are included to illustrate the theoretical results.
- Bifurcation analysis of the γ-Ricker population model using the Lambert W functionPublication . Rocha, J. Leonel; TAHA, Abdel-KaddousIn this work, we present the dynamical study and the bifurcation structures of the γ-Ricker population model. Resorting to the Lambert W function, the analytical solutions of the positive fixed point equation for the γ-Ricker population model are explicitly presented and conditions for the existence and stability of these fixed points are established. The main focus of this work is the definition and characterization of the Allee effect bifurcation for the γ-Ricker population model, which is not a pitchfork bifurcation. Consequently, we prove that the phenomenon of Allee effect for the γ-Ricker population model is associated with the asymptotic behavior of the Lambert W function in a neighborhood of zero. The theoretical results describe the global and local bifurcations of the γ-Ricker population model, using the Lambert W function in the presence and absence of the Allee effect. The Allee effect, snapback repeller and big bang bifurcations are investigated in the parameters space considered. Numerical studies are included.