Allee Effect Dynamics and Lambert <i>W</i> Functions on a 2D γ-Ricker Diffeomorphism

The purpose of this study is to examine the nonlinear dynamics and bifurcation structures of a 2D [Formula: see text]-Ricker diffeomorphism, with a particular focus on the variation of the Allee effect parameter [Formula: see text]. In this context, the Lambert W functions are employed as a pivotal instrument for identifying the nontrivial fixed points, which are expressed as analytic solutions of this class of transcendental functions. This study establishes certain properties on the Lambert W functions associated with this diffeomorphism, namely the variation of the number of nontrivial fixed points and the respective Allee effect dynamics. Furthermore, a detailed examination of the bifurcation structures of the 2D [Formula: see text]-Ricker diffeomorphism is conducted. In particular, the Allee effect bifurcation is defined and the existence of an extinction region is explored. Both of these phenomena are associated with the stability of the origin of the diffeomorphism under analysis. The extinction region is also characterized as the basin of attraction of the origin. The theoretical results are verified and illustrated by numerical simulations.