Generalized r-Lambert Function in the Analysis of Fixed Points and Bifurcations of Homographic 2-Ricker Maps

Abstract

This paper aims to study the nonlinear dynamics and bifurcation structures of a new mathematical model of the [Formula: see text]-Ricker population model with a Holling type II per-capita birth function, where the Allee effect parameter is [Formula: see text]. A generalized [Formula: see text]-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 [Formula: see text]-Ricker maps considered. The singularity points of the generalized [Formula: see text]-Lambert function are identified with the cusp points on a fold bifurcation of the homographic [Formula: see text]-Ricker maps. In this approach, the application of the transcendental generalized [Formula: see text]-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.