Bifurcation Analysis of the γ-Ricker Population Model Using the Lambert W Function

In this work, we present the dynamical study and the bifurcation structures of the gamma-Ricker population model. Resorting to the Lambert W function, the analytical solutions of the positive fixed point equation for the gamma-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 gamma-Ricker population model, which is not a pitchfork bifurcation. Consequently, we prove that the phenomenon of Allee effect for the gamma-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 gamma-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.