Stability and further analytical bifurcation behaviors of Moran–Ricker model with delayed density dependent birth rate regulation

Abstract

In this paper, we consider Moran–Ricker model with delayed density dependent birth rate presented by Neverova et al. (2016). It is well known that normal form of a dynamical system is easier to work with, while it has the same qualitative properties as those of the original one. Firstly, existence and local stability analysis of the model fixed points are addressed. Then, normal form theory and perturbation method are applied to investigate different types of bifurcations included in the model. It is demonstrated the presence of transcritical, flip and Neimark–Sacker bifurcations in the model. In addition, it is proved that the model has a chaotic behavior in the sense of Marotto. Finally, numerical simulations are provided to verify our analytic results compared with those obtained in Neverova et al. (2016), and explore further rich dynamics of the model as well. • Moran–Ricker model with delayed density dependent birth rate is considered. • Existence and local stability analysis of fixed points of the model are addressed. • Analytic bifurcation behaviors are investigated. • Existence of Marotto’s chaos is proved. • An effective technique, to discuss the complex dynamics of three dimensional discrete time models, is used.