Multiple bifurcations and managing chaos: A discretized ratio-dependent Holling–Tanner predator–prey model with Allee effect in prey

Abstract

This work introduces a ratio-dependent Holling–Tanner predator–prey model with the Allee effect in prey and then discretizes the introduced model through the Euler forward scheme. A brief discussion is held on the stability analysis for several fixed points in the discretized model. Several types of bifurcations, including codimension one and two bifurcations, are demonstrated in this study. Codimension-1 bifurcation, which covers Neimark–Sacker and flip bifurcations, and codimension-2 bifurcations, which include strong resonance 1:2, 1:3, and 1:4 at a positive fixed point. Various critical states under non-degeneracy conditions are computed using the critical normal form coefficient approach for each bifurcation. The model displays complex dynamical behaviours, like quasi-periodic orbits and chaotic sets. Additionally, the system’s chaos was managed by the development of control mechanisms, such as the OGY methodology. It has been established that bifurcation and chaos can be stabilized under certain circumstances. A thorough numerical simulation further supports our analytical findings, which include stability regions, bifurcation curves in 2D & 3D, phase plots, and the maximal Lyapunov exponent, etc.