Bifurcation Analysis in a Three-Dimensional Discrete Prey–Predator Model with Holling Type II Functional Response

In this study, we analyze a three-dimensional discrete prey–predator model with Holling type II functional response. The existence and local stability of its fixed points are obtained in terms of system parameters. By employing the center manifold theorem, bifurcation theory, and critical normal form coefficients methods, we prove the emergence of three codimension-2 bifurcations (fold–Neimark–Sacker, flip–Neimark–Sacker, and fold–flip) and three codimension-1 bifurcations (fold, flip, and Neimark–Sacker). To verify our theoretical findings, we provide bifurcation continuation curve, bifurcation diagrams, phase portraits, and diagrams of maximum Lyapunov exponents. The corresponding numerical simulations and numerical continuation results not only validate the proposed results, but also exhibit the interesting dynamical behaviors, such as limit cycle, periodic orbit, chaos and so on. Furthermore, we investigate the impact of initial conditions on the model by the local attraction basins and two-parameter space plots. Our results provide new insights into the complex dynamics of the three-dimensional discrete prey–predator model.