<p>This paper investigates the complex nonlinear behavior of the prey–predator model, with a detailed focus on global patterns, stability analysis, isoperiodic plots, bifurcation behavior, and chaos control. Particularly, the existence and uniqueness of the positive fixed point of the system and the topological classification of all the fixed points are calculated. The detailed discussion of complex global patterns highlights the novelty of this work. Using the detailed complex plots of the Lyapunov exponent, we have identified and characterized the panoramic view of multi-dynamic patterns across a wide parameter space. To quantify the system sensitivity, the largest Lyapunov exponent analysis is employed. In the largest Lyapunov exponent plots, we have obtained multi-dynamic regimes, including the Arnold tongues and shrimps, besides other periodic and chaotic patterns. Moreover, the bifurcation theory identified Hopf and flip bifurcations in this model. Due to Hopf bifurcation, limit cycles are also produced. A maximum Lyapunov exponent supports the existence of variation in dynamic oscillations. Chaos is controlled through the implementation of a hybrid control methodology. Numerical simulations are also provided to support the analytical findings. These simulation studies discovered long-term dynamics that were chaotic across various parameters.</p>