Exploring Stability Landscapes and Complex Dynamics in a Discrete Holling–Tanner Predator–Prey Model via Bifurcation Theory and Machine Learning

Discretization of continuous models can do more than approximate their dynamics; it can fundamentally transform their dynamical behavior, such as the complex dynamical behavior that translates the system to a chaotic state. In this study we investigated the discrete-time Holling–Tanner predator–prey model. We analytically developed conditions for the stable and nonhyperbolic state of coexistent fixed points and developed a critical threshold by using the center manifold theorem and normal form theory at which the period-doubling and Neimark–Sacker bifurcations exist. The analytical results are justified by the numerical examples along with the bifurcation diagrams and maximum Lyapunov exponent that clearly illustrate the transition from a stable coexistence to periodic oscillations and eventually to chaos. We have utilized the machine learning classifiers (random forest and decision tree) and concluded that the discretization step size is the most influential parameter, and it plays a decisive role in shaping the qualitative dynamics of the system. Furthermore, a hybrid control strategy has been used to control the chaos in the system generated by the period-doubling or Neimark–Sacker bifurcation. Overall, the findings highlight that discretization should not be regarded as a neutral numerical approximation; instead, it represents a crucial modeling decision that can significantly influence the predicted behavior of ecological systems.