Bifurcation Analysis and Qualitative Dynamics of the Time-Delayed Holling–Tanner Prey-Predator Model

In this study, we work on a Holling–Tanner prey-predator (HTPP) model by incorporating the impact of predator cannibalism and later introducing a delay in prey growth. We first examine its behavior without delay: we identify equilibrium points, describe their stability, and explain when the system remains stable or goes into oscillation. Then we add a time lag in the prey growth, which captures the natural lag in biological systems. We establish positivity, and by applying the Routh–Hurwitz criterion, we obtain explicit criteria for local stability and also prove global stability (GS) by applying a Lyapunov approach. Numerical simulations indicate that populations can converge in the ordinary model but oscillate indefinitely as soon as the delay passes a threshold. Aside from analysis, we also plot a detailed graph to show the findings. The results validate that delay significantly destabilizes the system, and cannibalism determines the long-term dynamics.