<p>In this paper, we explore the dynamic properties of discrete predator-prey models with diffusion on a coupled mapping lattice. We conducted a stability analysis of the equilibrium points, provided the normal form of the Neimark-Sacker and Flip bifurcations, and explored a range of Turing instabilities that emerged in the system upon the introduction of diffusion. Our numerical simulations aligned with the theoretical derivations, incorporating the computation of the maximum Lyapunov exponent to validate obtained bifurcation diagrams and elucidated the system's progression from bifurcations to chaos. By adjusting the self-diffusion and cross-diffusion coefficients, we simulated the shifts between different Turing instabilities. These findings highlight the complex dynamic behavior of discrete predator-prey models and provide valuable insights for biological population conservation strategies.</p>