Existence of diffusion-driven instability and Hopf bifurcation in a Spatio-temporal Predator–Prey system

This study presents a comprehensive dynamical analysis of a ratio-dependent reaction–diffusion prey–predator model, with an emphasis on nonlinear oscillatory behavior and the evolution of spatiotemporal patterns. The governing system of partial differential equations is investigated through local and global stability analysis to characterize the response of equilibrium states to small perturbations and to establish the conditions for long-term dynamic stability. Particular attention is devoted to Hopf-bifurcation analysis, which identifies the critical parameter values that govern the onset of self-sustained oscillations and periodic dynamics in the interacting populations. The emergence of diffusion-driven instability is further examined to reveal the formation of spatially distributed oscillatory patterns arising from the coupling between reaction kinetics and diffusion. Numerical simulations are performed to validate the theoretical findings and to illustrate the transition between stable, periodic, and spatially structured dynamic regimes. A coupled amplification-matrix analysis of the non-standard finite difference (NSFD) scheme, mesh-refinement tests, and a direct comparison with a classical explicit finite-difference method are used to assess discrete stability, consistency, positivity preservation, and small-step accuracy. The results demonstrate the proposed model’s ability to capture oscillatory responses and complex spatiotemporal behaviors governed by nonlinear interactions. The obtained results provide a comprehensive understanding of stability, bifurcation, and diffusion-driven pattern formation in the reaction-diffusion prey-predator model and may serve as a useful mathematical framework for future studies on nonlinear distributed dynamical systems.