Dynamics in a stochastic delayed predator-prey system with patch diffusion and Beddington-DeAngelis functional response

This paper investigates a class of predator-prey systems with time delays, stochastic perturbations, and patch diffusion, which incorporate the Beddington–DeAngelis (B-D) functional response. The proposed model integrates environmental noise, dispersal processes, and time delays, forming a comprehensive stochastic delayed diffusion framework that better reflects realistic ecological scenarios. By employing appropriate Lyapunov functions and the comparative principle of stochastic differential equations, we prove the existence of a unique global positive solution for any given positive initial value. Sufficient conditions are also established for uniform persistence of the system. Furthermore, through the construction of a novel Lyapunov functional, we derive sufficient conditions for the global attractivity of the system’s solutions, providing theoretical insights into the long-term dynamical behavior under the combined effects of stochasticity, delays, and diffusion. Finally, numerical simulations are conducted using MATLAB along with the Milstein discretization method, which corroborate the correctness and effectiveness of the theoretical results. This study offers a new theoretical framework and methodology for analyzing persistence and stability in stochastic delayed diffusive predator-prey systems.