Stability Analysis and PINN Approach on 2D Singular Reaction–Diffusion System

This study investigates the application of physics-informed neural networks (PINNs) to two-dimensional reaction–diffusion biological models with singular terms. Motivated by the classical prey–predator framework, we propose an improved model that incorporates human influence into the ecological environment. The stability of the reaction–diffusion system is analyzed, and an analytical solution is derived for a specific case to provide theoretical support for the numerical model. In addition, a PINN–Adam deep learning algorithm is developed to effectively handle the singular characteristics of the system. Unlike traditional finite element methods (FEMs), which rely on grid-based discretization, the proposed method utilizes random spatiotemporal sampling, leading to improved computational flexibility and prediction accuracy. The proposed approach is validated using real-world fish population data from coral reef ecosystems along the South Australian coast, demonstrating its effectiveness in modeling complex ecological dynamics.