In complex and dynamic environments, traditional pursuit–evasion studies may face challenges in offering effective solutions to sudden environmental changes. In this paper, a bio-inspired neural network (BINN) is proposed that approximates a pursuit–evasion game from a neurodynamic perspective instead of formulating the problem as a differential game. The BINN is topologically organized to represent the environment with only local connections. The dynamics of neural activity, characterized by the neurodynamic shunting model, enable the generation of real-time evasive trajectories with moving or sudden-change obstacles. Several simulation and experimental results indicate that the proposed approach is effective and efficient in complex and dynamic environments.