In complex and dynamic environments, traditional pursuit–evasion studies may face challenges in offering effective solutions. This paper aims to provide a novel approach that approximates a general pursuit–evasion game from a neurodynamics perspective instead of formulating the problem as a traditional differential game. In this paper, the neurodynamics-based approach aims to overcome the limitations of the traditional approach and improve the performance of the evaders in dynamic and uncertain environments. A bio-inspired neural network is proposed that approximates a general pursuit–evasion game from a neurodynamic perspective. The bio-inspired neural network is topologically organized to represent the environment with only local connections, and the dynamics of neural activity are characterized by a neurodynamic model. The pursuer has global effects on the whole neural network, while the obstacles only have local effects to guarantee the robot avoids collisions. The real-time collision-free evasion trajectories are generated through dynamic neural activities. Simulation results indicate that the proposed approach is able to guide evader robots to evade the pursuer in complex environments with static, moving, and sudden-change obstacles. In addition, the comparison studies illustrate that the proposed approach is effective and efficient in complex and dynamic environments. This paper brings new insights into the application of the bio-inspired neural network in the field of robotics and also presents many potential practical application scenarios.