Developing autonomous vehicles (AVs) helps improve the road safety and traffic efficiency of intelligent transportation systems (ITS). Accurately predicting the trajectories of traffic participants is essential to the decision-making and motion planning of AVs in interactive scenarios. Recently, learning-based trajectory predictors have shown state-of-the-art performance in highway or urban areas. However, most existing learning-based models trained with fixed datasets may perform poorly in continuously changing scenarios. Specifically, they may not perform well in learned scenarios after learning the new one. This phenomenon is called “catastrophic forgetting”. Few studies investigate trajectory predictions in continuous scenarios, where catastrophic forgetting may happen. To handle this problem, first, a novel continual learning (CL) approach for vehicle trajectory prediction is proposed in this paper. Then, inspired by brain science, a dynamic memory mechanism is developed by utilizing the measurement of traffic divergence between scenarios, which balances the performance and training efficiency of the proposed CL approach. Finally, datasets collected from different locations are used to design continual training and testing methods in experiments. Experimental results show that the proposed approach achieves consistently high prediction accuracy in continuous scenarios without re-training, which mitigates catastrophic forgetting compared to non-CL approaches. The implementation of the proposed approach is publicly available at https://github.com/BIT-Jack/D-GSM .