This paper proposes an adaptive model predictive control (MPC) method to enhance the accuracy and adaptability of lane keeping control under dynamic conditions. First, within the Koopman operator framework, dynamic mode decomposition (DMD) is performed in the original state space to linearize the three-degree-of-freedom nonlinear vehicle model, resulting in an initial linear model along with the associated system and covariance matrices. The model is then iteratively updated using an online dynamic mode decomposition (ODMD) algorithm with a forgetting factor based on real-time vehicle states and control inputs. The updated linear model is embedded into the MPC framework, where fuzzy rules are designed to adaptively adjust the forgetting factor according to model residuals. Validation on the CarSim-Simulink co-simulation platform shows that the proposed method achieves significantly lower lane-tracking errors than MPC based on a simplified linear vehicle model, and demonstrates continuous adaptive improvement compared with nonlinear MPC and DMD-based MPC under identical road conditions.