As a persistent neurodegenerative abnormality, Alzheimer's disease (AD) is affecting an increasing number of elderly people. The early identification of AD is critical for halting the disease progression at an early stage. However, the extraction and fusion of multi-modal features at different scales from brain images remains a challenge for effective AD recognition. In this work, a novel feature fusion long-range state space model (FF-LSSM) model is suggested for effective extraction and fusion of multi-level characteristics from scannings of MRI and PET. The FF-LSSM can extract whole-volume features at every scale and effectively decide their global dependencies via adopted 3D Mamba encoders. Moreover, a feature fusion block is employed to consolidate features of different levels extracted by each encoder to generate fused feature maps. A classifier is cascaded at the end, using the fused features to produce the predicted labels. The FF-LSSM model is optimized and evaluated using brain images of subjects from the ADNI dataset. The inference result on the testing set reveals the FF-LSSM accomplishes a classification ACC of 93.59% in CN vs. AD and 79.31% in sMCI vs. pMCI task, proving its effectiveness in disease classification. Finally, the introduction of the Grad-CAM method illustrates that the implied FF-LSSM can detect AD- and MCI-related brain regions effectively.