Automatic cell tracking in dense environments is plagued by inaccurate correspondences and misidentification of parent-offspring relationships. In this paper, we introduce a novel cell tracking algorithm named DenseTrack, which integrates deep learning with mathematical model-based strategies to effectively establish correspondences between consecutive frames and detect cell division events in crowded scenarios. We formulate the cell tracking problem as a deep learning-based temporal sequence classification task followed by solving a constrained one-to-one matching optimization problem exploiting the classifier's confidence scores. Additionally, we present an eigendecomposition-based cell division detection strategy that leverages knowledge of cellular geometry. The performance of the proposed approach has been evaluated by tracking densely packed cells in 3D time-lapse image sequences of bacterial biofilm development. The experimental results on simulated as well as experimental fluorescence image sequences suggest that the proposed tracking method achieves superior performance in terms of both qualitative and quantitative evaluation measures compared to recent state-of-the-art cell tracking approaches.