Motion models are vital for solving multiple object tracking (MOT), which makes instance-level position predictions of targets to handle occlusions and noisy detections. Recent methods have proposed the use of Single Object Tracking (SOT) techniques to build motion models and unify the SOT tracker with the object detector into a single network for high-efficiency MOT. However, three feature incompatibility issues in the required features of this paradigm are ignored, leading to inferior performance. First, the object detector requires class-specific features to localize objects of pre-defined classes. Contrarily, target-specific features are required in SOT to track the target of interest with an unknown category. Second, MOT relies on intra-class differences to associate targets of the same identity (ID). On the other hand, the SOT trackers focus on inter-class differences to distinguish the tracking target from the background. Third, classification confidence is used to determine the existence of targets, which is obtained with category-related features and cannot accurately reveal the existence of targets in tracking scenes. To address these issues, we propose a novel Task-specific Feature Encoding Network (TFEN) to extract task-driven features for different sub-networks. Besides, we propose a novel Quadruplet State Sampling (QSS) strategy to form the training samples of the motion model and guide the SOT trackers to capture identity-discriminative features in position predictions. Finally, we propose an Existence Aware Tracking (EAT) algorithm by estimating the existence confidence of targets and re-considering low-scored predictions to recover missed targets. Experimental results indicate that the proposed Discriminative Motion Model-based tracker (DMMTracker) can effectively address these issues when employing SOT trackers as motion models, leading to highly competitive results on MOT benchmarks.