We present the modeling and dynamic analysis of a Mobile Underwater Turbine System, a novel integration of Autonomous Underwater Vehicles and Hydrokinetic Turbines, for harvesting Marine Hydrokinetic Energy from the Gulf Stream. The Gulf Stream, an ocean current that flows off the coast of North Carolina, is a source of hydrokinetic energy. However, the meandering nature of the Gulf Stream makes it challenging to harvest the full energy potential of the stream using fixed turbine systems. One possible solution for increasing the amount of energy that can be extracted from the Gulf Stream involves using a mobile underwater energy harvester system that can follow the meandering stream so as to remain in regions of maximum energy potential. The framework for the conceptual design, and studies focusing on the feasibility of such a system, have been presented previously in (Divi, 2017). The focus of this paper is a mathematical model of the system which has been developed to analyze the dynamics of such a system, along with parametric studies utilizing this model to come up with a system with an optimized set of design parameters. A 6-DOF analytical model of the system is developed to gain an understanding of the system's dynamic behavior and stability. A bead-based tether model is further developed to analyze the behavior of the system when it is anchored and harvesting energy. A study regarding the effects of tether parameters such as the number of tether elements, the spring constant, and the damping coefficient of the tether on the tether behavior and computation time required for analysis, is put forth to help determine an optimal set of tether parameters. In addition, a set of system parameters such as turbine diameter, hull diameter, L/D ratio of the hull and ballast tank size, are analyzed to see how they affect the net energy produced and the maximum distance travelled by the system. Finally, three modes of power transfer to the shore are considered, and an optimization algorithm is presented and used to find the best set of parameters suited for maximum energy transfer for each mode.