The general public considers Fixed-Route Transit (FRT) to be inconvenient while Demand-Responsive Transit (DRT) provides much of the desired flexibility with a door-to-door type of service. However, FRT is typically more cost efficient than DRT to deploy. Therefore, there is an increased interest in flexible transit services including all types of hybrid services that combine FRT and pure DRT. The demand-responsive feeder transit, also known as Demand-Responsive Connector (DRC), is a flexible transit service because it operates in a demand-responsive fashion within a service area and moves customers to/from a transfer point that connects to a FRT network. In this research we develop analytical models, validated by simulation, to design the DRC system. Feeder transit services are generally operated with a DRC policy which might be converted to a traditional FRT policy for higher demand. By using continuous approximations, we provide an analytical modeling framework to help planners and operators in their choice of the two policies. We compare utility functions of the two policies to derive rigorous analytical and approximate closed-form expressions of critical demand densities. They represent the switching conditions, that are functions of the parameters of each considered scenario, such as the geometry of the service area, the vehicle speed and also the weights assigned to each term contributing to the utility function: walking time, waiting time and riding time. We address the problem faced by planners in determining the optimal number of zones for dividing a service area. We develop analytical models representing the total cost functions balancing customer service quality and vehicle operating cost. We obtain close-form expressions for the FRT and approximation formulas for the DRC to determine the optimal number of zones. Finally we develop a real-case application with collected customer demand data and road network data of El Cenizo, Texas. With our analytical formulas, we obtain the optimal number of zones, and the times for switching FRT and DRC policies during a day. Simulation results considering the road network of El Cenizo demonstrate that our analytical formulas provide good estimates for practical use.