The dynamics of demand responsive ride pooling (DRRP) systems is considered in a mean-field framework. The relevant dimensionless quantities determining the performance and viability of the system are identified. In the presence of an already established dominant market participant with comparable service quality (like, e.g., the private car), the mutual interaction of the actors (i.e., the customers sharing rides) by virtue of the route assignment algorithm gives rise to a discontinuous transition between two strongly different modes of operation. One of them represents the typical (unfavorable) performance of current ride pooling systems, while the other represents a new mode of operation in which virtually all customers use DRRP.