The earth’s climate is increasing in temperature and as a result many species’ habitat ranges are shifting. The shift in habitat ranges threatens the local persistence of many species. Mathematical models that capture this phenomena of range shift do so by considering a bounded domain that has a time dependant location on the real line. The analysis on persistence conditions has been considered in both continuous-time and -space, and discrete-time, continuous-space settings. In both model types density was considered to be continuous across the boundaries. However it has been shown that many species exhibit particular behaviour at habitat edges, such as biased movement towards the more suitable habitat. This behaviour should be incorporated into the analysis to obtain more accurate persistence conditions. In this thesis persistence conditions are obtained for generalized boundary conditions with a continuous-time and -space model for a range-shifting habitat. It is shown that a high preference for the suitable habitat at the trailing edge can greatly reduce the size of suitable habitat required for species persistence. As well, for fast shifting ranges, a high preference at the trailing edge is crucial for persistence.