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 can be captured mathematically by

generalizing the boundary conditions appropriately. The model presented is an

extension of previous models with such generalized boundary conditions and

persistance conditions are obtained. It is shown that trailing edge behaviour plays a

crucial role in persistance ability.