Linear stability criteria in a reaction-diffusion equation with spatially inhomogeneous delay

Time delays play an important role in many biological and ecological system s. However, they are usually incorporated into mathematical models in a way not explicitly dependent on space, even though the delay may be modelling some environmental aspect of the system. In this paper we study a scalar re action-diffusion equation with a spatially inhomogeneous delay, taken for s implicity in the form of a step function of the spatial coordinate. We deri ve the dispersion relation from which analytical results on the stability o r instability of the uniform steady states can be determined. We confirm an d extended these results by numerical simulations which confirm the possibi lity of qualitatively different types of behaviour on different parts of th e spatial domain.