Rapid Communication A stability theorem for unsymmetric Lienard equations

Time delays play an important role in many biological and ecological systems. 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 reaction-diffusion equation with a spatially inhomogeneous delay, taken for simplicity in the form of a step function of the spatial coordinate. We derive the dispersion relation from which analytical results on the stability or instability of the uniform steady states can be determined. We confirm and extended these results by numerical simulations which confirm the possibility of qualitatively different types of behaviour on different parts of the spatial domain.