Existence and uniqueness of a wavefront in a delayed hyperbolic-parabolic model

We consider a class of hyperbolic-parabolic equation with time delay and a non-local feedback due to the maturation for the adult population density of a single species population, and we show that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. We show the existence and uniqueness of a travelling wavefront of the hyperbolic-parabolic system in the so-called bistable case, by considering the same problem for a properly parametrized parabolic system, and then by considering the continuous dependence of the wave speed on the parameter involved.