Folds, canards and shocks in advection–reaction–diffusion models

Abstract

Tactically driven cell movement modelled by coupled advection–reaction–diffusion (ARD) equations typically exhibit smooth travelling waves, and less frequently sharp interfaces in the wave form. We study the existence of travelling waves with smooth and sharp interfaces in coupled ARD models by using geometric singular perturbation techniques. In particular, we show that a travelling wave analysis under an appropriate Liénard transformation reveals a generic fold condition to observe shock-like interfaces in the wave form. This geometric approach further explains automatically well-known jump and entropy conditions for shocks in hyperbolic PDE theory (Rankine–Hugoniot and Lax conditions). Our analysis also shows that canards, a special class of solutions within singular perturbation problems, play an important role in the construction of travelling waves with smooth and sharp interfaces.