Folds, Canards and Shocks in Advection-Reaction-Diffusion Models M. Wechselberger † and G.J. Pettet ‡ † School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia (corresponding author) ‡ Faculty of Science and Technology, Queensland University of Technology, Brisbane, QLD, Australia E-mail: wm@maths.usyd.edu.au 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 Li´ enard 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. AMS classification scheme numbers: 34C40, 34E15, 34E17, 35L67, 92C17