Front propagation in two-component reaction-diffusion systems with a cut-off The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with a cut-off was popularised by Brunet and Derrida in the 1990s as a model for many-particle systems in which concentrations below a given threshold are not attainable. While travelling wave solutions in cut-off scalar reaction-diffusion equations have since been studied extensively, the impacts of a cut-off on systems of such equations are less well understood. As a first step towards a broader understanding, we consider various coupled two-component reaction-diffusion equations with a cut-off in the reaction kinetics, such as an FKPP-type population model of invasion with dispersive variability due to Cook, a FitzHugh-Nagumo-style model with piecewise linear Tonnelier-Gerstner kinetics and, finally, a more general predator-prey model with a cut-off in both components that is motivated by standard Lotka-Volterra-type dynamics. Throughout, our focus is on the existence, structure, and stability of travelling fronts, as well as on their dependence on model parameters; in particular, we determine the correction to the front propagation speed that is due to the cut-off. Our analysis is for the most part based on a combination of geometric singular perturbation theory and the desingularisation technique known as “blow-up”. This work is joint with Zhouqian Miao (China Jiliang University), Mariya Ptashnyk (Heriot-Watt University), and Zak Sattar (University of Edinburgh). This article was published on 2025-04-22
