Curvature dependences of wave propagation in reaction–diffusion models

Abstract

Reaction–diffusion waves in multiple spatial dimensions advance at a rate that strongly depends on the curvature of the wavefronts. These waves have important applications in many physical, ecological and biological systems. In this work, we analyse curvature dependences of travelling fronts in a single reaction–diffusion equation with general reaction term. We derive an exact, non-perturbative curvature dependence of the speed of travelling fronts that arises from transverse diffusion occurring parallel to the wavefront. Inward-propagating waves are characterized by three phases: an establishment phase dominated by initial and boundary conditions, a travelling-wave-like phase in which normal velocity matches standard results from singular perturbation theory and a dip-filling phase where the collision and interaction of fronts create additional curvature dependences to their progression rate. We analyse these behaviours and additional curvature dependences using a combination of asymptotic analyses and numerical simulations.