Curvature dependences of wave propagation in reaction-diffusion models

TL;DR

This paper derives an exact expression for how wave front speed in reaction-diffusion systems depends on curvature, revealing complex behaviors across different propagation phases through analysis and simulations.

Contribution

It provides a non-perturbative, exact curvature dependence of wave speed in reaction-diffusion models, including analysis of multiple propagation phases.

Findings

Exact curvature dependence of wave speed derived

Identification of three propagation phases

Numerical simulations confirm theoretical predictions

Abstract

Reaction-diffusion waves in multiple spatial dimensions advance at a rate that strongly depends on the curvature of the wave fronts. 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 wave front. Inward-propagating waves are characterised 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.