A Hamilton-Jacobi approach to road-field reaction-diffusion models

TL;DR

This paper derives a Hamilton-Jacobi equation for a road-field reaction-diffusion model to analyze invasion fronts, revealing how different diffusivities on roads influence propagation speeds in ecological settings.

Contribution

It introduces a Hamilton-Jacobi framework for the road-field model and provides new insights into front speeds on conical domains with varying diffusivities.

Findings

Propagation speed can be computed from one-road problems when diffusivities are equal.

Speed along the faster road remains unchanged when diffusivities differ.

Speed along the slower road can be increased by the faster road's influence.

Abstract

We consider the road-field reaction-diffusion model introduced by Berestycki, Roquejoffre, and Rossi. By performing a "thin-front limit," we are able to deduce a Hamilton-Jacobi equation with a suitable effective Hamiltonian on the road that governs the front location of the road-field model. Our main motivation is to apply the theory of strong (flux-limited) viscosity solutions in order to determine a control formulation interpretation of the front location. In view of the ecological meaning of the road-field model, this is natural as it casts the invasion problem as one of finding optimal paths that balance the positive growth rate in the field with the fast diffusion on the road. Our main contribution is a nearly complete picture of the behavior on two-road conical domains. When the diffusivities on each road are the same, we show that the propagation speed in each direction in the cone can be computed via those associated with one-road half-space problem. When the diffusivities differ, we show that the speed along the faster road is unchanged, while the speed along the slower road can be enhanced. Along the way we provide a new proof of known results on the one-road half-space problem via our approach.