Shock-fronted travelling waves in a reaction-diffusion model with nonlinear forward-backward-forward diffusion

TL;DR

This paper analyzes shock-fronted travelling waves in a reaction-diffusion model with nonlinear forward-backward-forward diffusion, using geometric singular perturbation theory to prove their existence and explore their properties.

Contribution

It formalizes numerical observations by proving the existence of shock-fronted travelling waves in a complex reaction-diffusion model through advanced PDE analysis.

Findings

Shock-fronted travelling waves exist in the model.

Different embeddings produce waves with distinct properties.

The analysis confirms numerical predictions about wave behavior.

Abstract

Reaction-diffusion equations (RDEs) are often derived as continuum limits of lattice-based discrete models. Recently, a discrete model which allows the rates of movement, proliferation and death to depend upon whether the agents are isolated has been proposed, and this approach gives various RDEs where the diffusion term is convex and can become negative (Johnston et al., Sci. Rep. 7, 2017), i.e. forward-backward-forward diffusion. Numerical simulations suggest these RDEs support shock-fronted travelling waves when the reaction term includes an Allee effect. In this work we formalise these preliminary numerical observations by analysing the shock-fronted travelling waves through embedding the RDE into a larger class of higher order partial differential equations (PDEs). Subsequently, we use geometric singular perturbation theory to study this larger class of equations and prove the existence of these shock-fronted travelling waves. Most notable, we show that different embeddings yield shock-fronted travelling waves with different properties.