Propagating interface in a monostable reaction-diffusion equation with time delay

TL;DR

This paper studies the behavior of solutions to a time-delayed reaction-diffusion equation from population dynamics, showing convergence to a propagating interface using novel barrier constructions.

Contribution

It introduces a non-standard bistable approximation to analyze the monostable problem and proves the convergence to a propagating interface as the small parameter tends to zero.

Findings

Solutions converge to a propagating interface

Constructed accurate lower and upper barriers

Established convergence using bistable approximation

Abstract

We consider a monostable time-delayed reaction-diffusion equation arising from population dynamics models. We let a small parameter tend to zero and investigate the behavior of the solutions. We construct accurate lower barriers --- by using a non standard bistable approximation of the monostable problem--- and upper barriers. As a consequence, we prove the convergence to a propagating interface.