Monotone waves for non-monotone and non-local monostable reaction-diffusion equations

TL;DR

This paper establishes criteria for the existence and uniqueness of monotone wavefronts in non-monotone, non-local monostable reaction-diffusion equations, extending previous results and providing insights into wavefront monotonicity.

Contribution

It introduces a new criterion for monotone wavefront existence in non-local equations and proves their uniqueness within certain classes, generalizing prior local reaction results.

Findings

Criteria for monotone wavefront existence established

Uniqueness of wavefronts proved within specific classes

Monotonicity of minimal fronts confirmed under local reaction conditions

Abstract

We propose a criterion for the existence of monotone wavefronts in non-monotone and non-local monostable diffusive equations of the Mackey-Glass type. This extends recent results by Gomez et al proved for the particular case of equations with local delayed reaction. In addition, we demonstrate the uniqueness (up to a translation) of obtained monotone wavefront within the class of all monotone wavefronts (such a kind of conditional uniqueness was recently established for the non-local KPP-Fisher equation by Fang and Zhao). Moreover, we show that if delayed reaction is local then this uniqueness actually holds within the class of all wavefronts and therefore the minimal fronts under consideration (either pulled or pushed) should be monotone. Similarly to the case of the KPP-Fisher equations, our approach is based on the construction of an appropriate fundamental solution for associated boundary value problem for linear integral-differential equation.