Traveling waves in delayed reaction-diffusion equations in biology

Sergei Trofimchuk; Vitaly Volpert; Sergei Trofimchuk; Vitaly Volpert

doi:10.3934/mbe.2020339

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 6487-6514. doi: 10.3934/mbe.2020339

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Research article Special Issues

Traveling waves in delayed reaction-diffusion equations in biology

Sergei Trofimchuk ¹,
Vitaly Volpert ^{2,3,4
,
,}

1.
Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile
2.
Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne 69622, France
3.
INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne 69603, France
4.
Peoples' Friendship University of Russia (RUDN University), Miklukho-Maklaya St, Moscow 117198, Russia

Received: 22 July 2020 Accepted: 27 August 2020 Published: 25 September 2020

This paper represents a literature review on traveling waves described by delayed reactiondiffusion (RD, for short) equations. It begins with the presentation of different types of equations arising in applications. The main results on wave existence and stability are presented for the equations satisfying the monotonicity condition that provides the applicability of the maximum and comparison principles. Other methods and results are described for the case where the monotonicity condition is not satisfied. The last two sections deal with delayed RD equations in mathematical immunology and in neuroscience. Existence, stability, and dynamics of wavefronts and of periodic waves are discussed.
- traveling wave,
- reaction-diffusion equation,
- delay,
- stability,
- existence,
- dynamics
Citation: Sergei Trofimchuk, Vitaly Volpert. Traveling waves in delayed reaction-diffusion equations in biology[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6487-6514. doi: 10.3934/mbe.2020339

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Abstract

This paper represents a literature review on traveling waves described by delayed reactiondiffusion (RD, for short) equations. It begins with the presentation of different types of equations arising in applications. The main results on wave existence and stability are presented for the equations satisfying the monotonicity condition that provides the applicability of the maximum and comparison principles. Other methods and results are described for the case where the monotonicity condition is not satisfied. The last two sections deal with delayed RD equations in mathematical immunology and in neuroscience. Existence, stability, and dynamics of wavefronts and of periodic waves are discussed.

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