The sign of traveling wave speed in bistable dynamics

Jong-Shenq Guo; Ken-Ichi Nakamura; Toshiko Ogiwara; Chang-Hong Wu

doi:10.3934/dcds.2020047

Article Contents

2020, Volume 40, Issue 6: 3451-3466. Doi: 10.3934/dcds.2020047 open access

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The sign of traveling wave speed in bistable dynamics

1.
Department of Mathematics, Tamkang University, Tamsui, New Taipei City 25137, Taiwan
2.
Faculty of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, Japan
3.
Department of Mathematics, Josai University, Sakado, 350-0295, Japan
4.
Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan

^* Corresponding author: Jong-Shenq Guo
^* Corresponding author: Jong-Shenq Guo

Received: February 28, 2019

Revised: June 30, 2019

Published: March 2020

This work was partially supported by the Ministry of Science and Technology of Taiwan under the grants 105-2115-M-032-003-MY3 and MOST 108-2636-M-024-001 and by JSPS KAKENHI Grant Numbers JP15K04996 and JP18K03412

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Abstract

We are concerned with the sign of traveling wave speed in bistable dynamics. This question is related to which species wins the competition in multiple species competition models. It is well-known that the wave speed is unique for traveling wave connecting two stable states. In this paper, we first review some known results on the sign of wave speed in bistable two species competition models. Then we derive rigorously the sign of bistable wave speed for a special three species competition model describing the competition in two different circumstances: (1) two species are weak competitors and one species is a strong competitor; (2) three species are very strong competitors. It is interesting to observe that, under certain conditions on the parameters, two weaker competitors can wipe out the strongest competitor.

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Mathematics Subject Classification: Primary: 35K55, 35K57; Secondary: 92D25, 92D40.

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