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Travelling wave fronts of Lotka-Volterra reaction-diffusion system in the weak competition case

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  12 August 2021

Yang Wang ,
Hongliang Li  and
Xiong Li
Yang Wang
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006. People's Republic of China (ywang2005@sxu.edu.cn) (1574509811@qq.com)
Hongliang Li
Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006. People's Republic of China (ywang2005@sxu.edu.cn) (1574509811@qq.com)
Xiong Li
Affiliation:
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China (xli@bnu.edu.cn)
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Abstract

This paper is concerned with spreading phenomena of the classical two-species Lotka-Volterra reaction-diffusion system in the weak competition case. More precisely, some new sufficient conditions on the linear or nonlinear speed selection of the minimal wave speed of travelling wave fronts, which connect one half-positive equilibrium and one positive equilibrium, have been given via constructing types of super-sub solutions. Moreover, these conditions for the linear or nonlinear determinacy are quite different from that of the minimal wave speeds of travelling wave fronts connecting other equilibria of Lotka-Volterra competition model. In addition, based on the weighted energy method, we give the global exponential stability of such solutions with large speed $c$. Specially, when the competition rate exerted on one species converges to zero, then for any $c>c_0$, where $c_0$ is the critical speed, the travelling wave front with the speed $c$ is globally exponentially stable.

Keywords

Lotka-Volterratravelling wave frontsstability

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 92D25: Population dynamics (general) 92D40: Ecology
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 912 - 938
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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