Non-monotone traveling waves of the weak competition Lotka-Volterra system

Chiun-Chuan Chen; Ting-Yang Hsiao; Shun-Chieh Wang

doi:10.3934/dcdsb.2026049

Article Contents

2026, Volume 37: 206-228. Doi: 10.3934/dcdsb.2026049

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Non-monotone traveling waves of the weak competition Lotka-Volterra system

1.
National Center for Theoretical Science, Taipei, Taiwan
2.
Department of Mathematics, National Taiwan University, Taiwan
3.
Mathematics, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy

^*Corresponding author: Shun-Chieh Wang
^*Corresponding author: Shun-Chieh Wang

Dedicated to Professor Yoshihisa Morita on the occasion of his 70th birthday.

Received: September 7, 2025

Revised: January 26, 2026

Early access: March 6, 2026

Published online: March 6, 2026

Chiun-Chuan Chen is supported by the National Science and Technology Council, Taiwan (Grant Number 114-2115-M-002 -004 -MY3) and the National Center for Theoretical Sciences, Taiwan (NCTS). Ting-Yang Hsiao is supported by the ERC CONSOLIDATOR GRANT 2023 "Generating Unstable Dynamics in dispersive Hamiltonian fluids'', Project Number: 101124921. Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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Abstract

We investigate traveling wave solutions in the two-species reaction-diffusion Lotka–Volterra competition system under weak competition. For the strict weak competition regime $ (b<a<1/c, \, d>0) $, we construct refined upper and lower solutions combined with the Schauder fixed point theorem to establish the existence of traveling waves for all wave speeds $ s\geq s^*: = \max\{2, 2\sqrt{ad}\} $, and provide verifiable sufficient conditions for the emergence of non-monotone waves. Such conditions for non-monotone waves have not been explicitly addressed in previous studies. It is interesting to point out that our result for non-monotone waves also holds for the critical speed case $ s = s^* $. In addition, in the critical strong-weak competition case $ (b<a = 1/c, \, d>0) $, we rigorously prove, for the first time, the existence of front-pulse traveling waves.

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Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 34A34.

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Figure 1. When $ n = 26 $, $ \bar{v}(\xi) $(red line), $ \underline{v}(\xi) $(green line), $ v^* = \frac{2}{27} $(blue dash line) are all labeled on the figure. There exists $ v(\xi) $ lying between the red line and green line with $ \lim \limits_{\xi \to +\infty}v(\xi) = v^* $

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Figure 2. Set $ d = 1, a = 1, c = \frac{1}{2}, \underline{s} = 2.1, \mu_2 = 1.001 $. The horizontal axis is the wave speed $ s $, and the vertical axis represents the difference $ a-b>0 $. The blue area illustrates the region where non-monotone solutions $ v(\xi) $ exist

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