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Non-Monotone Traveling Waves of the Weak Competition Lotka-Volterra System

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

TL;DR

The paper addresses traveling-wave solutions in the two-species Lotka-Volterra competition-diffusion system under weak competition, proving existence for all speeds $s\ge s^* = \max\{2,2\sqrt{ad}\}$ via refined upper/lower solutions and Schauder's fixed point theorem. It provides verifiable criteria for non-monotone fronts and proves, for the first time, the existence of front-pulse waves at the critical weak competition regime, using degenerate reductions and a shrinking-box argument. The results extend the classical monotone-front theory (Tang–Fife) to non-monotone regimes and explore the degenerate critical case where $ac=1$, offering a richer description of invasion dynamics and pattern formation in competitive populations. Collectively, these findings advance understanding of wave shapes, speeds, and the coexistence invasion landscape in weakly competing species with diffusion, with potential ecological applications.

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-monotonic waves have not been explicitly addressed in previous studies. It is interesting to point out that our result for non-monotone waves also hold for the critical speed case $s=s^*$. In addition, in the critical weak competition case $(b<a=1/c,\,d>0)$, we rigorously prove, for the first time, the existence of front-pulse traveling waves.

Non-Monotone Traveling Waves of the Weak Competition Lotka-Volterra System

TL;DR

The paper addresses traveling-wave solutions in the two-species Lotka-Volterra competition-diffusion system under weak competition, proving existence for all speeds via refined upper/lower solutions and Schauder's fixed point theorem. It provides verifiable criteria for non-monotone fronts and proves, for the first time, the existence of front-pulse waves at the critical weak competition regime, using degenerate reductions and a shrinking-box argument. The results extend the classical monotone-front theory (Tang–Fife) to non-monotone regimes and explore the degenerate critical case where , offering a richer description of invasion dynamics and pattern formation in competitive populations. Collectively, these findings advance understanding of wave shapes, speeds, and the coexistence invasion landscape in weakly competing species with diffusion, with potential ecological applications.

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 , 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 , and provide verifiable sufficient conditions for the emergence of non-monotone waves. Such conditions for non-monotonic waves have not been explicitly addressed in previous studies. It is interesting to point out that our result for non-monotone waves also hold for the critical speed case . In addition, in the critical weak competition case , we rigorously prove, for the first time, the existence of front-pulse traveling waves.

Paper Structure

This paper contains 17 sections, 16 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assumptions and notations
  3. Main results
  4. Organization
  5. The existence of the Traveling wave solution
  6. Preliminaries
  7. The Existence Theorem
  8. The super-solution and sub-solution of $s>s^*$
  9. The super-solution and sub-solution of $s=s^*$
  10. The shrinking box argument
  11. Determination of the minimum speed
  12. The profile of the solution
  13. The monotone solution
  14. The non-monotone solution
  15. Front-pulse waves at critical weak competition
  16. ...and 2 more sections

Key Result

Theorem 1.1

Given $d>0$. Assume $a,b,c$ satisfy $(weak)$. For any $s \geq s^*$, there exists a traveling wave solution $(u,v)(\xi)$ satisfying maineq and asymptotic behavior.

Figures (2)

  • 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 $\lim_{\xi \to +\infty}v(\xi)=v^*$.
  • 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.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 23 more