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Propagation speed of traveling waves for diffusive Lotka-Volterra system with strong competition

Ken-Ichi Nakamura, Toshiko Ogiwara

Abstract

We study the propagation speed of bistable traveling waves in the classical two-component diffusive Lotka-Volterra system under strong competition. From an ecological perspective, the sign of the propagation speed determines the long-term outcome of competition between two species and thus plays a central role in predicting the success or failure of invasion of an alien species into habitats occupied by a native species. Using comparison arguments, we establish sufficient conditions determining the sign of the propagation speed, which refine previously known results. In particular, we show that in the symmetric case, where the two species differ only in their diffusion rates, the faster diffuser prevails over a substantially broader parameter range than previously established. Moreover, we demonstrate that when the interspecific competition coefficients differ significantly, the outcome of competition cannot be reversed by adjusting diffusion or growth rates. These findings provide a rigorous theoretical framework for analyzing invasion dynamics, offering sharper mathematical criteria for invasion success or failure.

Propagation speed of traveling waves for diffusive Lotka-Volterra system with strong competition

Abstract

We study the propagation speed of bistable traveling waves in the classical two-component diffusive Lotka-Volterra system under strong competition. From an ecological perspective, the sign of the propagation speed determines the long-term outcome of competition between two species and thus plays a central role in predicting the success or failure of invasion of an alien species into habitats occupied by a native species. Using comparison arguments, we establish sufficient conditions determining the sign of the propagation speed, which refine previously known results. In particular, we show that in the symmetric case, where the two species differ only in their diffusion rates, the faster diffuser prevails over a substantially broader parameter range than previously established. Moreover, we demonstrate that when the interspecific competition coefficients differ significantly, the outcome of competition cannot be reversed by adjusting diffusion or growth rates. These findings provide a rigorous theoretical framework for analyzing invasion dynamics, offering sharper mathematical criteria for invasion success or failure.
Paper Structure (6 sections, 7 theorems, 70 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Construction of time-independent supersolutions
  3. Determining the sign of the propagation speed of bistable traveling waves
  4. Symmetric nonlinearity case
  5. Nearly degenerate case
  6. Determinacy of the speed sign under strongly asymmetric competition

Key Result

Proposition 2.1

Set $(\phi_+(x),\psi_+(x))=(\sigma_p(ax)^p,\sigma_p(ax))$ for $a>0$. Then, $(\phi_+,\psi_+)$ is a time-independent supersolution of Co1 if all the following conditions hold:

Figures (2)

  • Figure 1: (Left) Parameter regions in the $(d,k)$-plane corresponding to negative speeds previously established. (Right) Additional regions (S1) and (S2) in Theorem \ref{['thm:negative-sym']} (highlighted in purple). The solid-line box $[1,10]\times[1,4]$ indicates the drawing area shown in the left figure.
  • Figure 2: (Left) Schematic representation of the regions of negative speed (blue), positive speed (red), and zero speed (solid curve) numerically suggested in ADD12. (Right) Regions of negative speed for $r=1$ and $k_2=2$ established by Theorem \ref{['thm:negative1']} (purple) and Theorem \ref{['thm:degenerate']} (blue).

Theorems & Definitions (19)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • proof
  • ...and 9 more