Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

Yeqing Xue ^[1] ; Zhaohai Ma ^[2] ; Zhihua Liu ^[3]
1. [1] Taiyuan University of Technology
  
  Taiyuan University of Technology
  
  China
2. [2] China University of Geosciences
  
  China University of Geosciences
  
  China
3. [3] Beijing Normal University
  
  Beijing Normal University
  
  China
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Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 4, 2023
Idioma: inglés
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Resumen
- In this paper, we consider the multidimensional stability of planar traveling waves for a class of Lotka–Volterra competition systems with time delay and nonlocal reaction term in n–dimensional space. It is proved that, all planar traveling waves with speed c > c∗ are exponentially stable. We get accurate decay rate t − n 2 e−τ σt , where constant σ > 0 and τ = (τ ) ∈ (0, 1) is a decreasing function for the time delay τ > 0. It is indicated that time delay essentially reduces the decay rate. While, for the planar traveling waves with speed c = c∗, we prove that they are algebraically stable with delay rate t − n 2 . The proof is carried out by applying the comparison principle, weighted energy and Fourier transform, which plays a crucial role in transforming the competition system to a linear delayed differential system.