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

Abstract

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 in the form of t − n 2 e −ϵτ σt , where constant σ > 0 and ϵ τ = ϵ(τ ) ∈ (0, 1) is a decreasing function for the time delay τ > 0. While, for the planar traveling waves with speed c = c * , we prove that they are algebraically stable in the form of t − n 2 . The Fourier transform plays a crucial role in transforming the competition systems to a linear delayed differential system. We establish the comparison principle and some estimates in weighted spaces L 1 w (R n ) and W 2,1 w (R n ) to obtain the main results.