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

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/2e^-ϵ _τσ t , where constant σ >0 and ϵ _τ=ϵ (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.