Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species

We investigate the multidimensional stability of planar traveling waves in competitive–cooperative Lotka–Volterra system of three species in n -dimensional space. For planar traveling waves with speed c > c * , we establish their exponential stability in L ∞ ( R n ) , which is expressed as t − n 2 e − ε τ σ t , where σ > 0 is a constant and ε τ ∈ ( 0 , 1 ) depends on the time delay τ > 0 as a decreasing function ε τ = ε ( τ ) . The time delay is shown to significantly reduce the decay rate of the solution. Additionally, for planar traveling waves with speed c = c * , we demonstrate their algebraic stability in the form t − n 2 . Our analysis employs the Fourier transform and a weighted energy method with a carefully chosen weight function.