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∞(Rn), which is expressed as t−n2e−ετσt, where σ>0 is a constant and ετ∈(0,1) depends on the time delay τ>0 as a decreasing function ετ=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−n2. Our analysis employs the Fourier transform and a weighted energy method with a carefully chosen weight function.