Stability of traveling wavefronts for a three-component Lotka-Volterra competition system on a lattice

This article concerns the stability of traveling wavefronts for a three-component Lotka-Volterra competition system on a lattice. By means of the weighted energy method and the comparison principle, it is proved that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as $j+ct \to -\infty$, where $j\in\mathbb{Z}$, $t>0$ and $c>0$, but the initial perturbation can be arbitrarily large on other locations