Stability of traveling wavefronts for a discrete diffusive Lotka–Volterra competition system

In this paper, we study a discrete diffusive Lotka–Volterra competition system. It is known that this system has traveling wavefronts. We prove that the traveling wavefronts are exponentially stable, when the initial perturbation around the traveling wavefronts decays exponentially as x→−∞x→−∞, but can be arbitrarily large in other locations. The approach we use here is the comparison principle and the weighted energy method.