We explore the stability of traveling waves in a class of non-cooperative reaction-diffusion systems with nonlocal dispersal, when the diffusivities are equal. Our stability criterion for monostable traveling waves is based on measuring initial perturbations by a weighted relative entropy. To demonstrate the practical applications of this theory, we will present several examples from ecology and epidemiology. This is a joint work with Jong-Shenq Guo (Tamkang University).