  In this work, we first prove a stability theorem for traveling waves in a
class of non-cooperative reaction-diffusion systems with nonlocal dispersal of
equal diffusivities. Our stability criterion is in the sense that the initial
perturbation is such that a suitable weighted relative entropy function is
bounded and integrable. Then we apply our main theorem to derive the stability
of traveling waves for some specific examples of non-cooperative systems
arising in ecology and epidemiology.
