  This paper is devoted to study the wave propagation and its stability for a
class of two-component discrete diffusive systems. We first establish the
existence of positive monotone monostable traveling wave fronts. Then, applying
the techniques of weighted energy method and the comparison principle, we show
that all solutions of the Cauchy problem for the discrete diffusive systems
converge exponentially to the traveling wave fronts when the initial
perturbations around the wave fronts lie in a suitable weighted Sobolev space.
Our main results can be extended to more general discrete diffusive systems. We
also apply them to the discrete epidemic model with the Holling-II type and
Richer type effects.
