# Wave propagation and its stability for a class of discrete diffusion systems

## TL;DR

This paper investigates wave propagation and stability in two-component discrete diffusive systems, establishing existence and exponential convergence of traveling wave fronts, with applications to epidemic models.

## Contribution

It introduces new methods for proving the existence and stability of traveling waves in discrete diffusive systems, extending to more general models.

## Key findings

- Existence of positive monotone traveling wave fronts.
- Exponential convergence of solutions to wave fronts.
- Application to discrete epidemic models with complex effects.

## Abstract

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.