# Stability and uniqueness of generalized traveling waves of lattice Fisher-KPP equations in heterogeneous media

## TL;DR

This paper studies the stability and uniqueness of generalized traveling wave solutions in lattice Fisher-KPP equations with heterogeneous media, establishing conditions for their existence, stability, and uniqueness in various media types.

## Contribution

It provides a comprehensive framework for proving the existence, stability, and uniqueness of generalized traveling waves in heterogeneous lattice Fisher-KPP equations, extending previous results.

## Key findings

- Existence of strictly positive entire solutions
- Stability and uniqueness of generalized traveling waves
- Applicability to periodic and heterogeneous media

## Abstract

In this paper, we investigate the stability and uniqueness of generalized traveling wave solutions of lattice Fisher-KPP equations with general time and space dependence. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability and uniqueness of generalized traveling waves connecting the unique strictly positive entire solution and the trivial solution zero. Applying the general stability and uniqueness theorem, we then prove the existence, stability and uniqueness of periodic traveling wave solutions of lattice Fisher-KPP equations in time and space periodic media, and the existence, stability and uniqueness of generalized traveling wave solutions of lattice Fisher-KPP equations in time heterogeneous media. The general stability result established in this paper implies that the generalized traveling waves obtained in many cases are asymptotically stable under well-fitted perturbation.