# American Institute of Mathematical Sciences

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May  2020, 25(5): 1757-1774. doi: 10.3934/dcdsb.2020001

## Stability analysis of traveling wave solutions for lattice reaction-diffusion equations

 1 Department of Mathematics, National Central University, Zhongli District, Taoyuan City 32001, Taiwan 2 General Education Center, National Taipei University of Technology, Taipei 10608, Taiwan

*Corresponding author: Jian-Jhong Lin

Received  March 2018 Revised  April 2019 Published  December 2019

Fund Project: The first author is supported by MOST (Grant No. 107-2115-M-008-009-MY3) and NCTS of Taiwan. The second author is supported by MOST (Grant No. 107-2115-M-027-002) of Taiwan.

In this work, we establish a framework to study the stability of traveling wave solutions for some lattice reaction-diffusion equations. The systems arise from epidemic, biological and many other applied models. Applying different kinds of comparison theorems, we show that all solutions of the Cauchy problem for the lattice differential equations converge exponentially to the traveling wave solutions provided that the initial perturbations around the traveling wave solutions belonging to suitable spaces. Our results can be applied to various discrete reaction-diffusion systems, e.g., the discrete multi-species Lotka-Volterra cooperative model, discrete epidemic model, three-species Lotka-Volterra competitive model, etc.

Citation: Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001
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