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

Cheng-Hsiung Hsu; Jian-Jhong Lin

doi:10.3934/dcdsb.2020001

Article Contents

2020, Volume 25, Issue 5: 1757-1774. Doi: 10.3934/dcdsb.2020001

This issue Previous Article A gradient-type algorithm for constrained optimization with application to microstructure optimization Next Article The two-grid and multigrid discretizations of the

$ C^0 $

IPG method for biharmonic eigenvalue problem

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

Cheng-Hsiung Hsu^1, and
Jian-Jhong Lin^2, ,

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
^*Corresponding author: Jian-Jhong Lin

Received: March 2018

Revised: April 2019

Early access: December 2019

Published: May 2020

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 34A12, 34A33, 34B15; Secondary: 34K10, 34K20.

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