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

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.

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

“…However, to the best of our knowledge, the stability of traveling wave solutions for multi-component discrete reaction diffusion systems is less reported. Recently, by comparison principles, Hsu and Lin [18] established a framework to study the stability of traveling wave solutions of the general system (1.8). Unfortunately, due to different type of diffusion terms, their results can not be applied to system (1.3).…”

“…Unfortunately, due to different type of diffusion terms, their results can not be applied to system (1.3). Motivated by these articles [17,18,22,29], we will prove the stability of traveling wave fronts for the 2-component discrete system (1.3) by establishing the L 1 w 1 , L 1 and L 2 -energy estimates for the perturbation system (see Theorem 2.2 and Section 4). Moreover, following the same proof arguments of the main theorem, we can extend the stability result to more general discrete diffusive system.…”