Traveling waves for a reaction-diffusion model with a cyclic structure

Tianran Zhang

doi:10.3934/dcdsb.2020006

Article Contents

2020, Volume 25, Issue 5: 1859-1870. Doi: 10.3934/dcdsb.2020006

This issue Previous Article On the limit cycles of a class of discontinuous piecewise linear differential systems Next Article Transition between monostability and bistability of a genetic toggle switch in Escherichia coli

Traveling waves for a reaction-diffusion model with a cyclic structure

Tianran Zhang^,

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

^* Corresponding author: Tianran Zhang
^* Corresponding author: Tianran Zhang

Received: December 2018

Revised: March 2019

Early access: December 2019

Published: May 2020

The author is supported by the National Natural Science Foundation of China (11871403), the Natural Science Foundation of Chongqing (cstc2018jcyjAX0439), and the Fundamental Research Funds for the Central Universities (XDJK2018C072)

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Abstract

In this paper, a reaction-diffusion model with a cyclic structure is studied, which includes the SIS disease-transmission model and the nutrient-phytoplankton model. The minimal wave speed $ c^* $ of traveling wave solutions is given. The existence of traveling semi-fronts with $ c>c^* $ is proved by Schauder's fixed-point theorem. The traveling semi-fronts are shown to be bounded by rescaling method and comparison principle. The existence of traveling semi-front with $ c = c^* $ is obtained by limit arguments. Finally, the traveling semi-fronts are shown to connect to the positive equilibrium by a Lyapunov function.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35C07, 92B05.

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