# American Institute of Mathematical Sciences

May  2020, 25(5): 1859-1870. doi: 10.3934/dcdsb.2020006

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

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

* Corresponding author: Tianran Zhang

Received  December 2018 Revised  March 2019 Published  December 2019

Fund Project: 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).

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.

Citation: Tianran Zhang. Traveling waves for a reaction-diffusion model with a cyclic structure. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1859-1870. doi: 10.3934/dcdsb.2020006
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