# American Institute of Mathematical Sciences

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September 2018, 38(9): 4329-4351. doi: 10.3934/dcds.2018189

## Traveling wave solutions for time periodic reaction-diffusion systems

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China 2 Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA

* Corresponding author: Guo Lin

Received  July 2017 Revised  April 2018 Published  June 2018

This paper deals with traveling wave solutions for time periodic reaction-diffusion systems. The existence of traveling wave solutions is established by combining the fixed point theorem with super- and sub-solutions, which reduces the existence of traveling wave solutions to the existence of super- and sub-solutions. The asymptotic behavior is determined by the stability of periodic solutions of the corresponding initial value problems. To illustrate the abstract results, we investigate a time periodic Lotka-Volterra system with two species by presenting the existence and nonexistence of traveling wave solutions, which connect the trivial steady state to the unique positive periodic solution of the corresponding kinetic system.

Citation: Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
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