# American Institute of Mathematical Sciences

May  2016, 21(3): 815-836. doi: 10.3934/dcdsb.2016.21.815

## On existence of wavefront solutions in mixed monotone reaction-diffusion systems

 1 Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403 2 Department of Mathematics, Computer Science, and Statistics, Purdue University Calumet, Hammond, IN 46323, United States 3 Department of Mathematics and Statistics, University of North Carolina in Wilmington, Wilmington, NC 28403

Received  July 2015 Revised  September 2015 Published  January 2016

In this article, we give an existence-comparison theorem for wavefront solutions in a general class of reaction-diffusion systems. With mixed quasi-monotonicity and Lipschitz condition on the set bounded by coupled upper-lower solutions, the existence of wavefront solution is proven by applying the Schauder Fixed Point Theorem on a compact invariant set. Our main result is then applied to well-known examples: a ratio-dependent predator-prey model, a three-species food chain model of Lotka-Volterra type and a three-species competition model of Lotka-Volterra type. For each model, we establish conditions on the ecological parameters for the presence of wavefront solutions flowing towards the coexistent states through suitably constructed upper and lower solutions. Numerical simulations on those models are also demonstrated to illustrate our theoretical results.
Citation: Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815
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