American Institute of Mathematical Sciences

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March  2010, 13(2): 393-414. doi: 10.3934/dcdsb.2010.13.393

Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models

Guo Lin 1, , Wan-Tong Li 2, and  Mingju Ma 1,

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China
2. 

School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received  December 2007 Revised  November 2008 Published  December 2009

This paper is concerned with the existence of traveling wave solutions in delayed reaction diffusion systems which at least contain multi-species competition, cooperation and predator-prey models with diffusion and delays. By introducing the mixed quasimonotone condition and the exponentially mixed quasimonotone condition, we reduce the existence of traveling wave solutions to the existence of a pair of admissible upper-lower solutions. To illustrate our main results, the existence of traveling wave solutions of multi-species competition, cooperation and predator-prey Lotka-Volterra systems with delays is considered. In particular, we show the precisely asymptotic behavior of the traveling wave solutions of these Lotka-Volterra systems.
Keywords: persistence of traveling wave solution., Delayed diffusion system, fixed point theorem, upper-lower solutions, mixed quasimonotonicity.
Mathematics Subject Classification: Primary: 35K57, 92D25; Secondary: 34C37, 34K1.
Citation: Guo Lin, Wan-Tong Li, Mingju Ma. Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 393-414. doi: 10.3934/dcdsb.2010.13.393
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