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Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

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Abstract

This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.

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Acknowledgments

The authors would like to thank an anonymous reviewer for his/her helpful comments and Yanli Huang for her valuable suggestions. This research was partially supported by the the National Natural Science Foundation of China (11101194) and the National Science Foundation (DMS-1022728).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Guo Lin

  2. Department of Mathematics, University of Miami, P. O. Box 249085, Coral Gables, FL, 33124-4250, USA

    Shigui Ruan

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  1. Guo Lin
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  2. Shigui Ruan
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Correspondence to Guo Lin.

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Lin, G., Ruan, S. Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays. J Dyn Diff Equat 26, 583–605 (2014). https://doi.org/10.1007/s10884-014-9355-4

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  • DOI: https://doi.org/10.1007/s10884-014-9355-4

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