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Journal of Dynamics and Differential Equations

, Volume 26, Issue 3, pp 583–605 | Cite as

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

  • Guo LinEmail author
  • Shigui Ruan
Article

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.

Keywords

Nonmonotone traveling wave solutions Contracting rectangle Invariant region Generalized upper and lower solutions 

Mathematics Subject Classification

35C07 37C65 35K57 

Notes

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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of MiamiCoral GablesUSA

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