TY - JOUR
T1 - Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays
AU - Lin, Guo
AU - Ruan, Shigui
N1 - Funding Information:
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).
Publisher Copyright:
© 2014, Springer Science+Business Media New York.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.
PY - 2014/11/4
Y1 - 2014/11/4
N2 - 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.
AB - 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.
KW - Contracting rectangle
KW - Generalized upper and lower solutions
KW - Invariant region
KW - Nonmonotone traveling wave solutions
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U2 - 10.1007/s10884-014-9355-4
DO - 10.1007/s10884-014-9355-4
M3 - Article
AN - SCOPUS:84912049267
VL - 26
SP - 583
EP - 605
JO - Journal of Dynamics and Differential Equations
JF - Journal of Dynamics and Differential Equations
SN - 1040-7294
IS - 3
ER -