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

Previous Article
On the time decay of solutions in porous-thermo-elasticity of type II
DCDS-B Home
This Issue
Next Article
Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay

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

Abstract
Related Papers

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

[1]	Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-13. doi: 10.3934/dcdsb.2019102
[2]	Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567
[3]	Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267
[4]	Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[5]	Wan-Tong Li, Guo Lin, Cong Ma, Fei-Ying Yang. Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 467-484. doi: 10.3934/dcdsb.2014.19.467
[6]	Jibin Li, Yi Zhang. On the traveling wave solutions for a nonlinear diffusion-convection equation: Dynamical system approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1119-1138. doi: 10.3934/dcdsb.2010.14.1119
[7]	Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101
[8]	João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217
[9]	Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014
[10]	Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575
[11]	Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159
[12]	Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006
[13]	Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155
[14]	Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[15]	Zhaosheng Feng, Goong Chen. Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 763-780. doi: 10.3934/dcds.2009.24.763
[16]	Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[17]	Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265
[18]	Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057
[19]	F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239
[20]	Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences