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Article

He, Yanli ; Li, Kun
Traveling wave solutions in a class of higher dimensional lattice differential systems with delays and applications. (English). Applications of Mathematics, vol. 66 (2021), issue 4, pp. 641-656
MSC: 34K10, 37L60, 39A10 | MR 4283307 | Zbl 07396171 | DOI: 10.21136/AM.2021.0159-19
Keywords:
higher dimensional lattice; traveling wave solution; delay; upper and lower solutions
Summary:
In this paper, we are concerned with the existence of traveling waves in a class of delayed higher dimensional lattice differential systems with competitive interactions. Due to the lack of quasimonotonicity for reaction terms, we use the cross iterative and Schauder's fixed-point theorem to prove the existence of traveling wave solutions. We apply our results to delayed higher-dimensional lattice reaction-diffusion competitive system.
References:
[1] Cahn, J. W., Mallet-Paret, J., Vleck, E. S. Van: Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math. 59 (1999), 455-493. DOI 10.1137/S0036139996312703 | MR 1654427 | Zbl 0917.34052
[2] Chen, X., Guo, J.-S.: Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations. J. Differ. Equations 184 (2002), 549-569. DOI 10.1006/jdeq.2001.4153 | MR 1929888 | Zbl 1010.39004
[3] Chen, X., Guo, J.-S.: Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math. Ann. 326 (2003), 123-146. DOI 10.1007/s00208-003-0414-0 | MR 1981615 | Zbl 1086.34011
[4] Cheng, C.-P., Li, W.-T., Wang, Z.-C., Zheng, S.: Traveling waves connecting equilibrium and periodic orbit for a delayed population model on a two-dimensional spatial lattice. Int. J. Bifurcation Chaos Appl. Sci. Eng. 26 (2016), Article ID 1650049, 13 pages. DOI 10.1142/S0218127416500498 | MR 3482808 | Zbl 1336.34106
[5] Chow, S.-N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equations 149 (1998), 248-291. DOI 10.1006/jdeq.1998.3478 | MR 1646240 | Zbl 0911.34050
[6] Guo, J.-S., Wu, C.-H.: Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system. Osaka J. Math. 45 (2008), 327-346. MR 2441943 | Zbl 1155.34016
[7] Guo, J.-S., Wu, C.-H.: Traveling wave front for a two-component lattice dynamical system arising in competition models. J. Differ. Equations 252 (2012), 4357-4391. DOI 10.1016/j.jde.2012.01.009 | MR 2881041 | Zbl 1251.34018
[8] Hankerson, D., Zinner, B.: Wavefronts for a cooperative tridiagonal system of differential equations. J. Dyn. Differ. Equations 5 (1993), 359-373. DOI 10.1007/BF01053165 | MR 1223452 | Zbl 0777.34013
[9] Huang, J., Lu, G., Ruan, S.: Traveling wave solutions in delayed lattice differential equations with partial monotonicity. Nonlinear Anal., Theory Methods Appl., Ser. A 60 (2005), 1331-1350. DOI 10.1016/j.na.2004.10.020 | MR 2112956 | Zbl 1084.34059
[10] Huang, J., Lu, G., Zou, X.: Existence of traveling wave fronts of delayed lattice differential equations. J. Math. Anal. Appl. 298 (2004), 538-558. DOI 10.1016/j.jmaa.2004.05.027 | MR 2086974 | Zbl 1126.34323
[11] Keener, J. P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47 (1987), 556-572. DOI 10.1137/0147038 | MR 0889639 | Zbl 0649.34019
[12] Li, K., Huang, J., Li, X., He, Y.: Traveling wave fronts in a delayed lattice competitive system. Appl. Anal. 97 (2018), 982-999. DOI 10.1080/00036811.2017.1295450 | MR 3777853 | Zbl 1400.34125
[13] Li, K., Li, X.: Traveling wave solutions in a delayed lattice competition-cooperation system. J. Difference Equ. Appl. 24 (2018), 391-408. DOI 10.1080/10236198.2017.1409222 | MR 3757175 | Zbl 1425.37050
[14] Li, W.-T., Lin, G., Ruan, S.: Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems. Nonlinearity 19 (2006), 1253-1273. DOI 10.1088/0951-7715/19/6/003 | MR 2229998 | Zbl 1103.35049
[15] Lin, G., Li, W.-T.: Traveling waves in delayed lattice dynamical systems with competition interactions. Nonlinear Anal., Real World Appl. 11 (2010), 3666-3679. DOI 10.1016/j.nonrwa.2010.01.013 | MR 2683821 | Zbl 1206.34099
[16] Mallet-Paret, J.: The Fredholm alternative for functional-differential equations of mixed type. J. Dyn. Differ. Equations 11 (1999), 1-47. DOI 10.1023/A:1021889401235 | MR 1680463 | Zbl 0927.34049
[17] Mallet-Paret, J.: The global structure of traveling waves in spatially discrete dynamical systems. J. Dyn. Differ. Equations 11 (1999), 49-127. DOI 10.1023/A:1021841618074 | MR 1680459 | Zbl 0921.34046
[18] Weng, P.: Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete Contin. Dyn. Syst., Ser. B 12 (2009), 883-904. DOI 10.3934/dcdsb.2009.12.883 | MR 2552078 | Zbl 1185.34114
[19] Wu, S.-L., Liu, S.-Y.: Travelling waves in delayed reaction-diffusion equations on higher dimensional lattices. J. Difference Equ. Appl. 19 (2013), 384-401. DOI 10.1080/10236198.2011.645815 | MR 3037281 | Zbl 1273.34074
[20] Wu, J., Zou, X.: Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations. J. Differ. Equations 135 (1997), 315-357. DOI 10.1006/jdeq.1996.3232 | MR 1441274 | Zbl 0877.34046
[21] Yu, Z.-X., Yuan, R.: Nonlinear stability of wavefronts for a delayed stage-structured population model on a 2-D lattice. Osaka J. Math. 50 (2013), 963-976. MR 3161423 | Zbl 1287.34061
[22] Yu, Z.-X., Zhang, W., Wang, X.: Spreading speeds and travelling waves for non-monotone time-delayed 2D lattice systems. Math. Comput. Modelling 58 (2013), 1510-1521. DOI 10.1016/j.mcm.2013.06.009 | MR 3143380
[23] Zhao, H.-Q.: Asymptotic stability of traveling fronts in delayed reaction-diffusion monostable equations on higher-dimensional lattices. Electron. J. Differ. Equ. 2013 (2013), Article ID 119, 15 pages. MR 3065072 | Zbl 1288.35095
[24] Zhao, H.-Q., Wu, S.-L.: Wave propagation for a reaction-diffusion model with a quiescent stage on a 2D spatial lattice. Nonlinear Anal., Real World Appl. 12 (2011), 1178-1191. DOI 10.1016/j.nonrwa.2010.09.011 | MR 2736300 | Zbl 1243.34013
[25] Zinner, B.: Stability of traveling wavefronts for the discrete Nagumo equation. SIAM J. Math. Anal. 22 (1991), 1016-1020. DOI 10.1137/0522066 | MR 1112063 | Zbl 0739.34060
[26] Zinner, B.: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Differ. Equations 96 (1992), 1-27. DOI 10.1016/0022-0396(92)90142-A | MR 1153307 | Zbl 0752.34007
[27] Zinner, B., Harris, G., Hudson, W.: Traveling wavefronts for the discrete Fisher's equation. J. Differ. Equations 105 (1993), 46-62. DOI 10.1006/jdeq.1993.1082 | MR 1237977 | Zbl 0778.34006
[28] Zou, X.: Traveling wave fronts in spatially discrete reaction-diffusion equations on higher-dimensional lattices. Electron. J. Differ. Equ. 1997 (1997), 211-222. MR 1672189 | Zbl 0913.34041
[29] Zou, X., Wu, J.: Local existence and stability of periodic traveling waves of lattice functional-differential equations. Can. Appl. Math. Q. 6 (1998), 397-418. MR 1668040 | Zbl 0919.34062
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