Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction

  • Received: 01 February 2021 Revised: 01 May 2021 Published: 22 July 2021
  • 35B40, 35K57, 35R20

  • This paper is concerned with the traveling wave fronts for a lattice dynamical system with global interaction, which arises in a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. We prove that all non-critical traveling wave fronts are globally exponentially stable in time, and the critical traveling wave fronts are globally algebraically stable by the weighted energy method combined with the comparison principle and the discrete Fourier transform.

    Citation: Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction[J]. Electronic Research Archive, 2021, 29(5): 3535-3550. doi: 10.3934/era.2021051

    Related Papers:

  • This paper is concerned with the traveling wave fronts for a lattice dynamical system with global interaction, which arises in a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. We prove that all non-critical traveling wave fronts are globally exponentially stable in time, and the critical traveling wave fronts are globally algebraically stable by the weighted energy method combined with the comparison principle and the discrete Fourier transform.



    加载中


    [1] A discrete convolution model for phase transitions. Arch. Rational. Mech. Anal. (1999) 150: 281-305.
    [2] Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math. (1999) 59: 455-493.
    [3] Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion. J. Differential Equations (2020) 268: 3449-3496.
    [4] Existence uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations. Adv. Differential Equations (1997) 2: 125-160.
    [5] Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete Contin. Dyn. Syst. (2009) 24: 659-673.
    [6] Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations. J. Differential Equations (2002) 184: 549-569.
    [7] Spreading speeds and traveling waves for a delayed population model with stage structure on a two-dimensional spatial lattice. IMA J. Appl. Math. (2008) 73: 592-618.
    [8] Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete Contin. Dyn. Syst. Ser. B (2010) 13: 559-575.
    [9] Traveling waves in lattice dynamical systems. J. Differential Equations (1998) 149: 248-291.
    [10] R. R. Goldberg, Fourier Transform, New York: Cambridge University Press, 1961.
    [11] Stability and uniqueness of traveling waves for a discrete bistable 3-species competition system. J. Math. Anal. Appl. (2019) 472: 1534-1550.
    [12] Local stability of traveling wave solutions of nonlinear reaction-diffusion equations. Discrete Contin. Dyn. Syst. (2006) 15: 681-701.
    [13] Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays. J. Math. Anal. Appl. (2019) 470: 388-400.
    [14] Stability for monostable wave fronts of delayed lattice differential equations. J. Dynam. Differential Equations (2017) 29: 323-342.
    [15] Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete Contin. Dyn. Syst. (2012) 32: 3621-3649.
    [16] Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. (1987) 47: 556-572.
    [17] Persistence and failure of complete spreading in delayed reaction-diffusion equations. Proc. Amer. Math. Soc. (2016) 144: 1059-1072.
    [18] Nonlinear stability of traveling wave fronts for nonlocal delayed reaction-diffusion equations. J. Math. Anal. Appl. (2012) 385: 1094-1106.
    [19] Asymptotic stability of traveling waves in a discrete convolution model for phase transitions. J. Math. Anal. Appl. (2005) 308: 240-256.
    [20] Global asumptotic stability of minimal fronts in monostable lattice equations. Discrete Cont. Dyn. Systems (2008) 21: 259-275.
    [21] Existence, uniqueness and stability of traveling wavefronts in a discrete reaction-diffusion monostable equation with delay. J. Differential Equations (2005) 217: 54-87.
    [22] Propagation and its failure in a lattice delayed differential equation with global interaction. J. Differential Equations (2005) 212: 129-190.
    [23] Traveling wavefronts for time-delayed reaction-diffusion equation: (I) Local nonlinearity. J. Differential Equations (2009) 247: 495-510.
    [24] Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity. J. Differential Equations (2009) 247: 511-529.
    [25] Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations. SIAM J. Math. Anal. (2010) 42: 2762-2790.
    [26] Stability of strong traveling waves for a non-local time-delayed reaction-diffusion equation. Proc. Roy. Soc. Edinburgh Sect. A (2008) 138: 551-568.
    [27] Asymptotic stability of traveling wavefronts for the Nicholson's blowflies equation with diffusion. Proc. Roy. Soc. Edinburgh Sect. A (2004) 134: 579-594.
    [28] Remark on stability of traveling waves for nonlocal Fisher-KPP equations. Int. J. Numer. Anal. Model. Ser. B (2011) 2: 379-401.
    [29] J. D. Murray, Mathematcial Biology: AN introduction, Third edition., in: Interdisciplinary Applied Mathematics, vol. 17, Springer, New York, 2002.
    [30] Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans. Amer. Math. Soc. (1987) 302: 587-615.
    [31] Global asymptotic stability of traveling waves in delayed reaction-diffusion equations. SIAM J. Math. Anal. (2000) 31: 514-534.
    [32] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.
    [33] Entire solutions in delayed lattice differential equations with monostable nonlinearity. SIAM J. Math. Anal. (2009) 40: 2392-2420.
    [34] Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J. Appl. Math. (2003) 68: 409-439.
    [35] Wave propagation in a two-dimensional lattice dynamical system with global interaction. J. Differential Equations (2020) 269: 4477-4502.
    [36] Solution of one heat equation with delay. Nonlinear Oscil. (2009) 12: 260-282.
    [37] Uniqueness and stability of traveling waves for cellular neural networks with multiple delays. J. Differential Equations (2016) 260: 241-267.
    [38] Global stability of traveling wave fronts for non-local delayed lattice differential equations. Nonlinear Anal. Real World Appl. (2012) 13: 1790-1801.
    [39] Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay. J. Math. Anal. Appl. (2019) 475: 605-627.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(548) PDF downloads(106) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog