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May  2020, 25(5): 1757-1774. doi: 10.3934/dcdsb.2020001

## Stability analysis of traveling wave solutions for lattice reaction-diffusion equations

 1 Department of Mathematics, National Central University, Zhongli District, Taoyuan City 32001, Taiwan 2 General Education Center, National Taipei University of Technology, Taipei 10608, Taiwan

*Corresponding author: Jian-Jhong Lin

Received  March 2018 Revised  April 2019 Published  May 2020 Early access  December 2019

Fund Project: The first author is supported by MOST (Grant No. 107-2115-M-008-009-MY3) and NCTS of Taiwan. The second author is supported by MOST (Grant No. 107-2115-M-027-002) of Taiwan.

In this work, we establish a framework to study the stability of traveling wave solutions for some lattice reaction-diffusion equations. The systems arise from epidemic, biological and many other applied models. Applying different kinds of comparison theorems, we show that all solutions of the Cauchy problem for the lattice differential equations converge exponentially to the traveling wave solutions provided that the initial perturbations around the traveling wave solutions belonging to suitable spaces. Our results can be applied to various discrete reaction-diffusion systems, e.g., the discrete multi-species Lotka-Volterra cooperative model, discrete epidemic model, three-species Lotka-Volterra competitive model, etc.

Citation: Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001
##### References:
 [1] J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta Metallurgica, 8 (1960), 554-562. [2] J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Applied Math., 59 (1999), 455-493.  doi: 10.1137/S0036139996312703. [3] J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5. [4] X. F. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153. [5] L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. [6] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction-diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I. [7] B.-S. Goh, Stability in models of mutualism, Am. Nat., 113 (1979), 261-275.  doi: 10.1086/283384. [8] J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009. [9] J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373. [10] C.-H. Hsu, J.-J. Lin and T.-H. Yang, Traveling wave solutions for Kolmogorov-type delayed lattice reaction-diffusion systems, IMA J. Appl. Math., 80 (2015), 1336-1367.  doi: 10.1093/imamat/hxu054. [11] C.-H. Hsu, J.-J. Lin and T.-S. Yang, Existence and stability of traveling wave solutions for multilayer cellular neural networks, Zeitschrift fur Angewandte Mathematik und Physik, 66 (2015), 1355-1373.  doi: 10.1007/s00033-014-0480-z. [12] C.-H. Hsu, J.-J. Lin and T.-S. Yang, Traveling wave solutions for delayed lattice reaction-diffusion systems, IMA J. Appl. Math., 80 (2015), 302-323.  doi: 10.1093/imamat/hxt039. [13] C.-H. Hsu, J.-J. Lin and T.-S. Yang, Stability for monostable wave fronts of delayed lattice differential equations, J. Dyn. Diff. Eqns., 29 (2017), 323-342.  doi: 10.1007/s10884-015-9447-9. [14] C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behavior of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139.  doi: 10.1088/0951-7715/26/1/121. [15] W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equations, Commun. Appl. Nonlinear Anal., 1 (1994), 23-46. [16] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038. [17] J. P. Laplante and T. Erneux, Propagation failure in arrays of coupled bistable chemical reactors, J. Phys. Chem., 96 (1992), 4931-4934.  doi: 10.1021/j100191a038. [18] G. Lin, W.-T. Li and M. J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Continuous Dyn. Syst. Ser. B, 13 (2010), 393-414.  doi: 10.3934/dcdsb.2010.13.393. [19] C.-K. Lin, C.-T. Lin, Y.-P. Lin and M. Mei, Exponential stability of non-monotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084.  doi: 10.1137/120904391. [20] S. W. Ma and X. F. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014. [21] M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction diffusion equation. Ⅰ. Local nonlinearity, J. Differential Equations, 247 (2009), 495-510.  doi: 10.1016/j.jde.2008.12.026. [22] M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction diffusion equation. Ⅱ. Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.  doi: 10.1016/j.jde.2008.12.020. [23] W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182. Readings in Mathematics, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9. [24] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speeds of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.  doi: 10.1093/imamat/68.4.409. [25] Z.-X. Yu and M. Mei, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Equations, 260 (2016), 241-267.  doi: 10.1016/j.jde.2015.08.037. [26] K. F. Zhang and X.-Q. Zhao, Spreading speed and travelling waves for a spatially discrete SIS epidemic model, Nonlinearity, 21 (2008), 97-112.  doi: 10.1088/0951-7715/21/1/005. [27] X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canadian Applied Mathematics Quarterly, 4 (1996), 421-444. [28] B. Zinner, Existence of traveling wavefront solution for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27.  doi: 10.1016/0022-0396(92)90142-A. [29] B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.  doi: 10.1006/jdeq.1993.1082.

show all references

##### References:
 [1] J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta Metallurgica, 8 (1960), 554-562. [2] J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Applied Math., 59 (1999), 455-493.  doi: 10.1137/S0036139996312703. [3] J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5. [4] X. F. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153. [5] L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. [6] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction-diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I. [7] B.-S. Goh, Stability in models of mutualism, Am. Nat., 113 (1979), 261-275.  doi: 10.1086/283384. [8] J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations, 252 (2012), 4357-4391.  doi: 10.1016/j.jde.2012.01.009. [9] J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373. [10] C.-H. Hsu, J.-J. Lin and T.-H. Yang, Traveling wave solutions for Kolmogorov-type delayed lattice reaction-diffusion systems, IMA J. Appl. Math., 80 (2015), 1336-1367.  doi: 10.1093/imamat/hxu054. [11] C.-H. Hsu, J.-J. Lin and T.-S. Yang, Existence and stability of traveling wave solutions for multilayer cellular neural networks, Zeitschrift fur Angewandte Mathematik und Physik, 66 (2015), 1355-1373.  doi: 10.1007/s00033-014-0480-z. [12] C.-H. Hsu, J.-J. Lin and T.-S. Yang, Traveling wave solutions for delayed lattice reaction-diffusion systems, IMA J. Appl. Math., 80 (2015), 302-323.  doi: 10.1093/imamat/hxt039. [13] C.-H. Hsu, J.-J. Lin and T.-S. Yang, Stability for monostable wave fronts of delayed lattice differential equations, J. Dyn. Diff. Eqns., 29 (2017), 323-342.  doi: 10.1007/s10884-015-9447-9. [14] C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behavior of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139.  doi: 10.1088/0951-7715/26/1/121. [15] W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equations, Commun. Appl. Nonlinear Anal., 1 (1994), 23-46. [16] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038. [17] J. P. Laplante and T. Erneux, Propagation failure in arrays of coupled bistable chemical reactors, J. Phys. Chem., 96 (1992), 4931-4934.  doi: 10.1021/j100191a038. [18] G. Lin, W.-T. Li and M. J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Continuous Dyn. Syst. Ser. B, 13 (2010), 393-414.  doi: 10.3934/dcdsb.2010.13.393. [19] C.-K. Lin, C.-T. Lin, Y.-P. Lin and M. Mei, Exponential stability of non-monotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084.  doi: 10.1137/120904391. [20] S. W. Ma and X. F. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014. [21] M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction diffusion equation. Ⅰ. Local nonlinearity, J. Differential Equations, 247 (2009), 495-510.  doi: 10.1016/j.jde.2008.12.026. [22] M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction diffusion equation. Ⅱ. Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.  doi: 10.1016/j.jde.2008.12.020. [23] W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182. Readings in Mathematics, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9. [24] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speeds of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.  doi: 10.1093/imamat/68.4.409. [25] Z.-X. Yu and M. Mei, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Equations, 260 (2016), 241-267.  doi: 10.1016/j.jde.2015.08.037. [26] K. F. Zhang and X.-Q. Zhao, Spreading speed and travelling waves for a spatially discrete SIS epidemic model, Nonlinearity, 21 (2008), 97-112.  doi: 10.1088/0951-7715/21/1/005. [27] X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canadian Applied Mathematics Quarterly, 4 (1996), 421-444. [28] B. Zinner, Existence of traveling wavefront solution for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27.  doi: 10.1016/0022-0396(92)90142-A. [29] B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discrete Fisher's equation, J. Differential Equations, 105 (1993), 46-62.  doi: 10.1006/jdeq.1993.1082.
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