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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  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 & 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.   Google Scholar

[2]

J. W. CahnJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156.   Google Scholar

[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.  Google Scholar

[7]

B.-S. Goh, Stability in models of mutualism, Am. Nat., 113 (1979), 261-275.  doi: 10.1086/283384.  Google Scholar

[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.  Google Scholar

[9]

J.-S. GuoY. WangC.-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.  Google Scholar

[10]

C.-H. HsuJ.-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.  Google Scholar

[11]

C.-H. HsuJ.-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.  Google Scholar

[12]

C.-H. HsuJ.-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.  Google Scholar

[13]

C.-H. HsuJ.-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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. LinW.-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.  Google Scholar

[19]

C.-K. LinC.-T. LinY.-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.  Google Scholar

[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.  Google Scholar

[21]

M. MeiC.-K. LinC.-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.  Google Scholar

[22]

M. MeiC.-K. LinC.-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.  Google Scholar

[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.  Google Scholar

[24]

P. X. WengH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

B. ZinnerG. 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.  Google Scholar

show all references

References:
[1]

J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta Metallurgica, 8 (1960), 554-562.   Google Scholar

[2]

J. W. CahnJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156.   Google Scholar

[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.  Google Scholar

[7]

B.-S. Goh, Stability in models of mutualism, Am. Nat., 113 (1979), 261-275.  doi: 10.1086/283384.  Google Scholar

[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.  Google Scholar

[9]

J.-S. GuoY. WangC.-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.  Google Scholar

[10]

C.-H. HsuJ.-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.  Google Scholar

[11]

C.-H. HsuJ.-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.  Google Scholar

[12]

C.-H. HsuJ.-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.  Google Scholar

[13]

C.-H. HsuJ.-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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. LinW.-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.  Google Scholar

[19]

C.-K. LinC.-T. LinY.-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.  Google Scholar

[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.  Google Scholar

[21]

M. MeiC.-K. LinC.-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.  Google Scholar

[22]

M. MeiC.-K. LinC.-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.  Google Scholar

[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.  Google Scholar

[24]

P. X. WengH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

B. ZinnerG. 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.  Google Scholar

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