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doi: 10.3934/era.2021051
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Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction

1. 

School of Science, Shanghai Institute of Technology, Shanghai 201418, China

2. 

School of Mathematics, University of Mining and Technology, Xuzhou, Jiangsu 221008, China

* Corresponding author: Cui-Ping Cheng

Received  February 2021 Revised  May 2021 Early access July 2021

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. Electronic Research Archive, doi: 10.3934/era.2021051
References:
[1]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Rational. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  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. Appl. Math., 59 (1999), 455-493.  doi: 10.1137/S0036139996312703.  Google Scholar

[3]

M. Chae and K. Choi, Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion, J. Differential Equations, 268 (2020), 3449-3496.  doi: 10.1016/j.jde.2019.09.061.  Google Scholar

[4]

X. Chen, Existence uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.   Google Scholar

[5]

F. Chen, Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity, Discrete Contin. Dyn. Syst., 24 (2009), 659-673.  doi: 10.3934/dcds.2009.24.659.  Google Scholar

[6]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.  Google Scholar

[7]

C.-P. ChengW.-T. Li and Z.-C. Wang, Spreading speeds and traveling waves for a delayed population model with stage structure on a two-dimensional spatial lattice, IMA J. Appl. Math., 73 (2008), 592-618.  doi: 10.1093/imamat/hxn003.  Google Scholar

[8]

C.-P. ChengW.-T. Li and Z.-C. Wang, 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, 13 (2010), 559-575.  doi: 10.3934/dcdsb.2010.13.559.  Google Scholar

[9]

S.-N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.  Google Scholar

[10]

R. R. Goldberg, Fourier Transform, New York: Cambridge University Press, 1961.  Google Scholar

[11]

J.-S. GuoK.-I. NakamuraT. Ogiwara and C.-C. Wu, Stability and uniqueness of traveling waves for a discrete bistable 3-species competition system, J. Math. Anal. Appl., 472 (2019), 1534-1550.  doi: 10.1016/j.jmaa.2018.12.007.  Google Scholar

[12]

X. Hou and Y. Li, Local stability of traveling wave solutions of nonlinear reaction-diffusion equations, Discrete Contin. Dyn. Syst., 15 (2006), 681-701.  doi: 10.3934/dcds.2006.15.681.  Google Scholar

[13]

C.-H. Hsu and J.-J. Lin, Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays, J. Math. Anal. Appl., 470 (2019), 388-400.  doi: 10.1016/j.jmaa.2018.10.010.  Google Scholar

[14]

C.-H. HsuJ.-J. Lin and T.-S. Yang, Stability for monostable wave fronts of delayed lattice differential equations, J. Dynam. Differential Equations, 29 (2017), 323-342.  doi: 10.1007/s10884-015-9447-9.  Google Scholar

[15]

R. HuangM. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst., 32 (2012), 3621-3649.  doi: 10.3934/dcds.2012.32.3621.  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.   Google Scholar

[17]

G. Lin and S.. Ruan, Persistence and failure of complete spreading in delayed reaction-diffusion equations, Proc. Amer. Math. Soc., 144 (2016), 1059-1072.  doi: 10.1090/proc/12811.  Google Scholar

[18]

G. Lv and M. Wang, Nonlinear stability of traveling wave fronts for nonlocal delayed reaction-diffusion equations, J. Math. Anal. Appl., 385 (2012), 1094-1106.  doi: 10.1016/j.jmaa.2011.07.033.  Google Scholar

[19]

S. Ma and Y. Duan, Asymptotic stability of traveling waves in a discrete convolution model for phase transitions, J. Math. Anal. Appl., 308 (2005), 240-256.  doi: 10.1016/j.jmaa.2005.01.011.  Google Scholar

[20]

S. Ma and X. Q. Zhao, Global asumptotic stability of minimal fronts in monostable lattice equations, Discrete Cont. Dyn. Systems, 21 (2008), 259-275.  doi: 10.3934/dcds.2008.21.259.  Google Scholar

[21]

S. Ma and X. Zou, Existence, uniqueness and stability of traveling wavefronts in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004.  Google Scholar

[22]

S. Ma and X. 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

[23]

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

[24]

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

[25]

M. MeiC. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790.  doi: 10.1137/090776342.  Google Scholar

[26]

M. Mei and J. W.-H. So, Stability of strong traveling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568.  doi: 10.1017/S0308210506000333.  Google Scholar

[27]

M. MeiJ. W.-H. SoM. Y. Li and S. S. P. Shen, Asymptotic stability of traveling wavefronts for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594.  doi: 10.1017/S0308210500003358.  Google Scholar

[28]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Numer. Anal. Model. Ser. B, 2 (2011), 379-401.   Google Scholar

[29]

J. D. Murray, Mathematcial Biology: AN introduction, Third edition., in: Interdisciplinary Applied Mathematics, vol. 17, Springer, New York, 2002.  Google Scholar

[30]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[31]

H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.  Google Scholar

[32]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.  Google Scholar

[33]

Z.-C. WangW.-T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.  Google Scholar

[34]

P. WengH. Huang and J. Wu, Asymptotic speed 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

[35]

Z. Xu, Wave propagation in a two-dimensional lattice dynamical system with global interaction, J. Differential Equations, 269 (2020), 4477-4502.  doi: 10.1016/j.jde.2020.03.041.  Google Scholar

[36]

D. Ya. KhusainovA. F. Ivanov and I. V. Kovarzh, Solution of one heat equation with delay, Nonlinear Oscil., 12 (2009), 260-282.  doi: 10.1007/s11072-009-0075-3.  Google Scholar

[37]

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

[38]

G.-B. Zhang, Global stability of traveling wave fronts for non-local delayed lattice differential equations, Nonlinear Anal. Real World Appl., 13 (2012), 1790-1801.  doi: 10.1016/j.nonrwa.2011.12.010.  Google Scholar

[39]

G.-B. Zhang, Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay, J. Math. Anal. Appl., 475 (2019), 605-627.  doi: 10.1016/j.jmaa.2019.02.058.  Google Scholar

show all references

References:
[1]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Rational. Mech. Anal., 150 (1999), 281-305.  doi: 10.1007/s002050050189.  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. Appl. Math., 59 (1999), 455-493.  doi: 10.1137/S0036139996312703.  Google Scholar

[3]

M. Chae and K. Choi, Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion, J. Differential Equations, 268 (2020), 3449-3496.  doi: 10.1016/j.jde.2019.09.061.  Google Scholar

[4]

X. Chen, Existence uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.   Google Scholar

[5]

F. Chen, Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity, Discrete Contin. Dyn. Syst., 24 (2009), 659-673.  doi: 10.3934/dcds.2009.24.659.  Google Scholar

[6]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.  Google Scholar

[7]

C.-P. ChengW.-T. Li and Z.-C. Wang, Spreading speeds and traveling waves for a delayed population model with stage structure on a two-dimensional spatial lattice, IMA J. Appl. Math., 73 (2008), 592-618.  doi: 10.1093/imamat/hxn003.  Google Scholar

[8]

C.-P. ChengW.-T. Li and Z.-C. Wang, 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, 13 (2010), 559-575.  doi: 10.3934/dcdsb.2010.13.559.  Google Scholar

[9]

S.-N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.  Google Scholar

[10]

R. R. Goldberg, Fourier Transform, New York: Cambridge University Press, 1961.  Google Scholar

[11]

J.-S. GuoK.-I. NakamuraT. Ogiwara and C.-C. Wu, Stability and uniqueness of traveling waves for a discrete bistable 3-species competition system, J. Math. Anal. Appl., 472 (2019), 1534-1550.  doi: 10.1016/j.jmaa.2018.12.007.  Google Scholar

[12]

X. Hou and Y. Li, Local stability of traveling wave solutions of nonlinear reaction-diffusion equations, Discrete Contin. Dyn. Syst., 15 (2006), 681-701.  doi: 10.3934/dcds.2006.15.681.  Google Scholar

[13]

C.-H. Hsu and J.-J. Lin, Stability of traveling wave solutions for nonlinear cellular neural networks with distributed delays, J. Math. Anal. Appl., 470 (2019), 388-400.  doi: 10.1016/j.jmaa.2018.10.010.  Google Scholar

[14]

C.-H. HsuJ.-J. Lin and T.-S. Yang, Stability for monostable wave fronts of delayed lattice differential equations, J. Dynam. Differential Equations, 29 (2017), 323-342.  doi: 10.1007/s10884-015-9447-9.  Google Scholar

[15]

R. HuangM. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst., 32 (2012), 3621-3649.  doi: 10.3934/dcds.2012.32.3621.  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.   Google Scholar

[17]

G. Lin and S.. Ruan, Persistence and failure of complete spreading in delayed reaction-diffusion equations, Proc. Amer. Math. Soc., 144 (2016), 1059-1072.  doi: 10.1090/proc/12811.  Google Scholar

[18]

G. Lv and M. Wang, Nonlinear stability of traveling wave fronts for nonlocal delayed reaction-diffusion equations, J. Math. Anal. Appl., 385 (2012), 1094-1106.  doi: 10.1016/j.jmaa.2011.07.033.  Google Scholar

[19]

S. Ma and Y. Duan, Asymptotic stability of traveling waves in a discrete convolution model for phase transitions, J. Math. Anal. Appl., 308 (2005), 240-256.  doi: 10.1016/j.jmaa.2005.01.011.  Google Scholar

[20]

S. Ma and X. Q. Zhao, Global asumptotic stability of minimal fronts in monostable lattice equations, Discrete Cont. Dyn. Systems, 21 (2008), 259-275.  doi: 10.3934/dcds.2008.21.259.  Google Scholar

[21]

S. Ma and X. Zou, Existence, uniqueness and stability of traveling wavefronts in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004.  Google Scholar

[22]

S. Ma and X. 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

[23]

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

[24]

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

[25]

M. MeiC. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790.  doi: 10.1137/090776342.  Google Scholar

[26]

M. Mei and J. W.-H. So, Stability of strong traveling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568.  doi: 10.1017/S0308210506000333.  Google Scholar

[27]

M. MeiJ. W.-H. SoM. Y. Li and S. S. P. Shen, Asymptotic stability of traveling wavefronts for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594.  doi: 10.1017/S0308210500003358.  Google Scholar

[28]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Numer. Anal. Model. Ser. B, 2 (2011), 379-401.   Google Scholar

[29]

J. D. Murray, Mathematcial Biology: AN introduction, Third edition., in: Interdisciplinary Applied Mathematics, vol. 17, Springer, New York, 2002.  Google Scholar

[30]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[31]

H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.  Google Scholar

[32]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.  Google Scholar

[33]

Z.-C. WangW.-T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.  Google Scholar

[34]

P. WengH. Huang and J. Wu, Asymptotic speed 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

[35]

Z. Xu, Wave propagation in a two-dimensional lattice dynamical system with global interaction, J. Differential Equations, 269 (2020), 4477-4502.  doi: 10.1016/j.jde.2020.03.041.  Google Scholar

[36]

D. Ya. KhusainovA. F. Ivanov and I. V. Kovarzh, Solution of one heat equation with delay, Nonlinear Oscil., 12 (2009), 260-282.  doi: 10.1007/s11072-009-0075-3.  Google Scholar

[37]

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

[38]

G.-B. Zhang, Global stability of traveling wave fronts for non-local delayed lattice differential equations, Nonlinear Anal. Real World Appl., 13 (2012), 1790-1801.  doi: 10.1016/j.nonrwa.2011.12.010.  Google Scholar

[39]

G.-B. Zhang, Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay, J. Math. Anal. Appl., 475 (2019), 605-627.  doi: 10.1016/j.jmaa.2019.02.058.  Google Scholar

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