## Taiwanese Journal of Mathematics

### Stability of Traveling Wavefronts for a Delayed Lattice System with Nonlocal Interaction

#### Abstract

In this paper, we mainly investigate exponential stability of traveling wavefronts for delayed $2D$ lattice differential equation with nonlocal interaction. For all non-critical traveling wavefronts with the wave speed $c \gt c_*(\theta)$, where $c_*(\theta) \gt 0$ is the critical wave speed and $\theta$ is the direction of propagation, we prove that these traveling waves are asymptotically stable, when the initial perturbation around the traveling waves decay exponentially at far fields, but can be allowed arbitrarily large in other locations. Our approach adopted in this paper is the weighted energy method and the squeezing technique with the help of Gronwall's inequality. Furthermore, from stability result, we prove the uniqueness (up to shift) of the traveling wavefront. Our results can apply to the discrete diffusive Mackey-Glass model and the dicrete diffusive Nicholson's blowflies model on $2D$ lattices.

#### Article information

Source
Taiwanese J. Math., Volume 21, Number 5 (2017), 997-1015.

Dates
Revised: 15 January 2017
Accepted: 15 January 2017
First available in Project Euclid: 1 August 2017

https://projecteuclid.org/euclid.twjm/1501599180

Digital Object Identifier
doi:10.11650/tjm/7964

Mathematical Reviews number (MathSciNet)
MR3707881

Zentralblatt MATH identifier
06871356

#### Citation

Pei, Jingwen; Yu, Zhixian; Zhou, Huiling. Stability of Traveling Wavefronts for a Delayed Lattice System with Nonlocal Interaction. Taiwanese J. Math. 21 (2017), no. 5, 997--1015. doi:10.11650/tjm/7964. https://projecteuclid.org/euclid.twjm/1501599180

#### References

• J. W. Cahn, J. Mallet-Parat and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math. 59 (1999), no. 2, 455–493.
• C.-P. Cheng, W.-T. Li and Z.-C. Wang, Spreading speeds and travelling waves in a delayed population model with stage structure on a $2D$ spatial lattice, IMA J. Appl. Math. 73 (2008), no. 4, 592–618.
• ––––, 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), no. 3, 559–575.
• I.-L. Chern, M. Mei, X. Yang and Q. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differential Equations 259 (2015), no. 4, 1503–1541.
• S. A. Gourley, J. W.-H. So and J. H. Wu, Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math. Sci. 124 (2004), no. 4, 5119–5153.
• S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations, 137–200, Fields Inst. Commun. 48, Amer. Math. Soc., Providence, RI, 2006.
• J.-S. Guo and C.-H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math. 45 (2008), no. 2, 327–346.
• C.-K. Lin, C.-T. Lin, Y. Lin and M. Mei, Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation, SIAM. J. Math. Anal. 46 (2014), no. 2, 1053–1084.
• C.-K. Lin and M. Mei, On travelling wavefronts of Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 1, 135–152.
• G. Lv and M. Wang, Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations, Nonlinearity 23 (2010), no. 4, 845–873.
• ––––, Nonlinear stability of traveling wave fronts for delayed reaction diffusion, Nonlinear Anal. Real World Appl. 13 (2012), no. 4, 1854–1865.
• S. Ma, P. Weng and X. Zou, Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation, Nonlinear Anal. 65 (2006), no. 10, 1858–1890.
• 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), no. 2, 495–510.
• ––––, Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity, J. Differential Equations 247 (2009), no. 2, 511–529.
• M. Mei, C. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal. 42 (2010), no. 6, 2762–2790.
• M. Mei and J. W.-H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), no. 3, 551–568.
• M. Mei, J. W.-H. So, M. Y. Li and S. S. P. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 3, 579–594.
• M. Mei and Y. Wong, Novel stability results for traveling wavefronts in an age-structured reaction-diffusion equation, Math. Biosci. Eng. 6 (2009), no. 4, 743–752.
• P. Weng, H. 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), no. 4, 409–439.
• P. Weng, J. Wu, H. Huang and J. Ling, Asymptotic speed of propagation of wave fronts in a $2D$ lattice delay differential equation with global interaction, Can. Appl. Math. Q. 11 (2003), no. 4, 377–414.
• S.-L. Wu, W.-T. Li and S.-Y. Liu, Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay, J. Math. Anal. Appl. 360 (2009), no. 2, 439–458.
• ––––, Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 1, 347–366.
• S.-L. Wu and T.-T. Liu, Exponential stability of traveling fronts for a $2D$ lattice delayed differential equation with global interaction, Electron. J. Differential Equations 2013 (2013), no. 179, 13 pp.
• Z.-X. Yu and M. Mei, Asymptotics and uniqueness of travelling waves for non-monotone delayed systems on $2D$ lattices, Canad. Math. Bull. 56 (2013), no. 3, 659–672.
• ––––, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Equations 260 (2016), no. 1, 241–267.
• Z.-X. Yu and R. Yuan, Nonlinear stability of wavefronts for a delayed stage-structured population model on a 2-D lattice, Osaka J. Math. 50 (2013), no. 4, 963–976.
• Z.-X. Yu, W. Zhang and X. Wang, Spreading speeds and travelling waves for non-monotone time-delayed $2D$ lattice systems, Math. Comput. Modelling 58 (2013), no. 7-8, 1510–1521.