Abstract
This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation.
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Bates, P.W., Chmaj, A.: A discrete convolution model for phase transitions. Arch. Ration. Mech. Anal. 150, 281–305 (1999)
Carr, J., Chmaj, A.: Uniqueness of traveling waves for nonlocal monostable equations. Proc. AMS 132, 2433–2439 (2004)
Chen, X., Guo, J.-S.: Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math. Ann. 326, 123146 (2003)
Chow, S.-N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 149, 248–291 (1998)
Chua, L.O.: CNN: A Paradigm for Complexity, World Scientific Series on Nonlinear Science, Series A, vol. 31. World Scientific, Singapore (1998)
Chua, L.O., Yang, L.: Cellular neural networks: theory. IEEE Trans. Circuits Syst. 35, 1257–1272 (1988)
Chua, L.O., Yang, L.: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35, 1273–1290 (1988)
Ellison, W., Ellison, F.: Prime Numbers, A Wiley-Interscience Publication. Wiley, New York (1985)
Erneux, T., Nicolis, G.: Propagation waves in discrete bistable reaction-diffusion systems. Physica D 67, 237–244 (1993)
Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1990)
Hsu, C.-H., Li, C.-H., Yang, S.-Y.: Diversity of traveling wave solutions in delayed cellular neural networks. Int. J. Bifurc. Chaos 18, 3515–3550 (2008)
Hsu, C.-H., Lin, S.-S.: Existence and multiplicity of traveling waves in a lattice dynamical system. J. Differ. Equ. 164, 431–450 (2000)
Hsu, C.-H., Lin, S.-S., Shen, W.: Traveling waves in cellular neural networks. Int. J. Bifurc. Chaos 9, 1307–1319 (1999)
Hsu, C.-H., Yang, S.-Y.: On camel-like traveling wave solutions in cellular neural networks. J. Differ. Equ. 196, 481–514 (2004)
Hudson, H., Zinner, B.: Existence of traveling waves for a generalized discrete Fisher’s equations. Commun. Appl. Nonlinear Anal. 1, 23–46 (1994)
Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572 (1987)
Lin, C.-K., Mei, M.: On travelling wavefronts of Nicholsons blowflies equation with diffusion. Proc. R. Soc. Edinb. 140A, 135152 (2010)
Mallet-Paret, J.: The global structure of traveling waves in spatial discrete dynamical systems. J. Dyn. Differ. Equ. 11, 49–127 (1999)
Ma, S., Zou, X.: Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay. J. Differ. Equ. 217, 54–87 (2005)
Ma, S., Zou, X.: Propagation and its failure in a lattice delayed differential equation with global interaction. J. Differ. Equ. 212, 129–190 (2005)
Mei, M., Ou, C., Zhao, X.-Q.: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations. SIAM J. Math. Anal. 42, 2762–2790 (2010)
Mei, M., Wang, Y.: Remark on stability of traveling waves for nonlocal Fisher-KPP equations. Int. J. Numer. Anal. Model. B 2, 379–401 (2011)
Nicholson, A.J.: Compensatory reactions of populations to stresses, and their evolutionary significance. Aust. J. Zool. 2, 1–8 (1954a)
Nicholson, A.J.: An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–65 (1954b)
Shi, Z.-X., Li, W.-T., Cheng, C.P.: Stability and uniqueness of traveling wavefronts in a two-dimensional lattice differential equation with delay. Appl. Math. Comput. 208, 484494 (2009)
Thieme, H., Zhao, X.: Asymptotic speed of spread and traveling waves for integral equations and delayed reaction-diffusion models. J. Differ. Equ. 195, 430–470 (2003)
Wang, Z.C., Li, W.-T., Ruan, S.: Travelling wave-fronts in reaction-diffusion systems with spatio-temporal delays. J. Differ. Equ. 222, 185–232 (2006)
Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)
Wu, J., Zou, X.: Asymptotical and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations. J. Differ. Equ. 135, 315–357 (1997)
Wu, S.-L., Liu, T.-T.: Exponential stability of traveling fronts for a 2D lattice delayed differential equation with global interaction. Electron. J. Differ. Equ. 179(13), 1–14 (2013)
Yang, Y., So, J.W.-H.: Dynamics for the diffusive Nicholson’s blowflies equation. In: Chen, W., Hu, S. (eds.) Dynamical Systems and Differential Equations, vol. 11, pp. 333–352. Southwest Missouri State University, Springfield (1998)
Zinner, B.: Existence of traveling wavefront solutions for discrete Nagumo equation. J. Differ. Equ. 96, 1–27 (1992)
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Cheng-Hsiung Hsu and Tzi-Sheng Yang have Partially supported by the Ministry of Science and Technology of Taiwan.
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Hsu, CH., Lin, JJ. & Yang, TS. Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations. J Dyn Diff Equat 29, 323–342 (2017). https://doi.org/10.1007/s10884-015-9447-9
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DOI: https://doi.org/10.1007/s10884-015-9447-9