Geodesic
Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models
Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 423-437

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we study a two-component Lotka–Volterra competition systemon a one-dimensional spatial lattice. By the comparison principle, together with the weighted energy, we prove that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as $j\,+\,ct\,\to \,-\,\infty $ , where $j\,\in \,\mathbb{Z}$ , $t\,>\,0$ , but the initial perturbation can be arbitrarily large on other locations. This partially answers an open problem by J.-S. Guo and C.-H.Wu.
DOI : 10.4153/CMB-2017-018-5
Mots-clés : 34A33, 34K20, 92D25, lattice dynamical system, competition model, traveling wavefront, stability 
Zhang, Guo-Bao; Tian, Ge. Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models. Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 423-437. doi: 10.4153/CMB-2017-018-5
@article{10_4153_CMB_2017_018_5,
     author = {Zhang, Guo-Bao and Tian, Ge},
     title = {Stability of {Traveling} {Wavefronts} for a {Two-Component} {Lattice} {Dynamical} {System} {Arising} in {Competition} {Models}},
     journal = {Canadian mathematical bulletin},
     pages = {423--437},
     year = {2018},
     volume = {61},
     number = {2},
     doi = {10.4153/CMB-2017-018-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-018-5/}
}
TY  - JOUR
AU  - Zhang, Guo-Bao
AU  - Tian, Ge
TI  - Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models
JO  - Canadian mathematical bulletin
PY  - 2018
SP  - 423
EP  - 437
VL  - 61
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-018-5/
DO  - 10.4153/CMB-2017-018-5
ID  - 10_4153_CMB_2017_018_5
ER  - 
%0 Journal Article
%A Zhang, Guo-Bao
%A Tian, Ge
%T Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models
%J Canadian mathematical bulletin
%D 2018
%P 423-437
%V 61
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-018-5/
%R 10.4153/CMB-2017-018-5
%F 10_4153_CMB_2017_018_5

[1] [1] Cheng, C., Li, W.-T., and Wang, Z.-C., Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice. IMA J. Appl. Math. 73 (2008), 592–618. http://dx.doi.org/10.1093/imamat/hxn003 Google Scholar

[2] [2] Cheng, C., Li, W.-T., and Wang, Z.-C., 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. http://dx.doi.Org/10.3934/dcdsb.2O10.13.559 Google Scholar

[3] [3] Guo, J.-S. and Liang, X., The minimal speed of traveling fronts for the Lotka-Volterra competition System. J. Dynam. Differential Equations 23 (2011), 353–363. http://dx.doi.Org/10.1007/s10884-011-9214-5 Google Scholar

[4] [4] Guo, J.-S. and Wu, C.-H., Wave propagation for a two-component lattice dynamical System arising in strong competition modeis. J. Differential Equations 250 (2011), 3504–3533. http://dx.doi.Org/10.1016/j.jde.2O10.12.004 Google Scholar

[5] [5] Guo, J.-S. and Wu, C.-H., Traveling wave front for a two-component lattice dynamical System arising in competition modeis. J. Differential Equations 252 (2012), 4357–4391. http://dx.doi.Org/10.1016/j.jde.2012.01.009 Google Scholar

[6] [6] Guo, J.-S. and Wu, C.-H., Recent developments on wave propagation in 2-species competition Systems. Discrete Contin. Dyn. Syst. Ser. B. 17 (2012), 2713–2724. http://dx.doi.org/10.3934/dcdsb.2012.17.2713 Google Scholar

[7] [7] Huang, R., Mei, M., and Wang, Y., Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete Contin. Dyn. Syst. 32 (2012), 3621–3649. http://dx.doi.org/10.3934/dcds.2012.32.3621 Google Scholar

[8] [8] Huang, R., Mei, M., Zhang, K.-J., and Zhang, Q.-F., Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete Contin. Dyn. Syst. 36 (2016), 1331–1353. http://dx.doi.org/10.3934/dcds.2016.36.1331 Google Scholar

[9] [9] Kan-on, Y. and Fang, Q., Stability of monotone traveling waves for competition-diffusion equations. Japan J. Indust. Appl. Math. 13 (1996), 343–349. http://dx.doi.org/10.1007/BF031 67252 Google Scholar

[10] [10] Li, W.-T., Zhang, L., and Zhang, G.-B., Invasion entire Solutions in a competition System with nonlocal dispersal. Discrete Contin. Dyn. Syst. 35 (2015), 1531–1560. http://dx.doi.org/10.3934/dcds.2015.35.1531 Google Scholar

[11] [11] Lin, C.-K., Lin, C.-T., Lin, Y.-P., and Mei, M., Exponential stability oj nonmonotone traveling waves for Nicholson's bbwflies equation. SIAM J. Math. Anal. 46 (2014), 1053–1084. http://dx.doi.Org/10.1137/120904391 Google Scholar

[12] [12] Lv, G. and Wang, M.-X., Nonlinear stability oj traveling wave fronts for nonlocal delayed reaction-diffusion equations. J. Math. Anal. Appl. 385 (2012), 1094–1106. http://dx.doi.Org/10.1016/j.jmaa.2O11.07.033 Google Scholar

[13] [13] Lv, G. and Wang, X.-H., Stability ojtraveling wave fronts for nonlocal delayed reaction diffusion Systems. J. Anal. Appl. 33 (2014), 463–480. http://dx.doi.Org/10.4171/ZAA/1 523 Google Scholar

[14] [14] Ma, S., and Duan, Y.-R., Asymptotic stability of traveling waves in a discrete convolution modelfor phase transitions. J. Math. Anal. Appl. 308 (2005), 240–256. http://dx.doi.Org/10.1016/j.jmaa.2005.01.011 Google Scholar

[15] [15] Ma, S. and Zhao, X.-Q., Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete Contin. Dyn. Syst. 21 (2008), 259–275. http://dx.doi.org/10.3934/dcds.2008.21.259 Google Scholar

[16] [16] Ma, S., and Zou, X., Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay. J. Differential Equations 217 (2005), 54–87. http://dx.doi.Org/10.1016/j.jde.2005.05.004 Google Scholar

[17] [17] Martin, R.-H. and Smith, H.-L., Abstract functional-differential equations and reaction-diffusion Systems. Trans. Amer. Math. Soc. 321 (1990), 1–44. http://dx.doi.Org/10.2307/2001590 Google Scholar

[18] [18] Mei, M., So, J.-W.-H., Li, M.-Y., and Shen, S.-S.-P., Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion. Proc. Royal Soc. Edinburgh. 134A(2004), 579–594. Google Scholar

[19] [19] Mei, M., and So, J.-W.-H., Stability ofstrong traveling waves for a nonlocal time-delayed reaction-diffusion equation. Proc. Royal Soc. Edinburgh. 138A(2008), 551–568. http://dx.doi.Org/10.1017/S0308210506000333 Google Scholar

[20] [20] Mei, M., Lin, C.-K., Lin, C.-T., and So, J.-W.-H., Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity. J. Differential Equations 247 (2009), 495–510. http://dx.doi.Org/10.1016/j.jde.2008.12.026 Google Scholar

[21] [21] Mei, M., Lin, C.-K., Lin, C.-T., and So, J.-W.-H., Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity. J. Differential Equations 247 (2009), 511–529. http://dx.doi.Org/10.1016/j.jde.2008.12.020 Google Scholar

[22] [22] Mei, M. and Wong, Y.-S., Novel stability resultsfor traveling wavefronts in an age-structured reaction-diffusion equation. Math. Biosci. Eng. 6(2009), 743–752. http://dx.doi.Org/10.3934/mbe.2009.6.743 Google Scholar

[23] [23] Mei, M., Ou, C., and Zhao, X.-Q., Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations. SIAM J. Math. Anal. 42 (2010), 2762–2790. http://dx.doi.Org/10.1137/090776342 Google Scholar

[24] [24] Smith, H.-L. and Zhao, X.-Q., Global asymptotic stability of traveling waves in delayed recation-diffusion equations. SIAM J. Math. Anal. 31 (2000), 514–534. http://dx.doi.Org/10.1137/S0036141098346785 Google Scholar

[25] [25] Tian, G. and Zhang, G.-B., Stabilty of traveling wavefronts for a discrete diffusive Lotka-Volterra competition System. J. Math. Anal. Appl. 447 (2017), 222–242. http://dx.doi.Org/10.1016/j.jmaa.2O16.10.012 Google Scholar

[26] [26] Yang, Y., Li, W.-T., and Wu, S., Exponential stability of traveling fronts in a diffusion epidemic System with delay. Nonlinear Anal. Real World Appl. 12 (2011), 1223–1234. http://dx.doi.Org/10.1016/j.nonrwa.2010.09.017 Google Scholar

[27] [27] Yang, Y., Li, W.-T., and Wu, S., Stability of traveling waves in a monostable delayed System without quasi-monotonicity. Nonlinear Anal. Real World Appl. 3 (2013), 1511–1526. http://dx.doi.Org/10.1016/j.nonrwa.2012.10.015 Google Scholar

[28] [28] Yu, Z.-X., Xu, F., and Zhang, W-G., Stability of Invasion traveling waves for a competition System with nonlocal dispersals. Appl. Anal. 96 (2017), 1107–1125. http://dx.doi.Org/10.1080/00036811.2016.1178242 Google Scholar

[29] [29] Yu, Z.-X. and Yuan, R., Nonlinear stability of wavefronts for a delayed stage-structured population model on 2-D lattice. Osaka J. Math. 50(2013) 963–976. Google Scholar

[30] [30] Yu, Z.-X. and Mei, M., Uniqueness and stability of traveling waves for cellular neural networks with multiple delays. J. Differential Equations 260, (2016), 241–267. http://dx.doi.org/10.1016/j.jde.2O15.08.037 Google Scholar

[31] [31] Zhang, G.-B., Global stability oftraveling wave fronts for non-local delayed lattice differential equations. Nonlinear Anal. Real World Appl. 13 (2012), 1790–1801. http://dx.doi.Org/10.1016/j.nonrwa.2011.12.010 Google Scholar

[32] [32] Zhang, G.-B. and Li, W.-T., Nonlinear stability oftraveling wavefronts in an age-structured population model with nonlocal dispersal and delay. Z. Angew. Math. Phys. 64 (2013), 1643–1659. http://dx.doi.Org/10.1007/s00033-013-0303-7 Google Scholar

[33] [33] Zhang, G.-B. and Ma, R., Spreading speeds and traveling wavesfor a nonlocal dispersal equation with convolution type crossing-monostable nonlinearity. Z. Angew. Math. Phys. 65 (2014), 819–844. http://dx.doi.Org/10.1007/s00033-013-0353-x Google Scholar

Cité par Sources :