Time periodic traveling waves for a periodic nonlocal dispersal model with delay

Wu, Shi-Liang; Huang, Ming-di

doi:10.1090/proc/15085

Time periodic traveling waves for a periodic nonlocal dispersal model with delay
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by Shi-Liang Wu and Ming-di Huang PDF

Proc. Amer. Math. Soc. 148 (2020), 4405-4421 Request permission

Abstract:

This paper is concerned with the time periodic traveling waves of a periodic nonlocal dispersal equation with delay. In the quasi-monotone case, we establish the exponential stability of all noncritical periodic traveling fronts. The exponential decay rate is also given. Then, we prove the uniqueness of all noncritical periodic traveling waves of the equation with or without quasi-monotone condition. Our results give an affirmative answer to a problem presented by Li, Wang, and Zhao.

References

Nicholas D. Alikakos, Peter W. Bates, and Xinfu Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2777–2805. MR 1467460, DOI 10.1090/S0002-9947-99-02134-0
Xinfu Chen and Jong-Shenq Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002), no. 2, 549–569. MR 1929888, DOI 10.1006/jdeq.2001.4153
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721
C.-H. Hsu and J.-J. Lin, Stability analysis of traveling wave solutions for lattice reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1757-1774.
Cheng-Hsiung Hsu, Jian-Jhong Lin, and Tzi-Sheng Yang, Stability for monostable wave fronts of delayed lattice differential equations, J. Dynam. Differential Equations 29 (2017), no. 1, 323–342. MR 3612977, DOI 10.1007/s10884-015-9447-9
Wan-Tong Li, Jia-Bing Wang, and Xiao-Qiang Zhao, Propagation dynamics in a time periodic nonlocal dispersal model with stage structure, J. Dynam. Differential Equations 32 (2020), no. 2, 1027–1064. MR 4094993, DOI 10.1007/s10884-019-09760-3
Chi-Kun Lin, Chi-Tien Lin, Yanping Lin, and Ming Mei, Exponential stability of nonmonotone traveling waves for Nicholson’s blowflies equation, SIAM J. Math. Anal. 46 (2014), no. 2, 1053–1084. MR 3174173, DOI 10.1137/120904391
Chi-Kun Lin and Ming Mei, On travelling wavefronts of Nicholson’s blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 1, 135–152. MR 2592717, DOI 10.1017/S0308210508000784
Ming Mei, Joseph W.-H. So, Michael Y. Li, and Samuel 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. MR 2068117, DOI 10.1017/S0308210500003358
Ming Mei, Chi-Kun Lin, Chi-Tien Lin, and Joseph W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity, J. Differential Equations 247 (2009), no. 2, 511–529. MR 2523689, DOI 10.1016/j.jde.2008.12.020
Zigen Ouyang and Chunhua Ou, Global stability and convergence rate of traveling waves for a nonlocal model in periodic media, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 3, 993–1007. MR 2873124, DOI 10.3934/dcdsb.2012.17.993
Hal L. Smith and Horst R. Thieme, Strongly order preserving semiflows generated by functional-differential equations, J. Differential Equations 93 (1991), no. 2, 332–363. MR 1125223, DOI 10.1016/0022-0396(91)90016-3
Zhi-Cheng Wang, Wan-Tong Li, and Shigui Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations 238 (2007), no. 1, 153–200. MR 2334595, DOI 10.1016/j.jde.2007.03.025
Xiang-Sheng Wang and Xiao-Qiang Zhao, Pulsating waves of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differential Equations 259 (2015), no. 12, 7238–7259. MR 3401596, DOI 10.1016/j.jde.2015.08.019
Dashun Xu and Xiao-Qiang Zhao, Dynamics in a periodic competitive model with stage structure, J. Math. Anal. Appl. 311 (2005), no. 2, 417–438. MR 2168406, DOI 10.1016/j.jmaa.2005.02.062
Guangyu Zhao and Shigui Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl. (9) 95 (2011), no. 6, 627–671 (English, with English and French summaries). MR 2802895, DOI 10.1016/j.matpur.2010.11.005
Guangyu Zhao and Shigui Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations 257 (2014), no. 4, 1078–1147. MR 3210024, DOI 10.1016/j.jde.2014.05.001