Exact asymptotic behavior of pulsating traveling waves for a periodic monostable lattice dynamical system
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- by Shi-Liang Wu, Guang-Sheng Chen and Cheng-Hsiung Hsu PDF
- Proc. Amer. Math. Soc. 149 (2021), 1697-1710 Request permission
Abstract:
This paper is concerned with the pulsating traveling waves of a periodic lattice dynamical system with monostable nonlinearity. We first establish the exponential upper and lower bounds of the pulsating wave profiles at minus infinity. Then we prove the uniqueness result and derive the asymptotic behavior of all non-critical monostable pulsating traveling waves. This might be the first time to obtain the exact asymptotic behavior of the pulsating traveling waves for periodic discrete systems towards the unstable steady state.References
- Henri Berestycki, François Hamel, and Lionel Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl. (9) 84 (2005), no. 8, 1101–1146 (English, with English and French summaries). MR 2155900, DOI 10.1016/j.matpur.2004.10.006
- Jack Carr and Adam Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2433–2439. MR 2052422, DOI 10.1090/S0002-9939-04-07432-5
- Xinfu Chen and Jong-Sheng Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann. 326 (2003), no. 1, 123–146. MR 1981615, DOI 10.1007/s00208-003-0414-0
- Xinfu Chen, Jong-Shenq Guo, and Chin-Chin Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal. 189 (2008), no. 2, 189–236. MR 2413095, DOI 10.1007/s00205-007-0103-3
- Shui-Nee Chow, John Mallet-Paret, and Wenxian Shen, Traveling waves in lattice dynamical systems, J. Differential Equations 149 (1998), no. 2, 248–291. MR 1646240, DOI 10.1006/jdeq.1998.3478
- Paul C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, vol. 28, Springer-Verlag, Berlin-New York, 1979. MR 527914, DOI 10.1007/978-3-642-93111-6
- R.A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 335–369.
- S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Nonlinear dynamics and evolution equations, Fields Inst. Commun., vol. 48, Amer. Math. Soc., Providence, RI, 2006, pp. 137–200. MR 2223351, DOI 10.1007/s00285-006-0050-x
- Jong-Shenq Guo and François Hamel, Front propagation for discrete periodic monostable equations, Math. Ann. 335 (2006), no. 3, 489–525. MR 2221123, DOI 10.1007/s00208-005-0729-0
- Jong-Shenq Guo and Chin-Chin Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations 246 (2009), no. 10, 3818–3833. MR 2514727, DOI 10.1016/j.jde.2009.03.010
- François Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math. Pures Appl. (9) 89 (2008), no. 4, 355–399 (English, with English and French summaries). MR 2401143, DOI 10.1016/j.matpur.2007.12.005
- François Hamel and Lionel Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 2, 345–390. MR 2746770, DOI 10.4171/JEMS/256
- A. N. Kolmogorov, I. G. Petrovsky, and N. S. Piskunov, Étude de l’équation de la diffusion avec croissance de la quantite de matière et son application à un problème biologique, Bull. Universite Etat Moscou, Ser. Int., Sect. A., 1 (1937), 1–25.
- Xing Liang and Xiao-Qiang Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal. 259 (2010), no. 4, 857–903. MR 2652175, DOI 10.1016/j.jfa.2010.04.018
- Grégoire Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl. (9) 92 (2009), no. 3, 232–262 (English, with English and French summaries). MR 2555178, DOI 10.1016/j.matpur.2009.04.002
- Wenxian Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5125–5168. MR 2657675, DOI 10.1090/S0002-9947-10-04950-0
- Wenxian Shen and Aijun Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations 249 (2010), no. 4, 747–795. MR 2652153, DOI 10.1016/j.jde.2010.04.012
- Wenxian Shen and Aijun Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1681–1696. MR 2869152, DOI 10.1090/S0002-9939-2011-11011-6
- Nanako Shigesada, Kohkichi Kawasaki, and Ei Teramoto, Traveling periodic waves in heterogeneous environments, Theoret. Population Biol. 30 (1986), no. 1, 143–160. MR 850456, DOI 10.1016/0040-5809(86)90029-8
- Yu-Juan Sun, Wan-Tong Li, and Zhi-Cheng Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations 251 (2011), no. 3, 551–581. MR 2802024, DOI 10.1016/j.jde.2011.04.020
- Zhi-Cheng Wang, Wan-Tong Li, and Shigui Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2047–2084. MR 2465829, DOI 10.1090/S0002-9947-08-04694-1
- Hans F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), no. 6, 511–548. MR 1943224, DOI 10.1007/s00285-002-0169-3
- Xiao Yu and Xiao-Qiang Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dynam. Differential Equations 29 (2017), no. 1, 41–66. MR 3612965, DOI 10.1007/s10884-015-9426-1
- Jack Xin, Front propagation in heterogeneous media, SIAM Rev. 42 (2000), no. 2, 161–230. MR 1778352, DOI 10.1137/S0036144599364296
- 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
- B. Zinner, G. Harris, and W. Hudson, Traveling wavefronts for the discrete Fisher’s equation, J. Differential Equations 105 (1993), no. 1, 46–62. MR 1237977, DOI 10.1006/jdeq.1993.1082
Additional Information
- Shi-Liang Wu
- Affiliation: School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi, 710071, People’s Republic of China
- ORCID: 0000-0002-0462-6161
- Email: slwu@xidian.edu.cn
- Guang-Sheng Chen
- Affiliation: School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi, 710071, People’s Repupblic of China; College of Mathematics and Computer Science, Guangxi Science & Technology Normal University, Laibin, 546199, People’s Republic of China
- Email: cgswavelets@126.com
- Cheng-Hsiung Hsu
- Affiliation: Department of Mathematics, National Central University, Zhongli District, Taoyuan City 32001, Taiwan
- MR Author ID: 624970
- ORCID: 0000-0001-7565-6352
- Email: chhsu@math.ncu.edu.tw
- Received by editor(s): May 5, 2020
- Received by editor(s) in revised form: September 14, 2020
- Published electronically: February 12, 2021
- Additional Notes: The first author was partially supported by the NSF of China (No. 11671315) and Natural Science Basic Research Program of Shaanxi (No. 2020JC-24). The third author was partially supported by the MOST (Grant No. 107-2115-M-008-009-MY) and NCTS of Taiwan
The first author is the corresponding author. - Communicated by: Wenxian Shen
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 1697-1710
- MSC (2020): Primary 34K05, 34A34, 34E05
- DOI: https://doi.org/10.1090/proc/15369
- MathSciNet review: 4242324