Bistable Periodic Traveling Wave Solutions to Lattice Competitive Systems

LI Jian

doi:10.21656/1000-0887.430071

Volume 44 Issue 4

Apr. 2023

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2023 > 44(4): 471-479

LI Jian. Bistable Periodic Traveling Wave Solutions to Lattice Competitive Systems[J]. Applied Mathematics and Mechanics, 2023, 44(4): 471-479. doi: 10.21656/1000-0887.430071

Citation:

LI Jian. Bistable Periodic Traveling Wave Solutions to Lattice Competitive Systems[J]. Applied Mathematics and Mechanics, 2023, 44(4): 471-479. doi: 10.21656/1000-0887.430071

Citation:

LI Jian. Bistable Periodic Traveling Wave Solutions to Lattice Competitive Systems[J]. Applied Mathematics and Mechanics, 2023, 44(4): 471-479. doi: 10.21656/1000-0887.430071

PDF( 391 KB)

Bistable Periodic Traveling Wave Solutions to Lattice Competitive Systems

doi: 10.21656/1000-0887.430071

LI Jian

School of Mathematics and Statistics, Xidian University, Xi'an 710071, P.R.China

Received Date: 2022-03-07
Rev Recd Date: 2022-04-11
Publish Date: 2023-04-01

Abstract

Abstract

The existence of bistable periodic traveling wave solutions to lattice competitive systems was studied. Firstly, the lattice competitive system of 2 species was transformed into a cooperative system. Then, the principle of comparison was established and a pair of upper and lower solutions were given to obtain the convergence of the solution at infinity, with the initial function satisfying certain conditions. By means of the vanishing viscosity method and the principle of comparison, the existence of the traveling wave solution connecting 2 stable periodic equilibrium points of the system, was proved.
- lattice competitive system,
- bistable periodic traveling wave,
- principle of comparison,
- vanishing viscosity method

FullText(HTML)

References(18)

References

[1]	BAO X X, WANG Z C. Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system[J]. Journal of Differential Equations, 2013, 255: 2402-2435. doi: 10.1016/j.jde.2013.06.024
[2]	FANG J, ZHAO X Q. Monotone wavefronts for partially degenerate reaction-diffusion systems[J]. Journal of Dynamics and Differential Equations, 2009, 21: 663-680. doi: 10.1007/s10884-009-9152-7
[3]	LI B. Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems[J]. Journal of Differential Equations, 2012, 252(9): 4842-4861. doi: 10.1016/j.jde.2012.01.018
[4]	ZHAO G Y, RUAN S G. Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion[J]. Journal de Mathématiques Pures et Appliquées, 2011, 95(6): 627-671. doi: 10.1016/j.matpur.2010.11.005
[5]	HAO Y X, LI W T, WANG J B. Propagation dynamics of Lotka-Volterra competition systems with asymmetric dispersal in periodic habits[J]. Journal of Differential Equations, 2021, 300: 185-225. doi: 10.1016/j.jde.2021.07.041
[6]	BAO X X, LI W T, WANG Z C. Time periodic traveling curved fronts in the periodic Lotka-Volterra competition diffusion system[J]. Journal of Dynamics and Differential Equations, 2017, 29: 981-1016. doi: 10.1007/s10884-015-9512-4
[7]	GUO J S, WU C H. Wave propagation for a two-component lattice dynamical system arising in strong competition models[J]. Journal of Differential Equations, 2011, 250: 3504-3533. doi: 10.1016/j.jde.2010.12.004
[8]	GUO J S, WU C H. Traveling wave front for a two-component lattice dynamical system arising in competition models[J]. Journal of Differential Equations, 2012, 252: 4357-4391. doi: 10.1016/j.jde.2012.01.009
[9]	WANG H Y, OU C H. Propagation direction of the traveling wave for the Lotka-Volterra competitive lattice system[J]. Journal of Dynamics and Differential Equations, 2021, 33: 1153-1174. doi: 10.1007/s10884-020-09853-4
[10]	SHEN W X. Traveling waves in time periodic lattice differential equations[J]. Nonlinear Analysis, 2003, 54(2): 319-339. doi: 10.1016/S0362-546X(03)00065-8
[11]	ZHANG K F, ZHAO X Q. Spreading speed and traveling waves for a spatially discrete SIS epidemic model[J]. Nonlinearity, 2008, 21(1): 97-112. doi: 10.1088/0951-7715/21/1/005
[12]	KRUXZKOV S N. First order quasilinear equations in several independent variables[J]. Mathematics of the USSR-Sbornik, 1970, 10: 217-243. doi: 10.1070/SM1970v010n02ABEH002156
[13]	GUO J S, HAMEL F. Front propagation for discrete periodic monostable equations[J]. Mathematische Annalen, 2006, 335: 489-525. doi: 10.1007/s00208-005-0729-0
[14]	CHEN X F, GUO J S, WU C C. Traveling waves in discrete periodic media for bistable dynamics[J]. Archive for Rational Mechanics and Analysis, 2008, 189: 189-236. doi: 10.1007/s00205-007-0103-3
[15]	陈妍. 时间周期的离散SIS模型的传播动力学[J]. 应用数学和力学, 2022, 43(10): 1155-1163. doi: 10.21656/1000-0887.420350 CHEN Yan. Propagation dynamics of a discrete SIS model with time periodicity[J]. Applied Mathematics and Mechanics, 2022, 43(10): 1155-1163. (in Chinese) doi: 10.21656/1000-0887.420350
[16]	郑景盼. 三物种竞争-扩散系统双稳行波解的波速符号[J]. 应用数学和力学, 2021, 42(12): 1296-1305. doi: 10.21656/1000-0887.420093 ZHENG Jingpan. The wave speed signs for bistable traveling wave solutions in 3-species competition-diffusion systems[J]. Applied Mathematics and Mechanics, 2021, 42(12): 1296-1305. (in Chinese) doi: 10.21656/1000-0887.420093
[17]	SHEN W X. Traveling waves in time almost periodic structures governed by bistable nonlinearities, Ⅱ: existence[J]. Journal of Differential Equations, 1999, 159(1): 55-101. doi: 10.1006/jdeq.1999.3652
[18]	WU S L, HSU C H. Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity[J]. Advances in Nonlinear Analysis, 2020, 9: 923-957.