Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

Traveling wave solutions in a higher dimensional lattice delayed cooperation system with nonlocal diffusion
  • Home
  • /
  • Traveling wave solutions in a higher dimensional lattice delayed cooperation system with nonlocal diffusion
  1. Home /
  2. Archives /
  3. Vol 62, No 1 (September 2023) /
  4. Articles

Traveling wave solutions in a higher dimensional lattice delayed cooperation system with nonlocal diffusion

Authors

  • Kun Li https://orcid.org/0000-0002-3799-8906
  • Yanli He https://orcid.org/0000-0002-8168-1527

DOI:

https://doi.org/10.12775/TMNA.2023.011

Keywords

Lattice, traveling wave solution, Schauder's fixed point theorem, upper and lower solutions

Abstract

This paper is concerned with the existence of traveling wave solutions of a higher dimensional lattice delayed cooperation system with nonlocal diffusion. For sufficiently small intraspecific cooperative delays, we construct upper and lower solutions under two different parameters conditions. And then, by using the monotone iterative and Schauder's fixed point theorem, we obtain the existence of traveling wave solutions. The lower bound of the wave speed is in accordance with the properties of linear determined.

References

J. Cahn, J. Mallet-Paret and E.S. Van Vleck, Traveling wave solutions for systems of ODE’s on a two-dimensional spatial lattice, SIAM J. Appl. Math. 59 (1998), 455–493.

X. Chen and J. Guo, Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics, Math. Ann. 326 (2003), 123–146.

X. Chen and J. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002), 549–569.

C. Cheng, W. Li, Z. Wang and S. Zheng, Traveling waves connecting equilibrium and periodic orbit for a delayed population model on a two-dimensional spatial lattice, Int. J. Bifurcat. Chaos Appl. Sci. Engrg. 26 (2016), 1650049, 1–13.

S. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations 149 (1998), 248–291.

J. Guo and C. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math. 45 (2008), 327–346.

J. Guo and C. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations 252 (2012), 4357–4391.

D. Hankerson and B. Zinner, Wave fronts for a cooperative triangonal system of differential equations, J. Dynam. Differential Equations 5 (1993), 359–373.

J. Huang and L. Huang, Traveling wavefronts in systems of delayed reaction diffusion equations on higher dimentional lattices, Acta Math. Appl. Sin. Ser. A 28 (2005), 100–113.

J. Huang and G. Lu, Travelling wave solutions in delayed lattice dynamical system, Chinese Ann. Math. Ser. A 25 (2004), 153–164.

J. Huang, G. Lu and S. Ruan, Existence of traveling wave fronts of delayed lattice differential equations, J. Math. Anal. Appl. 298 (2004), 538–558.

J. Huang, G. Lu and S. Ruan, Traveling wave solutions in delayed lattice differential equations with partial monotonicity, Nonlinear Anal. 60 (2005), 1331–1350.

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka–Volterra system with delays, J. Math. Anal. Appl. 271 (2002), 455–466.

W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka–Volterra competition model, J. Differential Equations 251 (2011), 1549–1561.

J. Keener, Propagation and its failure to coupled systems of discrete excitable cells, SIAM J. Appl. Math. 47 (1987), 556–572.

K. Li and X. Li, Traveling wave solutions in diffusive and competition-cooperation systems with delays, IMA J. Appl. Math. 74 (2009), 604–621.

K. Li and X. Li, Traveling wave solutions in a delayed diffusive competition system, Nonlinear Anal. 75 (2012), 3705–3722.

K. Li and X. Li, Traveling wave solutions in a reaction-diffusion competition-cooperation system with stage structure, Jpn. J. Ind. Appl. Math. 35 (2018), 157–193.

K. Li and X. Li, Traveling wave solutions in a delayed lattice competition-cooperation system, J. Difference Equ. Appl. 24 (2018), 391–408.

X. Li and G. Lin, Traveling wavefronts in nonlocal dispersal and cooperative Lotka–Volterra system with delays, Appl. Math. Comput. 204 (2008), 738–744.

G. Lin and W. Li, Traveling waves in delayed lattice dynamical systems with competition interactions, Nonlinear Anal. 11 (2010), 3666–3679.

A. Lotka, Elements of Physical Biology, Williams and Wilkins Company, Baltimore, 1925.

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations 217 (2005), 54–87.

J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differential Equations 11 (1999), 1–47.

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations 11 (1999), 49–127.

H. Thieme and X. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Dynam. Differential Equations 195 (2003), 430–470.

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature 118 (1926), 558–560.

P. Weng, Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), 883–904.

S. Wu and S. Liu, Travelling waves in delayed reaction-diffusion equations on higher dimensional lattices, J. Difference Equ. Appl. 19 (2013), 384–401.

J. Wu and X. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Differential Equations 135 (1997), 315–357.

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

Z. Yu, W. Zhang and X. Wang, Spreading speeds and travelling waves for non-monotone time-delayed 2D lattice systems, Math. Comput. Model. 58 (2013), 1510–1521.

H. Zhao, Asymptotic stability of traveling fronts in delayed reaction-diffusion monostable equations on higher-dimensional lattices, Electron. J. Differential Equations 2013 (2013), 119, 1–15.

H. Zhao and S. Wu, Wave propagation for a reaction-diffusion model with a quiescent stage on a 2D spatial lattice, Nonlinear Anal. 12 (2011), 1178–1191.

B. Zinner, Stability of traveling wave fronts for the discrete Nagumo equation, SIAM J. Math. Anal. 22 (1991), 1016–1020.

B. Zinner, Existence of traveling wave front solutions for the discrete Nagumo equation, J. Differential Equations 96 (1992), 1–27.

B. Zinner, G. Harris and W. Hudson, Traveling wave fronts for the discrete Fisher’s equation, J. Differential Equations 105 (1993), 46–62.

X. Zou, Traveling wave fronts in spatially discrete reaction-diffusion equations on higherdimensional lattices, Proceedings of the Third Mississippi State Conference on Difference Equations and Computational Simulations (Mississippi State, MS, 1997), Electron. J. Differ. Equ. Conf. 1, Southwest Texas State Univ., San Marcos, TX, 1998, pp. 211–221.

X. Zou and J. Wu, Local existence and stability of periodic traveling waves of lattice functional-differential equations, Can. Appl. Math. Q. 6 (1998), 397–418.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2023-09-30

How to Cite

1.
LI, Kun and HE, Yanli. Traveling wave solutions in a higher dimensional lattice delayed cooperation system with nonlocal diffusion. Topological Methods in Nonlinear Analysis. Online. 30 September 2023. Vol. 62, no. 1, pp. 203 - 218. [Accessed 29 June 2025]. DOI 10.12775/TMNA.2023.011.
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 62, No 1 (September 2023)

Section

Articles

License

Copyright (c) 2023 Kun Li, Yanli He

Creative Commons License

This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop