Traveling wave solutions in a higher dimensional lattice delayed cooperation system with nonlocal diffusion
DOI:
https://doi.org/10.12775/TMNA.2023.011Keywords
Lattice, traveling wave solution, Schauder's fixed point theorem, upper and lower solutionsAbstract
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.
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