Some remarks on lattice and nonlocal dispersal evolution systems

Xiao-Qiang Zhao

doi:10.3934/dcdss.2022137

Article Contents

2023, Volume 16, Issue 3&4: 787-796. Doi: 10.3934/dcdss.2022137

This issue Previous Article Qualitative structure of a discrete predator-prey model with nonmonotonic functional response Next Article

Some remarks on lattice and nonlocal dispersal evolution systems

Xiao-Qiang Zhao

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

Dedicated to Professor Jibin Li on the occasion of his 80th birthday

Received: March 2022

Revised: May 2022

Early access: July 12, 2022

Published online: July 12, 2022

Research is supported in part by the NSERC of Canada (RGPIN-2019-05648).

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Abstract

In this paper, we first characterize the continuity of a map in the space $ \mathcal{C} = BC(\mathcal{H},\mathbb{R}^m) $ equipped with the compact open topology. Then we show that linear lattice and nonlocal dispersal equations generate uniformly continuous semigroups in the Banach space $ \mathcal{B} = BC(\mathcal{H},\mathbb{R}^m) $ equipped with the supremum norm. Finally, we illustrate how to prove nonlinear lattice and nonlocal dispersal equations generate monotone semiflows with respect to the compact open topology.

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Mathematics Subject Classification: Primary: 34A33, 35K57; Secondary: 37C65.

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References

[1]	J. Fang and X.-Q. Zhao, Bistable travelling waves for monotone semiflows with applications, J. Eur. Math. Soc., 17 (2015), 2243-2288. doi: 10.4171/JEMS/556.
[2]	X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Communications on Pure and Applied Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.
[3]	R. Lui, Biological growth and spread modeled by systems of recursions, I. mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.
[4]	R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.
[5]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[6]	H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.
[7]	H. F. Weinberger, On spreading speeds and travelling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.
[8]	P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020.
[9]	K. F. Zhang and X.-Q. Zhao, Spreading speed and travelling waves for a spatially discrete SIS epidemic model, Nonlinearity, 21 (2008), 97-112. doi: 10.1088/0951-7715/21/1/005.