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doi: 10.3934/dcdss.2022137
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## Some remarks on lattice and nonlocal dispersal evolution systems

 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 2022

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

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.

Citation: Xiao-Qiang Zhao. Some remarks on lattice and nonlocal dispersal evolution systems. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022137
##### 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.

show all references

##### 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.
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