Propagation dynamics of a nonlocal reaction-diffusion system

Bang-Sheng Han; De-Yu Kong

doi:10.3934/dcds.2023028

Article Contents

2023, Volume 43, Issue 7: 2756-2780. Doi: 10.3934/dcds.2023028

This issue Previous Article Observable Lyapunov irregular sets for planar piecewise expanding maps Next Article Ergodic average of typical orbits and typical functions

Propagation dynamics of a nonlocal reaction-diffusion system

Bang-Sheng Han^1, , and
De-Yu Kong^1,

1.
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 611756, China

^*Corresponding author: Bang-Sheng Han
^*Corresponding author: Bang-Sheng Han

Received: September 2022

Revised: February 2023

Early access: March 2023

Published: July 2023

The first author is supported by [ Natural Science Foundation of China (12161052, 11801470, U22A20231), Natural Science Foundation of Sichuan Province (2022NSFSC1819), Central Government Funds for Guiding Local Scientific and Technological Development (2021ZYD0010) and the Fundamental Research Funds for the Central Universities (2682022ZTPY080)].

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Abstract

This paper is dedicated to investigate the propagation dynamics of a three-species reaction-diffusion system with nonlocal diffusion which includes diffusion kernel functions symmetry and asymmetry. First off, we give the asymptotic spreading speeds by using its partial commonality with the minimum propagation spreadings and construct a set $ \Pi $ to obtain their signs which just shows it is fundamentally different from local diffusion. The following main focus on the asymptotic behaviour with different initial values. The key point for asymmetry kernel is to construct suitable upper and lower solutions and obtain long-time asymptotic behavior by using comparison principle and our improved "forward-backward spreading" method which first proposed by Xu et al. (J Funct Anal 280(2021)108957). Especially, transforming the square scale, introducing new variables and using linear programming to obtain the existence of variables and other techniques are our improvement of this method which ensures it works for multi-population systems. Accordingly, the asymptotic behavior and some monotone property results with the symmetric kernel are all obtained by comparison principle.

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Mathematics Subject Classification: Primary: 35B51, 35K55; Secondary: 35K57, 92D25.

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References

[1]	F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, Real Sociedad Matemática Espa$\tilde{n}$ola, Madrid, 2010. doi: 10.1090/surv/165.
[2]	D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, 446 (1975), 5-49.
[3]	X. Bao, W.-T. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differ. Equ., 260 (2016), 8590-8637. doi: 10.1016/j.jde.2016.02.032.
[4]	P. W. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations, 48 (2006), 13-52.
[5]	H. Berestycki, J. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693-1745. doi: 10.1007/s00285-015-0911-2.
[6]	H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23 pp. doi: 10.1063/1.4764932.
[7]	F. Chen, Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Anal., 50 (2002), 807-838. doi: 10.1016/S0362-546X(01)00787-8.
[8]	J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differ. Equ., 244 (2008), 3080-3118. doi: 10.1016/j.jde.2007.11.002.
[9]	A. Ducrot, T. Giletti, J.-S. Guo and M. Shimojo, Asymptotic spreading speeds for a predator-prey system with two predators and one prey, Nonlinearity., 34 (2021), 669-704. doi: 10.1088/1361-6544/abd289.
[10]	P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, 2003,153–191. doi: 10.1007/978-3-662-05281-5_3.
[11]	J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974. doi: 10.1137/10080693X.
[12]	J. Garnier, T. Giletti and G. Nadin, Maximal and minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media, J. Dynam. Differential Equations., 24 (2012), 521-538. doi: 10.1007/s10884-012-9254-5.
[13]	S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333. doi: 10.1007/BF00160498.
[14]	F. Hamel and G. Nadin, Spreading properties and complex dynamics for monostable reaction-diffusion equations, Comm. Partial Differential Equations., 37 (2012), 511-537. doi: 10.1080/03605302.2011.647198.
[15]	C. Hu, Y. Kuang, B. Li and H. Liu, Spreading speeds and traveling wave solutions in cooperative integral-differential systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1663-1684. doi: 10.3934/dcdsb.2015.20.1663.
[16]	G. Jimmy, H. François and R. Lionel, Transition fronts and stretching phenomena for a general class of reaction-dispersion equations, Discrete Contin. Dyn. Syst., 37 (2017), 743-756. doi: 10.3934/dcds.2017031.
[17]	C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.
[18]	M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.
[19]	B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.
[20]	W.-T. Li, J.-B. Wang and X.-Q. Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, J. Nonlinear Sci., 28 (2018), 1189-1219. doi: 10.1007/s00332-018-9445-2.
[21]	W.-T. Li, W.-B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst., 37 (2017), 2483-2512. doi: 10.3934/dcds.2017107.
[22]	W.-T. Li, L. Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin, Dyn. Syst., 35 (2015), 1531-1560. doi: 10.3934/dcds.2015.35.1531.
[23]	X. Liang, Y. Yi and X-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differ. Equ., 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.
[24]	X. Liang and T. Zhou, Spreading speeds of nonlocal KPP equations in almost periodic media, J. Funct. Anal., 279 (2020), 108723, 58 pp. doi: 10.1016/j.jfa.2020.108723.
[25]	X. Liang and T. Zhou, Spreading speeds of nonlocal KPP equations in heterogeneous media, Acta Math. Sin., 38 (2022), 161-178. doi: 10.1007/s10114-022-0452-8.
[26]	G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differ. Equ., 244 (2008), 487-513. doi: 10.1016/j.jde.2007.10.019.
[27]	Q. Liu, S. Liu and K.-Y. Lam, Stacked invasion waves in a competition-diffusion model with three species, J. Differ. Equ., 271 (2021), 665-718. doi: 10.1016/j.jde.2020.09.008.
[28]	R. Lui, Biological growth and spread modeled by systems of recursions. Ⅱ. Biological theory, Math. Biosci., 93 (1989), 297-312. doi: 10.1016/0025-5564(89)90027-8.
[29]	F. Lutscher, E. Pachepsky and M.A. Lewis, The effect of dispersal patterns on stream populations, SIAM J. Appl. Math., 65 (2005), 1305-1327. doi: 10.1137/S0036139904440400.
[30]	D. E. Marco, M. A. Montemurro and S. A. Cannas, Comparing short and long-distance dispersal: Modelling and field case studies, Ecography., 34 (2013), 671-682. doi: 10.1111/j.1600-0587.2010.06477.x.
[31]	N. Wang and Z.-C. Wang, Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay, Discrete Contin. Dyn. Syst., 42 (2022), 1599-1646. doi: 10.3934/dcds.2021166.
[32]	H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.
[33]	H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.
[34]	C.-C. Wu, The spreading speed for a predator-prey model with one predator and two preys, Appl. Math. Lett., 91 (2019), 9-14. doi: 10.1016/j.aml.2018.11.022.
[35]	W.-B. Xu, W.-T. Li and S. Ruan, Spatial propagation in an epidemic model with nonlocal diffusion: The influences of initial data and dispersals, Sci. China Math., 63 (2020), 2177-2206. doi: 10.1007/s11425-020-1740-1.
[36]	W.-B. Xu, W.-T. Li and S. Ruan, Spatial propagation in nonlocal dispersal Fisher-KPP equations, J. Funct. Anal., 280 (2021), 108957, 35 pp. doi: 10.1016/j.jfa.2021.108957.
[37]	L. Zhang and X. Bao, Propagation dynamics of a three-species nonlocal competitive-cooperative system, Nonlinear Anal. Real World Appl., 58 (2021), Paper No. 103230, 17 pp. doi: 10.1016/j.nonrwa.2020.103230.