Traveling wavefront solutions in a nonlocal diffusion model with nonlocal delay effect

Dong Li; Xuechun He; Nengxing Tan; Rong Zou; Dong Li; Xuechun He; Nengxing Tan; Rong Zou

doi:10.3934/era.2026187

Electronic Research Archive

2026, Volume 34, Issue 6: 4172-4190. doi: 10.3934/era.2026187

Previous Article Next Article

Research article

Traveling wavefront solutions in a nonlocal diffusion model with nonlocal delay effect

1.
School of Mathematics and Statistics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2.
School of General Education, Chongqing Finance and Economics College, Chongqing 401320, China
3.
School of Mathematics and Quantitative Economics, Guangxi University of Finance and Economics, Nanning 530003, China

Received: 20 January 2026 Revised: 27 February 2026 Accepted: 14 April 2026 Published: 18 May 2026

This paper is devoted to investigating the existence of traveling wavefront solutions with a sufficiently large wave speed for a nonlocal diffusion model with nonlocal delay effect using a perturbation method. Our proof is based on an abstract formulation of the wave profile as a solution to an operator equation in a specific Banach space, combined with the Fredholm theory and the Banach contraction mapping principle. Several numerical simulations are presented to illustrate our main results.
- nonlocal diffusion,
- nonlocal delay,
- traveling wavefront solution,
- perturbation method
Citation: Dong Li, Xuechun He, Nengxing Tan, Rong Zou. Traveling wavefront solutions in a nonlocal diffusion model with nonlocal delay effect[J]. Electronic Research Archive, 2026, 34(6): 4172-4190. doi: 10.3934/era.2026187

Related Papers:

Abstract

This paper is devoted to investigating the existence of traveling wavefront solutions with a sufficiently large wave speed for a nonlocal diffusion model with nonlocal delay effect using a perturbation method. Our proof is based on an abstract formulation of the wave profile as a solution to an operator equation in a specific Banach space, combined with the Fredholm theory and the Banach contraction mapping principle. Several numerical simulations are presented to illustrate our main results.

References

[1]	J. Murray, Mathematical Biology. I. An Introduction, Springer-Verlag, New York, 2002. https://doi.org/10.1007/b98868
[2]	J. Coville, L. Dupaigne, Propagation speed of travelling fronts in nonlocal reaction-diffusion equations, Nonlinear Anal. Theory Methods Appl., 60 (2005), 797–819. https://doi.org/10.1016/j.na.2003.10.030 doi: 10.1016/j.na.2003.10.030
[3]	V. Hutson, M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 221–232. https://doi.org/10.1017/S0956792506006462 doi: 10.1017/S0956792506006462
[4]	L. Ignat, J. Rossi, A nonlocal convection-diffusion equation, J. Functional Anal., 251 (2007), 399–437. https://doi.org/10.1016/j.jfa.2007.07.013 doi: 10.1016/j.jfa.2007.07.013
[5]	C. Lee, M. Hoopes, J. Diehl, W. Gilliland, G. Huxel, E. Leaver, et al., Non-local concepts and models in biology, J. Theor. Biol., 210 (2001), 201–219. https://doi.org/10.1006/jtbi.2000.2287 doi: 10.1006/jtbi.2000.2287
[6]	K. Liu, S. Guo, Existence of periodic traveling waves in a nonlocal convection-diffusion model with chemotaxis and delay effect, Z. Angew. Math. Phys., 74 (2023), 179. https://doi.org/10.1007/s00033-023-02073-y doi: 10.1007/s00033-023-02073-y
[7]	F. Yang, W. Li, Z. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal. Real World Appl., 23 (2015), 129–147. https://doi.org/10.1016/j.nonrwa.2014.12.001 doi: 10.1016/j.nonrwa.2014.12.001
[8]	S. Busenberg, H. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differ. Equations, 1 (1996), 80–107. https://doi.org/10.1006/jdeq.1996.0003 doi: 10.1006/jdeq.1996.0003
[9]	S. Chen, J. Shi, Stability and Hopf bifurcation in a diffusive logistic population delay model with nonlocal delay effects, J. Differ. Equations, 253 (2012), 3440–3470. https://doi.org/10.1016/j.jde.2012.08.031 doi: 10.1016/j.jde.2012.08.031
[10]	J. Hale, S. Lunel, Introduction to Functional Differential Equations, Springer, New York, 2013. https://doi.org/10.1007/978-1-4612-4224-6
[11]	Y. Hao, G. Zhang, The dynamics of traveling wavefronts for a nonlocal delay competition system with local vs. nonlocal diffusions, Commun. Nonlinear Sci. Numer. Simul., 110 (2022), 106381. https://doi.org/10.1016/j.cnsns.2022.106381 doi: 10.1016/j.cnsns.2022.106381
[12]	Y. Hou, Y. Ding, Spatiotemporal dynamic analysis of delayed diffusive pine wilt disease model with nonlocal effect, SIAM J. Appl. Math., 84 (2024), 1312–1336. https://doi.org/10.1137/23M1575305 doi: 10.1137/23M1575305
[13]	J. Li, Y. Ding, New mechanism for spatial heterogeneity pattern revealed by nonlocal competition and host-taxis in a 2D pine wilt disease model, J. Math. Biol., 92 (2026), 19. https://doi.org/10.1007/s00285-025-02336-4 doi: 10.1007/s00285-025-02336-4
[14]	Y. Su, J. Wei, J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differ. Equations, 4 (2009), 1156–1184. https://doi.org/10.1016/0022-0396(84)90082-2 doi: 10.1016/0022-0396(84)90082-2
[15]	Z. Wang, W. Li, S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dyn. Differ. Equations, 20 (2008), 573–607. https://doi.org/10.1016/j.jde.2009.04.017 doi: 10.1016/j.jde.2009.04.017
[16]	X. Yan, W. Li, Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model, Nonlinearity, 6 (2010), 1413. https://doi.org/10.1142/S0218127409023342 doi: 10.1142/S0218127409023342
[17]	P. Bates, P. Fife, X. Ren, X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105–136. https://doi.org/10.1007/s002050050037 doi: 10.1007/s002050050037
[18]	X. Chen, Existence, uniqueness and asymptotic stability of travelling fronts in non-local evolution equations, Adv. Differ. Equations, 2 (1997), 125–160. https://doi.org/10.57262/ade/1366809230 doi: 10.57262/ade/1366809230
[19]	J. Coville, L. Dupaigne, On a non-local equation arising in population dynamics, Proc. R. Soc. Edinburgh Sect. A Math., 137 (2007), 727–755. https://doi.org/10.1017/S0308210504000721 doi: 10.1017/S0308210504000721
[20]	S. Guo, J. Zimmer, Travelling wavefronts in nonlocal diffusion equations with nonlocal delay effects, Bull. Malays. Math. Sci. Soc., 41 (2018), 919–943. https://doi.org/10.1007/s40840-017-0481-0 doi: 10.1007/s40840-017-0481-0
[21]	W. Li, Y. Sun, Z. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302–2313. 10.1016/j.nonrwa.2009.07.005 doi: 10.1016/j.nonrwa.2009.07.005
[22]	S. Pan, W. Li, G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377–392. https://doi.org/10.1007/s00033-007-7005-y doi: 10.1007/s00033-007-7005-y
[23]	H. Yagisita, Existence of traveling wave solutionsfor a nonlocal bistable equation: An abstract approach, Publ. Res. Inst. Math. Sci., 45 (2009), 955–979. http://dx.doi.org/10.9774/GLEAF.978-1-909493-38-4-2 doi: 10.9774/GLEAF.978-1-909493-38-4-2
[24]	Z. Yu, R. Yuan, Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response, Taiwan. J. Math., 17 (2013), 2163–2190. https://doi.org/10.1016/j.chaos.2012.07.002 doi: 10.1016/j.chaos.2012.07.002
[25]	G. Zhang, W. Li, Z. Wang, Spreading speeds and traveling waves for nonlocal dispersalequations with degenerate monostable nonlinearity, J. Differ. Equations, 252 (2012), 5096–5124. https://doi.org/10.1016/j.jde.2012.01.014 doi: 10.1016/j.jde.2012.01.014
[26]	Y. Zhou, S. Ji, Global stability of traveling wave fronts for a population dynamics model with quiescent stage and delay, Inter. J. Biomath., 15 (2022), 2250020. https://doi.org/10.1142/S1793524522500206 doi: 10.1142/S1793524522500206
[27]	T. Faria, W. Huang, J. Wu, Traveling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. London A, 462 (2006), 229–261. https://doi.org/10.1098/rspa.2005.1554 doi: 10.1098/rspa.2005.1554
[28]	D. Li, S. Guo, Traveling wavefronts in a reaction-diffusion model with chemotaxis and nonlocal delay effect, Nonlinear Anal. Real World Appl., 45 (2019), 736–754. https://doi.org/10.1016/j.nonrwa.2018.08.001 doi: 10.1016/j.nonrwa.2018.08.001
[29]	D. Li, X. He, X. Li, S. Guo, Traveling wavefronts in a two-species chemotaxis model with Lotka-Volterra competitive kinetics, Appl. Math. Lett., 114 (2021), 106905. https://doi.org/10.1016/j.aml.2020.106905 doi: 10.1016/j.aml.2020.106905
[30]	D. Li, N. Tan, H. Qiu, Traveling wave solutions in a modified Leslie-Gower model with diffusion and chemotaxis, Z. Angew. Math. Phys., 75 (2024), 165. https://doi.org/10.1007/s00033-024-02308-6 doi: 10.1007/s00033-024-02308-6
[31]	T. Faria, Global attractivity in scalar delayed differential equations with applications to population models, J. Math. Anal. Appl., 289 (2004), 35–54. https://doi.org/10.1016/j.jmaa.2003.08.013 doi: 10.1016/j.jmaa.2003.08.013
[32]	S. Chow, X. Lin, J. Mallet-Paret, Transition layers for singularly perturbed delay differential equations with monotone nonlinearities, J. Dyn. Differ. Equations, 1 (1989), 3–43. https://doi.org/10.1007/BF01048789 doi: 10.1007/BF01048789
[33]	K. Mischaikow, H. Smith, H. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Am. Math. Soc., 347 (1995), 1669–1685. https://doi.org/10.2307/2154964 doi: 10.2307/2154964
[34]	D. Li, S. Guo, Periodic traveling waves in a reaction-diffusion model with chemotaxis and nonlocal delay effect, J. Math. Anal. Appl., 467 (2018), 1080–1099. https://doi.org/10.1016/j.jmaa.2018.07.050 doi: 10.1016/j.jmaa.2018.07.050
[35]	T. Xu, S. Ji, R. Huang, M. Mei, J. Yin, Theoretical and numerical studies on global stability of traveling waves with oscillations for time-delayed nonlocal dispersion, Int. J. Numer. Anal. Modeling, 17 (2020), 68–86.

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)