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

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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.

Keywords

nonlocal diffusion / nonlocal delay / traveling wavefront solution / perturbation method

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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

[3] V. Hutson, M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 221-232. https://doi.org/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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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.