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Exponential stability of traveling wavefronts for a system modeling the geographic spread of black-legged tick Ixodes scapularis

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Abstract

This paper is concerned with the exponential stability of traveling wavefronts for a system modeling the geographic spread of black-legged tick Ixodes scapularis. It is shown in a recent work (Lai and Zou in J Differ Equ 269: 6400-6421, 2020) that this system admits traveling wavefronts when the basic reproduction number is greater than one and the wave speeds are larger than or equal to the asymptotic speed of spread. In this paper, we further study the asymptotic stability of traveling wavefronts. Applying the techniques of weighted energy method and the comparison principle, we prove that the traveling wavefronts with relatively large speed are exponentially stable, when the initial perturbation around the traveling wavefronts decays exponentially as \(x\rightarrow -\infty \), but can be arbitrarily large in other locations.

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References

  1. Bacon, R.M., Kugeler, K.J., Griffith, K.S., Mead, P.S.: Lyme disease-United States, 2003–2005. J. Am. Med. Assoc. 298, 278–279 (2007)

    Article  Google Scholar 

  2. Dennis, D.T., Nekomoto, T.S., Victor, J.C., Paul, W.S., Piesman, J.: Reported distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the United States. J. Med. Entomol. 35, 629–638 (1998)

    Article  Google Scholar 

  3. Caraco, S.G., Glavanakov, S., Chen, G., Flaherty, J.E., Ohsumi, T.K., Szymanski, B.K.: Stage-structured infection transmission and a spatial epidemic: a model for Lyme disease. Am. Nat. 160, 348–359 (2002)

    Article  Google Scholar 

  4. Cortinas, M.R., Kitron, U.: County-level surveillance of white-tailed deer infestation by Ixodes scapularis and Dermacentor albipictus (Acari: Ixodidae) along the Illinois River. J. Med. Entomol. 4(3), 810–819 (2006)

    Article  Google Scholar 

  5. Hamer, S., Hickling, G., Walker, E., Tsao, J.I.: Invasion of the Lyme disease vector Ixodes scapularis: implications for Borrelia burgdorferi endemicity. EcoHealth 7, 47–63 (2010)

    Article  Google Scholar 

  6. Madhav, N.K., Brownstein, J.S., Tsao, J.I., Fish, D.: A dispersal model for the range expansion of blacklegged tick (Acari: Ixodidae). J. Med. Entomol. 41, 842–852 (2004)

    Article  Google Scholar 

  7. Gourley, S.A., Lai, X., Shi, J., Wang, W., Xiao, Y., Zou, X.: Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis: analysis of a spatially nonlocal model. Math. Biosci. Eng. 15, 1033–1054 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lai, X., Zou, X.: Minimal wave speed and spread speed in a system modelling the geographic spread of black-legged tick Ixodes scapularis. J. Differ. Equ. 269, 6400–6421 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fang, J., Wei, J., Zhao, X.-Q.: Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system. J. Differ. Equ. 245, 2749–2770 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liang, X., Zhao, X.-Q.: Asymptotic speeds of spread and travelling waves for monotone semiflows with application. Commun. Pure Appl. Math. 60, 1–40 (2007)

    Article  MATH  Google Scholar 

  11. Chen, X., Guo, J.-S.: Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations. J. Differ. Equ. 184, 549–569 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, R., Mei, M., Zhang, K.J., Zhang, Q.F.: Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete Contin. Dyn. Syst. 36, 1331–1353 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lin, C.-K., Lin, C.-T., Lin, Y.-P., Mei, M.: Exponential stability of nonmonotone traveling waves for Nicholson’s blowflies equation. SIAM J. Math. Anal. 46, 1053–1084 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mei, M., Lin, C.-K., Lin, C.-T., So, J.W.-H.: Traveling wavefronts for time-delayed reaction-diffusion equation: (I) local nonlinearity. J. Differ. Equ. 247, 495–510 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mei, M., Lin, C.-K., Lin, C.-T., So, J.W.-H.: Traveling wavefronts for time-delayed reaction-diffusion equation: (II) nonlocal nonlinearity. J. Differ. Equ. 247, 511–529 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mei, M., Ou, C.-H., Zhao, X.-Q.: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations. SIAM J. Math. Anal. 42, 2762–2790 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mei, M., Zhang, K.J., Zhang, Q.F.: Global stability of traveling waves with oscillations for Nicholson’s blowflies equation. Int. J. Numer. Anal. Model. 16, 375–397 (2019)

    MathSciNet  Google Scholar 

  18. Sattinger, D.: On the stability of traveling waves. Adv. Math. 22, 312–355 (1976)

    Article  MATH  Google Scholar 

  19. Smith, H.L., Zhao, X.-Q.: Global asymptotical stability of traveling waves in delayed reaction-diffusion equations. SIAM J. Math. Anal. 31, 514–534 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tian, G., Zhang, G.-B.: Stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system. J. Math. Anal. Appl. 447, 222–242 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yu, Z., Mei, M.: Uniqueness and stability of traveling waves for cellular neural networks with multiple delays. J. Differ. Equ. 260, 241–267 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, G.-B.: Global stability of non-monotone traveling wave solutions for a nonlocal dispersal equation with time delay. J. Math. Anal. Appl. 475, 605–627 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mei, M., So, J.W.-H., Li, M.Y., Shen, S.S.P.: Asymptotic stability of traveling waves for the Nicholson’s blowflies equation with diffusion. Proc. Roy. Soc. Edinburgh Sect. A 134, 579–594 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wu, S.-L., Li, W.-T., Liu, S.: Asymptotic stability of traveling wave fronts in nonlocal reaction-diffusion equations with delay. J. Math. Anal. Appl. 360, 439–458 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wu, S.-L., Zhao, H.-Q., Liu, S.-Y.: Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability. Z. Angew. Math. Phys. 62, 377–397 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hao, Y.-C., Zhang, G.-B.: The dynamics of traveling wavefronts for a nonlocal delay competition system with local vs. nonlocal diffusions. Commun. Nonlinear Sci. Numer. Simul. 110, 106381 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hsu, C.-H., Yang, T.-S., Yu, Z.: Existence and exponential stability of traveling waves for delayed reaction-diffusion systems. Nonlinearity 31, 838–863 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hsu, C.-H., Lin, J.-J., Wu, S.-L.: Existence and stability of traveling wavefronts for discrete three species competitive-cooperative systems. Math. Biosci. Eng. 16, 4151–4181 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, C., Li, M., Weng, P.: Existence and stability of traveling wave fronts for a reaction-diffusion system with spatio-temporal nonlocal effect. Z. Angew. Math. Mech. 62, 1–24 (2017)

    Google Scholar 

  30. Wu, S.-L., Li, W.-T., Liu, S.-Y.: Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete Contin. Dyn. Syst. Ser. B 17, 347–366 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Yang, Y.-R., Li, W.-T., Wu, S.-L.: Exponential stability of traveling fronts in a diffusion epidemic system with delay. Nonlinear Anal. RWA 12, 1223–1234 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yu, Z., Xu, F., Zhang, W.G.: Stability of invasion traveling waves for a competition system with nonlocal dispersals. Appl. Anal. 96, 1107–1125 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yu, Z., Pei, J.W.: Stability of traveling wave fronts for a cooperative system with nonlocal dispersals. Jpn. J. Ind. Appl. Math. 35, 817–834 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, G.-B., Dong, F.-D., Li, W.-T.: Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete Contin. Dyn. Syst. Ser. B 24, 1511–1541 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Su, S., Zhang, G.-B.: Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity. Electron. J. Differ. Equ. 2020, 46 (2020)

    MathSciNet  MATH  Google Scholar 

  36. Xu, T.Y., Ji, S., Huang, R., Mei, M., Yin, J.X.: Theoretical and numerical studies on global stability of traveling waves with oscillation for time-delayed nonlocal dispersion equations. Int. J. Numer. Anal. Model. 17, 68–86 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Martin, R., Smith, H.: Abstract functional differential equations and reaction-diffusion systems. Trans. Am. Math. Soc. 321, 1–44 (1990)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are very grateful to the anonymous referees for their careful reading and helpful suggestions which led to an improvement of my original manuscript. This research was partially supported by NSF of China [12261081] and NSF of Gansu Province [21JR7RA121].

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  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, 730070, China

    Yu-Cai Hao, Guo-Bao Zhang & Juan He

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  1. Yu-Cai Hao
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  2. Guo-Bao Zhang
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Correspondence to Guo-Bao Zhang.

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Hao, YC., Zhang, GB. & He, J. Exponential stability of traveling wavefronts for a system modeling the geographic spread of black-legged tick Ixodes scapularis. Z. Angew. Math. Phys. 74, 116 (2023). https://doi.org/10.1007/s00033-023-02014-9

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  • DOI: https://doi.org/10.1007/s00033-023-02014-9

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