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Spreading Speeds of Nonlocal KPP Equations in Heterogeneous Media

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Abstract

This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media. The kernel K is assumed to depend on the media. First, we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues. Second, we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.

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References

  1. Aronson, D. G., Weinberger, H. F.: Multidimensional nonlinear diffusion arising in population genetics. Advances in Mathematics, 30(1), 33–76 (1978)

    Article  MathSciNet  Google Scholar 

  2. Bates, P. W., Fife, P. C., Ren, X., et al.: Traveling waves in a convolution model for phase transitions. Archive for Rational Mechanics and Analysis, 138(2), 105–136 (1997)

    Article  MathSciNet  Google Scholar 

  3. Berestycki, H., Coville, J., Vo, H. H.: Persistence criteria for populations with non-local dispersion. Journal of Mathematical Biology, 72(7), 1693–1745 (2016)

    Article  MathSciNet  Google Scholar 

  4. Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Communications on Pure and Applied Mathematics, 55, 0949–1032 (2002)

    Article  MathSciNet  Google Scholar 

  5. Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: II-Biological invasions and pulsating traveling fronts. J. Math. Pures Appl., 84, 1101–1146 (2005)

    Article  MathSciNet  Google Scholar 

  6. Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems. I. Periodic framework. Journal of the European Mathematical Society, 7(2), 173–213 (2005)

    Article  MathSciNet  Google Scholar 

  7. Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems. II. General domains. J. Amer. Math. Soc., 23, 1–34 (2010)

    Article  MathSciNet  Google Scholar 

  8. Berestycki, H., Nadin, G.: Spreading speeds for one-dimensional monostable reaction-diffusion equations. Journal of Mathematical Physics, 53(11), 115619, 23 pp. (2012)

    Article  MathSciNet  Google Scholar 

  9. Berestycki, H., Nadin, G.: Asymptotic spreading for general heterogeneous Fisher-KPP type equations. Memoirs Amer. Math. Soc., (2019)

  10. Bouin, E., Garnier, J., Henderson, C., et al.: Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels. SIAM Journal on Mathematical Analysis, 50(3), 3365–3394 (2018)

    Article  MathSciNet  Google Scholar 

  11. Coville, J., Dupaigne, L.: On a non-local equation arising in population dynamics. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 137(4), 727–755 (2007)

    Article  MathSciNet  Google Scholar 

  12. Coville, J.: Harnack type inequality for positive solution of some integral equation. Annali di Matematica Pura ed Applicata, 191(3), 503–528 (2012)

    Article  MathSciNet  Google Scholar 

  13. Ding, W., Liang, X.: Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media. SIAM Journal on Mathematical Analysis, 47, 855–896 (2015)

    Article  MathSciNet  Google Scholar 

  14. Fisher, R. A.: The wave of advance of advantageous genes. Annals of Eugenics, 7(4), 355–369 (1937)

    Article  Google Scholar 

  15. Gartner, J., Freidlin, M. I.: On the propagation of concentration waves in periodic and random media. Sov. Math. Dokl., 20, 1282–1286 (1979)

    MATH  Google Scholar 

  16. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin, 2006

    Google Scholar 

  17. Hudson, W., Zinner, B.: Existence of traveling waves for a generalized discrete Fisher’s equation. Comm. Appl. Nonlinear Anal., 1, 23–46 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Kolmogorov, A. N., Petrovsky, I. G., Piskunov, N. S.: Etude de l equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Moscow Univ. Math. Bull, 1, 1–25 (1937)

    Google Scholar 

  19. Liang, X., Zhao, X.: Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal., 259, 857–903 (2010)

    Article  MathSciNet  Google Scholar 

  20. Liang, X., Zhou, T.: Spreading speeds of KPP-type lattice systems in heterogeneous media. Communications in Contemporary Mathematics, 22(1), 1850083, 42 pp. (2020)

    Article  MathSciNet  Google Scholar 

  21. Liang, X., Zhou, T.: Spreading speeds of nonlocal KPP equations in almost periodic media. Journal of Functional Analysis, 279, 108723, 58 pp. (2020)

    Article  MathSciNet  Google Scholar 

  22. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, New York, Springer, 1983

    Book  Google Scholar 

  23. Shen, W.: Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models. Trans. Amer. Math. Soc., 362, 5125–5168 (2010)

    Article  MathSciNet  Google Scholar 

  24. Shen, W.: Existence of generalized traveling waves in time recurrent and space periodic monostable equations. J. Appl. Anal. Comput., 1, 69–93 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Shen, W., Zhang, A.: Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats. Journal of Differential Equations, 249(4), 747–795 (2010)

    Article  MathSciNet  Google Scholar 

  26. Shen, W., Zhang, A.: Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats. Proceedings of the American Mathematical Society, 140(5), 1681–1696 (2012)

    Article  MathSciNet  Google Scholar 

  27. Shigesada, N., Kawasaki, K., Teramoto, E.: Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol., 30, 143–160 (1986)

    Article  MathSciNet  Google Scholar 

  28. Souganidis, P. E., Tarfulea, A.: Front propagation for integro-differential KPP reaction—diffusion equations in periodic media. Nonlinear Differential Equations and Applications NoDEA, 26(4), Paper No. 29, 41 pp. (2019)

  29. Weinberger, H.: On spreading speeds and traveling waves forgrowthand migration models in a periodic habitat. J. Math. Biol., 45, 511–548 (2002)

    Article  MathSciNet  Google Scholar 

  30. Xin, J.: Existence of planar flame fronts in convective-diffusive periodic media. Archive for Rational Mechanics and Analysis, 121, 205–233 (1992)

    Article  MathSciNet  Google Scholar 

  31. Zhang, G. B., Li, W. T., Wang, Z. C.: Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity. Journal of Differential Equations, 252(9), 5096–5124 (2012)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences and Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, 230026, P. R. China

    Xing Liang & Tao Zhou

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  1. Xing Liang
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  2. Tao Zhou
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Correspondence to Xing Liang.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 11971454 and 12001514), the Fundamental Research Funds for the Central Universities and the Japan Society for the Promotion of Science

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Liang, X., Zhou, T. Spreading Speeds of Nonlocal KPP Equations in Heterogeneous Media. Acta. Math. Sin.-English Ser. 38, 161–178 (2022). https://doi.org/10.1007/s10114-022-0452-8

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  • DOI: https://doi.org/10.1007/s10114-022-0452-8

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