Skip to main content
SpringerLink
Log in

Spreading Speed in the Fisher-KPP Equation with Nonlocal Delay

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

This paper is concerned with the Fisher-KPP equation with diffusion and nonlocal delay. Firstly, we establish the global existence and uniform boundedness of solutions to the Cauchy problem. Then, we establish the spreading speed for the solutions with compactly supported initial data. Finally, we investigate the long time behavior of solutions by numerical simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Britton N F. Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model. SIAM J Appl Math, 1990, 50(6): 1663–1688

    Google Scholar 

  2. Gourley S A, Britton N F. A predator-prey reaction-diffusion system with nonlocal effects. J Math Biol, 1996, 34: 297–333

    Google Scholar 

  3. Gourley S A, Chaplain M A J. Travelling fronts in a food-limited population model with time delay. Proc Roy Soc Edinburgh Sect A, 2002, 132: 75–89

    Google Scholar 

  4. Gourley S A, Ruan S. Convergence and travelling fronts in functional differential equations with nonlocal terms: a competition model. SIAM J Math Anal, 2003, 35: 806–822

    Google Scholar 

  5. Gourley S A, So J H W, Wu J. Non-locality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics. J Math Sci, 2004, 124: 5119–5153

    Google Scholar 

  6. Gourley S A, Wu J. Delayed non-local diffusive systems in biological invasion and disease spread. Fields Inst Commun, 2006, 48: 137–200

    Google Scholar 

  7. Tian G, Wang Z-C. Spreading speed in a Food-limited population model with nonlocal delay. Appl Math Lett, 2020, 102: 106121

    Google Scholar 

  8. Zou X, Wu S. Traveling waves in a nonlocal dispersal SIR epidemic model with delay and nonlinear incidence. Acta Mathematica Scientia, 2018, 38A(3): 496–513

    Google Scholar 

  9. Jin Y, Zhao X-Q. Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure. SIAM J Math Anal, 2009, 40(6): 2496–2516

    Google Scholar 

  10. Liang X, Zhao X-Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm Pure Appl Math, 2007, 60: 1–40

    Google Scholar 

  11. Weng P, Zhao X-Q. Spatial dynamics of a nonlocal and delayed population model in a periodic habitat. Discrete Contin Dyn Syst, 2011, 29(1): 343–366

    Google Scholar 

  12. Hamel F, Ryzhik L. On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds. Nonlinearity, 2014, 27: 2735–3753

    Google Scholar 

  13. Lin G. Spreading speed of the delayed Fisher equation without quasimonotonicity. Nonlinear Anal Real World Appl, 2011, 12(6): 3713–3718

    Google Scholar 

  14. Liu X-L, Pan S-X. Spreading speed in a nonmonotone equation with dispersal and delay. Mathematics, 2019, 7(3): 291

    Google Scholar 

  15. Alfaro M, Coville J. Rapid traveling waves in the nonlocal Fisher equation connect two unstable states. Appl Math Lett, 2012, 25: 2095–2099

    Google Scholar 

  16. Alfaro M, Coville J, Raoul G. Traveling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. Comm Partial Differential Equations, 2013, 38: 2126–2154

    Google Scholar 

  17. Bao X, Li W-T. Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats. Nonlinear Anal Real World Appl, 2020, 51: 102975, 26

    Google Scholar 

  18. Berestycki H, Nadin G, Perthame B, Ryzhik L. The non-local Fisher-KPP equation: traveling waves and steady states. Nonlinearity, 2009, 22(12): 2813–2844

    Google Scholar 

  19. Ducrot A, Nadin G. Asymptotic behaviour of travelling waves for the delayed Fisher- KPP equation. J Differ Equ, 2014, 256: 3115–3140

    Google Scholar 

  20. Fang J, Zhao X-Q. Monotone wave fronts of the nonlocal Fisher-KPP equation. Nonlinearity, 2011, 24: 3043–3054

    Google Scholar 

  21. Gourley S A. Traveling front solutions of a nonlocal Fisher equation. J Math Biol, 2000, 41: 272–284

    Google Scholar 

  22. Gomez A, Trofimchuk S. Monotone traveling wavefronts of the KPP-Fisher delayed equation. J Differ Equ, 2011, 250: 1767–1787

    Google Scholar 

  23. Hasik K, Kopfová J, Nábělková P, Trofimchuk S. Traveling waves in the nonlocal KPP-Fisher equation: different roles of the right and the left interactions. J Differ Equ, 2016, 261: 1203–1236

    Google Scholar 

  24. Hasik K, Trofimchuk S. Slowly oscillating wavefronts of the KPP-Fisher delayed equation. Discret Contin Dyn Syst, 2014, 34: 3511–3533

    Google Scholar 

  25. Han B-S, Wang Z-C, Feng Z-S. Traveling waves for the nonlocal diffusive single species model with Allee effect. J Math Anal Appl, 2016, 443: 243–264

    Google Scholar 

  26. Kwong M-K, Ou C-H. Existence and nonexistence of monotone traveling waves for the delayed Fisher equation. J Differ Equ, 2010, 249: 728–745

    Google Scholar 

  27. Liu K, Yang Y, Yang Y. Stability of monostable waves for a nonlocal equation with delay and without quasi-monotonicity. Acta Mathematica Scientia, 2019, 39B(6): 1589–1604

    Google Scholar 

  28. Nadin G, Perthame B, Tang M. Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation. C R Math Acad Sci Paris, 2011, 349(9/10): 553–557

    Google Scholar 

  29. Schaff K. Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans Am Math Soc, 1987, 302: 587–625

    Google Scholar 

  30. Wang Z-C, Li W-T. Monotone travelling fronts of a food-limited population model with nonlocal delay. Nonlinear Anal Real World Appl, 2007, 8: 699–712

    Google Scholar 

  31. Wang Z-C, Li W-T, Ruan S. Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. J Differ Equ, 2006, 222: 185–232

    Google Scholar 

  32. Wu J, Zou X. Traveling wave fronts of reaction-diffusion systems with delay. J Dynam Diff Eqns, 2001, 13: 651–687

    Google Scholar 

  33. Yang Z, Zhang G. Global stability of traveling wavefronts for nonlocal reaction-diffusion equations with time delay. Acta Mathematica Scientia, 2018, 38B(1): 289–302

    Google Scholar 

  34. Daners D, Koch Medina P. Abstract Evolution Equations, Periodic Problems and Applications. Longman, Harlow, UK: Pitman Research Notes in Mathematics Series, 279, 1992

    Google Scholar 

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

    Google Scholar 

  36. Wu J. Theory and Applications of Partial Functional Differential Equations. New York, NY: Springer-Verlag, 1996

    Google Scholar 

  37. Aronson DG, Weinberger HF. Multidimensional nonlinear diffusions arising in population genetics. Adv Math, 1978, 30: 33–76

    Google Scholar 

  38. Gourley S A, Kuang Y. Wavefronts and global stability in a time-delayed population model with stage structure. R Soc Lond Proc Ser A Math Phys Eng Sci, 2003, 459: 1563–1579

    Google Scholar 

  39. Gourley S A, So J H W. Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain. J Math Biol, 2002, 44: 49–78

    Google Scholar 

  40. Xu D-S, Zhao X-Q. A nonlocal reaction-diffusion population model with stage structure. Can Appl Math Q, 2003, 11: 303–319

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Ge Tian  (田歌) & Zhicheng Wang  (王智诚)

  2. School of Information Science and Engineering, Lanzhou University, Lanzhou, 730000, China

    Haoyu Wang  (王浩雨)

Authors
  1. Ge Tian  (田歌)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haoyu Wang  (王浩雨)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhicheng Wang  (王智诚)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhicheng Wang  (王智诚).

Additional information

Supported by National Natural Science Foundation of China (12071193,11731005).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, G., Wang, H. & Wang, Z. Spreading Speed in the Fisher-KPP Equation with Nonlocal Delay. Acta Math Sci 41, 875–886 (2021). https://doi.org/10.1007/s10473-021-0314-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-021-0314-y

Key words

2010 MR Subject Classification

Search

Navigation