Journal of East China Normal University(Natural Science) >

2020 , Vol. 2020 >Issue 4: 18 - 25

DOI: https://doi.org/10.3969/j.issn.1000-5641.201911019

Mathematics

Travelling wave solutions of a diffusive single species model with a strong generic delay kernel

  • YANG Gaoxiang
Expand
  • 1. School of Mathematics and Statistics, Ankang University, Ankang, Shaanxi 725000, China;
    2. Institute of Mathematics and Applied Mathematics, Ankang University, Ankang, Shaanxi 725000, China

Received date: 2019-04-29

  Online published: 2020-07-20

Fold

Abstract

In this paper, the existence of travelling wave solutions of a diffusive single species model with a strong generic delay kernel is established in two steps. Firstly, in the case of a species model without time delay, the existence of travelling wave solutions of the species model is obtained by using qualitative theories of ordinary differential equations. Secondly, when the time delay is greater than zero and sufficiently small, the existence of travelling wave solutions of the species model is verified by using linear chain techniques and the geometric singular perturbation theory.

Key words： travelling wave solutions; strong generic delay kernel; geometric singular perturbation theory

Cite this article

YANG Gaoxiang . Travelling wave solutions of a diffusive single species model with a strong generic delay kernel[J]. Journal of East China Normal University(Natural Science), 2020 , 2020(4) : 18 -25 . DOI: 10.3969/j.issn.1000-5641.201911019

References

[1] WU J. Theory and Applications of Partial Functional Differential Equations [M]. New York: Springer-Verlag, 1996.
[2] MURRAY J D. Mathematical Biology: Spatial Models and Biomedical Applications [M]. New York: Springer, 2003.
[3] GOURLEY S A, CHAPLAM M A J, DAVIDSON F A. Spatiotemporal pattern formation in a nonlocal reaction-diffusion equation [J]. Dynamical Systems: An International Journal, 2001, 16: 173-192. DOI: 10.1080/14689360116914.
[4] ASHWIN P, BARTUCCELLI M V, BRIDGES T J, et al. Travelling fronts for the Kpp equation with spatiotemporal delay [J]. Z ANGEW MATH PHYS, 2002, 53: 103-122. DOI: 10.1007/s00033-002-8145-8.
[5] WANG Y F, YIN J X. Traveling waves for a biological reaction diffusion model with spatiotemporal delay [J]. J Math Anal Appl, 2007, 325: 1400-1409. DOI: 10.1016/j.jmaa.2006.02.077.
[6] WANG Z C, LI W T, RUAN S G. Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays [J]. J Differential Equation, 2006, 222: 185-232. DOI: 10.1016/j.jde.2005.08.010.
[7] WU C, LI M, WENG P. Existence and stability of traveling wave fronts for a reaction-diffusion system with spatio-temporal nonlocal effect [J]. ZAMM‐Journal of Applied Mathematics and Mechanics, 2017, 97(12): 1555-1578. DOI: 10.1002/zamm.201600170.
[8] ZHANG H T, LI L. Traveling wave fronts of a single species model with cannibalism and nonlocal effect [J]. Chaos, Solitons & Fractals, 2018, 108: 148-153.
[9] ZUO W J, SHI J P. Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay [J]. Communications on Pure & Applied Analysis, 2018, 17(3): 1179-1200.
[10] LI W T, RUAN S, WANG Z C. On the diffusive Nicholson’s blowflies equation with nonlocal delay [J]. J Nonlinear Science, 2007, 17: 505-525. DOI: 10.1007/s00332-007-9003-9.
[11] ZHANG J, PENG Y. Travelling waves of the diffusive Nicholson’s blowflies equation with strong generic delay kernel and non-local effect [J]. Nonlinear Analysis: Theory, Methods & Applications, 2008, 68(5): 1263-1270.
[12] ZHANG C, YAN X. Wavefront solutions in diffusive Nicholson’s blowflies equation with nonlocal delay [J]. Applied Mathematics and Mechanics, 2010, 31(3): 385-392. DOI: 10.1007/s10483-010-0311-x.
[13] BRITTON N F. Aggregation and the competitive exclusion principle [J]. Journal of Theoretical Biology, 1989, 136: 57-66. DOI: 10.1016/S0022-5193(89)80189-4.
[14] BRITTON N F. Spatial structures and periodic traveling wave in an integro-differential reaction diffusion population model [J]. SIAM Journal of Applied Mathematics, 1990, 50: 1663-1688. DOI: 10.1137/0150099.
[15] FENICHEL N. Geometric singular perturbation theory of for ordinary differential equations [J]. J Differential Equation, 1979, 31: 53-98. DOI: 10.1016/0022-0396(79)90152-9.
Options
Outlines

/

〈 〉