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一类具有强时滞核的单种群扩散模型的行波解

2020-08-04杨高翔

杨高翔

摘要: 本文中,建立了一類具有强时滞核的单种群扩散模型行波解的存在性. 首先, 在该模型没有时滞的情况下, 利用常微分方程的定性理论, 得到了该模型行波解的存在性. 然后, 在该模型中时滞非常小时, 结合线性链式法则和几何奇异摄动理论, 证明了该模型的行波解仍然存在.

关键词: 行波解; 强时滞核; 几何奇异摄动法

中图分类号: O175.26文献标志码: ADOI: 10.3969/j.issn.1000-5641.201911019

0 引言

近几十年以来, 反应扩散方程中行波解的存在性备受许多学者的关注, 尤其是生物领域中的反应扩散方程, 见文献[1-2]. 由于生物系统的复杂性, 各种因素被学者考虑到生物种群方程中去, 例如时滞因素、非局部效应等等[3-9]. 于是, 描述果蝇种群的Nicholson 方程也在考虑其耦合时空时滞[10-12] 的情况下, 该单种群模型中行波解的存在性也被证明.

早在1989 年, Britton [13-14] 给出如下一个耦合时空时滞的单种群模型:

3 总结

本文主要利用几何奇异摄动理论和链式法则讨论了一类耦合强时滞核的单种群模型行波解的存在性问题. 然而所得的结果只是在时滞量非常小的时候才能保证该行波解的存在, 至于随着时滞量的增加该行波解的波形是否会受到影响, 导致出现其他类型的行波解, 比如周期行波解[9] 等, 将在以后的工作中继续展开讨论.

[ 参 考 文 献]

[ 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]. ZANGEW 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 Nicholsons 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 Nicholsons blowflies equation with strong generic delay kernel and non-localeffect [J]. Nonlinear Analysis: Theory, Methods & Applications, 2008, 68(5): 1263-1270.

[12]ZHANG C, YAN X. Wavefront solutions in diffusive Nicholsons blowflies equation with nonlocal delay [J]. Applied Mathematics andMechanics, 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]. SIAMJournal 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.

(責任编辑: 林 磊)