具非局部扩散的 Leslie-Gower 种群模型的 Hopf-Hopf 分支研究

数学物理学报 ›› 2026, Vol. 46 ›› Issue (3): 1194-1217.

具非局部扩散的 Leslie-Gower 种群模型的 Hopf-Hopf 分支研究

刘玉英¹^,^*(), 段代凤², 魏俊杰³

¹ 中国矿业大学数学学院江苏徐州 221116
² 南京邮电大学理学院南京 210023
³ 哈尔滨工业大学数学学院哈尔滨 150001

收稿日期:2025-08-21 修回日期:2025-10-29 出版日期:2026-06-26 发布日期:2026-06-16
通讯作者: 刘玉英 E-mail:liuyuying@cumt.edu.cn
基金资助:
国家自然科学基金(12301643);国家自然科学基金(12171117);江苏省自然科学基金(BK20221106)

Hopf-Hopf Bifurcation Analysis in a Nonlocal Leslie-Gower Model

Yuying Liu¹^,^*(), Daifeng Duan², Junjie Wei³

¹ School of Mathematics, China University of Mining and Technology, Jiangsu Xuzhou 221116
² School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023
³ School of Mathematics, Harbin Institute of Technology, Harbin 150001

Received:2025-08-21 Revised:2025-10-29 Online:2026-06-26 Published:2026-06-16
Contact: Yuying Liu E-mail:liuyuying@cumt.edu.cn
Supported by:
NSFC(12301643);NSFC(12171117);Natural Science Foundation of Jiangsu Province(BK20221106)

摘要/Abstract

摘要：

该文研究了一类具非局部竞争和时滞的 Leslie-Gower 种群模型的动力学性质. 首先,借助模型中特征方程根的分布情况研究了各稳态解的稳定性. 其后, 以时滞为变化参数, 探究了系统 Hopf 分支的存在性; 以捕获率和时滞为双变化参数, 探究了模型中 Hopf-Hopf 分支的存在性. 通过中心流形理论, 推导出了具非局部项和时滞模型的 Hopf-Hopf 分支点附近的规范型. 最后, 利用数值模拟验证了所得的理论结果.研究发现,双参数的协同作用可激发系统产生稳定的空间非齐次周期解, 表明 Hopf-Hopf 分支点附近的动力学行为对 Leslie-Gower 系统的时空斑图形成及演化速度起着至关重要的作用.

关键词: 非局部扩散, Leslie-Gower 模型, 时滞, Hopf-Hopf 分支, 规范型.

Abstract:

In this paper, a nonlocal Leslie-Gower predator-prey model with delay and diffusion is investigated. Firstly, the local stability of the steady states in the model is studied with aid of the zeros in the characteristic equations. Besides, the existence of Hopf bifurcation is explored by taking $\tau$ as the varying parameter. Hopf-Hopf bifurcation singularity in the model was analyzed by choosing $tau$ and the capture rate $m$ as dual variable parameters. Furthermore, the normal form near the Hopf-Hopf singularity of the model is derived by using the central manifold theory. Finally, numerical simulations are carried out to illustrate the obtained theoretical results.The study reveals that the joint effect of two varying parameters can lead to stable spatially non-homogeneous periodic solutions in the system, which indicates that the dynamical behavior near the Hopf-Hopf singularity plays an important role in the formation and evolution of spatiotemporal patterns in the Leslie-Gower system.

Key words: nonlocal, Leslie-Gower model, delay, Hopf-Hopf bifurcation, normal form.

中图分类号:

O175

刘玉英, 段代凤, 魏俊杰. 具非局部扩散的 Leslie-Gower 种群模型的 Hopf-Hopf 分支研究[J]. 数学物理学报, 2026, 46(3): 1194-1217.

Yuying Liu, Daifeng Duan, Junjie Wei. Hopf-Hopf Bifurcation Analysis in a Nonlocal Leslie-Gower Model[J]. Acta mathematica scientia,Series A, 2026, 46(3): 1194-1217.

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