Stability switching and hopf bifurcation in a two-delay predation system with diffusion and network coupling

doi:10.1088/1674-1056/ae5c8a

中国物理B ›› 2026, Vol. 35 ›› Issue (8): 80204-080204.doi: 10.1088/1674-1056/ae5c8a

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Stability switching and hopf bifurcation in a two-delay predation system with diffusion and network coupling

Wenjie Yang(杨文杰)^1,2,3, Jingxin Hu(胡静欣)^1,4, Qianqian Zheng(郑前前)¹, Jianwei Shen(申建伟)^4,†, and Yiping Lin(林怡平)⁵

1 School of Science, Xuchang University, Xuchang 461000, China;
2 Complex Systems Research Center, Shanxi University, Taiyuan 030006, China;
3 Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis for Disease Control and Prevention, Taiyuan 030006, China;
4 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China;
5 Department of Applied Mathematics, Kunming University of Science and Technology, Kunming 650500, China

收稿日期:2026-01-18 修回日期:2026-04-02 接受日期:2026-04-08 发布日期:2026-07-23
通讯作者: Jianwei Shen E-mail:xcjwshen@gmail.com
基金资助:
This work is supported by the Key Scientific Research Project of Colleges and Universities in Henan Province(Grant No. 24B110017) and the Natural Science Foundation of Henan Province(Grant No. 242300420661).

Stability switching and hopf bifurcation in a two-delay predation system with diffusion and network coupling

Wenjie Yang(杨文杰)^1,2,3, Jingxin Hu(胡静欣)^1,4, Qianqian Zheng(郑前前)¹, Jianwei Shen(申建伟)^4,†, and Yiping Lin(林怡平)⁵

1 School of Science, Xuchang University, Xuchang 461000, China;
2 Complex Systems Research Center, Shanxi University, Taiyuan 030006, China;
3 Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis for Disease Control and Prevention, Taiyuan 030006, China;
4 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China;
5 Department of Applied Mathematics, Kunming University of Science and Technology, Kunming 650500, China

Received:2026-01-18 Revised:2026-04-02 Accepted:2026-04-08 Published:2026-07-23
Contact: Jianwei Shen E-mail:xcjwshen@gmail.com
Supported by:
This work is supported by the Key Scientific Research Project of Colleges and Universities in Henan Province(Grant No. 24B110017) and the Natural Science Foundation of Henan Province(Grant No. 242300420661).

摘要/Abstract

摘要： This paper studies stability switching and Hopf bifurcation in a two-delay predation system with diffusion and network coupling. By using the Kronecker-product formulation and Laplacian-mode decomposition, the high-dimensional characteristic equation of the coupled system is reduced to a family of mode-dependent characteristic equations. Based on this reduction, explicit stability-switching curves are derived in the $(\tau_1,\tau_2)$-plane, and the corresponding stability region of the positive equilibrium is determined. Furthermore, by applying the Hassard method and center-manifold reduction, the direction of the Hopf bifurcation, the stability of the bifurcating periodic solutions, and the variation of their periods are obtained. Numerical examples are presented to verify the theoretical results and illustrate the temporal and spatiotemporal dynamics generated by the two delays under network coupling. These results provide an effective analytical framework for studying delay-induced stability switching and oscillatory behavior in networked predator-prey systems.

关键词: predation system, network coupling, stability switching, time delays, Hopf bifurcation

Abstract: This paper studies stability switching and Hopf bifurcation in a two-delay predation system with diffusion and network coupling. By using the Kronecker-product formulation and Laplacian-mode decomposition, the high-dimensional characteristic equation of the coupled system is reduced to a family of mode-dependent characteristic equations. Based on this reduction, explicit stability-switching curves are derived in the $(\tau_1,\tau_2)$-plane, and the corresponding stability region of the positive equilibrium is determined. Furthermore, by applying the Hassard method and center-manifold reduction, the direction of the Hopf bifurcation, the stability of the bifurcating periodic solutions, and the variation of their periods are obtained. Numerical examples are presented to verify the theoretical results and illustrate the temporal and spatiotemporal dynamics generated by the two delays under network coupling. These results provide an effective analytical framework for studying delay-induced stability switching and oscillatory behavior in networked predator-prey systems.

Key words: predation system, network coupling, stability switching, time delays, Hopf bifurcation

中图分类号: (Function theory, analysis)

02.30.-f

02.30.Ks (Delay and functional equations) 02.30.Oz (Bifurcation theory) 02.30.Sa (Functional analysis)

Wenjie Yang(杨文杰), Jingxin Hu(胡静欣), Qianqian Zheng(郑前前), Jianwei Shen(申建伟), and Yiping Lin(林怡平). Stability switching and hopf bifurcation in a two-delay predation system with diffusion and network coupling[J]. 中国物理B, 2026, 35(8): 80204-080204.

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Stability switching and hopf bifurcation in a two-delay predation system with diffusion and network coupling

Stability switching and hopf bifurcation in a two-delay predation system with diffusion and network coupling

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