一类具有合作与自限效应的 Extended Fisher-Kolmogorov 系统的定态分歧

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1432-1443.

一类具有合作与自限效应的 Extended Fisher-Kolmogorov 系统的定态分歧

朱超¹(),郝清明¹(),潘志刚¹(),王艳华^2,^*()

¹西南交通大学数学学院成都 611756
²成都锦城学院通识教育学院成都 611731

收稿日期:2024-11-12 修回日期:2025-03-28 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 王艳华,E-mail: yhwang2021@yeah.net
作者简介:朱超, E-mail: zc1209931603@163.com|郝清明,E-mail: persist18784863089@163.com|潘志刚,E-mail: panzhigang@swjtu.edu.cn
基金资助:
国家自然科学基金(11901408);四川省自然科学青年基金(22NSFSC16338);中央高校理科创新培育项目(2682022ZTPY063)

Steady-State Bifurcation to a Class of Extended Fisher-Kolmogorov System with Cooperative and Self-Limiting Effects

Chao Zhu¹(),Qingming Hao¹(),Zhigang Pan¹(),Yanhua Wang^2,^*()

¹School of Mathematics, Southwest Jiaotong University, Chengdu 611756
²Chengdu Jincheng College, Chengdu 611731

Received:2024-11-12 Revised:2025-03-28 Online:2025-10-26 Published:2025-10-14
Supported by:
NSFC(11901408);Sichuan Provincial Natural Science Youth Fund(22NSFSC16338);Central University Basic Research Innovation Project(2682022ZTPY063)

摘要/Abstract

摘要：

该文研究了一类具有合作与自限效应的 Extended Fisher-Kolmogorov 系统的定态分歧. 采用拓展的 Lyapunov-Schmidt 约化方法和线性全连续场谱分解定理, 在 Dirichlet 边界条件下系统发生分歧, 给出了分歧解的具体表达式并讨论了其正则性, 揭示了生物种群出现周期性波动. 在 Neumann 边界条件下, 得到了发生超临界分歧与次临界分歧的完整判据, 讨论了分歧解的正则性. 当系统发生超临界分歧时, 种群数量缓慢扩大; 当系统发生次临界分歧时, 种群数量先急剧下降后逐渐稳定.

关键词: Extended Fisher-Kolmogorov 系统, Dirichlet 边界, Neumann 边界, 定态分歧, Lyapunov-Schmidt 约化, 正则性

Abstract:

This paper investigates the steady-state bifurcation to a class of Extended Fisher-Kolmogorov system with cooperative and self-limiting effects. By using the extended Lyapunov-Schmidt reduction method and the spectral decomposition theorem for linear completely continuous fields, the bifurcation of the system under Dirichlet boundary conditions are analyzed. Explicit expressions for the bifurcating solutions are provided, and the regularity of these solutions are discussed, revealing that biological populations exhibit periodic fluctuations. Under Neumann boundary conditions, complete criteria for supercritical and subcritical bifurcations are obtained, and the regularity of the bifurcating solution is explored. When the system undergoes a supercritical bifurcation, the population size gradually expands; when it undergoes a subcritical bifurcation, the population size initially plummets before gradually stabilizing.

Key words: Extended Fisher-Kolmogorov system, Dirichlet boundary, Neumann boundary, steady-state bifurcation, Lyapunov-Schmidt reduction, regularity

中图分类号:

O175.29

朱超, 郝清明, 潘志刚, 王艳华. 一类具有合作与自限效应的 Extended Fisher-Kolmogorov 系统的定态分歧[J]. 数学物理学报, 2025, 45(5): 1432-1443.

Chao Zhu, Qingming Hao, Zhigang Pan, Yanhua Wang. Steady-State Bifurcation to a Class of Extended Fisher-Kolmogorov System with Cooperative and Self-Limiting Effects[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1432-1443.

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