广西师范大学学报（自然科学版） ›› 2026, Vol. 44 ›› Issue (6): 122-133.doi: 10.16088/j.issn.1001-6600.2026031303

• 数学 • 上一篇    下一篇

具有交叉扩散捕食系统的周期解

袁海龙*, 党楠, 刘淑淑   

  1. 陕西科技大学 数学与数据科学学院, 陕西 西安 710021
  • 收稿日期:2025-11-18 修回日期:2026-03-08 出版日期:2026-11-05 发布日期:2026-11-10
  • 通讯作者: 袁海龙(1989—),男,陕西西安人,陕西科技大学讲师,博士。E-mail: yuanhailong@sust.edu.cn
  • 基金资助:
    国家自然科学基金(11901370);陕西省自然科学基础研究计划项目(2019JQ-516);陕西省教育厅专项科研计划项目(19JK0142);陕西省科协人才托举项目(20200508)

Periodic solutions of a predator system with cross-diffusion

Yuan Hailong*, Dang Nan, Liu Shushu   

  1. School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi’an Shaanxi 710021, China
  • Received:2025-11-18 Revised:2026-03-08 Online:2026-11-05 Published:2026-11-10

摘要： 本文研究一类具有Smith增长率与交叉扩散捕食系统的动力学行为。首先,分析系统在常微分情形下正平衡点的存在性与稳定性,并证明Hopf分岔的存在性;其次,利用规范型理论与中心流形定理,进一步分析系统Hopf分岔的方向及分岔所产生周期解的稳定性;再次,结合Floquet理论与特征值扰动方法,推导交叉扩散参数诱导周期解发生图灵不稳定性的条件,结果表明,当交叉扩散效应超越临界值时,稳定的空间周期解将失稳,从而激发空间斑图形成;最后,应用Matlab进行数值模拟,验证理论分析结果的正确性。

关键词: 捕食系统, 交叉扩散, 周期解, 图灵不稳, Hopf 分岔

Abstract: The dynamical behavior of a predator system with Smith growth and cross-diffusion is investigated. Firstly, the direction of Hopf bifurcation and the stability of periodic solutions for the ODE are analyzed using normal form theory and center manifold theorem. Secondly, an analytical criterion for cross-diffusion inducing the Turing instability of the periodic solutions is derived through Floquet theory and eigenvalue perturbation method. The results show that when cross-diffusion coefficients exceed a critical value, the originally stable spatially homogeneous periodic solutions lose stability under spatial perturbations, leading to spatial pattern formation. Finally, numerical simulations are performed using Matlab to verify the theoretical results.

Key words: predator system, cross-diffusion, periodic solution, Turing instability, Hopf bifurcation

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