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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2022, Vol. 39 ›› Issue (3): 495-501.doi: 10.3969/j.issn.1005-3085.2022.03.013

• • 上一篇    下一篇

一类具有食饵恐惧和避难所的反应扩散捕食模型的稳定性与 Hopf 分支

陈清婉,   柳文清   

  1. 闽南科技学院通识教育学院，泉州 362300
  • 出版日期:2022-06-15 发布日期:2022-08-15
  • 基金资助:
    福建省中青年教师教育科研项目 (JAT210616; JAT191035; JAT191044)；福建省教育科学“十四五”规划课题 (FJJKBK21-100).

Stability and Hopf Bifurcation of a Reaction Diffusion Predator-prey Model with Prey Fear and Refuge

CHEN Qingwan,   LIU Wenqing   

  1. College of General Education, Minnan Science and Technology Institute, Quanzhou 362300
  • Online:2022-06-15 Published:2022-08-15
  • Supported by:
    The Education and Scientific Research Project for Young and Middle-aged Teachers in Fujian Province (JAT210616；JAT191035; JAT191044); the Project of the 14$^{\rm th}$ Five-year Plan of Education Science in Fujian Province (FJJKBK21-100).

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摘要：

在捕食生态系统中，恐惧因子和食饵避难所都有重要的作用。为此，对一类带恐惧因子和食饵避难所的捕食-食饵反应扩散模型进行了研究。通过分析平衡点特征方程，得到了平衡点的局部渐近稳定性；将不受保护食饵比例作为分支参数，给出了正平衡点 Hopf 分支存在的条件。结果表明：避难所的存在会导致 Hopf 分支，产生空间齐次周期解。扩散的加入会产生新的Hopf分支点，产生空间非齐次周期解。这说明通过设立适当的食饵避难所或者减小捕食者的扩散，有助于物种共存。最后，利用 Matlab 进行数值模拟验证了所得的结论。

关键词: 恐惧因子, 避难所, Hopf 分支, 稳定性

Abstract:

Both fear factors and prey refuge have important effects in predation on the ecos-ystems. A class of reaction diffusion predator-prey models with fear effect and prey refuge is studied. We first provide the local asymptotic stability of the equilibrium point by using the linearization method and local bifurcation theory. Next, the existence of Hopf bifurcation and limit cycle is examined by choosing the ratio of un-protected prey as the bifurcation parameter. The results show that the existence of the refuge leads to Hopf bifurcation and spatially homogeneous periodic solution, and the addition of diffusion leads to new Hopf bifurcation points and inhomogeneous periodic solutions. This shows that the biological population can coexist by setting up an appropriate prey refuge or reducing the diffusion of predators. Finally, the conclusions are verified through numerical simulation.

Key words: fear factor, refuge, Hopf bifurcation, stability

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