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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 ›› 2021, Vol. 38 ›› Issue (1): 85-96.doi: 10.3969/j.issn.1005-3085.2021.01.008

• • 上一篇    下一篇

具有Allee效应的捕食-食饵扩散模型定性分析

李海侠   

  1. 宝鸡文理学院数学与信息科学学院，宝鸡 721013
  • 收稿日期:2018-08-27 接受日期:2020-03-06 出版日期:2021-02-15 发布日期:2021-04-15
  • 基金资助:
    国家自然科学基金 (61672021; 11801013)；陕西省自然科学基础研究计划项目 (2018JQ1066)；宝鸡市科技计划项目 (2018JH-20)；宝鸡文理学院博士科研项目 (ZK2018069).

Qualitative Analysis of a Diffusive Predator-prey Model with Allee Effect

LI Hai-xia   

  1. Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013
  • Received:2018-08-27 Accepted:2020-03-06 Online:2021-02-15 Published:2021-04-15
  • Supported by:
    The National Natural Science Foundation of China (61672021; 11801013); the Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ1066); the Science and Technology Program of Baoji (2018JH-20); the Doctoral Scientific Research Starting Foundation of Baoji University of Arts and Sciences (ZK2018069).

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摘要： 本文讨论一类具有 B-D 反应函数和 Allee 效应的捕食-食饵扩散模型正解的存在性、唯一性和多重性．首先运用不动点指数理论得到了正解存在的充分条件．接着利用特征值的变分原理给出了正解的唯一性条件．最后通过分析极限系统的正解，运用不动点指数理论、分歧理论和扰动理论确定了正解的确切重数和稳定性．讨论结果表明：只要 Allee 效应常量满足适当关系且捕食者的增长率较大，当参数满足适当条件时系统存在唯一正解，当食饵的增长率在一定范围内时系统恰好存在两个正解．

关键词: 捕食-食饵模型, Allee效应, 唯一性, 不动点指数理论, 扰动理论, 确切重数, 稳定性

Abstract: The existence, uniqueness and multiplicity of positive solutions to a diffusive predator-prey model with B-D functional response and Allee effect are discussed. By the fixed point index theory, the sufficient conditions for the existence of positive solutions are obtained. Secondly, the conditions for the uniqueness of positive solutions are given by the variational characterization of the lowest eigenvalue. Finally, based on the analysis of positive solutions to two limiting systems, the exact multiplicity and stability of positive solutions are determined by means of the combination of the fixed point index theory, bifurcation theory and perturbation theory of eigenvalues. When the Allee effect constants meet appropriate relationship and the growth rate of the predator is large, the results show that the system has only a unique positive solution when the parameters satisfy certain conditions, and has exactly two positive solutions when the growth rate of the prey lies in a certain range.

Key words: predator-prey model, Allee effect, uniqueness, fixed point index theory, perturbation theory, exact multiplicity, stability

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