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《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (9): 54-61.doi: 10.6040/j.issn.1671-9352.0.2018.667

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一类具有密度制约的捕食-食饵扩散模型的定性分析

李海侠   

  1. 宝鸡文理学院数学与信息科学学院, 陕西 宝鸡 721013
  • 出版日期:2019-09-20 发布日期:2019-07-30
  • 作者简介:李海侠(1977— ),女,博士,副教授,研究方向为偏微分方程及计算可视化. E-mail:xiami0820@163.com
  • 基金资助:
    国家自然科学基金资助项目(11801013);陕西省自然科学基础研究计划项目(2014JQ2-1003);陕西省教育厅专项(16JK1046);宝鸡市科技计划项目(2018JH-20);宝鸡文理学院博士科研项目(ZK2018069);渭南师范学院科研项目(16ZRRC05)

Qualitative analysis of a diffusive predator-prey model with density dependence

LI Hai-xia   

  1. Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, Shaanxi, China
  • Online:2019-09-20 Published:2019-07-30

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摘要： 研究了一类具有密度制约的捕食-食饵模型共存解的唯一性和渐近行为。利用不动点指数理论给出了共存解存在的充分条件。然后采用线性算子的扰动理论讨论了共存解的唯一性和稳定性。最后运用抛物方程的比较原理得到了系统的持久性和灭绝性条件,并通过数值模拟对其理论结果进行了验证。结果表明在一定条件下两物种共存且系统存在稳定的唯一共存解。

关键词: 捕食-食饵扩散模型, 密度制约, 不动点指数, 扰动理论, 唯一性, 渐近行为

Abstract: The uniqueness of coexistence solutions and asymptotic behavior for a diffusive predator-prey model with density dependence are studied. The sufficient conditions of the existence of coexistence solutions are given by means of the fixed point index theory. Then, by making use of the perturbation theory for linear operators, we discuss the stability and uniqueness of coexistence solutions. Finally, the conditions of the extinction and permanence for the system by the comparison principle for parabolic equations, and the theoretical results of asymptotic behavior for the system are verified by some numerical simulations. The results show that the two species can coexist and the system has a unique coexistence solution under certain conditions.

Key words: a diffusive predator-prey model, density dependence, fixed point index, perturbation theory, uniqueness, asymptotic behaviour

中图分类号: 

  • O175.26
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