在线咨询
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 ›› 2019, Vol. 36 ›› Issue (6): 658-666.doi: 10.3969/j.issn.1005-3085.2019.06.005

• • 上一篇    下一篇

一类具有交叉扩散的捕食-食饵模型正解的存在性

吕  杨1,  郭改慧1,  袁海龙1,2,  李书选1   

  1. 1- 陕西科技大学文理学院，西安  710021
    2- 西安交通大学数学与统计学院，西安  710049
  • 收稿日期:2017-10-27 接受日期:2018-05-08 出版日期:2019-12-15 发布日期:2020-02-15
  • 通讯作者: 郭改慧 E-mail: guogaihui@sust.edu.cn
  • 基金资助:
    国家自然科学基金(11901370; 61672021; 61872227)；中国博士后科学基金(2019M653578)；陕西省教育厅专项科学研究计划(19JK0142)；陕西省科技厅自然科学基础研究计划(2019JQ-516)；陕西科技大学博士科研启动基金资助项目(2017BJ-44).

Existence of Positive Solutions for a Predator-prey Model with Cross-diffusion

LV Yang1,  GUO Gai-hui1,  YUAN Hai-long1,2,  LI Shu-xuan1   

  1. 1- School of Arts and Sciences, Shaanxi University of Science & Technology, Xi'an 710021
    2- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
  • Received:2017-10-27 Accepted:2018-05-08 Online:2019-12-15 Published:2020-02-15
  • Contact: G. Guo. E-mail address: guogaihui@sust.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (11901370; 61672021; 61872227); the Postdoctoral Science Foundation of China (2019M653578); the Special Scientific Research Project of Education Department of Shaanxi Provience (19JK0142); the Fundamental Research of Natural Science in Shannxi Province (2019JQ-516); the Natural Science Foundation of Shaanxi University of Science and Technology (2017BJ-44).

PDF (PC)

361

摘要： 本文研究一类带有交叉扩散的捕食-食饵模型正解的存在性．首先，利用最大值原理得到了与交叉扩散系数无关的正解的先验估计；其次，建立了当交叉扩散系数充分大时的极限系统；最后，利用局部分歧理论得到了极限系统在半平凡解附近的局部分歧解的存在性，借助全局分歧理论说明了该极限系统的局部分歧解可以延拓为全局分歧解，并且该全局分歧解随着分歧参数在正椎内延伸至无穷．结论表明：当交叉扩散系数充分大时，两物种可以共存．

关键词: 捕食-食饵模型, 交叉扩散, 正解, 存在性

Abstract: The existence of positive solutions to a predator-prey model with cross-diffusion is considered. First, we derive some estimates which are independent of cross-diffusion by the maximum principle; second, the limiting behavior of positive solutions for large cross-diffusion is established; finally, we show the existence of local bifurcation solutions of the limiting system near the semi-trivial solution by the local bifurcation theorem, and we extend the local bifurcation solutions to the global bifurcation solutions by the global bifurcation theorem, and we show that the global bifurcation solutions can extend to infinity as the bifurcation parameter approaches infinity. The results demonstrate that the two species can coexist when the cross-diffusion is large.

Key words: predator-prey model, cross-diffusion, positive solutions, existence

中图分类号: