数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (5): 1385-1398.doi: 10.1016/S0252-9602(17)30080-2

• 论文 • 上一篇    下一篇

POSITIVE STEADY STATES OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH PREDATOR CANNIBALISM

王彪   

  1. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
  • 收稿日期:2016-01-20 修回日期:2017-01-24 出版日期:2017-10-25 发布日期:2017-10-25
  • 作者简介:Biao WANG,E-mail:wang.biao@stu.xjtu.edu.cn
  • 基金资助:

    This work was partially supported by the National Natural Science Foundation of China (11371286).

POSITIVE STEADY STATES OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH PREDATOR CANNIBALISM

Biao WANG   

  1. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
  • Received:2016-01-20 Revised:2017-01-24 Online:2017-10-25 Published:2017-10-25
  • Supported by:

    This work was partially supported by the National Natural Science Foundation of China (11371286).

摘要:

The purpose of this paper is to investigate positive steady states of a diffusive predator-prey system with predator cannibalism under homogeneous Neumann boundary conditions. With the help of implicit function theorem and energy integral method, non-existence of non-constant positive steady states of the system is obtained, whereas coexistence of non-constant positive steady states is derived from topological degree theory. The results indicate that if dispersal rate of the predator or prey is sufficiently large, there is no non-constant positive steady states. However, under some appropriate hypotheses, if the dispersal rate of the predator is larger than some positive constant, for certain ranges of dispersal rates of the prey, there exists at least one non-constant positive steady state.

关键词: Predator-prey, existence and nonexistence, pattern formation

Abstract:

The purpose of this paper is to investigate positive steady states of a diffusive predator-prey system with predator cannibalism under homogeneous Neumann boundary conditions. With the help of implicit function theorem and energy integral method, non-existence of non-constant positive steady states of the system is obtained, whereas coexistence of non-constant positive steady states is derived from topological degree theory. The results indicate that if dispersal rate of the predator or prey is sufficiently large, there is no non-constant positive steady states. However, under some appropriate hypotheses, if the dispersal rate of the predator is larger than some positive constant, for certain ranges of dispersal rates of the prey, there exists at least one non-constant positive steady state.

Key words: Predator-prey, existence and nonexistence, pattern formation