数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (6): 1961-1980.doi: 10.1007/s10473-020-0622-7

• 论文 • 上一篇    下一篇

EXISTENCE AND UNIQUENESS OF THE POSITIVE STEADY STATE SOLUTION FOR A LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH A CROWDING TERM

曾宪忠, 刘玲妤, 谢伟圆   

  1. School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China
  • 收稿日期:2019-04-27 修回日期:2019-09-15 出版日期:2020-12-25 发布日期:2020-12-30
  • 通讯作者: Xianzhong ZENG,E-mail:zxzh217@sohu.com E-mail:zxzh217@sohu.com
  • 作者简介:Lingyu LIU,E-mail:591283884@qq.com;Weiyuan XIE,E-mail:2310945546@qq.com
  • 基金资助:
    The work was supported by the Hunan Provincial Natural Science Foundation of China (2019JJ40079, 2019JJ50160), the Scientific Research Fund of Hunan Provincial Education Department (16A071, 19A179) and the National Natural Science Foundation of China (11701169)

EXISTENCE AND UNIQUENESS OF THE POSITIVE STEADY STATE SOLUTION FOR A LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH A CROWDING TERM

Xianzhong ZENG, Lingyu LIU, Weiyuan XIE   

  1. School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China
  • Received:2019-04-27 Revised:2019-09-15 Online:2020-12-25 Published:2020-12-30
  • Contact: Xianzhong ZENG,E-mail:zxzh217@sohu.com E-mail:zxzh217@sohu.com
  • Supported by:
    The work was supported by the Hunan Provincial Natural Science Foundation of China (2019JJ40079, 2019JJ50160), the Scientific Research Fund of Hunan Provincial Education Department (16A071, 19A179) and the National Natural Science Foundation of China (11701169)

摘要: This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value $\lambda_1^D(\Omega_0)$, and demonstrate that the existence of the predator in $\overline{\Omega}_0$ only depends on the relationship of the growth rate $\mu$ of the predator and $\lambda_1^D(\Omega_0)$, not on the prey. Furthermore, when $\mu<\lambda_1^D(\Omega_0)$, we obtain the existence and uniqueness of its positive steady state solution, while when $\mu\geq \lambda_1^D(\Omega_0)$, the predator and the prey cannot coexist in $\overline{\Omega}_0$. Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region $\overline{\Omega}_0$, which is different from that of the classical Lotka-Volterra predator-prey model.

关键词: Lotka-Volterra predator-prey model, crowding term, critical value, coexistence

Abstract: This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value $\lambda_1^D(\Omega_0)$, and demonstrate that the existence of the predator in $\overline{\Omega}_0$ only depends on the relationship of the growth rate $\mu$ of the predator and $\lambda_1^D(\Omega_0)$, not on the prey. Furthermore, when $\mu<\lambda_1^D(\Omega_0)$, we obtain the existence and uniqueness of its positive steady state solution, while when $\mu\geq \lambda_1^D(\Omega_0)$, the predator and the prey cannot coexist in $\overline{\Omega}_0$. Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region $\overline{\Omega}_0$, which is different from that of the classical Lotka-Volterra predator-prey model.

Key words: Lotka-Volterra predator-prey model, crowding term, critical value, coexistence

中图分类号: 

  • 35J57