具有Holling Ⅲ功能性反应的随机捕食食饵模型的平稳分布和周期解

数学物理学报 ›› 2018, Vol. 38 ›› Issue (5): 984-1000.

具有Holling Ⅲ功能性反应的随机捕食食饵模型的平稳分布和周期解

蓝桂杰,付盈洁,魏春金,张树文*()

集美大学理学院福建厦门 361021

收稿日期:2017-07-25 出版日期:2018-11-09 发布日期:2018-11-09
通讯作者: 张树文 E-mail:zhangsw_123@126.com
基金资助:
福建省自然科学基金(2016J05012);福建省自然科学基金(2016J01667)

Stationary Distribution and Periodic Solution for Stochastic Predator-Prey Systems with Holling-Type Ⅲ Functional Response

Guijie Lan,Yingjie Fu,Chunjin Wei,Shuwen Zhang*()

School of Sciences, Jimei University, Fujian Xiamen 361021

Received:2017-07-25 Online:2018-11-09 Published:2018-11-09
Contact: Shuwen Zhang E-mail:zhangsw_123@126.com
Supported by:
the Fujian Provincial Natural Science Foundation(2016J05012);the Fujian Provincial Natural Science Foundation(2016J01667)

摘要/Abstract

摘要：

该文研究了一类具有Holling Ⅲ功能性反应的随机捕食-食饵系统的动力学行为.对于自治系统，首先获得，对于任意的正初始值，系统都存在唯一的全局正解；第二，利用随机微分方程比较定理，得到系统的平均持续生存与灭绝的充分条件；第三，通过构造Lyapunov函数，证明了系统存在唯一的平稳分布且具有遍历性；而对于非自治系统，通过应用Has'minskii定理证明了，系统至少存在一个非平凡的正周期解；最后，给出数值模拟来验证主要结果

关键词: 捕食-食饵系统, 随机干扰, 平稳分布, 周期解

Abstract:

In this paper, we investigate the dynamics of stochastic predator-prey systems with Holling-type Ⅲ functional response. For the autonomous system, we firstly obtain that the system admits unique positive global solution starting from the positive initial value. Then, by comparison theorem for stochastic differential equation, sufficient conditions for extinction and persistence in mean are obtained. Thirdly, by constructing some suitable Lyapunov function, we prove that there are unique stationary distribution and they are ergodic. On the other hand, for the non-autonomous periodic system, we prove that there exists at least one nontrivial positive periodic solution according to the theory of Has'minskii. Finally, some numerical simulations are introduced to illustrate our theoretical result.

Key words: Predator-prey system, Random perturbation, Stationary distribution and ergodicity, Periodic solution

中图分类号:

O211.63

蓝桂杰,付盈洁,魏春金,张树文. 具有Holling Ⅲ功能性反应的随机捕食食饵模型的平稳分布和周期解[J]. 数学物理学报, 2018, 38(5): 984-1000.

Guijie Lan,Yingjie Fu,Chunjin Wei,Shuwen Zhang. Stationary Distribution and Periodic Solution for Stochastic Predator-Prey Systems with Holling-Type Ⅲ Functional Response[J]. Acta mathematica scientia,Series A, 2018, 38(5): 984-1000.

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参考文献 18

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