一类具有时滞及反馈控制的非自治非线性比率依赖食物链模型

数学物理学报 ›› 2022, Vol. 42 ›› Issue (1): 245-268.

一类具有时滞及反馈控制的非自治非线性比率依赖食物链模型

王长有¹(),李楠^2,*(),蒋涛^3,^4,*(),杨强¹

¹ 成都信息工程大学应用数学学院成都 610225
² 西南财经大学应用数学学院成都 610074
³ 成都信息工程大学控制工程学院成都 610225
⁴ 中国科学院沈阳自动化研究所机器人国家重点实验室沈阳 110169

收稿日期:2020-09-26 出版日期:2022-02-26 发布日期:2022-02-23
通讯作者: 李楠,蒋涛 E-mail:wangchangyou417@163.com;2972028881@qq.com;jiangtop@126.com
作者简介:王长有, E-mail: wangchangyou417@163.com
基金资助:
机器人国家重点实验室研究基金(2019-O13);四川省科技厅科学研究项目(2021YFH0069);成都信息工程大学人才引进项目(KYTZ201820);四川省科技厅项目(21ZYZYTS0158)

On a Nonlinear Non-Autonomous Ratio-Dependent Food Chain Model with Delays and Feedback Controls

Changyou Wang¹(),Nan Li^2,*(),Tao Jiang^3,^4,*(),Qiang Yang¹

¹ College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225
² Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074
³ Control Engineering College, Chengdu University of Information Technology, Chengdu 610225
⁴ State key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110169

Received:2020-09-26 Online:2022-02-26 Published:2022-02-23
Contact: Nan Li,Tao Jiang E-mail:wangchangyou417@163.com;2972028881@qq.com;jiangtop@126.com
Supported by:
the Robot Research Fund of State Key Laboratory(2019-O13);the Science Research Project of Sichuan Provincial Department(2021YFH0069);the Talent Introduction Project of Chengdu University of Information Engineering(KYTZ201820);the Sichuan Science and Technology Program(21ZYZYTS0158)

摘要/Abstract

摘要：

该文研究了一类具有时滞和反馈控制的三种群非线性非自治比率依赖的食物链模型.首先，基于时滞微分不等式理论，提出了一些新的分析方法，并构造了一个合适的李亚普诺夫函数.其次，得到了系统正解的持久性和全局吸引性的充分条件.第三，利用理论分析和不动点理论，讨论了相应的周期系统，建立了周期系统正周期解的存在性、唯一性和稳定性的充分条件.另外，给出了一些数值模拟，证明了我们的理论分析是正确的.最后，给出了相应的具有乘法噪声源的随机食物链模型的数值例子，并得到该模型一些新的有趣的解的变化过程.

关键词: 时滞, 比率依赖, 反馈控制, 持久性, 全局吸引性, 周期解

Abstract:

In this paper, we study a 3-species nonlinear non-autonomous ratio-dependent food chain system with delays and feedback controls. Firstly, based on the theory of delay differential inequality, some new analytical methods are developed and a suitable Lyapunov function is constructed. Secondly, sufficient conditions for the permanence and global attractivity of positive solutions for the system are obtained. Thirdly, by using the theoretical analysis and fixed point theory, the corresponding periodic systems are discussed, and the conditions for the existence, uniqueness and stability of positive periodic solutions of periodic systems are established. Moreover, we give some numerical simulations to prove that our theoretical analysis are correct. Finally, we still give an numerical example for the corresponding stochastic food chain model with multiplicative noise sources, and achieve new interesting change process of the solution for the model.

Key words: Delay, Ratio-Dependent, Feedback Control, Permanence, Global Attractive, Periodic solution

中图分类号:

O175.12

王长有,李楠,蒋涛,杨强. 一类具有时滞及反馈控制的非自治非线性比率依赖食物链模型[J]. 数学物理学报, 2022, 42(1): 245-268.

Changyou Wang,Nan Li,Tao Jiang,Qiang Yang. On a Nonlinear Non-Autonomous Ratio-Dependent Food Chain Model with Delays and Feedback Controls[J]. Acta mathematica scientia,Series A, 2022, 42(1): 245-268.

图/表 10

图 1

带有初值条件 $(5.2)$ 的系统 $(5.1)$ 的数值解"

图 1

图 2

带有不同初值的系统 $(5.1)$ 的数值解"

图 2

图 3

系统 $(5.1)$ 的动力学行为"

图 3

图 4

图 5

系统(5.3)–(5.4)在种群 $x_{1}$ 的噪声强度较低, 种群 $x_{2}$ 和 $x_{3}$ 的噪声强度很低时的数值解"

图 5

图 6

系统(5.3)–(5.4)在种群 $x_{2}$ 的噪声强度较低, 种群 $x_{1}$ 和 $x_{3}$ 的噪声强度很低时的数值解"

图 6

图 7

系统(5.3)–(5.4)在种群 $x_{3}$ 的噪声强度较低, 种群 $x_{1}$ 和 $x_{2}$ 的噪声强度很低时的数值解"

图 7

图 8

系统(5.3)–(5.4)在种群 $x_{1}$ 的噪声强度较高, 种群 $x_{2}$ 和 $x_{3}$ 的噪声强度很低时的数值解"

图 8

图 9

系统(5.3)–(5.4)在种群 $x_{2}$ 的噪声强度较高, 种群 $x_{1}$ 和 $x_{3}$ 的噪声强度较低时的数值解"

图 9

图 10

系统(5.3)–(5.4)在种群 $x_{3}$ 的噪声强度较高, 种群 $x_{1}$ 和 $x_{2}$ 的噪声强度较低时的数值解"

图 10

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