您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (10): 74-88.doi: 10.6040/j.issn.1671-9352.0.2023.547

•   • 上一篇    下一篇

一类随机捕食系统的平稳分布及其概率密度函数

赵玉凤1(),刘桂荣2   

  1. 1. 山西工商学院计算机信息工程学院, 山西 太原 030006
    2. 山西大学数学科学学院, 山西 太原 030000
  • 收稿日期:2023-12-26 出版日期:2024-10-20 发布日期:2024-10-10
  • 作者简介:赵玉凤(1985—), 女, 讲师, 硕士, 研究方向为生物数学. E-mail: 463920353@qq.com
  • 基金资助:
    山西省高等学校科技创新项目(2022L645);山西省高等学校教学改革创新项目(J20221313);山西省教育科学“十四五”规划课题项目(GH-220495)

Stationary distribution and probability density function of a stochastic predation system

Yufeng ZHAO1(),Guirong LIU2   

  1. 1. School of Computer Science and Information Engineering, Shanxi Technology and Business College, Taiyuan 030006, Shanxi, China
    2. School of Mathematical Sciences, Shanxi University, Taiyuan 030000, Shanxi, China
  • Received:2023-12-26 Online:2024-10-20 Published:2024-10-10

RichHTML

1

PDF (PC)

230

摘要：

建立一类具有捕食者阶段结构和比率依赖的Holling III型功能反应的随机捕食者-食饵模型。首先, 给出了随机模型全局正解的存在唯一性。其次, 通过构造合适的Lyapunov函数, 利用Has′Minskii的遍历性理论研究了模型的遍历平稳分布的存在唯一性。然后, 通过求解相应的三维Fokker-Planck方程的方法, 推导出随机捕食模型在正平衡点附近的概率密度函数的精确表达式。最后, 通过数值仿真验证了理论结果的合理性。

关键词: 随机捕食者-食饵模型, 平稳分布, 阶段结构, 比率依赖, 概率密度函数

Abstract:

A class of stochastic predator-prey models with predator-stage structure and rate-dependent Holling III type functional responses are developed. Firstly, the existence and uniqueness of global positive solutions for stochastic model are obtained. Secondly, the existence and uniqueness of the ergodic stationary distribution are studied by constructing a suitable Lyapunov function and using the ergodic theory of Has′Minskii. Next, by solving the corresponding three-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic predator-prey model near the positive equilibrium point is derived. Finally, the rationality of the theoretical results is verified by numerical simulation.

Key words: stochastic predator-prey model, stationary distribution, stage structure, ratio-dependent, probability density function

中图分类号: 

  • O175

图1

系统(3)的解在噪声强度σ12=0.2, σ22=0.2, σ32=0.12时的路径图"

图2

系统(3)的解在噪声强度σ12=0.2, σ22=0.2, σ32=0.12时的频率直方图和边缘概率密度函数曲线"

1 周翔宇, 吴建华. 一类带Michaelis-Menten收获项的Holling-Ⅳ型捕食-食饵模型的定性分析[J]. 应用数学学报, 2019, 42 (1): 32- 42.
ZHOU Xiangyu , WU Jianhua . The qualitative analysis of a Holling-IV type predator-prey model with Michaelis-Menten type prey harvesting[J]. Acta Mathematicae Applicatae Sinica, 2019, 42 (1): 32- 42.
2 蒋婷婷, 杜增吉. 带有脉冲和Holling-Ⅳ型功能反应函数的中立型捕食-食饵模型的周期解[J]. 数学物理学报, 2021, 41 (1): 178- 193.
doi: 10.3969/j.issn.1003-3998.2021.01.015
JIANG Tingting , DU Zengji . Periodic solutions of a neutral impulsive predator-prey model with Holling-type IV functional response[J]. Acta Mathematica Scientia, 2021, 41 (1): 178- 193.
doi: 10.3969/j.issn.1003-3998.2021.01.015
3 霍林杰, 张存华. 具有Holling-Ⅲ型功能反应的捕食扩散系统的稳定性和Hopf分支[J]. 山东大学学报(理学版), 2023, 58 (1): 16- 24.
doi: 10.6040/j.issn.1671-9352.0.2021.788
HUO Linjie , ZHANG Cunhua . Stability and hopf bifurcation of diffusive predator-prey system with Holling-Ⅲ type functional response[J]. Journal of Shandong University(Natural Science), 2023, 58 (1): 16- 24.
doi: 10.6040/j.issn.1671-9352.0.2021.788
4 王长有, 李楠, 蒋涛, 等. 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.
doi: 10.3969/j.issn.1003-3998.2022.01.020
WANG Changyou , LI Nan , JIANG Tao , et al. On a nonlinear non-autonomous ratio-dependent food chain model with delays and feedback controls[J]. Acta Mathematica Scientia, 2022, 42 (1): 245- 268.
doi: 10.3969/j.issn.1003-3998.2022.01.020
5 FU Jing , JIANG Daqing , SHI Ningzhong . Qualitative analysis of a stochastic ratio-dependent Holling-Tanner system[J]. Acta Mathematica Scientia, 2018, 38B (2): 429- 440.
doi: 10.3969/j.issn.0252-9602.2018.02.004
6 CHEN Mengxin , WU Ranchao , LIU Biao , et al. Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing-Hopf bifurcation point[J]. Communications in Nonlinear Science and Numerical Simulation, 2019, 77, 141- 167.
doi: 10.1016/j.cnsns.2019.04.024
7 GAO Xiaoyan , ISHAG Sadia , FU Shengmao , et al. Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator-prey model with predator harvesting[J]. Nonlinear Analysis: Real World Applications, 2020, 51, 102962.
doi: 10.1016/j.nonrwa.2019.102962
8 ROY Jyotirmoy , BARMAN Dipesh , ALAM Shariful . Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment[J]. Biosystems, 2020, 197, 104176.
doi: 10.1016/j.biosystems.2020.104176
9 LI Wenjie , JI Jinchen , HUANG Lihong . Global dynamic behavior of a predator-prey model under ratio-dependent state impulsive control[J]. Applied Mathematical Modelling, 2020, 77 (2): 1842- 1859.
10 李艳玲, 李瑜, 郭改慧. 一类混合比率依赖食物链扩散模型的Hopf分支[J]. 应用数学学报, 2019, 42 (1): 1- 14.
LI Yanling , LI Yu , GUO Gaihui . Hopf bifurcation for a class of hybrid ratio-dependent food chain model with diffusion[J]. Acta Mathematicae Applicatae Sinica, 2019, 42 (1): 1- 14.
11 李波, 梁子维. 一类具有Beddington-DeAngelis响应函数的阶段结构捕食模型的稳定性[J]. 数学物理学报, 2022, 42 (6): 1826- 1835.
LI Bo , LIANG Ziwei . Stability of stage-structured predator-prey models with Beddington-DeAngelis functional response[J]. Acta Mathematica Scientia, 2022, 42 (6): 1826- 1835.
12 张明扬. 两种具有Allee效应和阶段结构的捕食模型及最优捕获[D]. 武汉: 华中科技大学, 2021.
ZHANG Mingyang. Two predation models with Allee effect and stage structure and optimal capture[D]. Wuhan: Huazhong University of Science & Technology, 2021.
13 许洪菲. 带有阶段结构的交叉扩散捕食-食饵模型的动力学分析[D]. 哈尔滨: 哈尔滨师范大学, 2022.
XU Hongfei. Dynamics analysis in a cross-diffusion predators-prey model with stage structure[D]. Harbin: Harbin Normal University, 2022.
14 张钰珂, 孟新柱. 具有恐惧效应和Crowley-Martin功能反应的随机捕食模型的动力学[J]. 山东大学学报(理学版), 2023, 58 (10): 54- 66.
doi: 10.6040/j.issn.1671-9352.0.2022.635
ZHANG Yuke , MENG Xinzhu . Dynamies of a stochastic predation model with fear effect and Crowley-Martin functional response[J]. Journal of Shandong University(Natural Science), 2023, 58 (10): 54- 66.
doi: 10.6040/j.issn.1671-9352.0.2022.635
15 HAN Bingtao , JIANG Daqing , ZHOU Baoquan , et al. Stationary distribution and probability density function of a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth[J]. Chaos, Solitons and Fractals, 2020, 142 (5): 110519.
16 ZHAO Xin , ZENG Zhijun . Stationary distribution and extinction of a stochastic ratio-dependent predator-prey system with stage structure for the predator[J]. Physica A: Statistical Mechanics and its Applications, 2020, 545, 123310.
17 ZHANG Xinhong , YANG Qing , JIANG Daqing . A stochastic predator-prey model with Ornstein-Uhlenbeck process: characterization of stationary distribution, extinction and probability density function[J]. Communications in Nonlinear Science and Numerical Simulation, 2023, 122, 107259.
18 ZHOU Baoquan , JIANG Daqing , DAI Yucong , et al. Stationary distribution and probability density function of a stochastic SVIS epidemic model with standard incidence and vaccination strategies[J]. Chaos, Solitons and Fractals, 2021, 143, 110601.
19 KHASMINSKII Rafail . Stochastic stability of differential equations[M]. Beilin: Springer, 1980.
20 GARDINER C W . Handbook of stochastic methods for physics, chemistry and the natural sciences[M]. Berlin: Springer, 1983.
21 ROOZEN H . An asymptotic solution to a two-dimensional exit problem arising in population dynamics[J]. SIAM Journal on Applied Mathematics, 1989, 49 (6): 793- 1810.
22 HAN Bingtao , JIANG Daqing . Threshold dynamics and probability density functions of a stochastic predator-prey model with general distributed delay[J]. Communications in Nonlinear Science and Numerical Simulation, 2023, 128, 107596.
23 DESMOND J H . An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM Review, 2001, 43 (3): 525- 546.
[1] 丁佳鑫,郭永峰,米丽娜. 高斯噪声和Lévy噪声激励下欠阻尼周期势系统的相变行为[J]. 《山东大学学报(理学版)》, 2023, 58(8): 111-117.
[2] 张钰珂,孟新柱. 具有恐惧效应和Crowley-Martin功能反应的随机捕食模型的动力学[J]. 《山东大学学报(理学版)》, 2023, 58(10): 54-66.
[3] 程惠东,常正波. 脉冲存放食饵连续捕获捕食者阶段结构数学模型[J]. J4, 2010, 45(11): 83-87.
[4] 侯强，靳祯. 基于比率且包含食饵避难的HollingTanner模型分析[J]. J4, 2009, 44(3): 56-60 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] 郭乔进,丁轶,李宁. 一种基于上下文信息的乳腺肿块ROI检测方法[J]. J4, 2010, 45(7): 70 -75 .
[2] 付海艳,卢昌荆,史开泉 . (F,F-)-规律推理与规律挖掘[J]. J4, 2007, 42(7): 54 -57 .
[3] 刘洪华 . 色散方程的交替分组迭代方法[J]. J4, 2007, 42(1): 19 -23 .
[4] 刘昆仑. 变结构pair copula模型在金融危机传染分析中的应用[J]. 山东大学学报（理学版）, 2016, 51(6): 104 -110 .
[5] 汤晓宏1，胡文效2*，魏彦锋2，蒋锡龙2，张晶莹2，. 葡萄酒野生酿酒酵母的筛选及其生物特性的研究[J]. 山东大学学报（理学版）, 2014, 49(03): 12 -17 .
[6] 袁瑞强,刘贯群,张贤良,高会旺 . 黄河三角洲浅层地下水中氢氧同位素的特征[J]. J4, 2006, 41(5): 138 -143 .
[7] 郭文鹃,杨公平*,董晋利. 指纹图像分割方法综述[J]. J4, 2010, 45(7): 94 -101 .
[8] 张雯,张化祥*,李明方,计华. 决策树构建方法:向前两步优于一步[J]. J4, 2010, 45(7): 114 -118 .
[9] 宋 颖,张兴芳,王庆平 . 参数Kleene系统的α-反向三Ⅰ支持算法[J]. J4, 2007, 42(7): 45 -48 .
[10] 刘战杰1,马儒宁1,邹国平1,钟宝江2,丁军娣3. 一种新的基于区域生长的彩色图像分割算法[J]. J4, 2010, 45(7): 76 -80 .
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发