一类具有 Ornstein-Uhlenbeck 过程的随机捕食者-食饵模型的指数绝灭、平稳分布和概率密度函数

数学物理学报 ›› 2024, Vol. 44 ›› Issue (5): 1368-1379.

一类具有 Ornstein-Uhlenbeck 过程的随机捕食者-食饵模型的指数绝灭、平稳分布和概率密度函数

张雯雯,刘志军^*(),王清龙

湖北民族大学数学与统计学院湖北恩施 445000

收稿日期:2023-10-30 修回日期:2024-04-15 出版日期:2024-10-26 发布日期:2024-10-16
通讯作者: *刘志军, E-mail: zjliu@hbmzu.edu.cn
基金资助:
国家自然科学基金(12101211);湖北省自然科学基金(2023AFB1095)

Exponential Extinction, Stationary Distribution and Probability Density Function of A Stochastic Predator-Prey Model with Ornstein-Uhlenbeck Process

Zhang Wenwen,Liu Zhijun^*(),Wang Qinglong

School of Mathematics and Statistics, Hubei Minzu University, Hubei Enshi 445000

Received:2023-10-30 Revised:2024-04-15 Online:2024-10-26 Published:2024-10-16
Supported by:
National Natural Science Foundation of China(12101211);Natural Science Foundation of Hubei Province of China(2023AFB1095)

摘要/Abstract

摘要：

该文建立了一类具有 Ornstein-Uhlenbeck 过程、恐惧效应、Crowley-Martin 型和修正的 Leslie-Gower 型功能反应函数的捕食者-食饵模型. 首先通过构造合适的 Lyapunov 函数证明了全局解的存在唯一性, 随后获得了两物种指数绝灭和平稳分布存在的充分条件. 进一步通过求解相应的 Fokker-Planck 方程得到了概率密度函数的具体表达式. 最后通过三个数值例子验证了理论结果的可行性, 其研究表明随机干扰的波动强度和回复速率均会影响种群的生存.

关键词: 捕食者-食饵模型, Ornstein-Uhlenbeck 过程, 指数绝灭, 平稳分布, 概率密度函数

Abstract:

In the paper, a stochastic predator-prey model with Ornstein-Uhlenbeck process, fear effect, Crowley-Martin type and Leslie-Gower type functional responses is considered. Firstly, by constructing suitable Lyapunov functions, we prove that the existence and uniqueness of the global solution, and the sufficient conditions for the exponential extinction and the existence of stationary distribution are obtained. Secondly, we get access to the specific expression of probability density function via dealing with the corresponding Fokker-Planck equation. Finally, our theoretical results are verified by three numerical examples. The results show that the intensity of volatility and the reversion speed of stochastic disturbance will affect the survival of species.

Key words: Predator-Prey model, Ornstein-Uhlenbeck process, Exponential extinction, Stationary distribution, Probability density function

中图分类号:

O175

张雯雯, 刘志军, 王清龙. 一类具有 Ornstein-Uhlenbeck 过程的随机捕食者-食饵模型的指数绝灭、平稳分布和概率密度函数[J]. 数学物理学报, 2024, 44(5): 1368-1379.

Zhang Wenwen, Liu Zhijun, Wang Qinglong. Exponential Extinction, Stationary Distribution and Probability Density Function of A Stochastic Predator-Prey Model with Ornstein-Uhlenbeck Process[J]. Acta mathematica scientia,Series A, 2024, 44(5): 1368-1379.

图/表 4

参考文献 23

[1]	马知恩. 种群生态学的数学建模与研究. 合肥: 安徽教育出版社, 1996
	Ma Z E. Mathematical Modeling and Research of Population Ecology. Hefei: Anhui Education Press, 1996
[2]	Su W, Zhang X. Global stability and canard explosions of the predator-prey model with the sigmoid functional response. SIAM J Appl Math, 2022, 82(3): 976-1000 doi: 10.1137/21M1437755
[3]	Santra P K, Mahapatra G S, Phaijoo G R. Bifurcation and chaos of a discrete predator-prey model with Crowley-Martin functional response incorporating proportional prey refuge. Math Probl Eng, 2020, 2020: 1-18
[4]	Zhou Y, Sun W, Song Y F, et al. Hopf bifurcation analysis of a predator-prey model with Holling-II type functional response and a prey refuge. Nonlinear Dyn, 2019, 97: 1439-1450
[5]	Sarkar K, Khajanchi S. Impact of fear effect on the growth of prey in a predator-prey interaction model. Ecol Complex, 2020, 42: 100826
[6]	Liu Y Y, Cao Q, Yang W S. Influence of Allee effect and delay on dynamical behaviors of a predator-prey system. Comput Appl Math, 2022, 41(8): 396
[7]	Mishra S, Upadhyay R K. Spatial pattern formation and delay induced destabilization in predator-prey model with fear effect. Math Method Appl Sci, 2022, 11: 45
[8]	May R M. Stability and Complexity in Model Ecosystems. New Jersey: Princeton University Press, 2001
[9]	Chen X Z, Tian B D, Xu X, et al. A stochastic predator-prey system with modified LG-Holling type II functional response. Math Comput Simul, 2023, 203: 449-485
[10]	Tian B D, Yang L, Chen X Z, et al. A generalized stochastic competitive system with Ornstein-Uhlenbeck process. Int J Biomath, 2021, 14(1): 2150001
[11]	Zhang X H, Yang Q, Jiang D Q. A stochastic predator-prey model with Ornstein-Uhlenbeck process: Characterization of stationary distribution, extinction and probability density function. Commun Nonlinear Sci Numer Simul, 2023, 122: 107259
[12]	Mu X J, Jiang D Q, Hayat T, et al. A stochastic turbidostat model with Ornstein-Uhlenbeck process: Dynamics analysis and numerical simulations. Nonlinear Dyn, 2022, 107(3): 2805-2817
[13]	Zhou B Q, Jiang D Q, Hayat T. Analysis of a stochastic population model with mean-reverting Ornstein-Uhlenbeck process and Allee effects. Commun Nonlinear Sci Numer Simul, 2022, 111: 106450
[14]	Yang Q, Zhang X H, Jiang D Q. Dynamical behaviors of a stochastic food chain system with Ornstein-Uhlenbeck process. J Nonlinear Sci, 2022, 32(3): 34
[15]	Shi Z F, Jiang D Q. Environmental variability in a stochastic HIV infection model. Commun Nonlinear Sci Numer Simul, 2023, 120: 107201
[16]	Zuo W J, Jiang D Q. Stationary distribution and periodic solution for stochastic predator-prey systems with nonlinear predator harvesting. Commun Nonlinear Sci Numer Simul, 2016, 36: 65-80
[17]	Allen E. Environmental variability and mean-reverting processes. Discrete Contin Dyn Syst Ser B, 2016, 21(7): 2073-2089
[18]	Kutoyants Y A. Statistical Inference for Ergodic Diffusion Processes. London: Springer, 2003
[19]	Khasminskii R. Stochastic Stability of Differential Equations. New York: Springer Science Business Media, 2011
[20]	Gardiner C W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Berlin: Springer, 1985
[21]	Oksendal B. Stochastic Differential Equations:An Introduction with Applications. New York: Springer-Verlag Heidelberg, 2000
[22]	Ma Z E, Zhou Y C, Li C Z. Qualitative and Stability Methods for Ordinary Differential Equations. Beijing: Science Press, 2015
[23]	何晓群. 多元统计分析. 北京: 中国人民大学出版社, 2019
	He X Q. Multivariate Statistical Analysis. Beijing: China Renmin University Press, 2019