目前大多数的捕食模型都是用白噪声描述环境的随机性,针对真实情境下,模型中的参数可能满足Ornstein-Uhlenbeck过程的问题,提出具有避难所和Ornstein-Uhlenbeck过程的随机三种群捕食模型。利用伊藤公式、微分不等式及随机分析理论对模型进行动力学行为分析。证明了模型全局正解的存在唯一性,分别得到了每个种群平均持久生存和灭绝的充分条件,最后利用python进行数值模拟,验证了定理中所得到的结论,并进一步研究了避难所对种群数量的影响。
Most current predation models describe the stochastic nature of the environment in terms of white noise, for the problem that the parameters in the model may satisfy the Ornstein-Uhlenbeck process in a real situation, a stochastic three-species predation model with refuge and Ornstein-Uhlenbeck process is proposed. The dynamic behavior of the model is analyzed by its formula, differential inequality and stochastic analysis theory. The existence and uniqueness of the global positive solution of the model are proved, sufficient conditions for the average persistence and extinction of each population were obtained separately, finally, python is used to carry out numerical simulation to verify the conclusions obtained in the theorem, and further study the impact of refuge on population size.
[1]BERRYMAN A A. The orgins and evolution of predator-prey theory[J]. Ecology, 1992, 73(5): 1530-1535.
[2]HAQUE M. A detailed study of the Beddington-DeAngelis predator-prey model[J]. Mathematical Biosciences, 2011, 234(1): 1-16.
[3]XIE Y, WANG Z, MENG B, et al. Dynamical analysis for a fractional-order prey-predator model with Holling III type functional response and discontinuous harvest[J]. Applied Mathematics Letters, 2020, 106: 106342.
[4]LUO Y, ZHANG L, TENG Z, et al. Global stability for a nonautonomous reaction-diffusion predator-prey model with modified Leslie-Gower Holling-II schemes and a prey refuge[J]. Advances in Difference Equations, 2020, 2020(1): 1-16.
[5]YUE Q. Dynamics of a modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge[J]. SpringerPlus, 2016, 5(1): 1-12.
[6]ZHANG H, CAI Y, FU S, et al. Impact of the fear effect in a prey-predator model incorporating a prey refuge[J]. Applied Mathematics and Computation, 2019, 356: 328-337.
[7]QI H, MENG X. Threshold behavior of a stochastic predator-prey system with prey refuge and fear effect[J]. Applied Mathematics Letters, 2021, 113: 106846.
[8]DAS S, MAHATO P, MAHATO S K. Disease control prey-predator model incorporating prey refuge under fuzzy uncertainty[J]. Modeling Earth Systems and Environment, 2021, 7(4): 2149-2166.
[9]MUKHERJEE D. The effect of refuge and immigration in a predator-prey system in the presence of a competitor for the prey[J]. Nonlinear Analysis: Real World Applications, 2016, 31: 277-287.
[10] MUKHERJEE D, MAJI C. Bifurcation analysis of a Holling type II predator-prey model with refuge[J]. Chinese Journal of Physics, 2020, 65: 153-162.
[11] XU Y, LIU M, YANG Y. Analysis of a stochastic two-predators one-prey system with modified Leslie-Gower and Holling-typeⅡ schemes[J]. Journal of Applied Analysis & Computation, 2017, 7(2): 713-727.
[12] TIAN B, YANG L, CHEN X, et al. A generalized stochastic competitive system with Ornstein-Uhlenbeck process[J]. International Journal of Biomathematics, 2021, 14(1): 2150001.
[13] TONG Z, LIU A. A censored Ornstein-Uhlenbeck process for rainfall modeling and derivatives pricing[J]. Physica A: Statistical Mechanics and its Applications, 2021, 566: 125619.
[14] WANG W, CAI Y, DING Z, et al. A stochastic differential equation SIS epidemic model incorporating Ornstein-Uhlenbeck process[J]. Physica A: Statistical Mechanics and its Applications, 2018, 509: 921-936.
[15] ZHOU D, LIU M, LIU Z. Persistence and extinction of a stochastic predator-prey model with modified Leslie-Gower and Holling-type II schemes[J]. Advances in Difference Equations, 2020, 2020(1): 1-15.
[16] GAO Y, YAO S. Persistence and extinction of a modified Leslie-Gower Holling-typeⅡ predator-prey stochastic model in polluted environments with impulsive toxicant input[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 4894-4918.
[17] MAO X. Stochastic Differential Equations and Applications[M]. Chichester(UK):Horwood Publishing, 2007.
[18] IKEDA N, WATANABE S. Stochastic Differential Equations and Diffusion Processes[M]. Amsterdam: North-Holland: Elsevier, 2014.
[19] JIANG D, SHI N. A note on nonautonomous logistic equation with random perturbation[J]. Journal of Mathematical Analysis and Applications, 2005, 303(1): 164-172.
[20] LIU M, WANG K, WU Q. Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle[J]. Bulletin of mathematical biology, 2011, 73(9): 1969-2012.
[21] HIGHAM D J. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM Review, 2001, 43(3): 525-546.
