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《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    

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
  • 发布日期:2024-10-10
  • 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

ZHAO Yufeng1, LIU Guirong2   

  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
  • Published:2024-10-10

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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
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