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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (10): 24-31.doi: 10.6040/j.issn.1671-9352.0.2023.151

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带Lévy噪声的Lotka-Volterra竞争模型的参数估计

张诗苗(),吕艳*()   

  1. 南京理工大学数学与统计学院, 江苏 南京 210094
  • 收稿日期:2023-04-10 出版日期:2023-10-20 发布日期:2023-10-17
  • 通讯作者: 吕艳 E-mail:2813091458@qq.com;lvyan1998@aliyun.com
  • 作者简介:张诗苗(1999—), 女, 硕士研究生, 研究方向为随机微分方程的参数估计. E-mail: 2813091458@qq.com
  • 基金资助:
    国家自然科学基金资助项目(12371243)

Parameter estimation for competitive Lotka-Volterra model with Lévy noise

Shimiao ZHANG(),Yan LYU*()   

  1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China
  • Received:2023-04-10 Online:2023-10-20 Published:2023-10-17
  • Contact: Yan LYU E-mail:2813091458@qq.com;lvyan1998@aliyun.com

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摘要：

基于最小二乘原理, 考虑离散观测下Lévy噪声驱动的随机二种群Lotka-Volterra竞争系统的参数估计问题。讨论在噪声强度ε→0和样本量n→∞的情况下, 参数估计量的渐近一致性, 并得到了估计量的渐近分布。最后, 给出竞争模型参数估计的数值模拟, 结果与理论一致。

关键词: Lotka-Volterra模型, Lévy噪声, 最小二乘估计, 渐近一致性

Abstract:

Based on the principle of least square and the technique of discretization, parameter estimation problem for stochastic two-species Lotka-Volterra competitive system driven by Lévy noise is studied. The least squares estimator is proved to be asymptotic consistent in the case ε→0 and n→∞, and the asymptotic distribution of the estimator is obtained. Finally, the numerical simulation for the estimators of the competitive model is given, and the results are in line with the theory.

Key words: Lotka-Volterra model, Lévy noise, least squares estimator, asymptotic consistency

中图分类号: 

  • O211.63

表1

(ε1, ε2)=(0.05, 0.05)时, 未知参数的LSE和AE"

真值 n=500 n=2 000 n=5 000
LSE AE LSE AE LSE AE
b1=0.8 0.828 9 0.028 9 0.822 2 0.022 2 0.812 7 0.012 7
a11=0.7 0.795 3 0.095 3 0.736 2 0.036 2 0.716 1 0.016 1
a12=0.4 0.383 1 0.016 9 0.402 4 0.002 4 0.402 3 0.002 3
b2=0.9 0.983 0 0.083 0 0.922 4 0.022 4 0.910 7 0.010 7
a21=0.4 0.366 2 0.033 8 0.392 0 0.008 0 0.392 7 0.002 8
a22=0.6 0.682 6 0.082 6 0.620 1 0.020 1 0.608 2 0.008 2

表2

(ε1, ε2)=(0.01, 0.01)时, 未知参数的LSE和AE"

真值 n=500 n=2 000 n=5 000
LSE AE LSE AE LSE AE
b1=0.8 0.780 8 0.019 2 0.797 1 0.002 9 0.801 1 0.001 1
a11=0.7 0.715 0 0.015 0 0.710 4 0.010 4 0.707 1 0.007 1
a12=0.4 0.377 4 0.022 6 0.393 3 0.006 7 0.398 0 0.002 0
b2=0.9 0.939 4 0.039 4 0.917 2 0.017 2 0.907 1 0.007 1
a21=0.4 0.388 1 0.011 9 0.395 9 0.004 1 0.397 7 0.002 3
a22=0.6 0.638 0 0.038 0 0.616 0 0.016 0 0.606 8 0.006 8

表3

(ε1, ε2)=(0.005, 0.005)时, 未知参数的LSE和AE"

真值 n=500 n=2 000 n=5 000
LSE AE LSE AE LSE AE
b1=0.8 0.789 8 0.010 2 0.795 7 0.004 3 0.799 4 0.000 6
a11=0.7 0.704 8 0.004 8 0.703 5 0.003 5 0.703 2 0.003 2
a12=0.4 0.389 4 0.010 6 0.395 0 0.005 0 0.398 2 0.001 8
b2=0.9 0.917 6 0.017 6 0.910 6 0.010 6 0.906 2 0.006 2
a21=0.4 0.396 4 0.003 6 0.397 5 0.002 5 0.398 9 0.001 1
a22=0.6 0.616 3 0.016 3 0.609 8 0.009 8 0.605 6 0.005 6

图1

参数估计的正态QQ图"

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