JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (10): 24-31.doi: 10.6040/j.issn.1671-9352.0.2023.151

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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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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

CLC Number: 

  • O211.63

Table 1

LSE and AE of the unknown parameters in the case (ε1, ε2)=(0.05, 0.05)"

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

Table 2

LSE and AE of the unknown parameters in the case (ε1, ε2)=(0.01, 0.01)"

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

Table 3

LSE and AE of the unknown parameters in the case (ε1, ε2)=(0.005, 0.005)"

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

Fig.1

Normal QQ plot of the estimators"

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