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On the extinction route of a stochastic population model under heteroclinic bifurcation

异宿分岔下随机种群模型的灭绝路径

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Abstract

The noise-induced transition of the augmented Lotka-Volterra system is investigated under vanishingly small noise. Populations will ultimately go extinct because of intrinsic noise, and different extinction routes may occur due to the Freidlin-Wentzell large deviation theory. The relation between the most probable extinction route (MPER) and heteroclinic bifurcation is studied in this paper. The MPERs and the quasi-potentials in different regimes of parameters are analyzed in detail. Before the bifurcation, the predator goes extinct, and the prey will survive for a long time. Then, the heteroclinic bifurcation changes the MPER wherein both species go extinct. The heteroclinic cycle plays a role in transferring the most probable extinction state. Moreover, the analyses of the weak noise limit can contribute to predicting the stochastic behavior under finite small noise. Both the heteroclinic bifurcation and the rotational deterministic vector field can reduce the action necessary for the MPER.

摘要

本文在几近于零的小噪声情况下研究了增广Lotka-Volterra系统的噪声诱导跃迁. 由于固有噪声的存在, 种群最终将灭绝, 而Freidlin-Wentzell大偏差理论可能导致不同的灭绝路径. 本文研究了最有可能灭绝路径(MPER)与异宿分岔的关系, 详细分析不同参数区间的MPER和准势. 在分岔之前, 捕食者灭绝, 猎物将存活很长时间. 然后, 异宿分岔改变了两个物种灭绝的MPER. 异宿环起着转移最有可能灭绝状态的作用. 此外, 弱噪声极限分析有助于预测有限小噪声下的随机行为. 异宿分岔和旋转确定性向量场都可以减少MPER所需的作用量.

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control for Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Qing Yu  (虞晴) & Xianbin Liu  (刘先斌)

  2. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China

    Yang Li  (李扬)

Authors
  1. Qing Yu  (虞晴)
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  2. Yang Li  (李扬)
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  3. Xianbin Liu  (刘先斌)
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Corresponding author

Correspondence to Xianbin Liu  (刘先斌).

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11772149, and 12172167), A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and The Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Grant No. MCMS-I-19G01).

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Yu, Q., Li, Y. & Liu, X. On the extinction route of a stochastic population model under heteroclinic bifurcation. Acta Mech. Sin. 38, 221333 (2022). https://doi.org/10.1007/s10409-021-09062-x

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  • DOI: https://doi.org/10.1007/s10409-021-09062-x

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